Multifield theories in mechanics of solids

A D V A N C E S IN A P P L I E D M E C H A N I C S , V O L U M E 38

Multifield Theories in Mechanics of Solids* PAOLO MARIA MARIANO Dipartimento di lngegneria Strutturale e Geotecnica,' Universitgt di Roma "La Sapienza," 00184 Rome, Italy

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Structure of This Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 5

II. Configurations and Balance of Interactions . . . . . . . . . . . . . . . . . . . A. Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Measures of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Balance of Interactions from the Invariance of Outer Power . . . . E. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 21

III. Elastic Materials with Substructure . . . . . . . . . . . . . . . . . . . . . . . . A. Variational Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . B. S o m e Properties of Lagrangian Densities . . . . . . . . . . . . . . . . . . C. Influence of the Substructure on the Decay of Elastic Energy . . .

26 26 27 30

IV. Balance in Presence of Discontinuity Surfaces . . . . . . . . . . . . . . . . A. Interfaces: G e o m e t r i c Characterization . . . . . . . . . . . . . . . . . . . B. Balance at Discontinuity Surfaces . . . . . . . . . . . . . . . . . . . . . . .

33 33 35

V. Constitutive Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Constitutive Restrictions in the Bulk . . . . . . . . . . . . . . . . . . . . . B. Constitutive Restrictions at Discontinuity Surfaces . . . . . . . . . . . VI. Evolution of Defects and Interfaces in Materials with S u b s t r u c t u r e . . A. Configurational Forces in the Bulk . . . . . . . . . . . . . . . . . . . . . . B. Configurational Forces on a Discontinuity Surface . . . . . . . . . . . VII. Crack Propagation in Materials with Substructure . . . . . . . . . . . . . . A. Kinematics of Planar M o v i n g Cracks . . . . . . . . . . . . . . . . . . . . B. Balance of Standard and Substructural Interactions at the T i p . . . C. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Tip Balance of Configurational Forces . . . . . . . . . . . . . . . . . . . . E. C o n s e q u e n c e s of the Mechanical Dissipation Inequality . . . . . . . E Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A Modified Expression of J Integral . . . . . . . . . . . . . . . . . . . . . H. Energy Dissipated in the Process Zone . . . . . . . . . . . . . . . . . . .

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38 39 40 42 43 46 53 54 56 57 59 61 63 65 66

*To M. M., for simple and, at the same time, complicated reasons.

ISBN 0-12-002038-6

ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00

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VIII. Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Second-Gradient Theories as Special Cases of Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Examples of Specific Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Material with Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Phase (or Multiphase) Materials . . . . . . . . . . . . . . . . . . . . C. Cosserat Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Micromorphic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Nematic Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Ferroelectric Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Microcracked Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 70 73 74 75 76 78 80 81 83

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction A. GENERAL INTRODUCTION The term multifield theories indicates the wide range of models in which some graphic fields must be introduced to describe the influence of material substructures on the gross mechanical behavior of solids. In a certain sense these fields (order parameters, also called phase fields, microstructural fields, microdisplacements, microdeformations, etc., in different special cases) are models of the material structure. Solidification of metal alloys and their possible shape memory, damage states and evolution, and the influence of long chains of macromolecules on the behavior of polymers are examples of physical phenomena of great interest in engineering practice that can be analyzed fruitfully with the help of multifield theories. Usually, in continuum mechanics only the placement within the Euclidean space is assigned to each material patch (volume element), then changes in relative placements are evaluated to measure the crowding and the shearing of material patches (i.e., the deformation). In this way, however, the features of material texture, or, more generally, substructure, are overlooked. The starting idea of multifield theories is to assign to each material patch P at least one pair (x, qo) in which x(P) is the placement of P within the Euclidean space and qo(P) (order parameter l) furnishes information on the substructural configuration of the patch. i Of course, the word order is only conventional, qo can also describe disordered arrangements of macromolecules or crystalline grains. The term orderparameter arises from statistical physics,

Multifield Theories in Mechanics of Solids

3

R e m a r k 1 Many choices of qo can be made; some of them are considered natural, whereas others are of convenience. They depend on the physical circumstances and must be specified each time. 9 The simplest choice is to consider the field qo to be scalar valued. For example, qo may represent the void volume fraction in porous solids or the volume fraction of a certain material phase in a two-phase material, as in the case of austenite-martensite mixtures. Such a choice of the order parameter is not unique for this physical situation; it may be not sufficient, for example, to describe in some detail the directional distribution of grains of austenite or martensite at each point, and tensor-valued order parameters must be introduced. A scalar-order parameter may be also used in the case of mixtures of two fluids or to describe solidification phenomena. 9 qO can be a vector. Liquid crystals are the classical example of bodies modeled by such a special choice of the order parameter. Other situations in which the vector choice is made are direct models of rods and shells. A shell can be represented roughly speaking by the middle surface and a field of unit vectors orthogonal to it in the reference configuration (a Cosserat surface). Analogously, triads of vectors may be used to represent the behavior of cross sections of rods or of stratified rocks, such as gneiss. Vectororder parameters are also useful tools in describing the mechanics of defective crystals (they can be identified with the optical axes of the crystal) or microcrack systems (in this case, they may represent vectors orthogonal to plane microcracks or the perturbation of the displacement field induced by the presence of such defects), ferroelectrics, and magnetostrictive solids. 9 If the material patch is characterized by large molecules undergoing homogeneous deformations, second-order tensor-valued order parameters may be used, a choice that also describes the dipole approximation of some distribution of directional data; for example, the distribution of microcracks. Moreover, such a second-order tensor can represent the Nye's tensor in dislocated continua. 9 Distributions of microcracks can be also described by their quadrupole approximation. A fourth-order tensor must be introduced with the role of order parameter.

in which Landau introduced order parameters to study second-order phase transitions consisting of abrupt changes of symmetry in solids.

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Paolo Maria Mariano

Many other examples can be made, and some are presented in Section IX in a more detailed way. The order parameters quoted in the previous remark are elements of some finite dimensional manifold, here indicated with AA and usually considered compact and without boundary..A4 can be also infinite dimensional, in which case the order parameter field assigns to each material patch an entire distribution function that can represent the geometry of the material substructure or can be the distribution (not necessarily canonical) of different levels of energy within the patch. In the following discussion, I refer the developments presented in this article to the case in which .A/[ is finite dimensional. The characteristic features of each special model depend on the choice of M , and hence on the mathematical properties of it. The physical meaning (or, for instance, interpretation) of these properties should be specified each time, although they are often considered to be convenient devices only. For example, the metric on M determines the quadratic part of the kinetic energy (if any) associated to the order parameter, and therefore to the substructure of the material. In the case of crystalline materials (e.g., such a prominent role of kinetic energy can be recognized only at very high frequencies; however, although the case is rare, such a kinetic energy should be considered in these regimes and determines the metric of M . Basically, the order parameter is considered to be an observable quantity. An external spatial observer must take two different measures to evaluate both the position of each material patch and information on its substructure. In this way, x and ~ together characterize the physical configuration of the solid. Interactions are associated with ~: They are substructural interactions and depend on the nature of the material substructure. These interactions develop explicit power in the rate of the order parameter and perhaps of its gradient, and must be balanced. Consequently, new balance equations arise in addition to those of Cauchy and represent the balance of some sort of generalized momentum and moment of momentum. The latter balance implies an expression of the skew part of Cauchy stress in terms of substructural measures of interaction. The representation of the substructural interactions is a delicate problem. When the gradient of ~o can be evaluated in a covariant way through a connection (hopefully with a physical significance) on A/l, these interactions can be represented by appropriate tensors called microstress and self-force. This terminology is conventional only and evokes special situations in which these tensors are really "perturbations" (in some sense) of the macroscopic stress tensor and also represent some kind of internal forces. In addition, the question of the connection (by which the covariant X7~ can be expressed) is delicate from a conceptual point of view. There are situations in which a physically significant connection can clearly be recognized (e.g., for

Multifield Theories in Mechanics of Solids

5

nematic liquid crystals); however, there are other situations in which this is not so. The choice of the connection, in fact, influences not only the explicit representation of the gradient of r but also the representation of the power. When many connections can be defined indifferently, it is necessary to require the invariance of the power with respect to their choice. Such a requirement allows one to obtain results analogous (in terms of the structure of the balance equations) to the results assured by the existence of a natural and physically significant connection. In general, however, the interactions are represented by general functionals. This situation also occurs when nonlocal effects due to material substructure are considered in both time (memory) and space. Nonlocality can be represented by integrals in time or space. When these functionals must be introduced, retardation (memory) or myopia (space) theorems help develop them (to within some material constant) in terms of differential operators of the fields involved. In this case, one accounts for weak nonlocalities only. Alternatively, one can assign to each material patch P only its placement x(P) in the three-dimensional Euclidean space and decide to introduce internal variables describing microstructural effects. These variables are nonobservable objects by definition; no balance equations are associated with them. The derivatives of the free energy with respect to the internal variables (and, possibly, with respect to their gradients) are in fact not genuine interactions, but are rather only affinities that must satisfy only the second law of thermodynamics, and need not be balanced because they do not develop explicit mechanical power. However, models with internal variables can be derived from multifield theories by appropriate internal constraints. In this case, the substructure becomes latent. Only a possible kinetics (evolution rule) is associated to it. From a mathematical point of view, the assignment of a kinetics to qp, without considering the balance of genuine interactions, is tantamount to take some initial value of qo--say, corresponding to some point qo* on M - - a n d to give a rule selecting elements of the tangent space of M at qa*. Finally, the order parameter field can be chosen to be a stochastic field taking values on the manifold M . In this way, not only the descriptor of the substructure is associated with each material patch, but also the probability that such substructure is really present.

B. STRUCTURE OF THIS ARTICLE The aim of this article is to show that the multifield description of continua is a flexible framework to study many physical situations in which the analysis of substructures is important for both practical and theoretical reasons. The analytical

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tools (and, of course, the difficulties that one tackles when using them) appear to be necessary to describe with certain detail the behavior of material substructures. The general theory is presented first, followed by special cases and applications. The description of configurations and the deduction of balance equations for bodies lacking in discontinuity surfaces are in Section II. Section III is dedicated to the special case in which the behavior of the substructure is elastic, in which case the force exerted on an inclusion in the body by the surrounding medium is deduced from an appropriate version of Noether's theorem accounting for the order parameter field. The influence of the substructure on the axial decay of energy in linear elastic cylinders is also discussed. Section IV deals with the derivation of thermomechanical balance at discontinuity surfaces (interfaces) that are endowed by their own measures of interactions. Surface stress, surface microstress, and self-force are defined on the interfaces. Constitutive restrictions arising from a mechanical version the second law of thermodynamics and involving the measures of substructural interactions are treated in Section V. Section VI is dedicated to the analysis of the influence of substructures on configurational forces that drive the evolution of interfaces. The kinetic equation for interfaces is deduced from the balance of configurational forces and is expressed in terms of a generalized expression of Eshelby tensor. In Section VII, special attention is given to the evaluation of the influence of material substructures on macrocrack propagation. In Section VIII, the case in which the substructure becomes latent in presence of appropriate internal constraints is discussed. Finally, Section IX deals with the application of the general theory to special cases.

BIBLIOGRAPHIC NOTE The "Th6orie des corps d6formables" of Cosserat and Cosserat (1909) is the first known historical example of a special case of multifield theories treated systematically, even though the germinal idea was formulated by Voigt (1887). As is well known, Cosserat and Cosserat's point of view consists of considering each material patch as a rigid body (possibly described by its peculiar triad of vectors) that can rotate independently of the neighboring patches. Couple stresses are associated with these additional degrees of freedom 2 and are balanced. In 1958, Ericksen and Truesdell gave new insight to Cosserat and Cosserat's theory. Following the Cosserats, they consider such a theory to be a suitable tool to describe the 2They are considered "additional" degrees of freedom because, in the classic case of Cauchy materials, each point has only three degrees of freedom.

Multifield Theories in Mechanics of Solids

7

mechanics of rods and shells, which they represent as lines and surfaces, respectively, endowed at each point by triads of mutually perpendicular vectors (the triads describe the behavior of sections). Ericksen and Truesdell's (1958) seminal paper constitutes a generalization of the Cosserats' (1909) ideas because they consider such vectors (the order parameters) to be stretchable (for other contributions to the general theory of Cosserat and Cosserats' materials, see also Mindlin, 1965a,b; Toupin, 1964; Truesdell and Toupin, 1960; Aero and Kuvshinskii, 1960; Grioli, 1960; Mindlin and Tiersten, 1963; Marsden and Hughes, 1983; Povstenko, 1994; Epstein and de Leon, 1998; for related computational techniques, see also Simo et al., 1992). Such an approach to the mechanics of elastic structures have been used in many works since 1958 (see, for example, Amman, 1972, 1995; Amman and Marlow, 1993; Green et al., 1965; Green and Laws, 1966; DeSilva and Whitman, 1969, 1971; Ericksen, 1970; Naghdi, 1972; Simo and Vu-Quoc, 1988; Villaggio, 1997; for computational and stability aspects, see also Fox and Simo, 1992; Simo and Fox, 1989; Simo, et al., 1988, 1989, 1990). Various suggestions to adopt the Cosserats' scheme to describe dislocated structures in crystalline solids have been discussed. Triads of vector-order parameters have also been used by Davini (1986) to introduce a continuum theory of defective crystals (see also Davini and Parry, 1991). Within different settings, vector-order parameters have also been used by Ericksen to describe the behavior of macromolecules within a body (1960, 1962a) and to begin the modern continuum theory of liquid crystals ( 1961, 1962b,c, 1991) that has been further developed by Capriz (1988, 1994,) Capriz and Biscari (1994), and Virga (1994). Mindlin (1964) considers each material patch to be an elementary cell ("interpreted as a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material," p. 51) that can deform independently of the surrounding medium. A second-order symmetric tensor-valued order parameter is assigned to each cell. Such a continuum is usually called micromorphic (see also Grioli, 1960, 1990; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000). The approaches of Ericksen and Truesdell (1958), Grioli (1960, 1990) Mindlin (1964) and Toupin (1964, 1965) are special cases of continua with affine structure (see Capriz and Podio-Guidugli, 1976, 1977; Capriz et al., 1982). Higher-order micromorphic continua have been introduced by Green and Rivlin (1964) and further discussed by Germain (1973). A proposal of a nonlocal theory of micropolar continua can be found in Eringen (1973, 1976). For further studies, see Wang and Dhaliwal (1993). Scalar-order parameters were used in 1972 by Goodman and Cowin for describing granular flows. In 1979, Nunziato and Cowin used an analogous approach to

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introduce a nonlinear theory of elastic porous materials (further studies concerning this topic can be found in Cowin and Nunziato, 1983; Cowin, 1985; Dhaliwal and Wang, 1994; Nunziato and Walsh, 1978; Diaconita, 1987; Fr6mond and Nicolas, 1990; Mariano and Bemardini, 1998). Scalar-valued order parameters have been also used to describe phenomena of recrystallization (Gurtin and Lusk, 1999), general solid-solid phase transitions (e.g., Colli et al., 1990; Fr6mond, 1987; Fried and Gurtin, 1993, 1994, 1999; Fried and Grach, 1997), solidification phenomena (Anderson et al., 2000), and isotropic damage evolution (Markov, 1995; Fr6mond and Nedjar, 1996; Fr6mond et al., 1999). Anisotropic damage has also been studied from the point of view of multifield theories by Augusti and Mariano (1999; see also Mariano and Augusti, 1998; Mariano, 1999, and references therein) by using tensor- or vector-valued order parameters. Models with scalar-valued order parameters can be considered special cases of materials with "spherical" structure (Capriz and Podio-Guidugli, 1981). On the basis of classic Lagrangian dynamics of systems of particles, a first attempt to construct general framework for continua with substructures, at least in the case of holonomic-order parameters, has been proposed (Capriz and Podio-Guidugli, 1983). Capriz (1985) introduced the concept of latent microstructures, proving that some higher-order gradient theories of continua can be considered to be multifield theories with appropriate internal constraints. In 1989, Capriz proposed a general theory that is useful for establishing order parameter-based models of continua with substructure. This work opened the way to many theoretical questions, some of which are discussed in Capriz and Giovine (1997a, 1997b), Binz et al. (1998), Segev (1994, 2000), Capriz and Virga (1994), and Mariano and Capriz (2001). In 1990, Capriz and Virga adapted Noll's axioms on interactions in continuous bodies to account for self-interactions among microstructures described by order parameters in linear spaces. In principle, one may think that a body with a fine distribution of voids (or vacancies or microcracks) can be obtained from a mathematical point of view as a limit of a sequence of bodies. In other words, one takes the region occupied by the body in some configuration and at each step of the sequence considers different sets of discontinuities, assigned (perhaps) with some rules. Then one calculates the limit of the sequence and accepts the limit region of the Euclidean space obtained as a reasonable picture of the original finely fractured body. These limit processes are analogous to those used to reach the optimal shape of bodies under some optimum conditions (Kohn and Strang, 1986a,b,c). In 1993, Del Piero and Owen showed that an appropriate fabric tensor describing the influence of microcracks on the macroscopic deformation can be obtained as a consequence

Multifield Theories in Mechanics of Solids

9

of limit of sequences of bodies and corresponding deformations. In 2000, Del Piero and Owen showed that even the vector-order parameter describing liquid crystals (according to Ericksen's theory of nematic liquid crystalsmsee Ericksen, 1962b,c; Capriz, 1988, 1996) can be obtained with a procedure involving the limit of bodies. General results on the evolution of discontinuity surfaces and related configurational forces in continua with substructures during solid-solid phase transitions or during the evolutions of defects have been obtained (Mariano, 2000a, 2001).

II. Configurations and Balance of Interactions

A. CONFIGURATIONS As suggested in Section I, the complete placement of a material body B is described by mappings of the type K ' B ~ g3 x Ad

(1)

assigning to each material patch P of B the pair (placement, order parameter). Of course, g3 is the three-dimensional Euclidean point space, whereas Ad is the collection of all possible configurations of the substructure and is considered a finite-dimensional differentiable compact manifold without boundary. The mapping

KE3 "B ~ g3

(2)

assigning to each material patch its placement, defines the apparent configuration, (i.e., a representation of the body in which the substructure is forgotten). Moreover, KM'B

--+ .A4

(3)

defines the order parameter mapping. In this way, each K is a pair (Kg3, K.A4). For future use, an apparent reference configuration Kg3 is considered, with I(E3(B) being indicated by/3. It is assumed that B is a bounded connected regular region 3 (a fit region) of the Euclidean space and is endowed with a coordinate system {X}. 3Details about the minimal topological requirements necessary for/3 to develop continuum theories can be found in Noll and Virga (1988) and Del Piero and Owen (2000). For the purposes of this Section, one may thinkmroughly speaking--/3 as a bounded connected set that coincides with the interior of its closure and is endowed with a surface-like boundary with well-defined unit normal to within a finite number of corners and edges.

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For each K, the placement field is indicated by x(.) = (Ke3 o

~-l E~ j (.)

(4)

and the order-parameter field is indicated by

More precisely, given X 9 13, x(X) is the placement of a material patch P resting at X in/3, whereas qo(X) is a descriptor of the substructure of the same patch. At each X, the order parameter qo(X) is an element of.M, and .M itself is a nonlinear manifold, in most cases. 4 For each K, it is assumed that 9 x(/3) is also a fit region 9 x(.) is a one-to-one mapping of 13 into ~3 and is continuous and piecewise continuously differentiable 9 the gradient of deformation Vx, indicated with F, is such that detF > 0; that is, x(.) is orientation preserving 9 r

is continuous and piecewise continuously differentiable on/3

In this way the space of the configurations is the collection s of pairs (x(.), qo(.)), each one deriving from the corresponding K. s may be endowed by the structure of a manifold; its tangent space is indicated by T ff, whereas the cotangent space is indicated by T*s Example 1 A useful example to illustrate the statements presented previously is the direct modeling of plates. To this aim, consider an orthogonal frame of reference in s namely {Oele2e}, with O the origin, and a compact bounded set A in the plane ele2. If {X*} is a coordinate system in ele2, with X* = XTel + X~e2, the complete reference configuration of the plate is given by the set

X*+~elX*eA,~ 9

-~,~

(6)

where h is the thickness of the plate. In this picture, A is the apparent configuration. 4Some authors embed .M in a linear space and work consequently, using the handy properties of linear spaces. Their reasoning is based on the hypothesis that .M is finite-dimensional. Each finite-dimensional manifold may, in fact, be embedded into an appropriate linear space (Withney's theorem). However, such an embedding is not unique, and the question of what embedding is phisically significant is completely open. The only clear general results on this point of view are in Capriz and Virga (1990).

Multifield Theories in Mechanics of Solids

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To define the deformed configuration of the plate, two fields must be defined on A 9 x(.)" A ~ ~3, identifying the placement in ~3 of the points of the "midplane" of the deformed plate; it maps A onto a surface x(A) in ,f3 9 qO(') assigning to each X* a vector t belonging to the unit sphere S: = {t 6 ~3llt I -- 1}; i.e., ~(-) 9 (A) ~ S 2 Therefore, in this case, A// coincides with S 2. It is assumed that x(A) is a regular surface in ~.3 and (xl • x 2 ) . t > 0

(7)

everywhere in A where the fields are defined. In (7), x l(X*) and x z(X*) are tangent vectors of x(A) at x(X*); in particular, x,1 is the partial derivative of x with respect to X~ and X,e has the analogous meaning. Condition (7) imposes that t is never tangent to x(A) and excludes the physically unreasonable situation of infinite shearing deformations. Finally, provided the validity of (7), the current complete configuration of the plate is given by the set

x(X*)+~tlX*eA,

teS 2,~

-~,

(8)

Because t need not be normal to the deformed middle surface x(A), shear deformations are allowed in this description of plates. Moreover, the assumption that t is of unitary length precludes thickness changes and cannot take into account initial variable thickness. To account for thickness stretch and initial variable thickness, it suffices to require only Itl > 0 (not the more stringent condition Itl = 1).

B. MOTIONS

Motions are time-parameterized curves (xt, qPt) in the space of configurations, and at a given instant t ~ [0, d], the current placement and the order parameter, respectively, associated to each X ~ B are given by x(X, t)

~o(X, t)

(9)

Moreover, the velocity fields are given by ~(-, .)

~(., .)

(10)

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where the dot over x and qo means time derivative. Of course, qb belongs to the tangent space of .A4, and the pair (~(., t), ~(., t)) belongs to Tff. In the following, Vel indicates the set of pairs of fields (:~, ~). Consider two different observers differing by a rotation described by a proper orthogonal tensor Q with corresponding vector q (i.e., Q = exp(oq), (where e is Ricci's three-dimensional permutation indicator). These two observers evaluate two different values of qoBfor example, qOq and qp--connected by the following relation:

I

t.pq -- qg) -~- --~-q q=O

q + o(Iql)

(11)

If a time-parameterized family of rotations q(t) is now considered, inserting q(t) in (11) and evaluating the time derivative, it follows that, to within higher-order terms, ~q = d~oq dq

cl

(12)

q=0

where/1 is the angular velocity. The term (dqoq/dq)lq=0 is indicated in the sequel of this article with .,4; it is an operator mapping vectors of R 3 into elements of the tangent space of M . In terms of coordinates, ,4 is of the form .147, in which Greek indices denote (here and in the following discussion) the components of the atlas of coordinates on M , whereas Latin indices denote the coordinates in ,s In its matrix representation, .,4 is a (dim .A4 x 3) matrix (three columns and a number of lines equal to the dimension of AA). In the mathematical parlance, ,4 is the infinitesimal generator of the action of the orthogonal group S0(3) on .A4. With these premises, one can say that velocity fields (10) are rigid (and one indicates them with/~R and ~R) if XR - - c(t) +/1 • (x - x0);

r

-- ,,'d-Cl

(13)

C. MEASURES OF INTERACTION

Granted the possibility of defining a covariant gradient of the order-parameter field, indicated with Vqp, a set Jl (C) whose elements are of the form

(x, F, ~a, XT~a)

(14)

may be built up: in the geometric parlance, it is the first jet bundle on the manifold (space of the configurations).

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Analogously, J1 (Vel) indicates the set whose elements are of the type

(~5)

(~, F, ~, V~)

The power 79 is defined here as a real-valued functional on ,71 (Vel); that is,

79:Jl(Vel) ~ IR

(16)

and accounts for both ordinary and substructural interactions. In addition, it is assumed that, as usual, 79 is additively decomposed into external and internal contributions, for any part B* of the body T't3, = 79~xt - 79~t

(17)

where with the term part of B indicates any subset of B that is also a fit region. The basic problem is thus the representation of T'~t and T'~ t. Assuming the validity of (16), it is necessary to introduce measures of interaction acting on all the elements of (15) and developing power on them. Taking this into account, the following expressions for T'bxt and 79~t are assumed to hold for any part 13" of B:

~, (t 9i + 7- 9 qb) dA

~int

(s.~ +z.

dV + ft~ ( T . F + S . ~7~)dV

(18)

(19)

where the measures of interaction in (18) and (19) have the following meaning: 9 b

external bulk forces

9 /3

external bulk interactions on the substructure

9

boundary traction

9

7"

generalized boundary traction associated with the substructure

9 s

zero stress

9 z

internal self-force

9 T

first Piola-Kirchhoff stress tensor

9 S

microstress tensor

From this point, volume, area, and line differentials (namely dV, dA, and dl) are omitted in the integrals to render the formulas more schematic. Of course the

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Paolo Maria Mariano

reader will understand immediately the kind of differential he needs to use in developing explicit calculations by looking directly to the set on which the integral is calculated. Cauchy's theorem assures that Tn - t

on 0/3*

(20)

where n is the outward unit normal to the boundary of/3*, indicated with 0/3*; moreover, 5 S n - 7-

on 0/3*

(21)

One can write (19) when it is possible to define a connection by which the gradient of qo may be evaluated in covariant manner. This allows for the decomposition of, the substructural contributions to the power in terms of densities/3, qb, z . qb, and S . V gb. However, such a contribution could be expressed by some general complicated functionals of qo, qb, and their spatial derivatives, or could even disappear, as in the case of internal variable schemes. Note that, if necessary, one may define the power on the second jet bundle of ~s or the third, and so on. In (21), the nature of 7- is that of a generalized boundary traction that should be assigned at the external boundary of the overall body where S n - ~'. However, some microstructures, such as porous or microcracked solids, do not allow prescribed boundary data of the type ~'. A pore (or a microcrack) is determined by the surrounding medium and not by itself; therefore, it does not exist at the boundary (pores and microcracks can be considered "virtual" substructures). Hence the boundary data must be specified through a constitutive prescription, or are obtained as a result of some limit procedure based on shrinking some boundary layer at the boundary of the body. However, 7- can be completely prescribed as boundary datum in the case of liquid crystals or other material substructures. Analogous reasonings hold in the case of Dirichlet data (i.e., when values of the order parameter must be assigned on the boundary of the body). It may be that in some situations boundary layer data must be accounted for; however, the treatment of these situations is almost completely open.

5The existence of the microstress tensor by a Cauchy-like theorem is discussed by Capriz and Virga (1990), but in the case in which it is assumed that the manifold .A,4is embedded in some linear space. Probably a more general proof can be made by following the variational procedure in Fosdick and Virga (1989) for Cauchy continua because such a procedure can underline the need for the existence of the covariant gradient of ~o. A general proof of the existence of the microstress tensor on the basis of geometric measure theory might obtained by using the results of Degiovanni et al. (1999). A generalized Cauchy's theorem on manifolds has been obtained by Segev (2000).

Multifield Theories in Mechanics of Solids

15

Balance equations can be deduced from (17), (18), and (19) by assuming (as axioms) that at the equilibrium 9 the overall power T'B, vanishes for any choice of the velocity fields :~, qb and of any part B* of B 9 the internal power T'~t vanishes for any choice of rigid velocity fields and of any part 13" of/3 To this aim, equations (20) and (21) must be used together with Gauss theorem. In applying such a theorem, it is assumed that discontinuity surfaces, lines, and points for the fields involved in the integrals are absent. After some calculation, the overall power can be written in the following form:

T'B, = ft3, ((b

- s + DivT). i + (/3 - z + DivS). qb)

(22)

The requirement that ~e, = 0 or any choice of the pair (~:, qb) and of the part B* of B implies that b - s + DivT = 0

in B

(23)

/3 - z + DivS = 0

in B

(24)

Equation (23) is the standard Cauchy's balance to within the zero stress that must be formally introduced as a consequence of (16); equation (24) is the balance of substructural interactions, a sort of generalized balance of momentum. General information on the structure of s and T are obtained by exploiting the axiom requiring that T'~t vanishes when it is calculated on rigid velocity fields [defined by (13)] and on any part of B. By applying such an axiom, in fact, it follows that

eTF r = ATz +

(25)

inB

s=0

(vAT)S

in B

(26)

Of course, by (25), equation (23) reduces to the well-known balance b + DivT = 0

in 13

(27)

In addition, note that when the influence of the material substructure is negligible and order parameters are not considered, equation (26) reduces to skw(TF r) = 0

(28)

16

Paolo Maria Mariano

which is the standard property of symmetry of Cauchy stress tensor TF T [skw (.) extracts the skew-symmetric part of its argument]. Previous balances hold in general dissipative situations and must be supplemented by appropriate constitutive equations.

D. BALANCE OF INTERACTIONS FROM THE INVARIANCE OF OUTER POWER

The procedure discussed in Section II.C has been exploited in different cases to derive balance equations. One may question, however, if such a procedure needs too many hypotheses that may be relaxed. In particular, one might find a more general procedure in which the expression of inner power follows as a consequence and is not an axiom. This would be desirable because, in principle, it is possible to evaluate the outer power by experiments only. In Cauchy's solids, Noll's procedure requiring the invariance of the outer power with respect to changes of spatial observers allows one to obtain standard balance of forces and the symmetry of Cauchy stress. An analogous procedure can be followed in the case of materials with substructure. It underlines some delicate questions that may emerge when one selects some order parameter and tries to write balance equations. Basically, one writes the outer power in (18), taking into account (20) and (21), as

extf ~. --

(b. ~ +/3.

*

~b) +

~*

( T n . ~ + S n . ~b)

(29)

and requires the invariance of 7J~' with respect to all changes of observer. This is a request of invariance with respect to Galilean and rotational changes of spatial observers. Changes of spatial observers are typically given by ~* -- ~ + c(t) + ~l(t) x (x - x0)

(30)

(p* -- (p + A q ( t )

(31)

where ~* and @* are the fields evaluated by the observer after its change and e(t) is the translational velocity. As is common in multifield theories, in writing (31), it is assumed that rigid translations have no effects on the values of r Such an assumption applies even in the case in which the order parameter is a microdisplacement because it is always a "relative" microdisplacement. Galilean changes of observers are obtained from (30) and (31) with the choice ~ - 0, ~1 - 0, whereas rotational changes of observers are characterized by ~ - 0, ii - 0.

Multifield Theories in Mechanics of Solids

17

The requirement of invariance of (29) under transformations (30) and (31) implies

b + f ~, Tn)+ q. (f, , (x • b +

+ f ~, (x x Tn + AT$n))--0 (32)

for any choice of c and Cl and the part/3". The arbitrariness of c and ~1implies

f b+f Tn-0 *

dB*

(33)

O*

, ] 0 O*

Equation (33) is the standard integral balance of forces. The arbitrariness of/3* and the application of Gauss theorem imply from (33) that b + DivT = 0

(35)

From (34), taking into account the validity of (35), the following relation follows from the arbitrariness of/3* and the application of Gauss theorem: e T F T = .AT/3 + Div(.A TS)

(36)

or, by developing the divergence, e T F T - AT/3 + ATDivS + (VA T) S

(37)

To assure the validity of (37), two conditions must be satisfied. It is necessary that all the elements that are not multiplied directly by AT be equal to the product of A T with some generic element z of the cotangent space T*.A4 of .A4 (/3 and DivS are elements of the cotangent space of.A/l); that is, the existence ofz e T*.A// is necessary such that

ATz = e T F T - ( v A T ) s

in/3

(38)

which coincides with (26) and represents the generalized balance of "couples." Equation (37) can be thus written as

.AT (3 + Div8 - z) = 0

(39)

and is satisfied when the term in parentheses belongs to the null space of the linear operator A T6 /3 + Div8 - z 6 null space of AT

6AT is a matrix with three lines and a number of columns equal to dim .AA.

(40)

18

Paolo Maria Mariano

which is the second condition. In general, the null space of .Ar is the space orthogonal to the range of ,,4 (range of ,,4)• = null space of .,4r

(41)

when a concept of orthogonality is available on T.A4. The range of A at each qo is by definition a subset of the tangent space of .A4 (namely, T.A/[) at the same qo or it is coincident with the whole T.A4 at go. When the range of ,4 is coincident with the whole tangent space of .A4 at qo, the elements of its orthogonal in T.A/[ reduce to the singleton {0}; then the term/3 + DivS - z must be equal to the sole element of the orthogonal "of range of A." In other words,/3 + DivS - z vanishes identically /3 + DivS - z = 0

(42)

There are counterexamples in which the range of A does not coincide with the whole tangent space of .A4 at a certain qo. The prominent counterexample is the case in which the order parameter qo coincides with a stretchable vector cl; this happens in models of shells with through-the-thickness shear or of microcracked bodies. In this case, ,A = ocl, with o Ricci's tensor. It is obvious that at 0 = 0, the range of A coincides with the singleton {0}. Another counterexample is the model of porous bodies in which the order parameter is the scalar void volume fraction; in this case, .A4 coincides with some interval [0, a] of the real axis, and A vanishes identically. This follows by considering that for porous bodies, the value of qo remains unchanged by rigid rotations: then qgq = 99 and the derivative (dqgq/dq) Iq=o vanishes identically. Models of microcracked bodies also make use of second-order symmetric tensor-valued order parameters ~ij and the operator ,,4 coincides with (Oijr~rk -1- Eirerjk). Even in this case, at E = 0, the range of A coincides with the singleton {0}. In the case in which the range of ,A does not cover T.A/[ at any qo, the argument leading to (42) is not exhaustive, and one must write the differential inclusion (40) as

/3 + DivS - z' = - z "

(43)

where z' satisfies (38) and z" belongs to the null space of A r (i.e., A r z '' = 0). Consequently, (42) still holds, taking for z the difference z ' - z". This property cannot be derived by using the procedure in Section II.C. Example 2 An example of the occurrence of a term like z" in the balance of substructural interactions can be found in the theory of liquid crystals. The usual order parameter of liquid crystals is a vector d belonging to the unit sphere S 2 in/t~3.

Multifield Theories in Mechanics of Solids

19

A coincides with od, which can be also written as d • In balancing substructural interactions, it is required that the covector/3 + DivS - z at each material patch be parallel to the "averaged" direction d of the rodlike molecules of the liquid crystal at the same patch. Therefore, in this case, the balance of substructural interactions is written as

tic + divSC _ z c = c~d

(44)

with c~ some scalar constant and the operator div is calculated in the current configuration (see Ericksen, 1991). It is evident that otd is - z " because d x d = 0. Of course, (44) is written in the current configuration because one deals with a liquid. Therefore, tic, Sc, and z c are measures of interaction in the current configuration. The request of invariance under changes of spatial observers could perhaps not be conclusive in some cases (thanks to the arbitrariness of z'), and more stringent requests of invariance could be required to justify completely the balance equations. A possible way consists in requiting the invariance with respect to all possible representations of each type of substructure. Suggestions for such an invariance follow from the knowledge of situations in which some of the components of the order parameter can be chosen arbitrarily. If one requires in fact that some scalar function k of ~ - - a s , for example, the substructural kinetic e n e r g y - - b e invariant under the action of the rotation group S0(3), it is necessary that

.A T(O(,,k(~)) = 0

(45)

This is a system of partial differential equations in 2(dim .M) + 1 variables. Following a theorem of Tricomi (1954), it is possible to show that previous system admits 2(dim .M) + 1 - char A independent variables as solutions, where char.A is the characteristic of .,4 because .A is a matrix with three columns and a number of lines equal to dim A/[. Such independent variables can be chosen arbitrarily by an external observer. Under such a suggestion, one could think to map .A4 into other manifolds A/" by Cr-diffeomorphisms (r > 1), indicated with Jr, and call these diffeomorphisms 7 representations of the substructure when dim .M = dim A/'. If two elements on .M are related by a rigid rotation, then the corresponding elements on A/" (through the mapping zr) must be related by a rigid rotation as well. To each Jr : . M --+ A/', a mapping Trr is associated; it maps elements of the tangent space of M (i.e., TA4) into elements of the tangent space of A/" 7Roughly, a diffeomorphism is a one-to-one differentiable mapping such that its inverse is differentiable as well.

20

Paolo Maria Mariano

(i.e., TA/'). The mapping TJr may be then described by appropriate Jacobian matrices J, such that if qb is an element of TA/[, then Jqb belongs to TA/'. In other words, defining qo' - zr(qo), it follows that qb' - ~ and Jl - dqo'/dqo. The requirement of invariance of T'~xt under all possible representations of the substructure reduces to the invariance under changes qb --+ Jqb. To render T'~xt invariant under changes qb ~ ~ , it is necessary that

(46) The arbitrariness of qb and the application of Gauss theorem reduce (46) to (I)T(/3 + DivS) + (VJT)S = 0

(47)

where (I) = I - J. The validity of (47) implies the existence of some element i of the cotangent space of .A4 such that (I)Ti + (V2 T) S = 0. Thus (47) reduces to (I)v (/3 + DivS - ~) = 0 and the term in parentheses must vanish identically as a result of the arbitrariness of,II, thus of (I). However, this conclusion is only formal because the condition ~T~ + (VjIT)S = 0 may imply that S vanishes identically, as a result of the arbitrariness of dl, so as to obtain a reduced balance of the form /3 = 0

(48)

Among other things, such a conclusion could induce doubts about the expression of the external power 79~xt. There are situations in which nonlocal terms could appear in the expression of 79~,t through some general functional that may be expanded in series by means of some "myopia" theorems formally analogous to the theorems of "fading memory" used in standard continua with memory. In the former case, such theorems reduce the nonlocal influence of material patches, surrounding the one assigned, to a rather weak nonlocality in space, whereas in the latter, theorems of "fading memory" cut the influence of the events in the past on the present event (the nonlocality is thus in time). In any case, the validity of balances (35), (38), and (42) implies the proposition in the following. Proposition 1

By virtue of the balance equations (35), (38), and (42), it follows

that

(49)

and the last integral in (49) takes the name of internal power and is indicated with 79~t.

Multifield Theories in Mechanics of Solids

21

Note that (49) was used in Section II.C as an axiom [see (18) and (19)] to deduce balance equations (24), (26), and (27) by means of the "virtual power procedure." In this section, another procedure has been followed; only the explicit expression of the external power is assumed as an axiom, then the invariance with respect to changes of external observers is requested. This procedure is in essence more general than that in of Section II.C because it underline the need of elements of the type z" and the expression of the internal power has been deduced in this section as a theorem.

E. EFFECTS OF INERTIA One possible way to account for the effects of inertia pertaining to both the macroscopic motion and possible internal vibrations of the substructures consists of decomposing the volume forces b and/3 in their inertial (in) and noninertial (ni) parts as b = b in - [ - b ni

(50)

j~ _. j~ in ..[_ j~n i

(51)

and in assuming that, for any part B* of/3, d {kinetic energy of 13"} +

~ ) __ 0

(52)

rate of kinetic energy + power of inertial forces -- 0

(53)

dt

(b in x + j~in

9

In this way, one assumes the validity of the balance

and may interpret (53) as a constitutive prescription on the explicit expression of inertial terms once an expression of the kinetic energy has been selected. Although the kinetic energy density associated with the macroscopic motion is proportional to I/~l2, in fact, the kinetic contribution of substructures may have some complicated structure that depends on each special model and, in a certain sense, has constitutive nature. The kinetic energy of 13" is given by 1

ft3, (-~pi~. ~ + k(qo, (p))

(54)

where l p/~./~ is the standard kinetic energy density of material particles and

22

Paolo Maria Mariano

k(qo, qb) is the contribution of the substructures of the material to the kinetic energy density. The term k (., 9) is a nonnegative function such that k(., 0) = 0

(55)

02~k -r 0

(56)

where, here and in what follows, 0y means partial derivative with respect to the argument "y". Really, the symbol 0 was previously used before letters indicating sets. When 0 precedes a letter indicating a set (e.g., 0B), it indicates the boundary of B, whereas when 0 precedes any function, it means partial derivative. Moreover, k (-, 9) must be frame indifferent; that is, taking into account (11), it is necessary that k(qo, qb) - k(qo + Aq + o(Iql), qb + A/I + o(Iql))

(57)

for every q. The invariance condition (57) implies O~kA + O(ok(O~oA)(o - 0

(58)

Equation (58) can be obtained by developing in series the right-side term of (57) around k(qo, ~). Substituting (54) in (52) and taking into account that (52) must hold for any part B*, it follows that b in = -p:~ -

(59)

~-~-,geX - ,9~X

)

(60)

where X is called substructural kinetic coenergy density and is such that k -- O~X. O -

X

(61)

In the mathematical parlance, the kinetic coenergy is the Legendre transform with respect to ~ of the substructural kinetic energy density k. As a consequence of (59) and (60), the balances (35) and (42) become DivT +

b ni - - p ~

DivS - z + ~ni _

d dt

O(oX -- O~X

(62) (63)

It is noted that inertial effects ~in associated to the substructure of materials are very minute and not perceivable unless the substructure itself oscillates at high frequencies, as indicated by some experiments on the scattering of phonons within the lattices of crystalline materials, or on liquid crystals.

23

Multifield Theories in M e c h a n i c s o f Solids

The explicit expression of k is of constitutive nature; a possible quadratic form, 8 like the simplest one given by 1_ k - ~ D ( ~ , ~)

(64)

determines the metric on .M. In (64), the brackets and the comma (i.e., (.,-)), indicate the scalar product on TAd, the tangent space of Ad, and /) is some appropriate constant chosen to adjust eventually physical dimensions. When k is constitutively prescribed, the kinetic coenergy density can be obtained by solving the partial differential equation (61), whose solution X(qO, ~) is the sum of a special solution of the complete equation Xs and the homogeneous solution Xh, corresponding to k -- 0:

(65)

X = Xs + Xh

The explicit solution of (61) can be found in (Capriz and Giovine, 1997a). If k is homogeneous of second degree (hsd) in qb, then it coincides with ,~s. Finally, the inertial contributions of the material substructures can be written as follows:

[~in ( d d = -

- ~ 04, Xs - O~ X, + --~ Or

-- 0,r Xh

)

(66)

where the term d d-t 0r Xh - 0,r Xh

(67)

is powerless, that is,

(d

O~Xh - O~,Xh

)

9 0 -- 0

(68)

When k = O, the solution of equation (6 l) prescribes that the kinetic coenergy density X from (61) can be, at most, linear in O; in this case,/3 i" can also be, at most, linear in O, and situations of parabolic evolution can arise, as in the case of magnetostrictive solids or ferroelectrics. When one considers a relative velocity Oret of the order parameter and writes Orel + .,4(:1instead of the absolute velocity qb, it is possible to underline the presence 8Quadratic expressions for the substructural kinetic energy density can be found, for example, in the case of direct models of plates (Antman, 1995), multifield descriptions of ferroelectric (Da~i and Mariano, 2001), or microcracked bodies (Mariano, 1999) when microdisplacements are considered as order parameters. More general expressions must be considered when, for example, there is an intrinsic limit velocity for the propagation of perturbations in the materials. For example, in crystalline materials the dislocations cannot propagate with a velocity greater than the velocity of sound.

Paolo Maria Mariano

24 of a centrifugal term

a 2~ X ( ( O~Ail)(Ai~))

(69)

( O2cpX) .,461

(70)

- 2(0~,(.A/I)) TO~g

(71)

an entrainment term

and a Coriolis term

associated to substructural dynamics. 1 When k is a quadratic form in ~b (e.g., k(~, ~b) - 5~b. N~b, with some constant tensor N) and the velocities are only of rigid rotational type (/1 x x and A/l), l the total kinetic energy of the body becomes 5/1. (J -+- H)(1 where J is the standard moment of inertia given by f8 P (]x]21 - x | x) and H - f8 ATN'A" The final expression of the kinetic energy seems not to be compatible with the classical dynamics of rigid bodies. This paradox can be eliminated by imposing that

k(t,p, ~ ) -

k(qo, ~rel)-

Moreover, there are cases in which F is involved in the kinetic energy density. This is the case of liquid crystals when one chooses as order parameter an observerindependent vector given by F -l d, instead of an element d of the unit sphere S2 in R 3. A complete theory of liquid crystals that accounts for this has not yet been developed. Inertial interactions may also involve spatial derivatives of the acceleration fields. In this case, possible values of the order parameter are subjected to some internal constraint, so the substructure becomes latent (see Section VIII). A prominent example is given by models of capillary phenomena described by Korteweg's fluids, in which the order parameter is scalar and is coincident with det F (see Capriz, 1985). R e m a r k 1 Let the invariance of the kinetic energy density be required under Galilean changes of observers. Let also the invariance of the overall energy (sum of internal and kinetic energy) be required under rotational changes of observers. As a consequence, after some calculations not entirely trivial, the inertial contribution of the substructures results in d dtOCpk - O~k

(72)

When k is homogeneous of second degree (hsd) in ~, or when X is so, (72) may substitute the term in parentheses in (60). In this case, the two expressions of the inertial contributions are equivalent to within powerless terms. When the

Multifield Theories in Mechanics of Solids

25

substructural kinetic energy density requires more complicated expressions than hsd, the problem of the equivalence of the two procedures just explained is far from being completely clarified. Cases different from hsd may be necessary when a limit speed of the substructural perturbations (thus of ~) must be emphasized; invariance with respect to rules of changes in observers more general than the action of the rotation group could be necessary, and the consequences of the relevant gauge invariances examined. Remark 3 When cases in which the balance of substructural interactions reduces to/3 = 0 occur, from (51) and (60) it is possible to write only a kinetic equation of the form

t~ni__ d oqx(u q~rel) _ oqx(~o, q~grel) dt O0rel 0r

(73)

An example of absence of microstress in models of bodies with substructure is the theory of liquid with bubbles presented in Kiselev et al. (1999).

BIBLIOGRAPHIC NOTE Balance equations (24) and (26) are Capriz's (1989), their presentation in Section II.D follows that of Mariano (2000a). Mathematical details on the topics in Section II.D can be found in Di Carlo (1996). Additional remarks on the geometric nature of microstresses and self-forces can be found in Segev (1994, 2000) and Capriz and Giovine (1997a,b). Details on the examples before Remark 2 can be found in Capriz (1989, 2000). For a derivation of balance equations, see also Capriz and Podio-Guidugli (1983) and Capriz and Virga (1990, 1994). Discussions on the derivation of balance equations by means of virtual power arguments (involving the assumption a priori of the expression of the inner power)can be found in Germain (1973), but they are referred only to order parameters that are higher-order perturbations of the displacement field, then only first- and higher-order micromorphic materials are treated (see also Maugin, 1990). The procedure to obtain balance equations for substructural interactions on the basis of the invariance of the external power 7~! under changes of observers has been developed in Capriz and Virga (1994), Capriz (2000), and Mariano and Capriz (2001). Noll's classic procedure of invariance of external power can be found in the 1973 article. "Myopia" theorems for spatial nonlocalities and their applications to Cauchy's continua are in Capriz and Giovine (2000), whereas analogous "fading memory" theorems are in Coleman (1971).

Paolo Maria Mariano

26

All details about the relations between the substructural kinetic energy and the substructural kinetic coenergy are in Capriz and Giovine (1997a).

III. Elastic Materials with Substructure A. VARIATIONALCHARACTERIZATION

In this section, only hyperelastic materials with substructure (or, with some abuse, just "elastic") are examined. They are characterized by the existence of an elastic energy density, indicated with w, such that 6f,. w=f,.

(T 9~iF + S 93(Vqo) + z 96qo)

(74)

for all parts B* of/3, where ~ indicates the variation operator. 9 A consequence of this definition is the possibility to write the measures of interactions in terms of F, V qO, qO, once the expression of w has been selected. In general, in fact, one can assume w = t~(F, qo, XTqo)

(75)

Consequently, by calculating the variation of the first integral in (74), it follows that

f ( ( O v w - T). ~F + (~v~w - S)-~(XTqo) + (O~w - z). 6qO) = 0

(76)

Because variations can be chosen arbitrarily in (76), the following constitutive restrictions hold

Remark 4 structure:

0vw = T

(77)

Ov~w = S

(78)

O~ow = z

(79)

Consider the following special case of an elastic material with sub-

9 ~ni vanishes identically 9 t~ in

is only of powerless type and is given by Bqb, with B an appropriate tensor

9Of course, equation (74) can be written in terms of velocity fields.

Multifield Theories in Mechanics of Solids

27

9 w - t~(F, qo), thus the weakly non-local contribution of the order parameter due to the gradient of qo is neglected The balance (42) reduces to

B~o- O~w

(80)

and the self-force 0~,w becomes powerless. The order parameter assumes the character of an internal variable not satisfying balance of interactions that develop explicit power. Equation (80) is coincident with the evolution rule of an elastic internal variable. An analogous reduction of multifield models to internal variable ones can be obtained in nonconservative cases, as is shown in some parts of the following subsections.

B. SOME PROPERTIES OF LAGRANGIAN DENSITIES Before discussing some properties of Lagrangian densities for elastic materials with substructure, it is convenient to introduce a special symbol for a product between tensorial quantities that will be of future use both here and in following sections. This symbol is _,. The product ,_ is here defined as _," Lin(It~3, T.A4) x Lin(I~ 3, T*.A4) -+ Lin(It~3, R .3)

(81)

where Lin (It~3, •,3) is the space of linear forms associating three-dimensional covectors (belonging to the dual of R 3, which is usually indicated with R .3 and identified with R 3) to vectors in ]t~3 and Lin (It~3, T*.A4) is the space of linear forms o n R 3 taking values on the cotangent bundle ~~ T*.A4. Then, taking Vqo and the microstress S [which is a linear form associating elements of the cotangent space of.A//to vectors in ~;~3 thus an element of Lin(R 3, T'A//)], one writes by definition (Vqo r_,S)n 9v - S n . (Vqp)v

(82)

for any choice of vectors n and v. Note that when the order parameter is scalar valued, the product_, coincides with the standard tensor product | On the contrary, when qo is not scalar valued, the meaning of_, is not the one of a dyadic product. For example, if qo is a third-order ij tensor with components ~k ), one has (Vqo T,_S)jl n j 1) l - - Si~nJ(~rqo)i~l)vI. l~ cotangent bundle of.AAis the space of linear forms on the elements of the tangent space of .A.4.

Paolo Maria Mariano

28

Another product, indicated with +, is also of future use. It is defined as ,i, : T*.AA • Lin(R 3, T M )

~ ]~,3

(83)

thus its result is a covector, l~ Consequently, the products z,i,~'qo and/3+~'qo are covectors. For example, if qo is a fourth order tensor with components ~0/,nJ,,)one mn

ij

has (z+Vqo)l = zij (Vq~ Lagrangian densities for conservative dynamics in multifield theories are of the form

s -- s

x, ~, F, qo, qb, ~Tqo)

(84)

Of course, s depends on the metric y on the referential configuration 13 and on the metric on .A4 through the quadratic part of the kinetic energy k. Granted some regularity properties of the Lagrange density function, ~2 the following Euler-Lagrange equations hold: d

dtO~s - Oxs + DiVOFE -- 0

(85)

dtOCps - 0~,s + DivOv~,s - 0

(86)

d

To obtain a more compact form of equations (85) and (86), let the four-dimensional gradient V 4 be introduced. It is defined by

V4 -- ( ~ )

(87)

The four-dimensional divergence is indicated with V 4. , and then, by indicating with H and fI the derivatives 0V4xs and 0v4~,s respectively, equations (85) and (86) can be written as

V 4. lI - Oxl~ = 0

(88)

V 4" 1=I - 0qo,~ --" 0

(89)

I I With the same symbol +, a product + : Lin(R 3, T*.A/[) x Lin(R 3, Lin(R 3, T.M)) --> ]1~,3 is also indicated. It is not an abuse of notation, because the two products have the same structure. In this way, the product S+VV~0 determines a vector of I~3. 12See, for example, Renardy and Rogers (1993) for the discussion of regularity properties for s in the case of Cauchy materials. The extension to multifield theories is left to the reader as an exersize.

Multifield Theories in Mechanics of Solids

29

Let 1) be defined now by I~ -- s 4 -- V4x r FI -- V 4 (~T ,__1=i

(90)

where 14 is the four-dimensional unit tensor. Tensor 1) has physical dimensions of an energy density and is expressed in material coordinates. I) is also of second order, thus of the type ]?s, with indices r, ~ running 0, 1, 2, 3 (the coordinate 0 being the time). In the following, I? indicates the spacelike part of 1), is of the type ]?m with m, n running 1, 2, 3 and is given by P - s Proposition 2

FrT-

Vq~r_,S

(91)

If the equations of motion hold, then 4-r V~Ps + s

-0

(92)

where the comma as subscript represents in (92) the explicit partial derivative with respect to the coordinates. The proof of this proposition is based on the calculation of the derivative O~])~ and on a lemma stating that oy.....z ; - - ~ 1P m"

(93)

The proof of (93) is rather technical and can be found in (Mariano, 2000a) as well as the complete proof of previous proposition. An important Corollary is the following:

When the body is homogeneous and inertial effects can be neglected, for any closed sufficiently smooth surface within it, Corollary 3

fc,

Pn - fc, osed surface

( w I - F T T - vqoT . S ) n -- O

(94)

osed surface

Of course, n is the outward normal to the surface. The proof of Corollary 3 follows from the simple application of Gauss theorem. The integral (94) is the extended version of Rice's (1968) integral to multifield theories. Its importance in the study of crack propagation is explained in Section VII. The second-order tensor in the integrand, namely ( w I - F T T V qoT*,.q), is a modified version of Eshelby tensor that holds for elastic continua with substructure. If one inserts an inclusion in an elastic homogeneous body with substructure and defines the force ~ exerted on the inclusion by the surrounding medium

30

Paolo Maria Mariano

(following Eshelby, 1975) as the integral on a region 13" containing the inclusion in its interior, namely, -- -- f/3* /~,i

(95)

then, by (92) and (94)), in absence of inertial effects, one obtains - f ( w l - F T T - Vqpr_,S)n Ja B*

(96)

C. INFLUENCE OF THE SUBSTRUCTURE ON THE DECAY OF ELASTIC ENERGY

One of the central results of the linearized theory of elasticity is the proof of the longitudinal decay of elastic energy in cylinders loaded at one base only by equilibrated force systems. This phenomenon is usually known as Saint-Venant's effect. Here the analogous in multifield theories is shown and the influence of the material substructure on such a decay is evaluated. In this section, in which linearized situations are treated, reference and current configurations are identified with each other and the relevant measures of interaction are denoted with T, ,9, 2. The displacement field u - x(X) - X is introduced for convenience, and F replaced by Vu in the constitutive relations. Let (0X1X2X3) be an orthogonal coordinate system and D an open-bounded compact region in the plane X l X2. The body considered here is a semi-infinite cylinder if2 - / ) x [0, +cx~). In the following, n3 indicates the outward unit normal at /) (i.e., n 3 - - e 3 , with e3 the unit vector along X3), whereas nL indicates the outward unit normal at the lateral boundary 0/) • [0, +c~)./)(~'3) indicates the cross section at S(3 E [0, +c~), whereas f2(2"3, l) -/3(2"3) • [Yf3, Y(3 + l]. Moreover, it is understood in the following that

fa (.)=limf (X3)

l'---~~

(-'~3 ,/)

(.)

(97)

provided the existence of the limit. With these premises, the following assumptions apply" 1. External volume forces vanish: b - 0,/3 = 0; then DivT = 0

(98)

Div,9- ~ = 0

(99)

Multifield Theories in Mechanics o f Solids

31

2. The lateral boundary 0 b x [0, + e c ) is traction flee; boundary tractions are applied to b only and are self-equilibrated in the sense that Tn3 - 0

(100)

Sn3 - 0

(101)

x x 'l'n3 + ,,4T8n3) -- 0

(102)

fo fb( Moreover, it is assumed that lim f

(103)

(1"n3 9 u + ,Sn3 9 q)) = 0

J D(X3)

X3 ---~~

3. The elastic energy w(Vu, q0, Vqo) is a positive definite quadratic form in its variables. By indicating with y the triplet (Vu, qO, Vqo) in a way such that yl

= VII,

Y2 -- go, Y3 = V qo, the elastic energy density can be written as 1 w(Vu, qo, Vqo) -- -~aijYiYj

(104)

where aij is the ijth element of a matrix a expressed by

/

og~

a = (aij) =

Ogumlo 0g~

og .mlo

/

og mlo]

with o indicating a stress-flee state considered to be a natural (or reference) state of the body. Matrix a is such that aij = aji. It is assumed that there exists a > 0 such that aij Yi Yj <_ a ~'i f/i for any pair of constant ~7's. Assumption 3 also implies that w must be sufficiently regular to assure the validity of Taylor expansion around some "natural" state o, at least up to the second order, and that residual stresses are not accounted for; that is, it is assumed that Wo = 0;

8vuWlo = 0;

O~,wlo = 0;

Ov~oWlo= 0

(105)

32

Paolo Maria Mariano

Let U(X3) be the elastic energy of the part of the cylinder given by f2(X3) D(X3) x [X3, nt-oo). It is defined by

U(X3)- [ 113 J~ (x3)

(106)

The following proposition holds. It explains Saint-Venant's effect in elastic materials with substructure and is the main result of this section. Proposition 4

If the fields Tn3, ,.~n3, u, qp are square integrable, then

U ( X 3 ) < U ( 0 ) e x p ( X 3 - 1~ )_ _

(107)

for any 1 > O, X3 > I. The scalar )~ can be interpreted as the lowest non-zero characteristic value of the free vibrations of f2( f(3, l) with quadratic substructural kinetic energy.

When the ratio a/)~ is greater than the analogous ratio in the material considered without substructure, the presence of material substructures within an elastic body has a stabilizing effect and decreases the decay length of the elastic energy. Another kind of decay can be studied: the radial decay (Knops-Villaggio's effect) related to boundary conditions imposing that both u and qp vanish at the lateral boundary. It is possible to prove that the elastic energy of a cylindrical annulus at a given distance from/), of variable height and whose other surface coincides with the lateral surface of the cylinder, decays to zero algebraically at most.

BIBLIOGRAPHIC NOTE Section III.B is based on the first part of Mariano (2000a). With reference to Section III.C, the proof of Saint-Venant's decay in continua with substructure follows basically Toupin's (1965) proof in Cauchy's continua. Analogous proofs in the special cases of Cosserat continua and micromorphic continua have been developed in Berglund (1977) and Batra (1983), respectively. The Knops-Villaggio effect in the case of Cauchy's materials is proved in their 1998 article, whereas its possible discussion within the setting of multifield theories is in Mariano (2000b), along with the proof of Proposition 4.

Multifield Theories in Mechanics of Solids

33

IV. Balance in Presence of Discontinuity Surfaces The presence of discontinuity surfaces (also called interface) is a recurrent phenomenon in solids with material substructure. Interfaces between fluid and solid or solid and solid are evident in solidification phenomena and occur also in second-order phase-transitions between austenite and martensite. This section discusses how interactions can be balanced across discontinuity surfaces and at the junctions among them. Each discontinuity surface may or may not be considered to be endowed with its, own structure. In the former case, surface measures of interaction must be introduced because they take into account classic surface tension and a surface tension as a result of the material substructure.

A. INTERFACES: GEOMETRIC CHARACTERIZATION Within the body in its reference placement/3, an interface E is considered to be an oriented surface defined by E _= {X E cll3, f ( X ) =

0},

(~o8)

where the function f is smooth on/3. The orientation of Z is given with the normal vector field in, defined by m = V f / I V f l (Figure 1). Given any continuous and differentiable vector field e defined on Z, it is possible to define its surface gradient Vz by considering parametrized curves t on E and

V

FIG. 1. Geometric characterization of interfaces.

34

Paolo Maria Mariano

applying the chain rule e -- (Vze)i where i is the derivative of t with respect to the parametrization of the curve. The curvature tensor L of E is defined as the opposite of the surface gradient of the normal vector field, namely, L = -Vzm

(109)

whereas the overall curvature E is the trace of L KS = - t r ( V z m ) = - D i v z m

(110)

The previous formula allows one to define the surface divergence as the trace of the surface gradient. On each subsurface of E enclosed in a genetic part B* of B, a vector field u on the curve OB*NE is defined; it belongs to the tangent space of E at OB* A E m that is, v is the normal to 0B*N E in the direction of the tangent of E (see Figure 1). Given any field e that is continuous on B \ E, the limits e+ = lime(X + em, t),

X 6E

(111)

e---~0

define the jump [e] through [e] = e + - e -

(112)

when the difference makes sense. Given two such fields--for example, el and ezmthen, with some significance assigned to the product, [ele2] = [el](e2) + (el)[e2]

(ll3)

where (e) is the average given by (e)-

1 ~(e + + e - )

(114)

If the fields F and V q0 have the properties just mentioned for e, then a discontinuity surface E is called coherent if [F](I - m | m) = 0

(115)

[Vqo](l - m | m) = 0

(116)

where I is the second-order unit tensor, whereas (I - m | m) is the projector on E and condition (116) is taken here only for convenience in the sequel of this article. In the following, it is convenient to indicate with F the surface gradient of x, namely, F = V~x - (F)(I - m | m)

(117)

Multifield Theories in Mechanics of Solids

35

while with N the surface gradient of qo, namely N = Vzqo = (Vqo)(I - in | m)

(118)

Interfaces across which the order parameter is continuous are considered only in the sequel of this article; discontinuities are accounted for the gradients of the order parameter itself. Cases in which the order parameter itself can be discontinuous across E may occur. However, because qO is in general the element of a nonlinear manifold, the possible difference qo+ - qo- may not make sense. For example, if .A4 coincides with the unit sphere in the example in Section II.A, the difference of two arbitrary elements of such a sphere may not belong to the sphere itself. However, one can define the jump [qo] of qo even on a nonlinear manifold .A// when it is possible to find some group that acts on .A// as the translation group; that is, there exists on .A4 a given element @ such that it is possible to reach any other element of .A/[ by multiplying @ by elements of the group. In this case, the jump [qo] can be defined through the difference of the elements of the group. Alternatively, one may embed .A4 into a linear space by Whitney's theorem, even if the embedding is not unique. When qo is continuous across E, the jumps [~] and [V~o] are always defined because both qb and XTqoare elements of linear spaces.

B. B A L A N C E AT DISCONTINUITY SURFACES

1. Discontinuity Surfaces without Own Structure The simplest way to characterize interfaces is to imagine them only as purely mathematical surfaces free of any form of own structure. Balances of measures of interaction can be derived easily in this case. First, one must consider the integral form of (35) and (42) and for a generic part B*, crossed by the discontinuity surface E (i.e., B* A E -r 0), one writes

fBTn-0. b+f ~.

fB(~--Z)+] *

(119)

f Sn--0

(120)

,J0 B*

In equations (119) and (120), the bulk interactions b,/3, and z are assumed to be continuous throughout B*, whereas stress measures T and S may suffer jumps at E. Balances at E; can be obtained from (119) and (120) in different ways. Probably the simplest one consists of shrinking B* at the discontinuity surface E with a limit procedure prescribing that B* --+ B* A E. In making this, the bulk integrals vanish as a result of the continuity of their integrands, whereas the integrals on the boundary OB* reduce to integrals on B* A E of the jumps of their integrands.

36

Paolo Maria Mariano

By translating the previous words into symbols, it follows that

f,

ft3 0 b--+ , ,

0

as/3* ~ / 3 * n yz

(121)

Consequently, (119) and (120) reduce to f

IT]in - 0

(122)

[31m - 0

(123)

~*NZ

f 13*nz

The arbitrariness of/3* implies [T]m = 0

on Z

(124)

[S]m = 0

on Z

(125)

Proposition 5 In absence o f standard and substructural surface tensions on a discontinuity surface ]E, the standard and generalized tractions (t = Tm and 7" = 8 m , respectively) are continuous across Z. As a consequence of the invariance of the lower, equation (125) should be written as .,4 T [S]m -- 0. Thereof the assumption in Proposition 5 that surface tensions are absent implies that even surface self-forces of the type ~" such that .AT~" -- 0 vanish identically. 2. Discontinuity Surfaces with Own Structure

More complicated is the case in which ~ is endowed with its own structure. Then ]E may posses standard surface tension measured by means of a surface referential stress T and a surface tension induced by the material substructure and measured through a surface referential microstress S. By definition, T(I - in | m ) = T, S ( I - m |

m ) = S.

Let/3* be a genetic part of/3 intersected by the discontinuity surface ~:, as in Section IV.B.1, T v and St,, are, respectively, standard and generalized tractions at the curve 0/3" n Z (i.e., at the intersection of Y: with the boundary 0/3" of/3"). Vector u has been defined in Section IV.A: At each X belonging to 0/3* n Z, u(X) is the normal at the curve 0/3" N Y: at X in the direction of the tangent of Z at X. With these premises, the external power is written here as t3, --

(b. ~ + / 3 . ~) + *

f

(Tn. ~ + S n . ~) ~*

+ f ( T u . :~+ + •u. ~b+) da /3*OZ

(126)

Multifield Theories in Mechanics of Solids

37

By shrinking/3* at the discontinuity surface E as in Section IV.B. 1, 79~xt reduces to the external power "Dext ~*NE developed on E namely, r

T'~,t -+ T'~,tnr.

as 13" -+ 13" n E

(127)

where, thanks to the previously mentioned hypotheses of continuity of bulk interactions b and/3, 79~;'n$ =fB*n~: ([Tm /']

+ [Sm" ~]) + L t3* n:c (Tu

9~+ + ~u 9~ ~:)

(128)

Balance equations at the discontinuity surfaces can be obtained from (128) by applying the same procedure of Section II.D, that is, by requiring the invariance of 79ext B*NE with respect to all changes of observers After some calculations in which one accounts for the arbitrariness of k, ~,/3*, and uses Gauss theorem on the surface E, one finds the necessity of the existence of an element 3 of the cotangent space of .AA, called surface self-force, such that

ATz

=

eTF

T -

(vzAT)s

on E

(129)

and the validity of the following balance equations: [T]m + Divx T = 0 [S]m + DivES - 3 = 0

on E on E

(130) (131)

Equation (129) represents a sort of generalized "balance of couples" on E, whereas (130) and (131) are balances of interactions on E. The surface self-force must be considered to be the difference of two terms 3' and 3" such that 3' satisfies (129) and 3" belongs to the null space of .AT, that is.,

A r3'' = O. R e m a r k 5 Note that, in absence of surface substructural interactions ,9 and z, i.e. in absence of substructure, the standard surface Cauchy stress TF T is symmetric, i.e. ~F T = F~ T.

Proposition 6 By virtue of balance equations (35), (38), (42), (129), (130), and (131), it follows that

f (b 9 + ~ 9~) + f

~ (Tn 9k + Sn 9~)

BNE

+ f (/T/m. r*J+/S/m. ~e~)+ f(T. F + ~. N +~. e~)

(~32)

Paolo Maria Mariano

38

The last integral of equation (136) is thus the internal power "Dint in presence of a discontinuity surface endowed with its own structure within the body.

BIBLIOGRAPHIC NOTE Balances of interactions at interfaces free of own structure have been obtained in Capriz and Virga (1994). Special forms of balances (124) and (125) suitable for micromorphic (or higher-order micromorphic) continua are in Germain (1973). General balances of interactions at interfaces endowed with own structure have been obtained in Mariano (2000a). The matter of discontinuity surfaces in standard Cauchy continua is treated extensively in scientific literature (see, for example, Abeyaratne and Knowles, 1990, 1991, 1994; Pence, 1992; James, 1983; Gurtin, 1995, 2000, and references therein). Of course, in the case of Cauchy continua (see references mentioned previously), equation (131) is not present, and equation (129) reduces to the requirement of symmetry of the surface Cauchy's stress TF r. The scientific literature on discontinuity surfaces in standard Cauchy continua includes cases in which surface measures of interaction are considered as well as cases in which they are not. For example, the derivation of equation (130) and detailed comments on it can be found in Gurtin (1995, 2000).

V. Constitutive Restrictions Usually, in multifield theories one postulates that the first second laws of thermodynamics can be written in a form analogous to the one of Cauchy continua once one considers the extra power developed by the substructural measures of interaction. First, one formally assumes for any part 13" of 13 at each instant t the existence of the internal energy E1 (/3"), the entropy Ht(]3*), the heat flux Qt(]3*, ~,e), and the entropy flux QtI-I(B*, B *e) exchanged by B* with its exterior B *e. After the "natural" assumption that E1 and Ht are time differentiable, one then writes for the first law of thermodynamics d

d t E t ( B *) - Qt(13* ' 13*e) + "l)ext13* ,-

(133)

and for the second law d

d t H t ( B , ) >_ QH(B, ' ~,e)

(134)

(heat sources and entropy sources are neglected here for the sake of simplicity).

Multifield Theories in Mechanics of Solids

39

By introducing Helmoltz free energy density as {free energy density} = {internal energy density} - {temperature} {entropy density}

(135)

the second law may be written in terms of free energy, and is a tool that allows one to derive constitutive restrictions on the measures of interaction once the external power ~S)ext has been substituted by the internal power "pint. Then one must represent explicitly all the elements of (133) and (134). Basically one introduces bulk densities for internal energy and entropy; however, surface internal energy and entropy densities (together with surface heat and entropy fluxes) can be introduced in the presence of interfaces when relevant to describe different physical phenomena. Here thermodynamic phenomena related to variations of temperature are not treated (no details on the various explicit expressions of (133) and (134) are then given) and an isothermal version of the second law of thermodynamics is considered. It is called here mechanical dissipation inequality and prescribes that d

m dt

{free energy of 13"} - "Dint ,-t3, -< 0

(136)

A. CONSTITUTIVE RESTRICTIONS IN THE BULK In the bulk, it is assumed that {free energy of B*} - ft3, ~

(137)

where 7t is the bulk free-energy density. Consequently, the mechanical dissipation inequality becomes d--~ , ~ -

,(T'~'+z'q~+S'V~b)-<0

(138)

Usually, one assumes for ~ a constitutive structure of the form - ~(F, ~o, V~o)

(139)

By calculating the time derivative of the free energy in the mechanical dissipation inequality and collecting terms, it follows that

ft3((Ov~ - T)-~" + ,

1

z). q, +

- S)- V~b) _< 0

(140)

In principle, given any triplet (F, ~o, V~o) representing the isothermal state of any material patch placed at X, one can arbitrarily choose velocity fields F, ~b and

40

Paolo Maria Mariano

~'q~ from (F, qo, Vg~). Consequently, since the integrand of (140) is linear in the velocity fields, the following proposition follows. Proposition 7 interaction:

The following constitutive restictions hold for bulk measures of

T = 0v~(F, ~, Vq~)

(141)

z = 0~p(F, qa, Vqp)

(142)

$ = 0v~,~p(F, qo, Vqo)

(143)

Other expressions for #r could be assumed; for example one may choose #r = ~(F, ~, Vqo, F, ~b, Vgb). By inserting this expression in (138) and calculating the time derivative, one obtains t~,((0r~ - T). ~" +

- z).

+

+

- S ) . Vqb) 0

(144)

dr3 ,

Since one can arbitrarily choose in principle both velocity and acceleration fields, and the integrand is linear in the acceleration fileds, one obtains 0~.1# = 0, 0~ ~p = 0, 0v~ ~P = 0 that reduce ~p to the expression (139).

B.

C O N S T I T U T I V E R E S T R I C T I O N S AT D I S C O N T I N U I T Y S U R F A C E S

When a discontinuity surface E endowed with its own structure crosses the body, a surface excess energy must be accounted for. It assures the stability of the surface itself, in some sense its existence as a "thin layer" within the body. In this case, one writes {free energy of t3*} - f~, o + f B * n z 4~

(145)

where 4~ represents a surface free-energy density (i.e., the excess energy at the interface). Consequently, the mechanical dissipation inequality changes as

dt

V, + *

~ *NZ

)

-

( T . ~" + z. ~, + S . V~,) *

, ~ (m. [~l + m . [ ~ l ) - f ~ , ~ (T 9 F + ~ 9 Iq + 3 9 ~,•

_< 0 (146)

Multifield Theories in Mechanics of Solids

41

where, in writing (146), one uses the result in Proposition 6. By developing the time derivative of the integral ft3, ~P and taking into account the results of Proposition 7, the interfacial mechanical dissipation inequality follows:

afu

dt

*n~

fu *nz (
- { (ql". I~ + S. fil + 3" ~• Jr3*NE

< 0

(147)

Usually, one may accept for 4~ the following constitutive structure:

4, - ~(F, ~, N)

(148)

underlining in this way that the excess energy at the interface depends on the bulk deformational and substructural state immediately next to it. By developing the time derivative of 4~, the interfacial mechanical dissipation inequality (147) reduces to

f~

((aF4, - 7 ) . I~ + (a~4, - 3)" r177+ (aN4, - s). lq) *NZ

- f ((T)m. [k] + (S)m. [~]) < 0 Jr3*NE

(149)

Given any triplet (F, qa+, N) representing the isothermal state of any material patch at the interface, it is possible in principle to choose arbitrarily velocity fields F, ~+, Iq from (F, qo+, N). Because the integrand of previous inequality is linear in the velocity fields, the following proposition holds.

Proposition 8 of interaction:

The following constitutive restrictions hold for surface measures

qI' = OF4~(F, qo, N)

(150)

3 = 0~,#~(F, qp, N)

(151)

S = 0N4~(F, q#, N)

(152)

The arbitrariness of E implies also that (T)m. [k] + (S)m. [qb] > 0

at E

(153)

Once the explicit expression of the surface free energy has been selected, the surface measures of interaction follow from (150) to (152). Representation theorems for such energy expressions would be desiderable; they could assure that the derived measures of interaction possess the properties assigned by definition.

42

Paolo Maria Mariano

A research program with the aim of describing (or better, representing) energies such as 4~(F, qg, N) for interfaces is still far from being developed.

BIBLIOGRAPHIC NOTE

A general discussion on the expression of the principles of thermodynamics in multifield theories together with the derivation of bulk constitutive restrictions in absence of interfaces is in Capriz (1989). Standard references on thermodynamics of continua are Truesdell (1984), Coleman and Owen (1974), Ericksen (1998a), Silhav3~ (1997). The validity of a general mechanical dissipation inequality such as (136) has been proved in Mariano (1998) by requiting the boundedness from below the action functional along state transformations.

VI. Evolution of Defects and Interfaces in Materials with Substructure

When a body B is subjected to standard deformations, a material patch at X is mapped into x(X, t), the placement in the current configuration ~t at the instant t. Conversely, by the inverse motion x-l(X, t), one may associate the actual placement x of the material patch with its original place X. The picture becomes more complicated when the integrity of the body is altered by, for example, the evolution of defects or phase interfaces. When one observes such an evolution, one may describe it by taking time-varying parts of the reference configuration B. In the following, R(t) indicates any time-varying part of B. The loss of integrity of the body due to the evolution of defects or phase interfaces is basically considered as an additional "independent" kinematics in the reference configuration; this kinematics is associated to R(t). The time variation of R(t) generates new interactions between R(t) itself and the rest of the body; they drive or obstruct the motion of R(t) and are measured by means of a set of additional forces called configurationalforces. These forces are material forces in the sense that they live (or better, they are generated) in the reference configuration. In the standard picture of deformative processes, configurational forces do not exist. The first Piola-Kirchhoff stress T and the referential body forces b (as well as the referential measures of substructural interaction/3, z, S) are only mathematical pictures in B of the real interactions induced by deformation in the current configuration. They are obtained, in fact, by means of pull-back of the measures of interaction arising in the current configuration into B, which is considered fixed permanently.

Multifield Theories in Mechanics of Solids

43

When some part of 13 varies in time, the configurational forces generated by its evolution are measured through second-order tensors and internal forces. These tensors are introduced first as undeterminate terms that develop extra power in the kinematics of R(t), then are expressed explicitly in terms of standard and substructural measures of interaction. Inertial effects are neglected in this section for the sake of simplicity.

A. CONFIGURATIONAL FORCES IN THE BULK

It is assumed that the boundary 0 R(t) of R(t) is a regular surface in R 3, endowed with a outward unit normal n, and parametrized by a pair of parameters Ul and u2 in a way such that

X ~ OR(t)----> X -

X(ul,

U2,

t)

(154)

The velocity v of 0 R(t) in the reference configuration is ^

u -- 0 t X ( U l , //2, t)

(155)

Only the normal component V -- v 9n of v is independent of the parametrization (Ul, u2, t), and this property is crucial in the following. Let e be any continuous field of the time t and the place X. The time derivative of e following 0 R(t) is indicated with e ~ and defined by eO = ~ d e(]~(u i, u2, t'), t')lt,= t

dt'

(156)

and is the time derivative of e along trajectories crossing 0 R(t) at t' = t. When the previous definition is applied to x(X, t) and qo(X, t), it follows that x~ =

~ + Fv

(157)

qo~ = qb + (Vqo)v

(158)

The configurational interactions generated by the motion of R (t) in the reference configuration/3 are measured by means of 9 a

second-order tensor ~' representing the configurational stress

9 an internal configurational body force g 9 an external configurational body force e

44

Paolo Maria Mariano

The balance of bulk configurational interactions and their expression in terms of standard and substructural measures of interaction are shown in the following discussion. To obtain these results, first consider material observers that measure events in the reference (material) configuration. In evaluating the power -,pext - R ( t ) developed during the migration of R (t) in/3, a fixed material observer writes

ext_f R(t)

(b. z~+ / 3 . ~) + (t)

(Tn. x ~ + S n . ~p~ + ~ n . v)

(159)

R(t)

Forces g and e develop no power because they act on bulk points of/3 that are considered fixed when evaluated by a fixed material observer. Bulk interactions b and/3 develop power because they are mathematical pictures in/3 of interactions that in the current configuration act on points that move with respect to the material observer through x (X, t). When one considers a material observer moving with constant velocity v*, such an observer sees g and e as migrating with velocity v* relative to him; so g and e ,pext, mov. obs. evaluated by develop an extra power density (g + e) 9v*. The power --R~t) the moving observer can be thus written as R(,)mo

.

--

(b. ~ + / 3 . ~) + (t)

f.

(g + e). v* (t)

+ f (Tn. x ~ + S n . ~ ~ + ~ n . (v + v*)) J~ R(t)

(160)

However, a classic axiom of mechanics requires that the power be independent of the observer, namely, j~)ext

R~t)

~

,E)ext, mov. obs.

--R~t)

VV*

(161)

To satisfy this requirement, taking into account the arbitrariness of v*, it is necessary and sufficient that

f

R(t)

n+f, (g+e)-0

(162)

(t)

which is the integral balance of configurational forces on R (t). The arbitrariness of R (t) and the application of Gauss theorem imply DivY + g + e = 0

in 13

which is the local balance of configurational forces.

(163)

Multifield Theories in Mechanics of Solids

45

By taking into account the explicit expressions (157) and (158) of x ~ and qo~ the power "l:3ext --R(t) can be also written as 'Qext--fR R(t)

(t)

(b./~+/3.~)+f

R(t)

(Tn.~+Sn.qb)

+ f (~ + F r T + Vqor.S)n 9v Jo R(t)

(164)

As anticipated at the beginning of this section, only the component of v normal to OR(t) is independent of the parametrization of OR(t), whereas the tangential component of v depends on the parametrization itself. Physical plausibility requires that -"lgext - R ( t ) be independent of the parametrization of 0R(t). To satisfy this requirement, it is necessary that (~ + F r T + ~7qor*S) n be simply a vector normal to 0 R(t). In this way, it takes only the normal component of v when it is multiplied by v. Consequently, it is necessary and sufficient that + FTT + ~7~or,_S = wl

(165)

where co is a scalar function that is undetermined at this stage and I is the unit tensor. To determine co, the mechanical dissipation inequality is used. It requires here that md

"19ext ; ~---n(t)-
(166)

This inequality is conceptually analogous to the inequality (136). The sole difference is that here the part R of 13 varies in time. As a consequence of the timevariation of R, the following transport theorem holds:

"f,

-'~

(t)

(t)

++f

(167)

R(t)

where V is the normal component of v: V - v- n. By taking into account (165) and (167), the mechanical dissipation inequality may be written as

f,

(t)

(b.zt+/3.@)-f~ R(t)

(t)

(Tn.~:+Sn.qb)<0 R(t)

(168) This inequality must hold for any mechanical process involving R(t), that is, it must hold for any choice of velocity fields. Thus, the following identity must hold c o - 7z

(169)

Paolo Maria Mariano

46

Finally, this implies from (165) that ?-

fl-

F T T - VqoV._S

(170)

In this way it is shown that the configurational stress IF' can be expressed in terms of the free energy density, 7t and the stress measures T and S. When the setting is conservative, 7t reduces to the strain energy density w and has thus the expression of the generalized Eshelby tensor obtained in Section III. Here, the way followed to obtain the explicit expression of ~ assures that such an expression holds even for dissipative mechanical processes. This result guarantees the possibility of using the expression of ~ to describe dissipative processes like the evolution of macrocracks. As a result of the validity of the balance of configurational forces and standard and substructural balances of interactions, the expression of IP implies that g - -VTz + T 9 V F + S+VVqo + z+Vqo

(171)

e - - F T b - Vqo T+/3

(172)

B. CONFIGURATIONAL FORCES ON A DISCONTINUITY SURFACE Let E(t) be a moving interface within R (t) of normal m (t), as in Section IV. The boundary 0 R(t) N E(t) of R(t) n E(t) is a curve that can be parametrized by some parameter u in a way such that X 6 0 R(t) n E(t) is given by X - X(u, t). Thus, the velocity field u(X, t) of the curve OR(t) n E(t) is given by u(X, t) -- OtX(u, t)

(173)

Because E(t) moves within the body, points of it are described by X 6 E(t) so that X - X(vl, v2, t), where (vl, v2) is a parametrization different from (ul, u2) on OR(t). Consequently, the velocity ~ of E(t) may be defined as f(X, t)

-- OtX(Ul,

1)2,

t)

(174)

The attention is focused in the following discussion on normal velocities -~ - U m . Note that u-

~ + U0v

(175)

where, as in Section IV, u is the normal of the curve OR(t) N E(t) directed along the tangent to E(t) at OR(t) n E(t) and Uo = u . u. Two different time derivatives may be defined: the first is the time derivative following the curve 0 R(t) n E(t), and the second is the time derivative following

Multifield Theories in Mechanics o f Solids

47

E(t). Let e be any continuous field of the place X and the time t; the former type of time derivative is indicated with e ~, the latter with e -~. Velocity fields following the curve 0 R(t) N E(t) are given by x ~ =/~+ + F•

= (/~) + (F)u

qo~" : ~ + + Vqo+u : (~> + (Vqo)u

(176)

(177)

Velocity fields following E(t) are given by x ~ = ~+ + F+~ = (~) + (F)~

(178)

qo-~ = qb+ + V q o •

(179)

(~) + (Vqo)~

The following relations may be derived easily from previous definitions after some algebra: x ~" = x -~ + UoFu

(180)

qo~ = qo~- + Uo N u

(181)

The case in which a moving interface E(t) crosses R(t) introduces the need for additional configurational forces strictly related to the peculiar kinematics of the interface. They represent the forces driving or obstructing the interface: their balance allows one to derive a complete evolution equation for the interface itself. Surface configurational interactions are measured by means of 9 a surface configurational stress C 9 a surface internal configurational force gz The stress C is a second-order tensor that maps vectors tangent to E(t) into vectors of R 3. In evaluating the power "19ext --R(t) on R(t) taking into account the structure at the interface, a fixed material observer measures an extra power due to C v . u and then writes in the present situation

extf.

R(t)

:

( T n . x ~ + S n . q o ~ + IPn. v)

( b . x --}-/~. ~ ) --}(t)

R(t)

f + ] ( C u . u + ql'u. x c> + S t , . qoc>) Ja R(t)AE(t)

(182)

Taking into account the definitions of x ~ qo~ x ~, qo~', and the results on bulk

Paolo Maria Mariano

48

TQext may --R(t)

be written as

:~ + / 3 . ~) +

fOR(t)(Tn./~

configurational forces,

~ )R(t) e x t-f R (t)(b.

+ S n . qb + ~OV)

+ f ( T u . ~+ + S u . qb+ + (C + F+r• + Vqo+r_,S)u 9u) da R(t)AE(t) (183) Because TQext --R(t) must be independent of the parametrization of E as previously accepted in treating bulk configurational forces, here it needs to be independent even of the parametrization of the curve 0 R (t) 71 E (t). In other words, if t represents the tangent field to 0 R(t) @ E(t), because the component o f u tangential to 0 R(t) N E(t) depends on the parametrization of the curve, to assure independence of the parametrization of the curve 0 R(t) N E(t), it is necessary and sufficient that f~

(C + F+rqF + V ~ +r_,s)u 9t - 0

R(t)AE(t)

(184)

However, C may be decomposed into the sum of its tangential component Ctan and normal component m | e to the interface" Ctan maps tangent vectors to E into vectors tangent to the deformed interface, whereas e - Cm. Thanks to the arbitrariness of R(t), (184) reduces to (C + F+rqI ' + Vqp+r*S)u 9t - 0. In other words, the projection of the vector (C + F+rqr + Vqp + r * s ) u on E must vanish; in symbols, one writes that (I - m @ m)(C + F + r 7 + Vqp + r * g ) u - 0, which yields

Ctan + [::T~ + NT_,S __ coz(l - m | m)

(185)

with cozc a scalar function that is undetermined at this stage. Consequently, it follows that C + (F)r'II' + (Vqo)r_, S - cor~(I - m | m) + m | c

(186)

where c - (C + (F)rql" + (V~)r_,S)m. The following formula connects vectors e and c" e - c - T r (F)m - S(Vqo)m

(187)

When one writes the power as perceived by a moving material observer one must consider the contribution of the internal force gr.. As in the case of bulk configurational forces, the requirement of invariance of the power with respect to changes from fixed to moving observers and vice versa implies the validity of the following integral balance of configurational forces that holds in presence of

Multifield Theories in Mechanics of Solids

49

moving interfaces within the body: 0 R(t)

(t)

R(t)nz(t)

(t)nZ(t)

It is assumed that g and e are continuous on/3, whereas IP may suffer jumps at the interface. Consequently, by shrinking R(t) to R(t) n X(t), the arbitrariness of R(t) implies the following balance of configurational forces at the interface: on ]E

[I?]m + DivzC + gz = 0

(189)

The normal component of the interface balance of configurational forces plays an important role: It allows one to write the evolution equation for the interface itself. To obtain such a normal component, it is necessary only to multiply previous balance by the normal in at ]E. First one must consider that DivzC = Div~Ctan + Divz(m | e) and that 9

m.

DivzCtan

Cta n 9 t - -

--

T Divz (Ctan m)

-

Ctan 9 V z m

- D i v z ( C r ( I - In | m)m) +

Cta n . t

9 m . Divz (m | c) = Divz c As in Section IV, L is the curvature tensor of Z such that its trace/C = tr L is the overall curvature. With these premises, the normal configurational balance of forces can be written as Ill. [~]nl

+ Ctan " t -+-

Divz;z + gz = 0

on E

(190)

where gz = m . g z . To obtain an explicit evolution equation for the interface, it is necessary to exploit some consequences of the mechanical dissipation inequality written here considering the contribution of configurational forces as [cf. eq. (146)]

'(f,

dt

vext < 0

(t)

(191)

(t)nz~t)

Standard transport theorems prescribe that (192)

dt

(t)

(t)

R~t)

(t)NZ(t) (193)

dt

(t)nz(t)

(t)nZ(t)

where 4~-~ is the time derivative of 4~ following Z(t).

R(t)NZ(t)

PaoloMariaMariano

50

As a tool for developing future calculations, note that from (175), (178), (179), and (185), it follows that fa R(t)NE(t) (Cv. u + T v .

-~

R(t)nE(t)

X t>

coEua+f

+

~o~') (Cu.~+Tu.x-~+~;u.qo

-~)

(194)

R(t)nE(t)

By substituting (183), (192), (193), and (194) into (191), the mechanical dissipation inequality reduces to

fR(t)~-~(t)n~(t,[~P]U+L(t,nr~(t)(4~--c~ICU) R(t)nE(t)

(t)

R(t)

R(t)nE(t)

(195) This inequality must hold for any choice of the velocity fields, and for any choice of Ua. This implies as a first result wE -- 4'

(196)

The identity (196) allows one to express the tangential part of the surface configurational stress explicitly in terms of standard and substructural measures of interaction and the surface free energy, namely, Ctan - -

4~(I - in | m) - FTT - NT,_S

(197)

By shrinking R(t)to R(t) N E(t), as a result of the regularity properties assumed (and declared) in previous sections and to (196), in the limit R(t) --+ R(t) N E(t), the inequality (195) reduces to ([Tin. 5r + [ S m . ~]) (ONE

-{

Jo R(t)NE(t)

(t)nz(t)

(t)NY,(t)

(Cu. ~ + Tu. x -~ + Su. qo-~) _< 0

(198)

To reduce the inequality (198) further, one may use the following lemma, which is valid under the assumptions declared in this section up to this point.

Multifield Lemma

Theories

in M e c h a n i c s

51

of Solids

9

f

( C u . ~ + T u . x -~ + S t , . ~ o -<)

R(t)nE(t) f

(ql'. F -~ + S . N -< + 3" (~o) -< - [ T m . ~] - [,Sm. ~])

f

([~p]U + U g z

dR(t)nE(t) dR(t)NlE(t)

+ wzU/C + c . m -~)

(199)

The proof of this lemma is rather long and somewhat tedious. It is based substantially on the definitions (176)-(179), the use of Gauss theorem, and the auxiliary results 9 V z ~ - - V z ( U m ) -- - m | V z U + U V z m -- - m | m -~ - UL; 9

x -~ 9 Divz'~' -- - [ T m .

~] - U m .

[FTT]m;

9

~o-~ 9 D i v z ~ - 3" (~o) -~ - [ S m . ~b] - U m .

[V~pr,_S]m;

9 ql'- V z x -< = T . (F) -< - (T T ( F ) m ) . m -< - u(FTqr) 9 L; 9 S . Vzqo -< = S . (V~o) -< - (S r m) 9 m -< - u(NT_,s) 9 L. The results in this list are consequences of previous definitions in Section VI and the balances at the interface derived in Section IV. By introducing (199) into (198) and taking into account (196), one obtains the integral dissipation inequality at the interface

L(t)nE(t) ~)~- -~-L(t)nz(t) UgZ + f

dR(t)nz(t)

(c. m -~ - T . F -~ - 3" (~o) -~ - ~" N -~) 5 0

(200)

The arbitrariness of R ( t ) implies the following local version of the mechanical dissipation inequality, namely, ck <- + e . m <- - T . F <- - 3 . (~o) ~- -

S . N <- + U g z

<_0

(201)

Two results follow from (201). First, it is possible to generalize the results of Proposition 8 to take into account the possible anisotropy of ~2(t). To this aim, one may choose a constitutive relation of the form 4~ = ~(m, F, q0, N)

(202)

52

Paolo Maria Mariano

where the dependence on the normal m accounts for the anisotropy of the interface itself. In further steps, one calculates the time derivative 4~-~ of 4~ following E(t), inserts it into (201) (by collecting analogous terms) and observes that, given any state (m, F, qo, N), velocity fields m -~, F -~, (qo)-~, and N -~ may be chosen arbitrarily from (m, F, qo, N). Consequently, because (201) must hold for any choice of velocity fields, the following relations must hold:

"IP - aFt(m, F, ~, N)

(203)

,~ = a ~ ( m , F, ~, N)

(204)

,~ -

aN~(m,

(205)

c -

- a m P ( m , F, ~ , N)

F, qo, N )

(206)

where, relative to the results in Proposition 8, there is here an explicit dependence on the normal of the interface and vector c represents the force generated by the anisotropy of E during the motion. Consequently, as a result of (203)-(207), the mechanical dissipation inequality reduces to gz U _< 0

(207)

gz - - ~ z ( m , F, ~o, N, U)U

(208)

A solution of (207) is given by

where gz is a positive scalar function of arguments that are the same as in (203)(206) plus the normal velocity U. Note that gz is the normal component of the dissipative force gz driving the interface [inequality (207) states the dissipative nature of gz]. Because gz is dissipative, it depends on the velocity U and may depend on the other state variables: gz must be assigned by a constitutive prescription. When one assigns ~z, one determines the evolution of E(t). The evolution equation of E(t) is, in fact, the normal configurational balance (190) once one substitutes in (190) the explicit expression of ~ [identity (170)], the expression of Cta n [identity (197)], the constitutive restriction (206) [taking into account (187)] and finally (208), which is the fundamental ingredient for writing the evolution equation. Remark 6 With the previous results, it is possible to prove that the tangential component of the configurational force balance (189) is identically satisfied once gz is purely normal" namely, once gz(I - m | m) - 0.

Multifield Theories in Mechanics of Solids

53

BIBLIOGRAPHIC NOTE A rather general treatment of the influence of material substructures on the expression of bulk and surface configurational forces can be found in Mariano (2000a). Some of the proofs discussed here are developed following different methods in some aspects from the ones of the previously mentioned paper. Results on the influence of scalar order parameters on the influence of bulk configurational forces can be found in Fried and Gurtin (1993, 1994, 1999) and Fried and Grach (1997). A detailed treatment of configurational forces in standard Cauchy materials can be found in Gurtin (1993, 1995, 2000), Gurtin and Struthers (1990), Angenent and Gurtin (1989), Abeyaratne and Knowles (1991), Gurtin and Podio-Guidugli (1996), Maugin (1993, 1995), and references therein. The term configurational is due to Nabarro (1985) (see Ericksen, 1998). The seminal ideas on this subject are in Eshelby (1951, 1975).

VII. Crack Propagation in Materials with Substructure One application of the framework of configurational forces discussed in Section VI is the analysis of the propagation of macrocracks. Evolving cracks may be considered to be evolving curves (for two-dimensional bodies) in the reference configuration. By inverse motion, in fact, one could pull back the actual opened and deformed crack into a curve in the reference configuration, which is a picture of the crack taken as closed. Of course, when the crack evolves in the actual configuration, its picture in the reference configuration evolves in time: configurational forces driving the crack must be considered. By using some of the results presented in the previous discussion, it is possible to discuss the influence of material substructures on the force driving the crack tip. Experiments show the influence of material texture on crack propagation (see Herrmann and Roux, 1990). When one models some material as a Cauchy continuum, such an influence is considered only indirectly, through constitutive equations, and often models of the local cohesive structure at the tip of the crack are proposed. They offer interesting special solutions; however, these models can be difficult to manage when one considers the crack propagation to be a global bifurcation problem, because in this case it is often necessary to know the overall distribution of material texture rather than the situation near the crack tip. In the framework of multifield theories, one may derive the driving force at the crack tip in a way accounting for the explicit influence of the perturbations induced

Paolo Maria Mariano

54

on standard interactions by material substructures. This is the result shown in this section, in which for simplicity, all calculations are restricted to two-dimensional bodies.

A. KINEMATICSOF PLANAR MOVING CRACKS

In this section, the body B is considered as an open compact region of the two-dimensional point space ,re. A crack in B is identified as a regular curve C represented by a function r : [0, ~] ---> B (with [0, ~] some bounded interval of the real axis ~) defined as follows: ' ( 0 , ~ ] --+ B r(0) ~ 0B

{r

(209)

This definition states that one considers the crack as a curve in B starting from the boundary and ending within/3, without intersecting further on the boundary of/3. The crack tip is, of course, r(g), and is indicated by Z; it is a point in oc2 belonging to the interior of/3. The curve C is also characterized by its tangent vector t - 0~r, 5 ~ (0, g) and the lateral normal m (thus such that t. m = 0). When the crack evolves in a time interval [0, d], C is a function C(t) of time, defined as

C(t) = {r(s, t) E 1315 E [0, ~], t E [0, d], r(0, t) ~ 0B, r(0, t) -----r(0, 0), Vt} (210) with the condition

C(tl) c C(t2), u

< t2

(211)

It is assumed that C evolves without intersecting further on the boundary of B: the tip Z becomes a function of time Z(t) and Z(t) 6 B, whenever t 6 [0, d]. In other words, one imagines studying the crack propagation during a time interval in which the crack does not completely cut the body B (Figure 2). The velocity of the crack tip Vtip is defined as dZ Vtip = dt

(212)

When one assigns the tangent of C at the tip Z--namely, t (Z(t))--one assigns the direction of propagation of the crack. It is thus possible to write Vtip = f'(t)t (Z(t)). From this definition, it follows that m(X) 9Vtip ~ O, as X --+ Z(t). Special parts of B are of future use; in particular, it is necessary to consider a disc D centered at the tip. The boundary 0 D of the disc intersects C at only one point

Multifield Theories in Mechanics of Solids

55

11

B t(z(o) FIG. 2. Geometric characterization of a cracked body. denoted with X A . The outward unit normal at 8 D is indicated by n, whereas the radius of D is indicated by O. When the crack evolves, the disc is considered to be time varying to follow the evolution of the crack; thus D and XA become D(t) and XA (t). Let f be the velocity of the curve C(t). The velocity of XA (t) can be written as ra(t) -~ Xa(t) -- Ua(t)ta(t), where/~a is the scalar amplitude of the velocity and ta is the tangent of C(t) at the point XA. Because t (Z(t)) is prescribed, it is assumed that ta(t) ~ t (Z(t)) Cta(t) - ~ V(t)

as O ~ 0 as O --~ 0

(213) (214)

Let e represent any continuous field of the place X and the time t. The time derivative of e following the crack tip is defined as e~(X, t) = 8te(Y, t)lv=x-z(t)

(215)

where Y denotes a genetic point in s By applying this definition to x and qo, the velocity fields following the tip can be written as

x <>- ~ + Fvtip qD<>= ~ + (V ~)u

(216) (217)

Assume now the existence of a tip velocity field ~tip and a tip order parameter ~lti p when the crack is deformed. To be more precise, first assume that if X 6 C and x(X, t) is the position of X at the instant t, then x(X, t) ~ x(Z(t), t) and qo(X, t ) ~ qo(Z(t), t), as X ~ Z(t) uniformly in time. The assumption of the existence of Vtip and Vitip i s tantamount to requiting that x<>(X, t) ~ Vtip and 9:,<>(X, t) ~ @tip, as X ~ Z(t) uniformly in time. rate

Paolo Maria Mariano

56

The boundary of the time-varying disc D(t) may be parametrized by some parameter v in such a way that X 6 0 D ( t ) =~ X - fK(v, t). The velocity vo ofthe boundary 0 D(t) of the disc is given by VD(X, t) = OtfK(v, t)

(218)

Only the normal component of v D - - n a m e l y , / 2 -- VD.n--is independent of the parametrization of 0 D(t). Velocity fields x ~ and ~~ following OD(t) may be defined in the same way as in (157) and (158) as X~ -- ~ + FVD

(219)

qa~ - qb + (Vqp)VD

(220)

The physics of the crack imposes a kinematical condition, namely a condition of impenetrability" [x] 9m >_ 0

(221)

In other words, when the body deforms, the margin of the cracks do not penetrate one into another. In presence of a crack, the mapping x(.) is no longer pointwise one-to-one and becomes a piecewise bijection" x(.) is one-to-one to within the curve C; that is, a set of zero volume measure.

B. BALANCE OF STANDARD AND SUBSTRUCTURAL INTERACTIONS AT THE TIP The following assumptions on standard and substructural measures of interaction apply: 9 bulk measures of interaction b and/3 are continuous on/3 9 the self-force z is continuous on/3 9 stresses T and S may be singular at the crack tip and suffer jumps across C; they are also continuous and continuously differentiable outside C 9 surfaces stresses 7i" and S and surface self-force 3 are not considered Two new measures of interactions are assumed acting at the crack tip; they represent external actions acting at the crack tip as "concentrated f o r c e s "

Multifield Theories in Mechanics of Solids 9 tip force

57

btip

9 tip substructural measure of interaction

~tip

The previous assumptions guarantee that balance equations (35), (38), and (42) hold in the bulk, and that balance equations (128) and (129) hold on C. Additional balance equations hold at the tip; they may be derived by writing the integral balances of interactions with respect to the disc D and shrinking the disc to the tip. Integral balances accounting for tip interactions are

fo

b+f

(Tn) + btip -- 0

(222)

D

fD(~ -- Z)-l- fo (Sn)-~- ~tip --

(223)

To write them, it is necessary only to add btip and Otip to (119) and (120). Because T and 8 may be singular at the crack tip, by shrinking the disc to the tip--that is, letting O ~ 0 u t h e s e balances reduce to

where

btip + ftip Tn - 0

at tip

(224)

/3tip + ftip 8n - 0

at tip

(225)

ftip must be interpreted as limo_,o fo"

Remark 7 The assumption that the pair (x <>, ~o<>)tends to (Vtip, ~Ntip) a s X ~ Z(t) implies that

retip) - 0

(226)

Sn . (~<> - fVt~p) = 0

(227)

L T n . (x ~ -

fp

C. EFFECTS OF INERTIA

To evaluate the effects of inertia on standard and substructural balances one may follow the technique adopted in Section II. To this aim, first assume the following

Paolo Maria Mariano

58

decompositions of tip interactions into inertial and noninertial components: bti p - blnp -~- btnip

(228)

fl tnip

(229)

fl tip -- fl iTp +

These decompositions must be used together with relations b = b in + b ni and f l _ flin+ flni. The noninertial part bt'~/p of btip is a possible standard external force applied at the tip of the crack, whereas flt~iipcan be interpreted as a possible external substructural action at the tip. The terms bt'~/pand fltnip are not essential, and are quoted here only for the sake of completeness. On the contrary, the inertial terms may be identified explicitly. To obtain this identification, it is first necessary to evaluate the production of standard momentum and the production of substructural momentum, indicated with pr and pr., respectively, during the evolution of the D(t). Because the disc D evolves in the reference configuration, there is an inflow of momentum across OD(t). The inflow of standard momentum is -faD(t)pi~bl, whereas the inflow of substructural momentum may expressed as -f~o(t)OcpX Lt, where L/ is the normal velocity of OD(t) (i.e., /d = yD. n). Consequently, one may write by definition

dfo p -f

pr-

~

(t)

(230)

D(t)

pr. ---

(t)

(t)

D(t)

The basic step to identify bitp and fli~ is to consider a balance analogous to (53) and write

blnp +

fo

(t)

bin -- --pr

I' I~ITP + I t~in -- --]Or. dD (t)

(232)

(233)

By shrinking D(t) to the tip, the assumptions of regularity of bulk measures of interactions b and fl and the regularity properties of motions imply

fD(t) bin; fo(t) flin; m

d f

dt Jo(t)

p~:

d -~fD(t) ~ --fo(,)~

all tend to0

asD--+Z

(234)

Multifield Theories in Mechanics of Solids Because the previously listed terms vanish in the limit D ~ identifications hold because v o 9 n = U --+

Vtip 9n)

bi~p - ft/p(pfc)(vt/p, n) =

as D ~

ftip(p~ |

59 Z, the following

Z

n)Vtip

~'~p : ftip(~X)(Vtip "n) ~ ~.p(~,x Q n)Vtip

(235)

(236)

As a consequence, the tip balances (224) and (225) reduce to

Remark 8

btn.p+ ftipTn : - ftt.p(px | n)vtip

at tip

(237)

tip "3t- Sn -- -- fti p (O(oX Q n)Vtip t~niftl.p

at tip

(238)

Physical plausibility suggests that standard and generalized tractions

Tn and Sn are bounded by up to the tip as D ~ Z. When bt~i%a n d fltnp vanish identically (as usual), the hypothesis of boundedness of the stresses implies fi Tn -- O ==~fi n| P

P

. Sn=O==~ fi n| p

(239)

P

D. TIP BALANCE OF CONFIGURATIONAL FORCES

When the crack evolves, its mathematical picture C varies in time and configurational forces intervene to obstruct or drive the crack. The following assumptions apply to the framework developed in Section VI:

9 ~ may be singular at the crack tip and suffer jumps across C; it is also continuous and continuously differentiable outside C 9 g and e are continuous and on I~ 9 C is now a vector c and is continuous along C 9 gz is continuous on C

Paolo Maria Mariano

60

New configurational forces are associated with the tip and are peculiar of tip singularity. They are 9 internal tip configurational force gtip 9 external tip configurational force etip of inertial nature The integral balance of configurational forces (188) on D must be written here accounting for the tip configurational forces. It may be deduced from work invariance arguments such as those, discussed in, Section VI and is expressed by

D(t)

IPn+ fo (t)

(g+e)+fo

(t)NC(t)

(gz)

+ gtip -k- etip -- CA - - 0

(240)

By shrinking D at the tip of the crack, equation (240) reduces to (241)

gtip + etip -- Ctip + ftip ]?n -- 0

which is the configurational tip balance. It is now necessary to characterize the tip configurational forces explicitly. First, the attention is focused on etip, which is of inertial nature. To derive an explicit expression for it in terms of velocity fields, one may follow a procedure analogous to that discussed in Section II.A and may derive etip by requiring that the rate of kinetic energy E of D(t) plus the power of all inertial forces on D(t) vanish identically. In symbols, one writes

d ~ + fo 0 -- dt

(bin "x + t~in "(P) + btip" in reap + ,~tip" in ,Tvtip+ etip 9 Vtip

(t)

(242)

Because D(t) varies in time, in computing the rate of the kinetic energy it is necessary to consider its inflow through the boundary of D(t), because this inflow is due to the motion of D(t) itself. Then the rate of kinetic energy of D(t) is given by

dt

~

fo(1 (t)

)

2(P~" ~) + k(~, @) -

D(t)

(p~. ~) + k(~, @) H (243)

The next step is the introduction of (243), (235), and (236) into (242) and the shrinkage of D(t) at the tip Z. Note that f~o(t)(.)H ~ ftip(.)(vtip, n) because

Multifield Theories in Mechanics of Solids

61

D(t) ~ Z(t). At the end of calculations, one finds that etip " u

-- u

" ftip (~(,o~, " x)-'F- k(qo, qo)) n

- vti p

f

Vtip)n-- u

. ftip(OCpX . Wtip)n

(244)

To reduce (244) further, one defines the relative kinetic energy ~.rel as ~.rel-1 1 1 "~plX-~ltip 12 and observes that ~.rel- -~p~Itip " Vtip -- "~pX" X - pX" reap. Moreover, Ytip PVtip " Vtip -- limos0 fad Pretip " ~r but retip is independent of spatial coordinates, then lim0~0 fad pretip 9retip -- lim0~0 retip 9 r4tipfao P -- 0 because the density of mass p is a continuous function. By taking into account these auxiliary results on the relative kinetic energy, and because (244) must hold for any choice of the velocity of the tip, it follows that

etip--ftt.pl~reln-+-ftipk(qo,@)n-ftip(O(oX.f~Ctip)n

(245)

The identity (245) characterizes completely the external configurational force etip and specifies its inertial nature.

E. CONSEQUENCES OF THE MECHANICAL DISSIPATION INEQUALITY To give some characterization of the internal configurational tip force gtip, it is necessary to exploit the mechanical dissipation inequality, which is now written as

d re{free energy of D(t)} - {power developed on D(t)} < 0

dt

(246)

A line free energy density 4~ along the crack C is considered besides the bulk free energy density ~p. The line free energy accounts for surface tension c along the faces of the crack. It is assumed that ~p is continuous on/3 and may suffer jumps at C, whereas 4~ is continuous along C. The line free energy density 4~ does not depend on the time t. From (203) to (206), 4~ may depend only on the normal m of C because standard and substructural surface measures of interaction are not considered here. However, m does not depend on t because there is no motion of C along its normal. Possible phenomena of aging are not considered; thus 4~ does not depend explicitly on t. With these assumptions, d--td{free energy of D(t)} -- --~

(t) ~ + -~

(t)nc(t) 4~

(247)

62

Paolo Maria Mariano

Because the normal velocity of C is zero, the transport theorem (167) holds and is written here as --

(t)

dt

r =

(t)

r +

D(t)

eL/

(248)

whereas for the line free energy one finds -dt

(t)NC(t)

~) -- ~tip ~/ -- ~)A bl A

(249)

because the integral in (249) is a line integral. By inserting (248) and (249) into (247) and writing explicitly the power developed on D ( t ) , it follows that fD

(t)

~-s .qt_ f

-fo

(t)

D(t)

(!~r~[) '~ ~)tip (/ -- ~)A~IA

(b"/~ +/3" r

-- btip 9 ~r

fo

D(t)

(Tn. k + S n .

r + CU)

- ~tip " fVtip - etip 9 Vtip + U AIA 9 CA < 0

(250)

The last integral in (250) is an alternative manner to write f

(Tn. D(t)

Xo

o

+ Sn.qo + I?n. vo)

and the expression used in (250) can easily be obtained by using (219), (220), and (170) [see also the analogous integral in (183)]. A first result can be obtained directly from (250). Because the inequality (250) must hold for any choice of the velocity fields and--among othersmfor any choice of/~a, as a result of the arbitrariness of D ( t ) , the following identity must hold:

= t. c

(251)

it characterizes completely the tangent part of the configurational stress c: this tangential part is the surface tension along the faces of the crack. By shrinking D ( t ) at the tip, one obtains from (250) the tip mechanical dissipation inequality: ~)tip V -- btip " Vtip - ~tip " Wtip - etip " u

- ft, Tn. ~ - ft/pSn 9~, _< 0

-

lP(Vtip . n )

(252)

63

Multifield Theories in Mechanics of Solids

The use of the tip configurational balance (241) and the identity (251) allow one to reduce the tip mechanical dissipation inequality (252) to

ftip 1 .fti,o fti o o

gtip " Vtip -- btip

" Vtip -- tOtip "~ltip

--

l[f(Vtip

"

n)

Note that from (170) it follows trivially that

so that inequality (253) reduces to gtip 9 vtip - btip 9 Vtip - ,Otip 9 Wtip - ftip T n . (~, + Fvtip) - ftip,..~n . ( ~ -4c-(~Tqo)Vtip) < O

(255)

From (216) and (217) and the assumption previously stated that (x<>, qo<>) (Vtip, wtip) as X ~ Z, it follows that ft.pTn'(~'+Fvtip)+fttipSn'(~(Vq~

Sn

(256) By substituting (256) into (255) and using the tip balances (232) and (233), the mechanical dissipation inequality reduces to gtip " Vtip < 0

(257)

which characterizes the dissipative nature of the internal configurational force gtip. It is necessary to prescribe constitutively only gtip rather than the energy release rate (as usual in technical literature). The constitutive prescription of gtip is subjected only to the condition (257).

F. DRIVING FORCE

Previous results allow one to derive an expression of the driving force at the tip of the crack, which accounts for the presence of material substructures and the interactions they generate (and consequently the expression of the energy release rate during crack propagation). To simplify the developments in the following, it is first assumed that btnp and/3t]p vanish identically.

64

Paolo M a r i a M a r i a n o

By using the identities (170) and (245), the tip balance of configurational forces can be written as ftip((lP "at- ~.rel "-[- k ( ~ , (a) - O~X 9 qVtip) I - F r T - ( V ~ ) r * S ) n - (.tip -- -grip

(258) The integral in (258) is indicated here with jm for compactness of notation, jm represents the tip traction exerted by the material on an infinitesimal neighborhood around the tip of the crack. The component of the tip balance of configurational forces along the direction of propagation of the crack is obtained by multiplying the balance (258) by t (Z(t)) t ( Z ( t ) ) . jm -

t (Z(t))

9 C,ip -

-gtip.

t (Z(t))

(259)

By denoting with Jm the product t (Z(t)) 9jm and using (251), equation (259) may be written as (260)

Jm -- Ck,ip -- --grip" t (Z(t))

In equation (260), the term grip 9 t(Z(t)) is the internal force exerted by molecular bonds that opposes motion of the tip, whereas f - Jm -dPtip -- O(oX " W t i p )

t (z(t)) 9 f,/p((~ +

ereZ +

~:(~,, ~)

I - FTT + ( v ~ ) T * S ) n -

~)tip

(261)

is the driving force at the tip accounting for the influence of material substructures. Because Vtip = ~'(t) t (Z(t)), the internal dissipation inequality (257) and the tip balance (258) imply f'~' >_ 0

(262)

which represents a version of the internal dissipation inequality. From (262), when the crack grows (i.e., when V > 0), 9 the driving force must be nonnegative f >_ 0

(263)

9 the tip traction must form an acute angle with the direction of propagation

t(Z(t)), j,, >_ 4~tip > 0

(264)

Multifield Theories in Mechanics of Solids

65

The results (263) and (264) coincide with the analogous results in Cauchy continua.

G. A MODIFIED EXPRESSION OF J INTEGRAL

As a consequence of (239), the tip traction jm reduces to jm --

(( ~ + ~Pls 1 2 + k(qo, ~) ) I - F T T - ( V ~ ) T,,S ) n

(265)

Consequently, it follows that Jm V -- jm " Vtip = jm " V ( t ) t (Z(t))

= ftip ( T n . i~+Sn. ~ + (Tz + ~ Pl~lZ + k(~p, (o)) (Vtip . n))

(266)

The product Jm V is the flow of energy into an infinitesimal neighborhood of the crack tip. Jm is thus the dynamic energy release rate accounting for substructures because it has the physical dimensions of an energy. When the influence of substructures is not considered, Jm coincides with t(Z(t)).

' )

~p + ~Pl~l 2 I - FTT

)n

which is the standard dynamic energy release rate in Cauchy continua (see, e.g., Freund, 1990; Gurtin, 2000; Maugin, 1992). If inertial effects are absent, the tip traction jm reduces to its "quasi-static", counterpart jm,qs given by

jm,qs -- ftip(~l - FTT-(Vqo)T ,__S)n -- ftip~n

(267)

If the body forces b and ~ are absent, the material is homogeneous [in particular, ~ = ~(F, qo, Vqo), so ~p does not depend explicitly on X], the faces of the crack are free of standard and substructural tractions (T• = 0; S • = 0), the "quasi-static" energy release rate Jm,qs - t (Z(t)) . jm,qs i$ path independent; that is, Proposition 10

Jm,qs -- t

(Z(t)) . fr Pn

(268)

where F is any closed, regular, nonintersecting path beginning and ending at the crack.

66

Paolo Maria Mariano To sketch the proof of previous proposition, let intF denote the closed region

of,f2 with boundary 1-'. By Gauss theorem, fr ]?n = fintF Div]]3) + fintI'nC []l~]m" By using the configurational force balance (163), Div? may be substituted by the sum - (g + e). The absence of body forces implies e = 0 [see (172)], whereas the homogeneity of the material implies g - 0 [one calculates the gradient of 7t, inserts it into (171), and uses Proposition 7]. In addition, by using (170), one realizes that the hypothesis concerning the faces of the crack implies that [/~]m = [~]m. As a consequence,

t (Z(t)) 9 f~trnc [~p]m -- f~trnc [~p] t 9m - 0 and the validity of Proposition 10 is proved.

H. ENERGY DISSIPATED IN THE PROCESS ZONE

When a crack propagates in an elastic-plastic material, a critical zone around the crack tip occurs: the process zone. It is highly unstable (in the sense that any increment of the loads may alter its coherence even drastically), in a certain sense "fragmented," so that one may doubt that basic axioms of continuum mechanics (e.g., the continuity of the material) do not work well within it. When one evaluates the energy dissipated during the evolution of the crack, hence of the process zone, one realizes that J integral is no longer sufficient to describe the energy dissipated into the process zone and other path integrals must be introduced. Let P indicate the process zone around the crack tip. Assume that the boundary 0 P of P is a closed regular curve without self-intersection that admits normal n. During the evolution of the crack, P is considered time dependent [i.e., P = P(t)]. The curve 0 P(t) is described by some function X = X(vp, t), with Ve an appropriate parameter along 0 P(t). The velocity Vp of 0 P is given by ve(X, t) -OtX(ve, t). In a local frame {X*} of coordinates centered at the tip Z, the velocity Vp may be written as V p = Vp,tr

+ CI X X* + a ' X * + a

where Vp,tr and/1 are the rigid translational and rotational components, respectively; a'X* is the component of the velocity associated to the self-similar

Multifield Theories in Mechanics of Solids

67

expansion of P(t); and d is the component associated to the distortion of the process zone. Of course Vp,tr, (~, and a* are independent of the space coordinates. Proposition 11

The energy dissipated into the process zone, ~(P), is given by dp(p)

--

YP,tr

" jm(P)

+/1" L + a*M + I

(269)

where i n ( P ) --

1 (( ~pl:~l 1 2 ) 1 (( 1 ) ( ( ,~pl:~l) 2 + k(~o, ~) I 1 (( 1 ) r +

L =

!/r + ~PlRI 2 + k(qo, qb) I - FTT - (gqo)r_,s

P

M =

r +

n x X*

- FTT - (Vqo) T_,S n . X*

P

I =

) ) ) )

+ k(cp, ~) I - F r T - (Vqo)~_,S n

P

r + ~plfr 2 + k(qo, qb) I - F r T - (Vqo)r_,S

P

- f (Tn. ~+Sn-~b) aa P

(270) (271)

(272)

n-a

(273)

Note that when the results of this subsection are applied to the direct modeling of plates (i.e., to Cosserat surfaces) it is possible to express both J integral and the basic laws of the evolution of cracks (224) and (225) directly in terms of normal and shear stresses and bending moments. This result allows one to obtain a strong reduction of the computational burden in numerical calculations involving cracks that cut the thickness of the plate completely.

BIBLIOGRAPHIC N O T E

This section is based mainly on some unpublished notes of the writer. Proposition 11 may be proved by adapting to the present situation the general results of Proposition 5 in Mariano (2000a); a special case of Proposition 11 can be found in Mariano (1995). In the case of Cauchy materials, the evolution of cracks has been treated with the framework of configurational forces in Gurtin and Podio-Guidugli (1996), Gurtin (2000) and Maugin (1992), and some classic results on crack propagation (see Freund, 1990) on fracture have been reobtained within such a theoretical setting.

68

Paolo Maria Mariano

Detailed discussions on the process zone around the crack tip can be found in Aoki et al. (1981, 1984), Curtin and Futamura (1990), Hutchinson (1987), Freund and Hutchinson (1985) and Lam and Freund (1985). The concept of energy release rate has been introduced in Atkinson and Eshelby, (1968) and Freund (1972), whereas the original motivation of the J integral can be found in Rice (1968).

VIII. Latent Substructures Material substructures are called latent when there is a set of holonomic or anholonomic constraints relating the order parameter to the descriptors of the macroscopic motion and deformation. In defining the concept of latence, Capriz (1985) writes: "I say that the microstructure is latent when, though its effects are felt in the balance equations, all relevant quantities can be expressed in terms of geometric and kinematic quantities pertaining to apparent placements" (p. 49). First, one assumes that 9 there is no substructural inertia: k(qp, qb) - 0 9 substructural bulk interactions are absent:/3 = 0 An immediate consequence is that the balance of substructural interactions (63) reduces to DivS = z

(274)

Consequently, the generalized balance of couples (38) changes in eTF T = Div(.A TS)

(275)

whereas the density of internal power of substructural interactions changes in z. ~ + S . ~'~b - Div(S r qb)

(276)

It has a divergence form only and the product sT ~b can be interpreted as a substructural flux of power that corresponds to the interstitial work flux, which is necessary to consider in the special case of higher gradient elastic materials (as shown in the following). Another crucial assumption is the following: 9 the substructural flux of power is objective

Multifield Theories in Mechanics o f Solids

69

This condition requires that ST ~b must not change when qb changes into ~ + A~I (i.e., S T ~ -- S r (qb + .A~I) for any choice of the rigid rotational velocity el)- This implies that ST A = 0

(277)

Condition (277) further reduces the balance of couples to TF T = FT T

(278)

which is the standard symmetry condition of symmetry Cauchy stress TF T. The last assumption that defines completely latent substructures is the following: 9 the order parameter is constrained by a set of frictionless holonomic and anholonomic constraints that express it in terms of the deformation gradient F and, perhaps, of its gradients R e m a r k 9 When Div(S ~ ~b) -- 0 for some special choice of the order parameter, substructural interactions become powerless. In this case, the order parameter appears on constitutive equations only, and an evolution equation for it must be considered instead of the balance of substructural interactions. This is another case in which multifield theories reduce to internal variable schemes. R e m a r k 10 The balance of substructural measures of interactions gives rise to evolution equations for the order parameter even in situations more general than those occurring in the case of latent substructures. The constitutive prescriptions of Proposition 7 are peculiar of thermodynamic equilibrium, or of a reasonable 13 neighborhood of it. In principle, nonequilibrium parts of the interaction measures, depending on the velocity fields, could be considered along nonequilibrium thermodynamic processes. This happens, for example, when viscosity phenomena occur. An interesting case occurs when one considers a decomposition of the self-force z into its equilibrium and nonequilibrium components. The equilibrium part of z is given by (142), whereas the nonequilibrium part z ne may depend on the rate of the order parameter, that is, z -- 0~o~(F, qo, Vq~) + z"e(F, qa,Vq~; qb)

(279)

The nonequilibrium part of the self-force z ne is dissipative and is such that z "e. qb > 0 13The physical meaning of reasonable is currently a matter of open discussion.

(280)

70

Paolo M a r i a M a r i a n o

A solution of (280) is

Zne --zne~o

(281)

with i ne an appropriate definite positive tensor (possibly scalar in some special model) such that ine is a function

i n e : zne(F, qo, Vr

~0)

(282)

In common special cases, a decomposed free energy of the form

-- ~l(V, ~) + @2(qO,V~) [with 61(I, qg) -- 0, I the unit tensor] may be selected and

(283)

lp2 chosen as

1

~ 2 = -~b V qo . V qo + cr(qo)

(284)

with b an appropriate constant and cr(qo) a double-well coarse-grained potential, as in cases of solidification or solid-to-solid phase transitions. When this happens, the bulk balance of substructural interactions (274), as a result of (143), changes into

A ~ - b A q o - 0~0o'(~o) - 0~o~l(F, ~)

(285)

which is a generalized form of the Ginzburg-Landau equation. When, in fact, both A and qO are scalar valued and the body does not undergo deformations, (285) reduces to A~b = bAqo - 0~ocr(~o)

(286)

which is the standard Ginzburg-Landau equation with kinetic coefficient A.

A. SECOND-GRADIENT THEORIES AS SPECIAL CASES OF LATENT SUBSTRUCTURES An important case of latence is characterized by the internal frictionless constraint = ~(F)

(287)

which expresses the order parameter as a function of the macroscopic gradient of deformation F. From (287), by time differentiation one obtains that along the motion qb = ( ~ ) F "

= (~)(grad~)F

(288)

Multifield Theories in Mechanics of Solids

71

where grad indicates the gradient calculated with respect to x. When the velocity i is rigid [see (13)] gradi = e/l

(289)

@R --- e (OqF~) Fc 1

(290)

and (288) changes into

consequently, from (12) it follows that (291)

A = e(o~)F

This relation reduces (277) to (292)

F = o

eS r (~)

which implies that the third-order tensor A, whose elements are given by (293)

t~AB C -- ( s T ) ~ A ( O F ~ ) i B F i C

is symmetric in the last indices, that is, AABC = AACB

(294)

In the standard treatment of internal constraints [such as (287)] in Cauchy continua, it is necessary todecompose the stress T in its "active" and "reactive" parts, indicated with T and T, respectively (as standard in scientific literature on internal constraints). The latter is assumed to be powerless. Analogously, here it is prescribed that r

T=

+T;

a

z--z+z;

r

~

S=

+

(295)

~.F+~.~+~.V~=0 v~', r

(296)

and

By substituting (288) into (296), one finds two conditions. The first condition is that r

+ ~(aF~ + S(V(aF~)) = 0

(297)

The second condition is that third-order tensor S(0v~) is symmetric in the last two indices

(298)

72

Paolo Maria Mariano

Another basic assumption here is that the free energy density 7r has the following structure: A

~p - ~p (F, V F )

(299)

When one uses the mechanical dissipation inequality to obtain constitutive restrictions on the measures of interaction, one finds that (140) reduces to

f

,

v + (~vv~P

+ +

vv) _< o (300)

Given any state (F, V F ) , velocity fields ~" and V~" can be chosen arbitrarily from (F, VF). This arbitrariness implies that a

a

I" + ~ + V(SOF~) -- (VS)OF~ -- 01~

(301)

a

S Ov~ - Ovv ~

(302)

By inserting (297) and (301) in (299) and using (302), one proves the validity of the following proposition"

Proposition 12 (Capriz, 1985) -- Ov~ - Div(Ovr~) - Div(F skw(OvF~F-l))

(303)

or, in components, Tiaa -- OFiAff/ -- (O(VF)iA, O ) , , --(Fis(O(VF)j,c ff/ Faj' -- O(VF)jBAO ) F ~ I ) , c

(304)

where capital indices refer to the reference configuration, whereas the other indices refer to the current configuration. Note that (302) is the standard constitutive restriction of second-gradient elastic materials, which are thus special cases of continua with latent substructure (i.e., special cases of multifield theories).

BIBLIOGRAPHIC NOTE

This subsection is based on Capriz (1985). Many other remarks on latent substructures can be found in Capriz (1989), whereas the special case of smectic liquid

Multifield Theories in Mechanics of Solids

73

crystals is treated in Capriz (1994). The standard theory of second-gradient Cauchy materials can be found in Dunn and Serrin (1985), where the necessity of the introduction of a rate of supply of mechanical energy, called interstitial working in the balance of energy, is proved to be necessary to eliminate the incompatibility with the second law of thermodynamics shown in Gurtin (1965) for these models of materials. The fundamental result of Capriz (1985) is that the interstitial working is not an object whose existence is assumed without any explicit reference to some types of interactions; rather, it is a consequence of the existence of substructural interactions due to a substructure that generates the oscillations of deformations that are measured by VF.

IX. Examples of Specific Cases The framework discussed in previous sections allows one to describe many material substructures. Detailed special theories can be found in the references listed at the end of Section I. Here some prominent examples are summarized briefly. To build up any special multifield theory describing some particular phenomenon, one must 9 choose a suitable order parameter and then M to model the substructure of the material 9 choose an appropriate form of the free energy ~p 9 evaluate the possible occurrence of latence of substructures induced by the need of some internal constraints motivated by the physical experience After these steps, the constitutive equations for the measures of interaction follow from Proposition 7 and, in the presence of discontinuity surfaces, are supplemented by the results in Proposition 8. In this way, one can write field equations (35), (39), and (46) [with the addition of (129), (130), and (131) in presence of interfaces] in terms of x(.) and qo(-) and then attempts to solve them. In particular, when one chooses to solve (35), (39), and (43) by means of finite element schemes and then must analize integral (relaxed) forms of the field equations, one obtains stiffness matrices more articulated than those of Cauchy continua and an array containing both the components of the placement (or the displacement) of each material patch and the components of the order parameter.

74

Paolo Maria Mariano A. MATERIAL WITH VOIDS

The model of materials with voids is the simplest multifield model. When pores are finely distributed throughout the body, one way to describe them is to choose the order parameter qo as a scalar that associates to each point X the void volume fraction of the material patch at X. In this case, .Ad reduces to the interval of the real axis [0, 1] and A is identically zero [see discussions before (43)]. Substructural bulk measures of interaction/3 and z reduce to scalar, whereas the microstress S becomes a vector as a consequence of Proposition 7. An interesting case is the one of linear elastic materials with voids (or elastic porous materials), with perhaps some damping effects in the pores. By indicating with e the infinitesimal strain tensor e = symVu (u is the displacement), linear constitutive equations relevant for this case (and written with respect to a reference state free of stress) can be obtained by taking for the free energy ~p a quadratic form in e, qO, and Vq9 and applying Proposition 7. In addition, one can consider small viscous effects due to the surface tension at pores by using the procedure discussed in Remark 8 before (283). At the end of calculations, one obtains for elastic porous materials the following constitutive relations with damping: g-,(1)

,,-,(2)

1"ij --Cijhk~hk + ,-.ijkq),k + Gij (D

(305)

-- C(3)~ - C(4)~ - Ci(5)~ij - C~ 6)q),i

(306)

t.~i _ ,,-,(7) c ij qg,i -Jr-_(8) I-.ijk~jk -Jr- C~9)q)

(307)

where 1", ~, and S represent (as in Section III) linearized measures of interaction; Cijhk is the usual stiffness tensor; and C (i) are appropriate constitutive constants. Of course, ~0,i denotes the derivative Ox, q). In the isotropic case, (305)-(307) reduce to

Tij = ~.~ij~,hh + 21zeij -I- ~ ( l ) ~ i j

(308)

= _ ~ ( 2 ) ~ _ ~:(3)q9 _ ~(l)6h h

(309) (310)

~--~i = ~(4)q9,i

where ~ij is the unit tensor and X and # are the standard Lam6 constants and are related to the other constants by the following inequalities: /z>O; 3X + 2/z ~> O;

~(4)/>0;

~(3)>~0

(3X + 2/z)~ <3) >~ 12~
(311) (312)

Multifield Theories in Mechanics of Solids

75

which assure uniqueness and weak stability of solution of the dynamic balance equations. When one analyzes the dynamics arising from (62) and (63) with constitutive equations (308)-(310), one finds the validity of the proposition in the following. Proposition 13 (Cowin and Nunziato, 1983) Transverse waves propagate at a constant speed without affecting the porosity of the material and without attenuation. Longitudinal acoustic waves are both attenuated and dispersed as a result of the changes in material porosity that accompany the wave. Note that the damping term in (306) arises from a nonequilibrium component z ne of the self-force. In this sense, these linear materials are not purely elastic.

BIBLIOGRAPHICNOTE The nonlinear theory of materials with voids has been introduced by Nunziato and Cowin (1979). Proposition 13 is discussed in Cowin and Nunziato (1983). Further studies on this topic can be found in Cowin (1985), Dhaliwal and Wang (1994), Nunziato and Walsh (1978), Diaconita (1987), Fr6mond and Nicolas (1990), and Mariano and Bernardini (1998).

B. TwO-PHASE (OR MULTIPHASE)MATERIALS In two-phase materials, possible phase transitions may change one phase into another. A scalar-order parameter may be used to associate to each point X information on the phases; that is, it may associate to X the volume fraction of one phase, or it may be the indicator of same phase. In the latter case, ~0 is zero if there is not a certain phase and is equal to 1 if the contrary is true. Then interfaces (across which phase transition occurs) are the boundaries of sets that admit ~0 as an indicator function. When ~0 is interpreted as the fraction of a certain phase, .A4 reduces to [0, 1], and the interfaces are considered to be smeared throughout the whole body. Transition layers are identified throughout the body as the thin layers in which V~o undergoes large oscillations. The use of such a kind of order parameter allows one to approximate the nonconvex variational problems arising in the evaluation of two phase bodies. If order parameters are not introduced, one manages, in fact,

76

Paolo Maria Mariano

potentials that have two wells (one well for each phase). During the evolution of phase transitions, the order parameter is ruled by an evolution law, such as (186), which derives from the balance of substructural interactions. In the case of multiphase materials, one may choose as order parameter a list ~ = (991 . . . . .

qgN)

(313)

whose entries take values on [0, 1] and are subjected to the constraint N Z(/9 i -

(314)

1

i-I

thus only N - 1 are independent. In this case, .A4 becomes the cube [0,1] x . . . x [0,1]

Ntimes

(315)

BIBLIOGRAPHIC NOTE

Two-phase materials have been treated with the help of scalar-order parameters in Penrose and Fife (1990), Colli et al. (1990) Fr6mond (1987), Fried and Gurtin (1993, 1994, 1999), and Fried and Grach (1997), and references therein; the original idea of using order parameters to describe phase transitions is from L. D. Landau.

C. COSSERAT CONTINUA

Each patch is considered to be a rigid body that can rotate independently from the neighboring patches. Such a rotation can be described by using as order parameter an orthogonal tensor Q (X, t). In this case, M reduces to the space of orthogonal tensors Orth + such that QTQ = I and det Q = 1. Simple calculations show that z" is always zero because .A4 coincides with Orth +. The strain tensor E, which in Cauchy solids is given by I ( F T F - I), is here given by E = 2MTM- M T - M

(316)

where 1

M - :(QTFZ

I)

(317)

Multifield Theories in Mechanics of Solids

77

In the setting of small deformations, the infinitesimal strain tensor e becomes = V'u - eq

(318)

with q a vector describing the rotation. If one chooses the free energy as a function of F, Q, VQ, by applying Proposition 7, one realizes that z is a second-order tensor, S a third-order tensor, and, consequently,/3 is a second-order tensor. Moreover, after a straightforward calculation, one realizes that ,4 is a third-order tensor given by .A - - e Q ( . A i j k eijh Qhk), and then (39) becomes -

e(jOQ r - zQ T + (DivS)Q r)

-

0

-

(319)

It is then possible to put 1

- ~e(/3Q T)

(320)

-- ~e(zQ r + S V Q v)

(321)

1

,.~- IoSQT

(322)

2 where/3 and ~ are vectors whose components are 1

~i - -~Oijkl~jlO T

(323)

Zi = ~Oijk (Zjl O~ + Silm Qrlm,k)

(324)

1

whereas S is a second-order tensor given by 1

8ij -- -~OijkSklm O/rm

(325)

The generalized balance of couples (38) reduces to 1 e T F r = e~ 2

(326)

which eliminates the self-force from the balance of substructural interactions. Consequently, the balance of substructural interactions becomes /3 - 1 T F r + Div,~ - 0 2

(327)

To account for inertial effects, one must consider the right-side term of (63), in which the kinetic coenergy X density must depend only on the product Q v Q to be objective. Simple algebra allows one to write tensor Q T 0 in terms of a vector

78

Paolo Maria Mariano

a -- e ( Q TI)), and X may depend thus only on a. In this way, the explicit dynamic expression of (327) becomes

/ 3 - ~ T1 F T + D i v , ~ -

( p OX(a))" Oct

(328)

The framework of Cosserat materials is widely used to model structural elastic elements such as beams, plates, and shells. In addition, some proposals for a Cosserat plasticity have been developed with the aim of analizing certain aspects of localization of deformations.

BIBLIOGRAPHIC NOTE

A detailed list of references on Cosserat materials is given at the end of Section I. Here I followed some notes in Capriz (1989), Ericksen and Truesdell (1958), Antman (1972), and Villaggio (1997). For Cosserat plasticity, see Steinmann, (1994).

D. MICROMORPHICMATERIALS The model of micromorphic materials is useful to describe the influence of large molecules on gross mechanical behavior, as happens in some polymers. Each material patch is considered to be a cell that may undergo additional deformations independently of the neighboring patches. The order parameter qo is a second-order invertible tensor (the gradient of microdeformations) and A4 reduces to Lin + [i.e., the space of linear forms between vector spaces], with positive determinant. By using Proposition 7, one realizes that the self-force z becomes a second-order tensor whereas the microstress S is a third-order tensor. Nonlinear measures of deformation are (cf. Capriz, 1989)

E--~I(FTF-I); M- ~(qoVF-l); ~-qo -~Vqo

(329)

By assuming that r = Qg~ after rigid changes of spatial observers (where Q is an orthogonal tensor describing the rotation of the observers), one obtains that A is a third-order tensor given by eg~, or in indices, ~ i j k = eijl~Olk. Of course, because ~o ~ Lin § it never vanishes; then z" in (43) is identically zero, as in the case of Cosserat continua.

Multifield Theories in Mechanics of Solids

79

An interesting aspect of micromorphic material appears evident in the setting of small deformations. It is possible to introduce the relative infinitesimal strain ~r as

er = V u -

qo

(330)

and associate to each patch the state (e, er, Vqo). In this way, the internal power becomes f13 (T " ~ "~- Z " ~r -'~"S " V qo)

(331)

,

If one assumes an internal constraint prescribing that ~r = 0

(332)

the inner power (329) reduces to

fB

(T.e+S.

VVu)

(333)

,

which is the inner power that one writes when dealing with second-gradient materials. From the multifeld scheme of micromorphic materials with the internal constraint (332), strain gradient plasticity has been formulated to account for length scale-dependent effects in plasticity, such as the Hall-Petch effect (the strength of polycristalline aggregates increases with decreasing grain size) or the experimental results showing strong size effects in the case of torsion tests on copper wires. Basically, one realizes that the presence of VV u in (333) accounts for weak nonlocal effects as a result of the latence induced by the constraint (332). However, each multifield theory is "weakly nonlocal" in nuce as a result of the presence of the gradient of the order parameter in the list of constitutive variables.

BIBLIOGRAPHIC NOTE

Micromorphic materials have been introduced in Mindlin (1964; see also Grioli, 1960, 1990; Bofill and Quintanilla, 1995; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000; Capriz and Podio-Guidugli, 1976). For the strain gradient plasticity, see Fleck and Hutchinson (1997).

Paolo Maria Mariano

80

E. NEMATICLIQUID CRYSTALS Nematic liquid crystals are characterized by the presence of rodlike molecules smeared throughout the liquid. A commonly accepted choice of the order parameter is a second-order tensor Do representing a second-order approximation of an unknown orientation distribution function 14 of the molecules9 In some sense, DD(X) represents the macroscopic average of the orientation of the molecules in the patch at X. Do is also such that

DD -- DTD

trDo = 1

(334)

and has nonnegative eigenvalues. Usually, the deviatoric part D of Do is adopted instead of Do, and one has D = Do - lI. Nematic liquid crystals are characterized by the coincidence of two of the three eigenvalues of D, which is represented in the form

D=~' (

d|

')

9

d.d-1

9

2' < ~ < 1

(335)

The scalar ~"is interpreted as degree of orientation. Vector d (which is the averaged direction of the molecules of each material element) and the parameter ~" are taken as order parameters. Consequently, one has from Proposition 7 measures of interaction z0 and So associated to d (which is the averaged direction of the molecules of each material element) and measures of interactions zc and Sc associated to ~'. There are two manifolds A/l; the first manifold coincides with the unit sphere in It~3 and the second manifold is the interval [ - 21, 1]. Two different substructural balances of the same form of (42) must be satisfied. As pointed out in Example 2 in Section II.D, because one deals with a liquid, one must have measures of interaction in the current c c C configuration and denotes them with/3~, z~, $~,/3~, z~, S~. In this way, one writes two balances of substructural interactions [because one has at his disposal two order parameters; see (44)] /3~ - z~ + divS~ - otd

(336)

/3~ - z~ + divS} - 0

(337)

the Cauchy balance of standard interactions divT C + b' = 0

(338)

where T r is Cauchy stress and b c denotes bulk forces in the current configuration, and only one balance of torques because r on [ - ~1, 1] vanishes identically skw (T ~ + z~ | d + (gradd) rS~) - 0 14An analogous choice can be made in the case of microcracked bodies.

(339)

Multifield Theories in Mechanics of Solids

81

where grad is calculated with respect to coordinates in the current configuration. _ct! m 0. Note that in (337), zc The free energy depends on cl and its gradient and ot and its gradient, in addition to the measures of deformation. The use of Proposition 7 allows one to obtain constitutive prescription after pushing forward the relevant constitutive variables. The degree of orientation is not the sole characteristic parameter that can be derived from D. To describe the possible emergence of optic biaxiality of the nematic, two other parameters must be used and may be derived from D. In particular, one realizes that the tensor Do generates an ellipsoid. Then the degree of prolation dg and the degree of triaxiality dt are useful parameters to describe the nematic distribution and are defined by I

dg -- -~

(340)

(3)~i - 1 ) I

dt =

6~/-3

I)~i - ~.i+11

(341)

i=1

where ~-i a r e the eigenvalues of Do, i = 1, 2, 3. They can also be considered to be order parameters.

BIBLIOGRAPHIC NOTE

The theory of liquid crystals with variable degree of orientation has been introduced by Ericksen (1991) and the derivation of balance equations as well as the constitutive restrictions can be found in his paper. The degree of prolation dg and the degree of triaxiality dt have been introduced by Capriz and Biscari (1994).

F. FERROELECTRIC SOLIDS

Some crystalline materials as barium-titanate experience spontaneous polarization associated to the rearrangements at the crystalline temperature called Curie temperature. To describe polarization, one may choose as order parameter a vector p of ~3 such that [Pl < Ps with Ps a material constant. In this way, M becomes the ball {p 6/~3 I Ip[ < Ps}. In writing the overall power 79 , one must consider not only the external power j'Qext and the inner power TAint of mechanical interaction, but also a self-power TAself of electrical nature. Then, 79 = j-Qext _ ~')int _ TAself. In particular, with reference to

Paolo Maria Mariano

82

any part B~' of the current configuration ~t of the ferroelectric body, one writes

pext(B'~)-faB: ( b C ' v + j O c ' o ) + f

B~ ( (

Tc -- 2(P" 1

n)2 ) n . v +

SCn 90 ) (342)

/,

~s)int(]3t) -- [

dB

79self(Bt) --

(T C. gradv +

(pC(grade)p. v + ;

S c.

gradl~ + z c. 1~)

fie. P)Jr f

l

~ ( p - n ) 2 n 9v B;

(343)

(344)

where v is the velocity ~, e is the electric field, and the superscript c indicates that the relevant fields are taken in the current configuration. Moreover, one may disregard gravitational forces and consider the noninertial parts of bulk interactions b c and/3 c to be measures of the electrostatic interactions of B t with its exterior. Taking into account such a hypothesis, the procedure of Section II.C allows one to obtain balance equations in the current configuration divT C + pC(grade)p = divSC - zC +

PCe --

pC~r

Prop + B p

(345) (346)

where Pm is a material constant and B is an appropriate second-order tensor such that B p . p = O. In the reference configuration, the balances (344) and (345) become DivT + p(grade)p = p~

(347)

DivS - z + pe - (det F)(pmli -+- BI~)

(348)

T = (det F)TCF - r

(349)

S = (det F)SCF - r

(350)

z = (det F)z c

(351)

F)p c

(352)

where

p = (det

To derive constitutive equations one chooses in this case ~ / - - it(F, p) + W ( p ) + 1 ff2Vp 9 V p

(353)

where 9 is a material constant and s is such that Lt(Q, .) = 0

(354)

Multifield Theories in Mechanics of Solids

83

for any orthogonal tensor Q. Taking into account (353), Proposition 7 allows one to derive constitutive equations. During the polarization, the walls of ferroelectric domains evolve; in addition, the coupling of mechanical deformation and electric polarization can cooperate and generate localization of domain walls on some "macroscopic" discontinuity surface that may evolve and can be considered as an "emerging" domain wall. To study the evolution of standard and emerging domain walls, one can use the general results of Section VI, taking into account that here the configurational stress I? assumes the form ~ = ~pl - F r T - (Vp) TS. In the case of rigid ferroelectrics, I? reduces to

I~rigid--

W ( p ) + ~ff V p . Vp

I-

ff V p ) r V p

(355)

and the evolution equation (190) reduces to an equation describing an isotropic motion by curvature (see Dav] and Mariano, 2001) which fits, for example, the experimental results on barium titanate.

BIBLIOGRAPHIC NOTE

The model of ferroelectric materials summarized in Section IX.F is discussed in Dav] (2001) and in Dav] and Mariano (2001) and allows one to deduce expressions for the evolution rules for domain walls and comers in accord with experimental evidences. Basic remarks on ferroelectrics can be found in Jona and Shirane (1962), Little (1955), Hwang et al. (1995), Rosakis and Jiang (1995), and references therein.

G. MICROCRACKEDMATERIALS In bodies free of microcracks, any given deformation maps each patch at X into a point x(X) of the Euclidean space and the displacement u(X) from X to x(X) is u(X) = x(X) - X. When microcracks are diffused within the body, under the same given deformation the patch at X undergoes a displacement u + d, where d is the kinematical perturbation due to the enlargement or the closure of microcracks. At each x, d(x) is a relative displacement; it is the difference between the real placement x'(X) of the patch at X and the theoretical placement x(X) occurring at X if microcracks were absent i.e. d = x' - x. It is thus possible to take the vector d as order parameter; therefore, A//reduces to I~3. By using Proposition 7, one realizes that the self-force z is a vector and the microstress S is a second-order tensor.

84

Paolo M a r i a M a r i a n o

The standard gradient of deformation F is given by F - I + Vu (where I is the unit tensor), whereas the overall gradient of deformation Ftot is given by Ftot F+Vd. Overall measures of deformations associated to Ftot are the overall right CauchyGreen Ctot given by Ctot --FtotFtot T and the overall deformation tensor Etot = 1 ~ (Cto, - I). Two linearized measures of deformation may be derived by taking IVul << 1 and IVcll << 1. They are indicated with eu and ed and defined by eu - symVu

ed -- symVd

(356)

Balance equations may be deduced in the manner indicated in Section II. When one considers the equilibrium of a microcracked body without the help of order parameters, one writes the energy and tries to minimize it over a domain endowed with an unknown set of discontinuities for the displacement field: the microcracks. Then one must find the solution of the variational problem

minlftsw(X,Vu)+fG(ts)r(X,[u]| }

(357)

where w(., .) is the bulk strain energy density, r(., .) is the surface energy density of microcracks, and nm is the normal to each microcrack face. Problem (357) has two unknowns: the displacement field u(. ) and the set G(/3) of microcracks. Thus the equilibrium problem becomes a free boundary variational problem and is very difficult to solve. By using the just described order parameter, one may regularize this free boundary problem and reduce it to a more simple one. In particular, it is possible to "approximate" (357) with the problem

min{fw(x, vu, ct,vcl)]

(358)

where/3 is considered to be free of microcracks and cl represents the kinematical perturbation induced to u by microcracks. When irreversible behavior occurs and the microcracks may grow, one considers the free energy r instead of w and may use the procedure in Section II.D and Proposition 7 to write balance equations of the form Div(OvCr) + b = 0

(359)

Div(OvdTr) -- 0d~ = 0

(360)

s k w ( O v ~ F r + OdTZ|

cl + Vd T0vd~) = 0

(361)

where Ov~ is the Piola-Kirchhoff stress T, 0vo r is the referential microstress S, and Od~/ris the self-force z'. A basic problem is thus to find an explicit expression for ~r (i.e., constitutive equations for the measures of interaction). One way to obtain

Multifield Theories in Mechanics of Solids

85

constitutive equations is to use an identification procedure based on the equivalence of the density of power in the continuum and the power of a cell of a periodic discrete model of the microcracked body. This procedure may always be used when the order parameter has the physical meaning of a displacement, a rotation, or a deformation, provided a suitable choice of the discrete model. The physics of microcracked bodies suggests the adoption (at least in the case of elastic behavior) of a discrete model made of two periodic lattices connected by elastic links. The first lattice (macrolattice) is made of indistinct material points (spheres) and describes the material at molecular level. The second lattice (mesolattice) is made of empty shells (lakes) connected to one another by elastic links and describes the material at the mesoscopic level of microcracks; each shell is forced to deform along a plane orthogonal to its major axis only. This is a case of "virtual" substructures because the stiffness of each shell is the one of the material surrounding each microcrack. Because the lattice is assumed to be periodic, the attention is focused on a cell of the lattice: a representative volume element (RVE). Chosen the RVE of the discrete system, it has N spheres and M shells; in the macrolattice of the RVE there are L links: LN is the number of interlattice links, and LM is the number of links in the mesolattice. It is assumed that the elastic links in the RVE can carry only axial forces: ti denotes the force in the ith link of the macrolattice, zt the force in the lth interlattice link, Zj the force in the jth link in the mesolattice, and the force on the shell located at h is indicated with z0h. Spheres located at a and b in the RVE undergo displacements u a and u b, whereas a shell located at h (or at k) undergoes a relative displacement clh (or clk) of its margins along the plane of deformation. In this way, appropriate measures of deformation in the discrete RVE are Oh, clh - clk, Illa -- d h, and u a - u b, where the last three measures of deformation make sense only when the spheres at a and b and the shells at h and k are connected. There are two steps to obtain constitutive equations for the measures of interaction in the continuum from the discrete model in the case of infinitesimal deformations.

1. First one equalizes the density of the internal work in the continuum with the internal work developed in the discrete RVE and in the case of linearized kinematics writes

T . Vu + ~,. cl + , S . V d -

1 Vol(RVE)

LN

M

l=l

h=l

i=1

LM j=l

ti 9 (u a - u b) \

)

(362)

Paolo Maria Mariano

86

where L is the number of links of the macrolattice in the RVE, LN is the numbers of interlattice links, M is the number of shells, and LM is the number of links in the mesolattice. 2. Then one must write the measures of deformation in the discrete RVE in terms of the ones of the continuum. A reasonable choice is to take U a --" U ( X ) +

Vu(x)(a - x)

(363)

d h -- O(X) +

Vd(x)(h

(364)

-

x)

with x a point in the RVE chosen such that ua

-

u b =

O h -- O k : Ua -- O h :

Vu(x)(a

Vu(x)(a

-

(365)

b)

Vd(x)(h - k) -

x) -

(366) (367)

V d ( x ) ( h - x)

By inserting (363)-(367) in (361), one obtains

1

ti | (a - b) + Z

Vol(RVE) i=1

(368)

M

Vol(RVE) ~

h=l

)

1=1

1

1 Vol(RVE)

zt | (a - x)

z~

z0~ | (h - x ) + ~ zj | (h - ~ j=l

(369)

~ z, | (h - x~ t=l

(370) At this point, it is necessary only to choose appropriate constitutive relations for the links in the RVE to find explicitly the measures of interaction in the linearized kinematies T, ~, ,S in terms of Vu, d, Vcl. The simplest choice is ti - IK(ua - ub), Z~ - - Q d h, z l - ]~-]~(Ua - dh), and zj - - D l d h l l d h l ( ( h - k ) / I h - k12), with K, Q, H, D appropriate stiffnesses. Note that the first three constitutive relations are forms of the Hook's law, whereas zj (the interaction between two neighboring shells) is nonlinear is due to the fact that two neighboring microcracks interact like neighboring dislocations located at a distance ( h - k ) / I h - kl 2 from each other. When one inserts the previous constitutive relations into (368)-(370), one finds explicit constitutive equations for the measures of interaction in the continuum. The simplest form, which is valid in the case of linear elastic behavior

Multifield Theories in Mechanics of Solids

87

(microcracks do not grow) and when the material is centrosymmetric, is the following: T = AVu + A'Vd - Cd S - A'Vu + GVcl

(371) (372) (373)

where the fourth-order tensors A, A', C, G have major symmetries and can be computed explicitly by the just described procedure (a detailed calculation of them can be found in Mariano and Stazi, 2001). Consequently, balance equations (359) and (360) become, in the case of linear elastic behavior of the material, b + div(AVu + A'Vd) = 0

(374)

div(A'Vu + GVd) - Cd = 0

(375)

When one solves with standard finite element procedures the equilibrium problem with constitutive equations (371)-(373), one finds as a basic result the occurrence of localization phenomena of the deformation. This result cannot be obtained by using standard linear elasticity in the setting of Cauchy continua and has an immediate experimental evidence (to make an experiment, the reader can take any rectangular sheet of paper, load it with a traction in the middle point of one of its sides and fix only the opposite side; localization occurs). These localization phenomena can be obtained from (374) and (375) as a result of the presence of the self-force Cd and of the gradient of the order parameter. Basically, I conjecture that one may find localization phenomena of the order parameter (i.e., of the substructure) in a wide range of possible multifield models as a result of the analytical structure of the balance of substructural interactions. From a physical point of view, I interpret these localization phenomena of the order parameter as the description of cooperative patterns of material substructures.

BIBLIOGRAPHIC NOTE

Details about this multifield model of microcracked bodies can be found in Mariano (1999), Mariano and Stazi (2001), and Mariano et al. (2001); localization phenomena of deformation within the setting of linearized constitutive equations are described in Mariano and Stazi (2001) and Mariano et al. (2001).

88

Paolo Maria Mariano

Acknowledgments I t h a n k G i u l i a n o A u g u s t i , G i a n f r a n c o Capriz, and E r i k van der G i e s s e n for p r o f o u n d and d e t a i l e d d i s c u s s i o n s a b o u t the topics in this article. M y friend and f o r m e r student F u r i o L o r e n z o Stazi was f u n d a m e n t a l in m a k i n g c o r r e c t i o n s and t y p i n g this article.

References Abeyaratne, R., and Knowles, J. K. (1990). On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Sol. 38, 345-360. Abeyaratne, R., and Knowles, J. K. (1991). Kinetic relations and the propagation of phase boundaries in solids. Arch. Rat. Mech. Anal. 114, 119-154. Abeyaratne, R., and Knowles, J. K. (1994). Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction. Arch. Rat. Mech. Anal. 126, 203-230. Aero, E. L., and Kuvshinskii, E. V. (1960). Fundamental equations of the theory of elastic media with rotationally interacting particles. Fizika Tverdogo Tela 2, 1399-1409. Anderson, D. M., McFadden, G. B., and Wheeler, A. A. (2000). A phase-field model of solidification with convection. Physica D 135, 175-194. Angenent, S., and Gurtin, M. E. (1989). Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface. Arch. Rat. Mech. Anal. 108, 323-391. Antman, S. S. (1972). The theory of rods. In Handbuch der Physic, Vol. 2 (C. Truesdell, ed.), pp. 641703. Springer Verlag, Berlin. Antman, S. S. (1995). Nonlinear Problems of Elasticity. Springer Verlag, Berlin. Antman, S. S., and Marlow, R. S. (1993). New phenomena in the buckling of arches described by refined theories. Int. J. Sol. Struct. 30, 2213-2241. Aoki, S., Kishimoto, K., and Sakata, M. ( 1981 ). Energy release rate in elastic plastic fracture problems. ASME--J. Appl. Mech. 48, 825-829. Aoki, S., Kishimoto, K., and Sakata, M. (1984). Energy flux into process region in elastic plastic fracture problems. Eng. Fract. Mech. 19, 827-836. Atkinson, C., and Eshelby, J. D. (1968). The flow of energy into the tip of a moving crack. Int. J. Fract.

4, 3-8. Augusti, G., and Mariano, R M. (1999). Stochastic evolution of microcracks in continua. Comp. Meth. Appl. Mech. Eng. 168, 155-171. Batra, R. C. (1983). Saint-Venant's principle in linear elasticity with microstructure. J. Elast. 13, 165-173. Berglund, K. (1977). Generalization of Saint-Venant's principle to micropolar continua. Arch. Rat. Mech. Anal. 64, 317-326. Binz, E., de Leon, M., and Socolescu, D. (1998). On a smooth geometric approach to the dynamics of media with microstructure. C. R. Acad. Sci. Paris t. 326, 227-232. Bofill and Quintanilla (1995). Some qualitative results for the linear theory of thermo-microstritch elastic solids. Int. J. Engng Sci. 14, 2115-2125. Capriz, G. (1985). Continua with latent microstructure. Arch. Rat. Mech. Anal. 90, 43-56. Capriz, G. (1988). Some observations on the dynamics of Cosserat continua and the biaxial nematic liquid crystals. Atti Sere. Mat. Fis. Uni~: Modena 36, 281-288.

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