Variational formulations for the linear viscoelastic problem in the time domain

Accepted Manuscript Variational formulations for the linear viscoelastic problem in the time domain A. Carini, O. Mattei PII:

S0997-7538(15)00051-0

DOI:

10.1016/j.euromechsol.2015.05.007

Reference:

EJMSOL 3188

To appear in:

European Journal of Mechanics / A Solids

Received Date: 2 December 2014 Accepted Date: 12 May 2015

Please cite this article as: Carini, A., Mattei, O., Variational formulations for the linear viscoelastic problem in the time domain, European Journal of Mechanics / A Solids (2015), doi: 10.1016/ j.euromechsol.2015.05.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A. Carini, O. Mattei∗

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Variational formulations for the linear viscoelastic problem in the time domain

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DICATAM, University of Brescia, Via Branze 43 - 25123 Brescia, Italy

Abstract

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Under the assumption of small displacements and strains, we formulate new variational principles for the linear viscoelastic hereditary problem, extending the well-known Hu-Washizu, Hellinger-Reissner, Total Potential Energy, and Complementary Energy principles related to the purely elastic problem. In addition, a new global minimum formulation is derived, giving an energetic interpretation. The new formulations are based on a convolutive bilinear form of the Stieltjes type and on the division of the time domain into two equal parts, with the resulting decomposition of the variables and of the equations governing the problem. In particular, the global minimum principle is achieved by virtue of the positive definiteness of a part of the split constitutive law operator and by means of a partial Legendre transform, and is then used to provide bounds of the overall mechanical properties of viscoelastic composite materials.

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Keywords: Viscoelasticity, Variational principles, Composite materials, Bounds

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1. Introduction

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The earliest variational formulations for the viscoelastic problem, although under restrictive assumptions, date back to Biot (1956), Freudenthal and Geiringer (1958), Olszak and Perzyna (1959), and Onat (1962), but ∗

Corresponding author Email addresses: [email protected] (A. Carini), [email protected] (O. Mattei)

Preprint submitted to European Journal of Mechanics - A/Solids

March 30, 2015

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the first true formulation has to be ascribed to Gurtin (1963), who generalized the classical elasticity principles to the linear viscoelastic case, using implicitly a convolutive bilinear form. Then, Gurtin, using a method able to transform initial-boundary value problems into equivalent boundary value problems, governed by integro-differential equations, formulated variational principles for the elastodynamics (Gurtin, 1964a) and for other linear initial value problems (Gurtin, 1964b). Subsequently, Tonti (1973) highlighted the crucial role played by the choice of a suitable bilinear form, in order to provide a variational formulation for the given problem. In particular, he showed how the use of a bilinear form of the convolutive type allows one to provide a variational formulation for initial value problems, making unnecessary their transformation into problems with only boundary conditions. The ideas of Gurtin and Tonti have been exploited by many authors (Schapery (1964), Leitman (1966), Taylor et al. (1970), Brilla (1972), Reddy (1976), just to name a few) and they have also been extended to the method of boundary integral equations (see Carini et al. (1991)). Hlav´aˇcek (1966) proposed extremum formulations for isotropic viscoelastic materials with Poisson’s ratio invariant in time and, under the same assumptions, a minimum formulation has been proposed also by Srinatha and Lewis (1982). Christensen (1968, 1971), using state functions such as the free energy, proposed an extremum variational formulation, valid under restrictive assumptions. Applications of his results have been carried out by Kulejewska (1984). Rafalski (1969, 1972, 1979) formulated extremum principles based on a bilinear form with respect to which the operator of nth derivation, with the initial conditions, proves to be self-adjoint and positive definite. Breuer (1973) established minimum principles for incompressible viscoelastic solids. For the non-linear thermo-viscoelastic problem, new variational principles have been developed by Biot (1976). Reiss and Haug (1978), expounding the ideas of Rafalski, formulated extremal principles for problems with initial values, including the problem of hereditary viscoelasticity. Huet (1992), through the use of pseudo-convolutive and pseudo-biconvolutive bilinear forms, although under restrictive assumptions, obtained two principles, extensions of the minimum principles of Total Potential Energy and Complementary Energy related to the linear elasticity.

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Tonti (1972) provided a general criterion of “potentiality”, in search of variational principles. Magri (1974) illustrated the way to contrive suitable bilinear forms, with respect to which a given linear operator proves to be symmetric, and then stated a constructive method to endow any linear problem with a variational formulation. Tonti (1984) generalized the method to any non-linear problem. The latter approach was applied also to the linear viscoelasticity case (see Carini et al. (1995) and Carini and De Donato (2004)). Many efforts have been carried out with the purpose of extending, to the viscoelastic case, the results obtained within the framework of the homogenization theory elaborated for the elastic case, with particular attention to the bounds of the “overall properties”. This has been done, primarily, through quasi-elastic approximations, for special temporal values, in the domain of the Laplace transform and for methods based on complex moduli (see Hashin (1965), Schapery (1964), Minster (1973), Roscoe (1969, 1972), Ch´etoui (1980), Ch´etoui et al. (1986), Huet (1995), Cherkaev and Gibiansky (1994), Gibiansky and Milton (1993), Vinogradov and Milton (2005)). Few results have been achieved regarding the bounds of the creep and relaxation viscoelastic functions in the time domain, and only under restrictive assumptions (see Christensen (1969)) or as pseudo-elastic approximations (see Schapery (1974)). The work of Milton (1990), thanks to its generality, can be easily extended to time-dependent problems, although it has not been formulated explicitly to that purpose, leading to a variational formulation also for the viscoelasticity case. Finally, Huet (1995), using the concept of pseudo-convolutive bilinear form, derived useful unilateral and bilateral bounds for the relaxation function tensor. In this paper we formulate five new principles: four are of the variational type and one of the minimum type. The new formulations are gathered from the decomposition of the time interval into two subintervals of equal length. Separating the variables defined over the first subinterval from those related to the second one, we obtain a formal doubling of the unknowns. Accordingly, the constitutive law operator is split into sub-operators, arranged into a two-by-two matrix that is symmetric with respect to a bilinear form of the convolutive type in the time variable. On the main diagonal, we have two operators: one is null and the other is positive definite, since the related quadratic form physically represents a free energy, positive by virtue of the results obtained in the thermodynamics field by Staverman and Schwarzl (1952a,b), Coleman (1964), Mandel (1966), Coleman and Mizel (1967), Brun

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2. The linear viscoelastic problem Let us consider a body Ω ⊂ R3 made of a linear viscoelastic material, that may be heterogenous and anisotropic. An orthogonal Cartesian reference system is used, with coordinates xr , r = 1, 2, 3. The components of vectors, second order and fourth order tensors are indicated with the usual indicial notation. Einstein’s convention over repeated indices is adopted. The aim is to determine the displacement, strain and stress fields at every point of the material, for every time t in the interval [0, 2T ], with T > 0, being the solid undisturbed for t < 0. Let us denote by V the volume of the region Ω and by Γ = Γu ∪ Γp the external surface, with unit outward normal ni (xr ). Let ui (xr , t), ij (xr , t) and σij (xr , t) be, respectively, the displacement, strain and stress fields at the point xr ∈ Ω, at the time t ∈ [0, 2T ]. The stress field σij (xr , t), with σij (xr , t) = 0 for t < 0, satisfies the equilibrium equations:

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(1969), Del Piero and Deseri (1996, 1997) and Amendola et al. (2012). Nevertheless, the quadratic convolutive form associated with the whole constitutive law is not convex but, applying a partial Legendre transform, it is possible to reformulate the constitutive law so that the associated quadratic form is convex. This is a well-known technique (see Callen (1960)), used also by Cherkaev and Gibiansky (1994). The resulting minimum formulation allows one to seek bounds of the mechanical properties of a homogenized solid, given those of the viscoelastic constituents of the heterogeneous medium. Such inequalities are formally similar to those obtained by Cherkaev and Gibiansky (1994), and Milton (1990) in the frequency domain. The paper is organized as follows. In section 2, the linear viscoelastic problem is presented. In section 3 and 4, respectively, the constitutive law (in the Boltzmann form) and the whole problem are rephrased on the basis of the decomposition of the time domain. Furthermore, in section 4, the five variational formulations are provided. In section 5, the problem is written for an RVE of a composite material made of viscoelastic phases and, in section 6, bounds of the homogenized mechanical properties of the composite are shown. In section 7, similar results are obtained in case the constitutive law is written in the Volterra form. Finally, in section 8, the concluding remarks are presented.

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σij/j (xr , t) + bi (xr , t) = 0 in Ω × [0, 2T ] σij (xr , t) nj (xr ) = pi (xr , t) on Γp × [0, 2T ] 4

(1)

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where bi (xr , t) are the volume forces, pi (xr , t) the surface forces imposed on Γp , and the symbol / indicates the partial derivative operation. The displacement field ui (xr , t) and the strain field ij (xr , t), with ui (xr , t) = 0 and ij (xr , t) = 0 for t < 0, fulfill the strain-displacement relations: (2)

where u0i (xr , t) is the displacement field imposed on Γu . In this paper we deal only with non-aging materials (i.e., we consider only the hereditary viscoelasticity case), for which the direct constitutive law, that relates the strain field ij (xr , t) to the stress field σij (xr , t), in the Boltzmann form, reads as follows: Z t σij (xr , t) = Rijhk (xr , t − τ ) dhk (xr , τ ) (3)

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1 ui/j (xr , t) + uj/i (xr , t) in Ω × [0, 2T ] 2 ui (xr , t) = u0i (xr , t) on Γu × [0, 2T ]

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ij (xr , t) =

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0−

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where the integral has to be meant in the Stieltjes sense, and Rijhk (xr , t), with t > 0 (we assume that Rijhk (xr , t) = 0 for t < 0), is the relaxation kernel. In particular, the latter is a tensor of the fourth rank whose components, functions of time and location, are obtained for a unit-step strain history. Equation (3) derives directly from the Boltzmann superposition principle (see Boltzmann (1874, 1878)), and it provides the stress field at the time t, in the fixed point xr ∈ Ω, due to the strain increments dij (xr , τ ), for τ ∈ [0, t]. Suppose that the relaxation tensor satisfies the following symmetry properties, as in elasticity: Rijhk (xr , t) = Rjihk (xr , t) = Rijkh (xr , t) = Rhkij (xr , t) ∀xr ∈ Ω, ∀t ∈ [0, 2T ] (4) and that the following inequalities

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∞ Rijhk (xr ) γij γhk > 0

0 (xr ) γij γhk > 0 Rijhk

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hold, for every xr ∈ Ω and for every non-vanishing symmetric second order 0 ∞ tensor γij , Rijhk (xr ) and Rijhk (xr ) being defined, respectively, as ∞ Rijhk (xr ) := lim Rijhk (xr , t)

0 Rijhk (xr ) := lim Rijhk (xr , t) t→0

t→+∞

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where, for every fixed xr ∈ Ω, we suppose that

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where Cijhk (xr , t) is called creep kernel. The existence of the inverse constitutive law (7) depends on the choice of the functional space D and on the properties of the function R˙ ijhk (xr , t). In particular, if we assume that  1. D = tij (xr , τ ); tij (xr , τ ) ∈ L2 (0, 2T ) ;

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can be locally integrated. This implies that Rijhk (xr , t) is absolutely contin0 uous and bounded and, therefore, Rijhk (xr ) = Rijhk (xr , 0). From (6), it is clear that, fixed xr ∈ Ω, the stress field σij (xr , t) depends on the strain ij (xr , t) at the time t and on the strain history tij (xr , τ ) = ij (xr , t−τ ), for τ ∈ [0, t]. Given tij (xr , τ ), by virtue of (6), we can determine not only the stress field σij (xr , t), but also its history, σijt (xr , t) = σij (xr , t−τ ), for τ ∈ [0, t]. Let us denote by D and Dσ the spaces of the strain and stress histories, respectively. Consider, now, the problem of the invertibility of equation (6), that is the determination of the conditions under which we can express the strain field ij (xr , t) in terms of the stress field σij (xr , t), as follows Z t ij (xr , t) = Cijhk (x, 0) σhk (xr , t) + C˙ ijhk (xr , τ ) σhk (xr , t − τ ) dτ (7)

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∂Rijhk (xr , t) R˙ ijhk (xr , t) := ∂t

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The direct constitutive law (3) can be written, by virtue of a formal integration by parts, in the Volterra form (see Volterra (1909, 1912, 1913)) as follows: Z t R˙ ijhk (xr , τ ) hk (xr , t − τ ) dτ (6) σij (xr , t) = Rijhk (xr , 0) hk (xr , t) +

2. R˙ ijhk (xr , t) ∈ L1 (0, 2T );

3. conditions (4) and (5) hold; 4. the sine Fourier transform of R˙ ijhk (xr , t), that is Z ∞ ˆ R˙ ijhk (xr , ω) := R˙ ijhk (xr , t) sin(ω t) dt 0

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is such that

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ˆ˙ R ijhk (xr , ω) γij γhk < 0 ∀ω > 0 (Graffi’s inequality (Graffi, 1928))

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3. Reformulation of the constitutive law

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γij being any symmetric second order tensor;  then, Dσ = σijt (xr , τ ); σijt (xr , τ ) ∈ L2 (0, 2T ) , and the inverse constitutive law exists and takes the form (7). For further details on the invertibility of the equation (6), see for instance Fabrizio (1992) and Bozza and Gentili (1995). For the sake of simplicity, throughout the following the dependence on the variable xr will be omitted, unless strictly necessary.

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3.1. Direct constitutive law The direct constitutive law (3), written in a compact form as

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Hereafter, the subscript 1 will refer to quantities defined over the time interval [0, T ], while the subscript 2 will be used in reference to quantities defined over [T, 2T ].

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Let us split the time interval [0, 2T ] into two equal subintervals, [0, T ] and [T, 2T ]. Accordingly, the strain and stress fields can be written, respectively, as:  1ij (t) for t ∈ [0, T ] ij (t) = (8) 2ij (t) for t ∈ [T, 2T ]  σ1ij (t) for t ∈ [0, T ] (9) σij (t) = σ2ij (t) for t ∈ [T, 2T ]

σij (t) = L ij (t)

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L(.) :=

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Rijhk (t − τ ) d(.) 0−

by virtue of (8) and (9) and thanks to the Boltzmann superposition principle, turns into: Z t σ1ij (t) = Rijhk (t − τ ) d1hk (τ ) for t ∈ [0, T ] (10) 0−

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Z σ2ij (t) =

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Z Rijhk (t − τ ) d1hk (τ ) +

Rijhk (t − τ ) d2hk (τ ) for t ∈ [T, 2T ] T

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A B C 0



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 −Rijhk (t − τ ) d(.)  0 :=   Z t  Rijhk (t − τ ) d(.)

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Rijhk (t − τ ) d(.)  for t ∈ [T, 2T ]     for t ∈ [0, T ] 0

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σ := the operatorial formulation

1ij (t) 2ij (t)



σ2ij (t) σ1ij (t)



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proves to be equivalent to the constitutive laws (10)-(11). Now, fixed a material point xr of the viscoelastic solid, and T > 0, consider the following convolutive bilinear form of the Stieltjes type: Z 2T 0 00 0 00 hσij , ij ic := σij (2T ) ∗ ij (2T ) := σij0 (2T − t) d00ij (t) (16) 0−

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where the symbol ∗ means the time convolution product, in the Stieltjes sense, over the interval [0, 2T ], σij0 (t) := L0ij (t), and 0ij (t) and 00ij (t) are symmetric tensors, the histories of which belong to D . Let us observe that, by means of decompositions (8)-(9) and of definitions (12)-(13), it follows that

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Setting

hL 0ij , 00ij ic = hA 01ij , 001ij ic + hB 02ij , 001ij ic + hC 01ij , 002ij ic = hL 0 , 00 ic (17)

The well-known symmetry of the operator L with respect to the bilinear form (16) (see, for instance, Gurtin (1963) and Tonti (1973)) implies the symmetry of the operator L. In particular, it is worth noting that the operator A is symmetric, i.e. Z TZ T 0 00 Rijhk (2T −t−τ ) d01hk (τ ) d001ij (t) = hA 001ij , 01ij ic (18) hA 1ij , 1ij ic = 0−

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while B is the adjoint operator of C, that is Z T Z 2T −t 0 00 Rijhk (2T − t − τ ) d02hk (τ ) d001ij (t) hB 2ij , 1ij ic = 0− T Z 2T Z 2T −t Rijhk (2T − t − τ ) d001hk (τ ) d02ij (t) = hC 001ij , 02ij ic = 0−

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e For the sake of simplicity, througout the following, we will indicate C = B, where the symbol e means adjoint operator. In order to derive further properties of the operator A, it is useful to highlight its relation to the Helmholtz free energy Ψ (xr , T ), at the point xr ∈ Ω and at time T , in isothermal conditions. In the literature several definitions and according expressions for the free energy functional have been proposed (see, for example, Volterra (1928), Staverman and Schwarzl (1952a,b), Breuer and Onat (1964), Coleman and Owen (1975), Dill (1975), Graffi (1974, 1982, 1986), Gurtin and Hrusa (1988, 1991), Graffi and Fabrizio (1989), Morro and Vianello (1989, 1990)). It has also been shown that the expression of the free energy may not be unique and that a certain functional may represent a free energy only for specific classes of relaxation tensors (see, for instance, Del Piero and Deseri (1996)). Within the framework of linear viscoelastic materials, a single-integral expression of the free energy has been suggested for the first time by Volterra (1928), subsequently followed by the results of Staverman and Schwarzl (1952a,b), Bland (1960), and Hunter (1961), who considered a particular simplified model of the viscoelastic material, consisting of a network of linear viscous and elastic elements, i.e., dampers and springs. They independently provided the same double-integral expressions of the free energy Ψ and of the power dissipated density Φ that prove to be, for the case in point, the following quadratic functionals, respectively: Z Z 1 T T 0 Ψ(1ij ) = Rijhk (2T − t − τ ) d01hk (τ ) d01ij (t) (20) 2 0− 0− Z TZ T 0 Φ(1ij ) = − R˙ ijhk (2T − t − τ ) d01hk (τ ) d01ij (t) (21)

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where 01ij is a generic strain tensor with a history defined over the interval 9

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[0, T ]. By comparison with the bilinear form (18), one evicts

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A remarkable aspect of the functional Ψ, often called Dill free energy, is that it depends on the given relaxation function but not on the type of model chosen to derive its expression. In fact, Breuer and Onat (1964) found the same expression of the free energy without using a model representation and they showed that such an expression cannot be determined uniquely. Subsequently, Christensen and Naghdi (1967), by assuming a continuity hypothesis on the strains and using the Stone-Weierstrass theorem, approximated the free energy by a polinomial in a set of real, continuous, linear functionals of the strains and, using the Riesz rapresentation theorem, expressed these linear functionals in terms of Stieltjes integrals. Neglecting the constant term and the initial stress effect, the polinomial expansion of the free energy, truncated to the second order term, gives the form (20) with the more general Rijhk (T − t, T − τ ) in place of Rijhk (2T − t − τ ). Coleman (1964) introduced the free energy as a primitive concept and adopted the second principle of the thermodynamics, in the form of ClausiusDuhem inequality, as a starting point. As it is well known, under isothermal conditions, the Clausius-Duhem inequality combined with the local balance energy equation becomes

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(22)

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1 Ψ(01ij ) = hA 01ij , 01ij ic 2

˙ + σij (t)˙ij (t) ≥ 0 Φ = −Ψ

(23)

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This means that the rate of dissipation energy Φ must be non-negative. Integrating (23) from a given time t0 to another given time t1 (with t0 < t1 ), we obtain the following integrated dissipation inequality: Z t1 σij (τ )˙ij (τ ) dτ ≥ Ψ(t1 ) − Ψ(t0 ) (24) t0

Coleman (1964), using restrictions imposed by the second law of thermodynamics, has proved that the free energy must be chosen in such a way that: 1) the integrated dissipation inequality (24) holds; 2) among all deformation histories ending with a given value, the constant deformation history gives the least free energy; 3) for every deformation history, the gradient of the instantaneous free energy is equal to the instantaneous stress (generalized 10

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(n)

(−1)n Rijhk (t) γij γhk ≥ 0 n = 0, 1, 2, ...

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for t > 0 and for every symmetric tensor of the second order γij , where the superscript (n) indicates the n-th time derivative. The above inequality holds if and only if Rijhk (t) admits of a continuous spectral representation (for further details, see Del Piero and Deseri (1996, 1997)): Z ∞ Rijhk (t) = φijhk (α) e−α t dα (26) 0

with φijhk (α) positive definite tensor of the fourth rank. This assumption is not restrictive since, as pointed out by Huet (1995), it is valid for most real viscoelastic materials, and it leads to the strict positivity of the free energy (20). In effect, if (26) holds, the quadratic form (20) turns into: Z 1 ∞ φijhk (α) ghk (α) gij (α) dα Ψ= 2 0

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where

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stress relation), and 4) the equilibrium free energy at the equilibrium deformation eij at time t = ∞ for the linear viscoelastic material has the form ∞ eij ehk . Ψ∞ (eij ) = 21 Rijhk A remarkable result found by Del Piero and Deseri (1996) states that the functional of Staverman and Schwarzl (20) fulfills the integrated dissipation property if and only if the relaxation function is symmetric and completely monotonic. This means that without the completely monotonic assumption functional (20) cannot be interpreted as a free energy. For this reason, let us suppose, now, that the relaxation tensor Rijhk (t) is completely monotonic, that is, differentiable to any order with

Z

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gij (α) :=

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e−α(T −t) d01ij (t)

Due to the positive definiteness of φijhk (α), it follows that φijhk (α) ghk (α) gij (α) ≥ 0

where the equality to zero can never occur simultaneously for all values of the parameter α, if 01ij (t) 6= 0 for every t ∈ [0, T ] (Mandel, 1966). Thus, 1 Ψ(01ij ) = hA 01ij , 01ij ic > 0 ∀01ij (t) 6= 0 2 11

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where



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e −1 B −BA e −1 BA S= −1 −A B A−1   2ij (t) θ2 = σ2ij (t)

and

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θ1 =

−σ1ij (t) 1ij (t)



(28) (29)

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and, therefore, the operator A is positive definite. It is noteworthy to point out that in case the relaxation tensor Rijhk (t) admits of a discrete spectrum, instead of the continuous one given by (26), the kernel of the quadratic form (20) becomes degenerate, the quadratic form (20) non-negative and therefore the operator A positive semi-definite. In spite of the assumption of completely monotony of the relaxation functions and the consequent positive definiteness of the operator A, the overall operator L is not positive definite. In order to transform L into an operator endued with such a property, let us apply a partial Legendre transform to the saddle-shaped quadratic functional associated with the constitutive law (15). We recall that such an operation allows one to convert saddle-point functionals into convex ones (see, for instance, Callen (1960)). In particular, for the case in point, it is equivalent to a partial inversion of the constitutive law (15), that consists in the exchange of the variable 2ij (t) with σ1ij (t) in the second line of (15). This yields to the following new form of the constitutive law: S θ2 = θ1 (27)

with S symmetric and positive semi-definite operator with respect to the convolutive bilinear form (16). The symmetry of S derives directly from e while its positive the symmetry properties of the operators A, B and B, semi-definiteness derives from the positive definiteness of the operator A. In particular, let us point out that the quadratic form associated with the operator S through the bilinear form (16), that is

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hθ 02 , S θ 02 ic = θ 02 (2T ) ∗ S θ 02 (2T )

where θ 02 is the vector of components 02ij (t) and σ20 ij (t), symmetric tensors with histories belonging, respectively, to D and Dσ , takes the following form:     θ 02 (2T ) ∗ S θ 02 (2T ) = A−1 B 02ij (2T ) − σ20 ij (2T ) ∗ B 02ij (2T ) − σ20 ij (2T ) 12

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It is worth noting that the knowledge of the solution of the viscoelastic problem in the time subinterval [T, 2T ], that is, σ2ij (t) and 2ij (t), allows one to determine, through equation (27), the stress and strain fields, σ1ij (t) and 1ij (t), in the time subinterval [0, T ], without any further information. This means that the constitutive law (27) contains in itself the necessary information to obtain the solution of the viscoelastic problem in the first subinterval, given that on the second subinterval. Contrariwise, given σ1ij (t) and 1ij (t), the sole constitutive equation (27) is not sufficient to determine univocally σ2ij (t) and 2ij (t) because of the non-invertibility of the operator S. The possibility of determining θ 1 known θ 2 and not vice versa descends from the triangular shape of the constitutive operator L (12), typical of the initial values problems, for which the solution at a certain time depends only on the past history (the so-called causality principle). In fact, it is clear from the equations of the constitutive law (15) that the knowledge of 1ij (t) and σ1ij (t) does not allow to determine σ2ij (t), whereas the knowledge of 2ij (t) and σ2ij (t) allows, through the first equation, to determine 1ij (t) and, then, through the second equation, σ1ij (t), provided that A is invertible. We stress the fact that the positive definiteness (and the consequent invertibility) of the operator A is, therefore, an essential ingredient since it expresses the request that two different deformation histories on [0, T ] do not produce the same effects in terms of stresses on [T, 2T ]. Furthermore, if A is invertible, the state of the material at time T is fully defined by the deformation history on [0, T ] and the current deformation at time T (Del Piero and Deseri (1997)), whereas if A is only semi-definite, as happens when relaxation functions have discrete spectrum, the concept of state should be changed, by introducing the notion of minimal state, in order to define a new expression for the free energy that proves to be positive definite even for non completely monotonic relaxations functions (Fabrizio et al. (2010)).

AC C

205

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Therefore, θ 02 (2T ) ∗ S θ 02 (2T ) vanishes not only when θ 02 (t) = 0 for every t ∈ [T, 2T ], i.e. 02ij (t) = 0 and σ20 ij (t) = 0, but also when σ20 ij (t) = B 02ij (t). Then, the kernel of the operator S is n o ker S = (02ij , σ20 ij ) : σ20 ij = B 02ij

3.2. Inverse constitutive law Under the assumptions regarding the relaxation kernel Rijhk (t) and the strain histories, the direct constitutive law (3) is invertible. Then, due to (8) 13

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and (9) and by means of the Boltzmann superposition principle, we have: Z t 1ij (t) = Cijhk (t − τ ) dσ1hk (τ ) for t ∈ [0, T ] (31) 0−

Z 2ij (t) =

T

0−

Z

t

Cijhk (t − τ ) dσ1hk (τ ) +

Cijhk (t − τ ) dσ2hk (τ ) for t ∈ [T, 2T ] T

(32)

SC

and, in a compact form: L−1 σ = 

T 235

240

241

EP

239

(34)

It is clear that the symmetric operator A is negative definite, due to the positive definiteness of the operator A with respect to the bilinear form (16). We remark that by proceeding as in subsection 3.1 to obtain the equation equivalent to (27), one obtains again equation (27).

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 for t ∈ [0, T ]     for t ∈ [T, 2T ]

As the notation suggests, the operator L−1 has to be meant as the inverse operator of L. The new operators A and B are related to the operators A and B, previously introduced in (12), by means of the following relations: B = B −1 e −1 A = −B −1 A B

237



0−

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236

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where the operator L−1 has the following expression:  Z t Cijhk (t − τ ) d(.) 0    0−  e 0 B L−1 := :=   Z t Z T B A  Cijhk (t − τ ) d(.) Cijhk (t − τ ) d(.)

(33)

3.3. On the decomposition of the time domain into two unequal subintervals In subsection 3.1, by virtue of the positive definiteness of the operator A, we transformed the non-definite operator L into the positive semi-definite operator S. The same procedure cannot be applied when the time domain [0, 2T ] is split into two unequal subintervals, [0, Ts ] and [Ts , 2T ], with Ts 6= T . Let us consider the following two situations.

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1) Ts < T . For this case, the constitutive law operator L may be split as follows (see Fig.1b):   B1 B2 A11 A12  A B3 0    e12 A22 (35) L= e  e3 0 0   B1 B e2 B 0 0 0

TE D

For the sake of clarity, we report the explicit expression of each operator: Z Ts A11 : [0, Ts ] → [2T − Ts , 2T ] A11 (.) := Rijhk (t − τ ) d(.) 0− Z T Rijhk (t − τ ) d(.) A12 : [Ts , T ] → [2T − Ts , 2T ] A12 (.) := Ts T

Z

A22 : [Ts , T ] → [T, 2T − Ts ] B1 : [T, 2T − Ts ] → [2T − Ts , 2T ]

Rijhk (t − τ ) d(.)

A22 (.) := Z

Ts 2T −Ts

Rijhk (t − τ ) d(.)

B1 (.) :=

EP

B2 : [2T − Ts , 2T ] → [2T − Ts , 2T ]

T

Z

t

Rijhk (t − τ ) d(.)

B2 (.) := 2T −Ts Z t

B3 : [T, 2T − Ts ] → [T, 2T − Ts ]

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242

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SC

where the operators A and B, defined in equation (12), have been decomposed in the following way:     A11 A12 B1 B2 B= (36) A= e B3 0 A12 A22

Rijhk (t − τ ) d(.)

B3 (.) := T

The operator L in equation (35) is still symmetric with respect to the bilinear form (16). To streamline let us introduce the operators: Ba =   the notation,  B1 B2 and Bb = B3 0 so that the operator L can be written as 

 A11 A12 Ba e21 A22 Bb  L= A ea B eb 0 B

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With the aim to highlight the decomposition of the constitutive law operator over the two subintervals [0, Ts ] and [Ts , 2T ], let us rearrange the submatrices of the operator L as follows:     A12 Ba A11 X Y e   L = A12 (37) A22 Bb = ˜ Y Z e e Ba Bb 0

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SC

In the case the operator X = A11 is not definite, the application of the Legendre transformation proves to be completely useless. Therefore, let us suppose that X is positive definite. Therefore, by applying a partial Legendre transform, the constitutive law may be written in terms of the following operator S:   Y˜ X −1 Y − Z −Y˜ X −1 (38) S= −X −1 Y X −1

TE D

In order to assert that the whole operator S is at least positive semi-definite, we need, first, to verify the possible positive definiteness of the operators on the main diagonal: X −1 and Y˜ X −1 Y − Z. The former is positive definite, X being supposed to be positive definite, while the latter reads " # −1 −1 e e A A A − A A A B − B 12 12 22 12 a b 11 11 Y˜ X −1 Y − Z = ea A−1 e e B A − B B A−1 12 b a 11 11 Ba

243 244

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Note that, due to the assumed positive definiteness of the operator A11 , the ea A−1 operator B 11 Ba is, at least, positive semi-definite and it is positive definite e12 A−1 in case Ba is injective. The sign of the operator A 11 A12 − A22 may be determined by exploiting the positive definiteness of the overall operator A, defined in (36). In particular, let us consider the quadratic form h1 , A1 ic > 0 (the subscript 1 still refers to the time interval [0, T ]). It follows that      A11 A12 I h1 , A1 ic = I II ∗ e = I ∗A11 I +2I ∗A12 II +II ∗A22 II II A12 A22

where the subscripts I and II refer, respectively, to the subintervals [0, Ts ] and [Ts , T ]. By setting I = −A−1 11 A12 II

we have

  e12 A−1 h1 , A1 ic = −II ∗ A A − A 22 II > 0 11 12 16

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e12 A−1 Hence, the operator A 11 A12 − A22 is not positive definite and, therefore, we can conclude that also the operator Y˜ X −1 Y − Z is not, as well as the overall operator S (see equation (38)).

SC

2) Ts > T . In this case, the constitutive law operator split over the two subintervals [0, Ts ] and [Ts , 2T ] reads (see Fig.1c):     Bd A Bc X Y ec 0 0 = e L= B Y 0 ed 0 0 B

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In case the operator X is definite, we can apply a partial Legendre transformation in order to transform the non-definite operator L into a semi-definite one. X, as it is also clear from its expression, is not definite. In effect, consider the quadratic form associated with the operator X:      A Bc 1 II hI , XI ic = 1  ∗ ˜ = 1 ∗ A1 + 21 ∗ Bc II II Bc 0

TE D

where the subscripts I and 1 refer, respectively, to the time intervals [0, Ts ] and [0, T ], whilst the superscipt II refers to the time interval [T, Ts ]. By choosing II = 0

246 247

248

1 = −A−1 Bc II

we have hI , XI ic = −h1 , A1 ic < 0. Therefore, since X is not a definite operator, it is not possible to transform the operator L into a semi-definite operator, through a partial Legendre transformation.

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245

EP

it follows that hI , XI ic = h1 , A1 ic > 0, due to the positive definiteness of the operator A, whereas, by choosing

4. Reformulation of the linear viscoelastic problem

Let us denote by E and C the equilibrium and kinematic operators, respectively, i.e., E represents the divergence operator and C the symmetric part of the gradient operator. Therefore, by virtue of the decompositions (8)(9), the problem (constituted by the equilibrium equations (1), the straindisplacement relations (2) and the constitutive law (3), written in the form 17

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with the following boundary  0 0 0 0 0 nj  0 0 0 0 nj 0   0 0 0 0 0 0   0 0 0 0 0 0   0 −nj 0 0 0 0 −nj 0 0 0 0 0

conditions:    p2i u1i   u2i   p1i      1ij   0      2ij  =  0      σ1ij   −nj u02 i −nj u01i σ2ij

       





      =      

b2i b1i 0 0 0 0

       

in in in in in in

Ω × [T, 2T ] Ω × [0, T ] Ω × [T, 2T ] Ω × [0, T ] Ω × [T, 2T ] Ω × [0, T ] (39)

SC

      

u1i u2i 1ij 2ij σ1ij σ2ij

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−E 0 −I 0 0 0



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249

(15)) reads  0 0 0 0 0  0 0 0 0 −E   0 0 A B 0   0 0 B e 0 −I   0 C 0 −I 0 C 0 −I 0 0

      

on Γp × [T, 2T ] on Γp × [0, T ]

on Γu × [T, 2T ] on Γu × [0, T ] (40) or, in a compact form, with obvious meaning of the symbols:

252

EP

251

It is worth noting that the submatrix of MI corresponding to the constitutive law operator coincides with the operator L (symmetric but not positive definite with respect to (16)), given by (12). Considering, instead, the inverse constitutive law (33), we get the following four-fields formulation:      0 0 0 E u1i −b2i in Ω × [T, 2T ]  0      0 E 0   u2i   −b1i  in Ω × [0, T ]  (42)  0 −C A B   σ1ij  =  0  in Ω × [T, 2T ] e 0 in Ω × [0, T ] σ2ij 0 −C 0 B

AC C

250

(41)

TE D

MI zI = bI in Ω × [0, 2T ] TI zI = gI on Γ × [0, 2T ]



 0 0 0 −nj u1i  0 0 −nj   0   u2i   0 nj 0 0   σ1ij nj 0 0 0 σ2ij





 −p2i   −p1i  =    nj u02  i nj u01i 18

on on on on

Γp × [T, 2T ] Γp × [0, T ] Γu × [T, 2T ] Γu × [0, T ]

(43)

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or, in a compact form: MII zII = bII in Ω × [0, 2T ] TII zII = gII on Γ × [0, 2T ]

     

255

0 0 0 0 0 0 0 nj nj 0

 0 0 −nj u2i   0 −nj 0   u1i  0 0 0    2ij 0 0 0   σ2ij σ1ij 0 0 0





    =    

−p1i −p2i 0 nj u01i nj u02i

Ω × [0, T ] Ω × [T, 2T ] Ω × [0, T ] Ω × [0, T ] Ω × [T, 2T ] (45) Γp × [0, T ] Γp × [T, 2T ] (46) Γu × [0, T ] Γu × [T, 2T ]     

in in in in in

     

on on

on on

obtained taking into account the constitutive law written as in (27). The latter equations can be written in the following compact form:

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254



SC

    

Finally, consider the following five-fields formulation:    0 0 0 0 E u2i −b1i     0 0 0 E 0   u1i   −b2i −1 −1    e e 0 0 BA B −B A I    2ij  =  0 −1 −1   σ2ij   0 0 −C −A B A 0 σ1ij 0 −C 0 I 0 0

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

(44)

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253

MIII zIII = bIII in Ω × [0, 2T ] TIII zIII = gIII on Γ × [0, 2T ]

256

(47)

The three problems (41), (44) and (47) can be condensed as follows

EP

Mi zi = bi in Ω × [0, 2T ] Ti zi = gi on Γ × [0, 2T ]

(48)

257

AC C

or, in a compact form, as

Ni zi = fi

where i = I, II, III. Consider the following non-degenerate bilinear form: Z Z 0 00 0 00 hhNi zi , zi iic = Mi zi (2T ) ∗ zi (2T ) dΩ + Ti z0i (2T ) ∗ z00i (2T ) dΓ (50) Ω

258 259

(49)

Γ

where z0i and z00i are arbitrary vectors, belonging to the domain of the operator Ni , i = I, II, III. 19

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261

The symmetry of the operators Ni , i = I, II, III with respect to (50) can be easily proved (for the sake of brevity, we omit such a proof). Then, the compact problem (49) proves to be equivalent to the following variational problem (the equivalence is guaranteed by the non-degeneracy of the chosen bilinear form over the domain and over the range of the operator Ni , see Magri (1974), for instance): Fi (zi ) = stat Fi (z0i ) 0

(51)

SC

zi

RI PT

260

where

1 hhNi z0i , z0i iic − hhfi , z0i iic 2Z Z 1 1 0 0 Mi zi (2T ) ∗ zi (2T ) dΩ + Ti z0i (2T ) ∗ z0i (2T ) dΓ = 2 Ω 2 Γ Z Z 0 − bi (2T ) ∗ zi (2T ) dΩ − gi (2T ) ∗ z0i (2T ) dΓ

M AN U

Fi (z0i ) =

Ω

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264

z0i being any admissible vector, that is, any vector belonging to the domain of Ni , and zi the solution of the problem, with i = I, II, III. In particular, for i = I we obtain a variational formulation of the Hu-Washizu type, while for i = II we get a formulation of the Hellinger-Reissner type. Finally, for i = III we obtain a new variational formulation. Let us impose the strain-displacement relations into the Hu-Washizu formulation, i.e. into (52) with i = I. Thus, we get the following functional of the Total Potential Energy type:  Z  1 0 0 0 0 0 0 e C u (2T ) ∗ C u (2T ) dΩ A C u1i (2T ) ∗ C u1i (2T ) + 2B TPE(u1i , u2i ) = 1i 2i 2 Ω   Z − b2i (2T ) ∗ u01i (2T ) + b1i (2T ) ∗ u02i (2T ) dΩ  ZΩ  0 0 − p2i (2T ) ∗ u1i (2T ) + p1i (2T ) ∗ u2i (2T ) dΓ

EP

263

Γ

AC C

262

(52)

Γp

(53)

Imposing, instead, the equilibrium equations into the functional of the Hellinger-Reissner type, i.e. into (52) with i = II, one obtains the following

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functional of the Total Complementary Energy type:  Z  1 0 0 0 0 0 0 e σ (2T ) ∗ σ (2T ) dΩ TCE(σ1ij , σ2ij ) = A σ1ij (2T ) ∗ σ1ij (2T ) + 2 B 1ij 2ij 2 Ω  Z  0 0 0 0 nj u2i (2T ) ∗ σ1ij (2T ) + nj u1i (2T ) ∗ σ2ij (2T ) dΓ − Γu

(54)

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Both the above two-fields functionals are of the saddle-point type, due to the non-positive definiteness of the submatrix corresponding to the constitutive law operator. In fact, both TPE and TCE functionals fulfill, respectively, the following variational principles: TPE(u1i , u2i ) = min stat TPE(u01i , u02i ) 0 0 u1

i

u2

(55)

i

u01i and u02i being compatible displacement fields (i.e. displacement fields that satisfy equations (2)), and stat TCE(σ10 ij , σ20 ij ) TCE(σ1ij , σ2ij ) = max 0 0 σ1

270

σ10 ij and σ20 ij being equilibrated stress fields (i.e. stress fields that satisfy equations (1)). The TPE functional is minimum with respect to u01i and the TCE functional is maximum with respect to σ10 ij due to the positive definiteness of the operator A. In order to obtain a global minimum principle, let us impose the straindisplacement relations for the variable u02i and the equilibrium equations for the variable σ20 ij into the five-fields functional FIII . In this way, we get the following two-fields mixed functional: Z  1 0 0 e A−1 B C u0 (2T ) ∗ C u0 (2T ) − A−1 B C u0 (2T ) ∗ σ 0 (2T ) F(σ2ij , u2i ) = B 2i 2i 2i 2ij 2 Ω  −1 0 0 −1 0 0 e − B A σ2ij (2T ) ∗ C u2i (2T ) + A σ2ij (2T ) ∗ σ2ij (2T ) dΩ Z Z 0 + b1i (2T ) ∗ u2i (2T ) dΩ + p1i (2T ) ∗ u02i (2T ) dΓ Ω Γp Z − nj u01i (2T ) ∗ σ20 ij (2T ) dΓ

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269

(56)

ij

EP

268

σ2

AC C

267

ij

Γu

(57) 21

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which fulfills the following minimum principle: σ2

i

(58)

SC

e A−1 B u02 − BA e −1 σ20 B − b1i = 0 in Ω × [0, T ] i/jj ij/j  =0 ⇒ e A−1 B u0 e −1 σ 0 − B nj = p1i on Γp × [0, T ] BA 2i/j 2ij (59)  (  1 u01i/j + u01j/i = −A−1 B u02i/j + A−1 σ20 ij in Ω × [0, T ] 2 δσ20 F(σ20 ij , u02i ) = 0 ⇒ u01i = u01i on Γu × [0, T ] (60) It is worth noting that equations (59) coincide with the equilibrium equations written for σ1ij , while equations (60) correspond to the strain-displacement relations written for u1i . This proves that the stationarity of the functional (57) gives the solution of the problem. In order to show that the solution is the minimum point of (57), consider the second variation of the functional with respect to u02i and σ20 ij : (

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δu02 F(σ20 ij , u02i )

δu202 F(σ20 ij , u02i )

EP

272

u2

σ20 ij being any equilibrated stress field and u02i any compatible displacement field. In order to prove this result, note that the first variation of functional (57) with respect to u02i and σ20 ij leads to:

Z

=

δσ220 F(σ20 ij , u02i )

Ω

e A−1 B C δu02 ∗ C δu02 dΩ > 0 B i i Z

= Ω

A−1 δσ20 ij ∗ δσ20 ij dΩ > 0

Both the above inequalities hold thanks to the positive definiteness of the operator A with respect to the bilinear form (16). Hence, functional (57) is minimum in the solution. In particular, it can be proved that, due to the positive semi-definiteness of the operator S, the minimum of (57) is a global minimum, i.e. F(σ2ij , u2i ) = 0min0 F(σ20 ij , u02i ) (61)

AC C

271

ij

RI PT

F(σ2ij , u2i ) = min min F(σ20 ij , u02i ) 0 0

u2 ,σ2 i

22

ij

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SC

RI PT

Let Ω be the RVE of a heterogenous material made of n phases, whose mechanical behavior is fully described by the constitutive law (3). Henceforth, given a generic function f (x, t), we use f (t) to indicate the volume average of f (x, t) over Ω: Z 1 f (t) = f (x, t) dΩ V Ω

M AN U

Suppose one applies kinematic boundary conditions of the “affine” type all over the surface Γ and that there are no volume forces. Hence, the problem on the RVE reads  σij/j = 0 in Ω × [0,  
2T ]  in Ω × [0, 2T ] ij = 21 ui/j + uj/i (62) ui = ij xj on Γ × [0, 2T ]    σij = Lij in Ω × [0, 2T ] It must be noted that the constitutive law (62-d) assumes the explicit expression given by (10) for t ∈ [0, T ], and by (11) for t ∈ [T, 2T ], respectively, and that  1ij (t) for t ∈ [0, T ] ij (t) = (63) 2ij (t) for t ∈ [T, 2T ]

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275

Let us consider the homogenized constitutive law, given by Z t h σ ij (t) = Rijhk (t − τ ) dhk (τ ) for t ∈ [0, 2T ]

EP

274

5. The problem on the RVE

(64)

0−

where

AC C

273

σ ij (t) =



σ 1ij (t) for t ∈ [0, T ] σ 2ij (t) for t ∈ [T, 2T ]

(65)

h We suppose that the tensor Rijhk (t), called homogenized relaxation kernel, is endowed with the same properties of the tensor Rijhk (t). The determination h of the exact expression of Rijhk (t) can be achieved via direct methods which, usually, can be applied only for very specific cases. Therefore, in section 6, we provide bounds on the constitutive law operator by applying the minimum variational principle (61). For this purpose, let us consider decompositions

23

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(63) and (65). Then, using the Boltzmann superposition principle, equation (64) can be written as Z t h Rijhk (t − τ ) d1hk (τ ) for t ∈ [0, T ] (66) σ 1ij (t) = 0−

Z σ 2ij (t) =

T

0−

h (t − τ ) Rijhk

Z

t

d1hk (τ ) +

h (t − τ ) d2hk (τ ) for t ∈ [T, 2T ] Rijhk

T

SC

(67)

and, in a compact form, as Lh  = σ T

 Z Lh :=



Ah B h eh 0 B



− τ ) d(.)  −  0 :=   Z t  h Rijhk (t − τ ) d(.)

278 279



− τ ) d(.)  for t ∈ [T, 2T ]     for t ∈ [0, T ] 0

h Rijhk (t

and  and σ are the volume averages of the vectors  and σ, defined, respectively, in (13) and (14). The overall operator Lh is the homogenized counterpart of the operator L and, therefore, it is endued with the same properties, i.e. it is symmetric but not positive definite with respect to the bilinear form (16). Following, then, the same procedure used in subsection 3.1, i.e. applying a partial Legendre transform to obtain a positive semi-definite formulation, one obtains (69) Sh θ 2 = θ 1

EP

280

T

t

TE D

277

Z

h Rijhk (t

0−

276

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where

(68)

AC C

where θ 2 and θ 1 are the volume averages of the vectors θ 2 and θ 1 , defined in (29) and (30), respectively, and   e h (Ah )−1 B h −(B) e h (Ah )−1 (B) h S := −(Ah )−1 B h (Ah )−1 The operator Sh is the homogenized counterpart of S. In effect, applying the volume average operation to the equation (27), that is S θ2 = θ1

281

and comparing the latter with (69), it is clear that S θ 2 = Sh θ 2 . 24

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6. Bounds

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For the problem on the RVE (62), the mixed functional (57), by applying the divergence theorem, reads Z  1 0 0 e A−1 B 02 (2T ) − 2 σ20 (2T ) ∗ A−1 B 02 (2T ) F(σ2ij , 2ij ) = 02ij (2T ) ∗ B ij ij ij 2 Ω  + σ20 ij (2T ) ∗ A−1 σ20 ij (2T ) dΩ − V σ 0 2ij (2T ) ∗ 1ij (2T )

(70)

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In the solution, the integral term in the above equation, by virtue of definitions (28) and (29) of the operator S and the vector θ 2 , respectively, can be written as Z  1 e A−1 B 2 (2T ) − 2 σ2 (2T ) ∗ A−1 B 2 (2T ) 2ij (2T ) ∗ B ij ij ij 2 Ω  Z 1 + σ2ij (2T ) ∗ A−1 σ2ij (2T ) dΩ = θ2 (2T ) ∗ S θ2 (2T ) dΩ 2 Ω

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Thanks to equation (27) and by means of Hill’s principle (both θ 2 and θ 1 = S θ 2 are vectors containing equilibrated stress fields and compatible strain fields), the latter turns into Z V 1 θ2 (2T ) ∗ S θ2 (2T ) dΩ = θ 2 (2T ) ∗ θ 1 (2T ) 2 Ω 2 Finally, recalling that θ 1 = Sh θ 2 and using the definition of the operator Sh , functional (70), in the solution, takes the following form:  V e h (Ah )−1 B h 2 (2T ) − 2 σ 2 (2T ) ∗ (Ah )−1 B h 2 (2T ) F(σ2ij , 2ij ) = 2ij (2T ) ∗ B ij ij ij 2  h −1 + σ 2ij (2T ) ∗ (A ) σ 2ij (2T ) − V σ 2ij (2T ) ∗ 1ij (2T )

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Consider now, as admissible fields for functional (70), the averaged fields, that is σ20 ij = σ 2ij and 02ij = 2ij . Thus, 

e −1 B 2 (2T ) − 2 σ 2 (2T ) ∗ A−1 B 2 (2T ) 2ij (2T ) ∗ BA ij ij ij  + σ 2ij (2T ) ∗ A−1 σ 2ij (2T ) − V σ 2ij (2T ) ∗ 1ij (2T )

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(72)

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Hence, by virtue of the minimum principle (61), the following noteworthy inequality holds: e h (Ah )−1 B h 2 (2T ) − 2 2 (2T ) ∗ B e h (Ah )−1 σ 2 (2T ) 2ij (2T ) ∗ B ij ij ij e −1 B 2 (2T ) + σ 2ij (2T ) ∗ (Ah )−1 σ 2ij (2T ) ≤ 2ij (2T ) ∗ BA ij e −1 σ 2 (2T ) + σ 2 (2T ) ∗ A−1 σ 2 (2T ) (73) − 2 2ij (2T ) ∗ BA ij ij ij

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It is interesting to observe that the above inequality resembles those provided by Milton (1990) (formula 18.18) and Cherkaev and Gibiansky (1994) (formulas 5.5 and 5.6) in the frequency domain for the case of complex moduli, whereas inequality (73) holds in the time domain. From the homogenized constitutive law (67), written for t ∈ [T, 2T ], i.e.

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σ 2ij (t) = Ah 1ij (t) + B h 2ij (t)

(74)

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if 1ij = −(Ah )−1 B h 2ij , then σ 2ij = 0, and, therefore, inequality (73) turns into (75)

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e h (Ah )−1 B h 2 (2T ) ≤ 2 (2T ) ∗ BA e −1 B 2 (2T ) 2ij (2T ) ∗ B ij ij ij

Considering, instead, 1ij = (Ah )−1 σ 2ij , it follows that 2ij = 0, and then (73) yields to σ 2ij (2T ) ∗ (Ah )−1 σ 2ij (2T ) ≤ σ 2ij (2T ) ∗ A−1 σ 2ij (2T )

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7. On the Volterra approach The bounds obtained in section 6 are derived from the constitutive law written in the Boltzmann form (3) and from the consequent “natural” choice 26

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of the bilinear form (16). In this section, we consider, instead, the constitutive law written in the Volterra form (6), that is Z t R˙ ijhk (x, t − τ ) hk (x, τ ) dτ σij (x, t) = LV ij (x, t) = Rijhk (x, 0) hk (x, t) +

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(76) where the integral has to be meant in the Lebesgue sense. The operator LV is symmetric with respect to the following bilinear form of the convolutive type (fixed x ∈ Ω and T > 0): Z 2T 0 0 0 0 hij , σij icV := ij (2T ) ? σij (2T ) := 0ij (t) σij0 (2T − t) dt (77) 0

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where the symbol ? indicates the convolution product with respect to the time variable, over the interval [0, 2T ], in the Lebesgue sense. Due to decompositions (8)-(9), equation (76) can be written in a compact form as LV  = σ (78)

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where



defined as 

T



(79)

t

 ˙ Rijhk (0)(.) + Rijhk (t − τ ) (.) dτ  for t ∈ [T, 2T ]  T 0   Z t  for t ∈ [0, T ] 0 Rijhk (0)(.) + R˙ ijhk (t − τ ) (.) dτ Z

R˙ ijhk (t − τ ) (.) dτ

Z

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  LV :=   

BV 0

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LV :=

AV eV B

0

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eV is a symmetric operator with respect to (77), since AV is symmetric and B is the adjoint of BV . Furthermore, AV is negative definite, thanks to the completely monotonic property of the relaxation kernel, for which the first time derivative of Rijhk (t) is negative (see equation (25)). Contrariwise to the quadratic form associated with A, the one associated with AV does not represent a free energy. In particular, it does not have the dimension of an energy but of an energy times a time. The operator LV is not negative definite, but applying a partial Legendre transform to (78), the latter can be rephrased as follows

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SV θ 2 = θ 1 27

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 SV :=

eV A−1 BV B V −A−1 V BV

eV A−1 −B V A−1 V

 (81)

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where

Following the same procedure illustrated in subsection 3.1, it is possible to give a variational formulation also to the linear viscoelastic problem written with the Volterra constitutive law in the forms (78) and (80). In particular, the following variational principles hold: u1

i

u2

i

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TPEV (u1i , u2i ) = max stat TPEV (u01i , u02i ) 0 0

TCEV (σ1ij , σ2ij ) = min stat TCEV (σ10 ij , σ20 ij ) 0 0 ij

σ2

ij

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σ1

and

FV (σ2ij , u2i ) = max FV (σ20 ij , u02i ) 0 0 u2 ,σ2 i

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(83)

(84)

ij

where functionals TPEV , TCEV , and FV have the same expressions of TPE, TCE, and F, given by equations (53), (54), and (57), respectively, provided to replace the operators A and B with AV and BV , and the convolutive product ∗ with ?. Considering the problem (62) on the RVE, with the Volterra constitutive law, and following the same arguments presented in section 6, it is possible to derive bounds on the homogenized constitutive law operator. In particular, by means of the global maximum principle (84) related to the mixed functional, we deduce the following inequality:

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eVh (AhV )−1 BVh 2 (2T ) − 2 2 (2T ) ? B eVh (AhV )−1 σ 2 (2T ) 2ij (2T ) ? B ij ij ij eV A−1 BV 2 (2T ) + σ 2ij (2T ) ? (AhV )−1 σ 2ij (2T ) ≥ 2ij (2T ) ? B ij V

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304 305 306

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eV A−1 σ 2 (2T ) + σ 2 (2T ) ? A−1 σ 2 (2T ) − 2 2ij (2T ) ? B ij ij ij V V

e h are the homogenized counterparts of AV , BV and B eV , where AhV , BVh and B V respectively. 8. Conclusions

In the present note, we derive five new variational formulations, one of which is of the minimum type, for media with a linear hereditary viscoelastic behavior. By virtue of the division of the time domain into two equal 28

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subintervals, the original unknowns of the viscoelastic problem are doubled. The consequent decomposition of the viscoelastic constitutive law operator allows one to isolate a symmetric and positive definite operator. The positive definiteness of such an operator has a precise mechanical and physical meaning, being strictly related to the expression of the free energy first derived by Staverman and Schwarzl (1952a,b) and it leads to the formulation of a minimum principle. The latter seems to be the first minimum principle valid on a finite period of time and related to the thermodynamics of the viscoelastic problem. As it is well-known, minimum formulations can be successfully used for many practical purposes, for example, to obtain bounds of the homogenized mechanical properties of composite materials, as shown in this note. In particular, the inequalities provided are formally similar to those obtained by Cherkaev and Gibiansky (1994), and Milton (1990) in the frequency domain. Splitting the time domain into two equal subintervals is the only way to obtain variational formulations that are also extremum. This derives from a physical property of the linear viscoelastic solids, as noted by Staverman and Schwarzl (1952a) (p.474): “In simple relaxation or creep experiments the thermodynamic functions at time t depend on the relaxation or creep function at time 2t.”, or, as reported by Mandel (1966) (p.801)“... si la connaissance de la matrice fij (ou rij ) entre 0 et t suffit pour definir le comportement purement mecanique d’un corps viscoelastique (a temperature constante) dans le meme intervalle de temps, elle ne definit pas son comportement energetique. Ce dernier ne serait defini dans l’intervalle 0,t que lorsqu’on connait fij (ou rij) dans l’intervalle double 0, 2t.”. The use of a convolutive bilinear form and the decomposition of the time domain into two equal subintervals are the necessary ingredients to reformulate the viscoelastic problem in terms of a minimum variational problem, with evident advantages, both theoretical and computational.

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Acknowledgment Work done within a research project financed by the Italian Ministry of Education, University, and Research (MIUR). The authors are especially thankful to Professors F. Genna, G. Giorgi, G.W. Milton and G. Del Piero for helpful comments and stimulating discussions.

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Figure 1: Decomposition of the time domain and consequent decomposition of the constitutive law operator for the following cases: (a) Ts = T , (b) Ts < T , (c) Ts > T .

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Highlights • We use a bilinear form of the convolutive type in the time variable. • The time domain is decomposed into two subintervals of equal length, with the consequent decomposition of the linear viscoelastic constitutive law operator.

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• From the split constitutive law operator, we isolate a symmetric and positive definite sub-operator and by virtue of a Legendre transform we rephrase the constitutive law so that the associated quadratic form is convex. • Five new variational formulations, one of which is of the minimum type, are derived.

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• Elementary bounds of the homogenized mechanical properties of viscoelastic composite materials are obtained by using the minimum variational principle here derived.