Strain rate tensors and constitutive equations of inelastic micropolar materials

Accepted Manuscript Strain rate tensors and constitutive equations of inelastic micropolar materials H. Altenbach, V.A. Eremeyev PII: DOI: Reference:

S0749-6419(14)00114-4 http://dx.doi.org/10.1016/j.ijplas.2014.05.009 INTPLA 1798

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International Journal of Plasticity

Please cite this article as: Altenbach, H., Eremeyev, V.A., Strain rate tensors and constitutive equations of inelastic micropolar materials, International Journal of Plasticity (2014), doi: http://dx.doi.org/10.1016/j.ijplas.2014.05.009

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Strain rate tensors and constitutive equations of inelastic micropolar materials H. Altenbacha , V.A. Eremeyeva,b,∗ a

Institut f¨ur Mechanik, Fakult¨at f¨ur Maschinenbau, Otto-von-Guericke-Universit¨at Magdeburg, D-39106 Magdeburg, Germany b South Scientific Centre of RASci & South Federal University, Rostov on Don, Russia

Abstract Nonlinear micropolar continuum model allows to describe complex micro-structured media, for example, polycrystals, foams, cellular solids, lattices, masonries, particle assemblies, magnetic rheological fluids, liquid crystals, etc., for which the rotational degrees of freedom of material particles are important. The constitutive equations of the hyperelastic nonlinear micropolar continuum can be expressed using the strain energy density depending on two strain measures. In the case of inelastic behavior the constitutive equations of the micropolar continuum have more complicated structure, the stress and couple stress tensors as well as other quantities depend on the history of strain measures. In what follows we discuss the constitutive equations of the nonlinear micropolar continuum using strain rates. Keywords: Micropolar continuum, Cosserat continuum, strain rate tensors, constitutive relations, hypoelasticity

1. Introduction Recently the interest to generalized models of continuum mechanics is growing with respect to the necessity to describe of media with microstructure and more complex behaviour of materials. Among the generalized media there are strain-gradient, micropolar (Cosserat) and micromorphic models of continua. In particular, the model of micropolar or Cosserat continuum is used for the description of such micro-structured media as, for example, polycrystals, foams, cellular solids, lattices, masonries, particle assemblies, soils, magnetic rheological fluids, liquid crystals, etc., for which the rotational degrees of freedom of material particles and the couple stresses are important. Specific but very perspective for engineering are so-called nanocrystalline materials investigated over the past couple of decades, see Gleiter (2000); Meyers et al. (2006). The latter are ultrafine-grained materials with a grain size under 100 nm with special properties. So these materials contain an extremely large fraction of grain boundaries with properties different from the ones of bulk materials. In other words, such microstructure of nanostructured materials is determined by characteristic length scale of order of few or tenth nanometers. Among various methods of manufacturing of nanostructured materials are electrodeposition, crystallization, and after severe plastic deformations, see Valiev et al. (2000); Valiev (2004). Another example of nanostructured materials are metal nanolaminates, see Mara et al. (2004, 2008); Misra and Gibala (2000); Misra et al. (2004). There are thousands of papers that appeared on this topic since few last decades. One of the peculiarities of nanostructured materials is the size-effect that is dependence of material properties on the microstructure characteristic length. Modelling of the elasto-plastic behaviour of nanostructured materials is based on various approaches, see review by Gates et al. (2005) on multiscale simulations. The influence of non-homogeneities including the grain-size effect on the material properties discussed by Valiev et al. (2000); Valiev (2004); Meyers et al. (2006); Berbenni et al. (2007); McDowell (2008, 2010); Misra and Singh (2014). McDowell (2008, 2010); Luscher et al. (2010) highlighted that for description of such non-classical materials some extended theories of plasticity can be applied that is the micropolar, micromorphic, strain gradient plasticity ∗ Corresponding

author Email addresses: [email protected] (H. Altenbach ), [email protected]; [email protected] (V.A. Eremeyev) Preprint submitted to Journal of Plasticity

June 10, 2014

theories. The micropolar theory of plasticity begin since the works by Sawczuk (1967); Lippmann (1969). Lippmann (1969) has published paper entitled “A Cosserat Theory of Plastic Flow” where he generalized the flow rule taking into account the couple stresses, see also further discussion on possible applications by Becker and Lippmann (1977); Bogdanova-Bontcheva and Lippmann (1975); Diepolder et al. (2001) and the review Lippmann (1995). The idea of using micropolar kinematics for plastic flow is based, in particular, on the possibility to take into account independent grain or other subdomains rotations. As in classical theories of plasticity, the micropolar plasticity under small and finite deformations can be developed with the use of multiplicative or/and additive decompositions, see Vardoulakis (1989); de Borst (1993); Steinmann (1994a,b); Forest et al. (2000); Forest and Sievert (2006); Grammenoudis and Tsakmakis (2001, 2005a,b, 2009); Neff (2006); Salehi and Salehi (2014). Development of the micropolar plasticity was highly motivated by proper description of strain localization observed during plastic deformations, see original paper by de Borst (1991) and Dietsche et al. (1993); Peri´c et al. (1994); Ehlers and Volk (1997); Dietsche et al. (1993); Dietsche and Willam (1997); Forest (1998); Forest et al. (2001); Bauer et al. (2012). Keller and Trusov (2002) suggested incremental model with independent spin for description of plastic deformations. Mathematical study of problems of the micropolar elastoplasticity are performed by Neff and Chełmi´nski (2005, 2007); Chełmi´nski and Neff (2008); Neff et al. (2007, 2009). Description of plastic deformations of nanostructured materials within the framework of inelastic Cosserat continuum is similar to modelling of behaviour of granular and porous media given by Bogdanova-Bontcheva and Lippmann (1975); Becker and Lippmann (1977); Ehlers and Volk (1997); Mori et al. (1998); Matsushima et al. (2003), of discrete structures such as masonries, see Trovalusci and Masiani (1997); Ehlers et al. (2003); Masiani and Trovalusci (1996); Besdo (2010); Trovalusci and Pau (2013). The Cosserat plasticity is strongly related with strain gradient plasticity and theory of dislocations, see Sievert et al. (1998); Gurtin (2002); Forest and Sedl´acˇ ek (2003); Forest and Sievert (2003); Forest (2008); Clayton et al. (2006); Besson et al. (2010); Clayton (2011); Polizzotto (2014). In particular, Mayeur and McDowell (2014) recently presented comparison between gradient plasticity of Gurtin-type and micropolar plasticity under small deformations. For strain gradient plasticity we refer also to the landscape works by M¨uhlhaus and Aifantis (1991); Fleck and Hutchinson (1997); Gao et al. (1999); Huang et al. (2000); Gurtin (2002); Bertram and Forest (2013). The strain gradient theories find widely applications in other fields of mechanics, in elasticity (Aifantis, 2003; dell’Isola et al., 2009b), poromechanics (dell’Isola et al., 2000; Sciarra et al., 2007, 2008), for description capillarity-related phenomena (dell’Isola and Seppecher, 1997, 1995; dell’Isola et al., 2009a, 2012), and even for bone remodelling (Madeo et al., 2012). The extended models of continua such as the second-gradient models require extended boundary conditions considered for example by dell’Isola and Seppecher (1995); dell’Isola et al. (2012); Srinivasa and Reddy (2013); Baek and Srinivasa (2003). The micropolar plasticity can be considered as a part of more wide micromorphic theory of plasticity for which microstrains are related not only with microrotations but with complete microdeformations, see Eringen (1999); Forest (2013). The micromorphic plasticity is presented by Forest and Sievert (2006); Forest (2009); Grammenoudis et al. (2009); Grammenoudis and Tsakmakis (2010); Regueiro (2010); Forest et al. (2014). Since the micropolar and micromorphic theories can be considered as theories with internal degrees of freedom it is worth to mention the works on the inelasticity theories based on internal variables by Rice (1971); Maugin and Muschik (1994); Hirschberger and Steinmann (2009); Horstemeyer and Bammann (2010). Indeed, the microrotation tensor used in the micropolar continuum can be interpreted as a tensor-valued internal variable describing the rotational degrees of freedom of nano-sized grains, particles in suspensions, etc., while the balance of angular momentum as a corresponding balance equation for such internal variable, see Eringen (1999); Capriz (1989). For the proper formulation of the constitutive equations one has to formulate strain measures and corresponding strain rates. Within the framework of classical Cauchy continuum the discussion of strain measures and strain rates is presented in thousands of papers, see for example Xiao et al. (1997b); Bruhns et al. (1999); Xiao et al. (1997a, 1998b,a, 2000); Surana et al. (2013) and references therein. The comparison of stress-rate-type of constitutive equations is given by Szab´o and Balla (1989). The specific logarithmic strain rate and the corresponding constitutive equations were introduced by Xiao et al. (1997b,a). A review on the state of the art is given in the recent paper by Bruhns (2014). The discussion of strain measures for the polar elastic materials is presented by Pietraszkiewicz and Eremeyev (2009a,b), where the natural Lagrangian and Eulerian strain measures are introduced as a result of three possible ways, that from pure geometrical considerations, from the requirement of invariance of the strain energy density under superposed rigid body motions and as measures conjugate to corresponding stress tensors. Ramezani and Naghdabadi (2007) discussed strain–stress pairs within the framework of micropolar mechanics. Using strain rate 2

tensors Ramezani and Naghdabadi (2010); Ramezani et al. (2008) consider incremental equations of the micropolar hypo-elasticity. Trovalusci and Masiani (1997) established the interrelations between strain rates in discrete and continual models. The aim of the paper is to present the foundations of a constitutive theory for nonlinear micropolar inelastic media. For that purpose we extend in a systematic way the material modeling approach settled by Noll and Trusdell to Cosserat materials. For the first time we introduce history variables including generalized Rivlin-Ericksen tensors and, based on these tools, the formulation of a constitutive theory for inelastic Cosserat media. The paper is organized as follows. In Section 2 we recall the basic equations of the micropolar kinematics and dynamics. The constitutive equations for inelastic micropolar media are discussed in Section 3. These equations are expressed via operator dependence of the Cauchy stress and couple stress tensors on the history of strain measures. Objectivity principle and symmetry requirements are applied to obtain reduced forms of the constitutive equations. In particular the consequences of change of orientation of the observer are systematically highlighted. In Section 4 we introduce the Rivlin–Ericksen tensors for the micropolar media. We formulate constitutive equations for Cosserat media with memory which has not been done in that systematic way in previous contributions. We formulate new constitutive micropolar models based on implicit form constitutive equations similar as proposed by Rajagopal (2003, 2006); Prˆusˇa and Rajagopal (2012). Finally, in Section 5 we discuss constitutive equations for inelastic micropolar media using the introduced strain rate tensors. The application of the objectivity and symmetry principles enable us to give some new constraints on the form of the constitutive equations. Examples of explicit constitutive equations are provided in the isotropic case in the form of viscoelastic or hypo-elastic constitutive equations. For modelling of isotropic behaviour we apply the theory of invariants presented by Spencer (1971); Zheng (1994). Here we derived new representations of various constitutive equations including restrictions following from the fact that some strain-rate tensors and couple stress tensor are pseudotensors (not true ones) and change sign for mirror reflection transformation of the space. We generalize the concept of viscoelastic fluids and solids, and hypo-elasticity to micropolar materials. The Section ends with some elements on the formulation of plasticity yield surfaces and flow rules. Appendix A contains necessary information on the invariants of one and two non-symmetric tensors used for formulation of constitutive equations. In what follows we use the direct tensor calculus in the sense of Gibbs, see Wilson (1901). For basic rules of the direct tensor notation the reader may also consult Lebedev et al. (2010); Truesdell (1966, 1991); Truesdell and Noll (2004), among others. In particular, we define the gradient and divergence operators in the actual and reference configurations as follows grad (•) = Grad (•) =

∂(•) ⊗ rk , ∂xk

∂(•) ⊗ Rk , ∂Xk

div (•) =

∂(•) k ·r , ∂xk

ri =

Div (•) =

∂(•) · Rk , ∂Xk

Ri =

∂r , ∂xi ∂R , ∂xi

ri · rk = δki , Ri · Rk = δki ,

where xi and Xi are the Eulerian and Lagrangian coordinates, respectively, and δij is the Kronecker symbol. 2. Kinematics and motion equations of micropolar media The description of motion of material particles of a micropolar continuum is based on the assumption that every particle has six degrees of freedom as in the case of rigid body dynamics. Three of the degrees of freedom are translational as in classical elasticity, and other three degrees are orientational or rotational. In the actual configuration χ at instant t, the position of a particle of micropolar continuum is given by the position vector r(t). The particle orientation is defined by an orthonormal trihedron dk (t) (k = 1, 2, 3) whose vectors are called directors, dk · dm = δkm , see Kafadar and Eringen (1971); Eringen and Kafadar (1976); Eringen (1999); Eremeyev et al. (2013). Two vector fields r and dk define the translational and rotational motions of a material particle, respectively. To describe the medium relative deformation, we use some fixed position of the body that may be taken at t = 0 or another fixed instant t = τ; we call this position the reference configuration κ. Here the position of particle is defined by the position

3

vector R, whereas its orientation by directors Dk . Hence, the motion of a micropolar continuum can be described by two following vectorial fields r = r(R, t), dk = dk (R, t). The change of the directors is described by an orthogonal tensor H(t) = dk (t) ⊗ Dk called the microrotation tensor. As a result we describe the micropolar continuum deformation by the following relations: r = r(R, t),

H = H(R, t).

(1)

Vector r(t) describes the position of the particle of the continuum at time t, whereas H(t) defines its rotation. The linear velocity is given by the relation dr v= , dt d where denotes the material derivative with respect to t. Differentiation of dk leads to introducing of the angular dt velocity vector ω ddk = ω × dk , k = 1, 2, 3, dt where × is the vector (cross) product. ω can be also expressed using the derivative of H as follows ! 1 dH ω=− · HT , (2) 2 dt × where the dot denotes the dot (inner) product and (. . .)T - transposed. The symbol (...)× stands for the vector invariant of a second-order tensor. In particular, for a dyad a ⊗ b we have (a ⊗ b)× = a × b. Relation (2) means that ω is the axial vector associated with the skew-symmetric tensor dH/ dt · HT . We restrict ourselves by pure mechanical theory. In micropolar mechanics in addition to classical stresses the couple stresses are introduced. The Eulerian equations of motion of micropolar media are ρ

dv = div T + ρf, dt

j

dω = div M − T× + ρm, dt

(3)

where T and M are the stress and couple stress tensors of Cauchy type which are non-symmetric, in general,ρ is the density in the actual configuration, j is the measure of rotatory inertia of particles of micropolar medium, f and m are the external forces and couples, respectively. The motion equations (3) should be complemented by corresponding boundary and initial conditions. Static boundary conditions consist of external forces and couples t∗ and µ∗ given on the body boundary a or on some its part a f ⊂ a (in the actual configuration): T · n = t∗ ,

M · n = µ∗

on a f .

(4)

On the rest part of the boundary a we use two kinematic conditions r = r∗ ,

H = H∗

(H∗ · HT∗ = I)

on

au ,

(5)

where r∗ and H∗ are given functions describing the translations and microrotations of the body particles on au = a\a f , and I is the unit tensor. For example, if au is fixed then we have r = R,

H = I on au .

(6)

Other types of boundary conditions are also possible in the micropolar mechanics. For rate-dependent solid materials and fluids instead of (6) one uses the kinematic boundary conditions in the forms v = v0 ,

ω = ω0 4

on

au ,

(7)

where v0 and ω0 are given functions. Obviously, for micropolar materials there are various types of mixed boundary conditions. For example, in the micropolar hydrodynamics on a free surface they use the following boundary condition for the angular velocity, see Migoun and Prokhorenko (1984), 1 ω = − rot v, 2

on au .

In the case of micropolar materials with constrained rotations the boundary conditions are different, in general, see Nowacki (1986); Srinivasa and Reddy (2013). For dynamic problems we assign the initial conditions r = r0 ,

v = v0 ,

H = H0

T

(H0 · H0 = I),

ω = ω0

at

t = t0

(8)

with given r0 , v0 , H0 , and ω0 . 3. Constitutive equations of micropolar solids with memory In pure mechanical theory of the micropolar continuum with memory the constitutive equations consist of dependence of the stress and the couple stress tensors of the history of deformations. As a result, T and M take the following form: T(t) = A1 [Ft (s), Ht (s), Grad Ht (s)],

M(t) = A2 [Ft (s), Ht (s), Grad Ht (s)],

(9)

where we introduced the histories of the deformation gradient Ft (s) = F(t − s),

F(t) = Grad r(t),

s ≥ 0,

of the microrotation tensor Ht (s) = H(t − s),

s ≥ 0.

Here A1 and A2 are operators describing the micropolar material behaviour. According to the principle of material frame-indifference (Truesdell, 1966; Truesdell and Noll, 2004) T and M are indifferent (objective) quantities. To introduce the notion of frame indifference we recall the definition of equivalent motions. In classical mechanics, two motions r and r∗ are called equivalent if they relate as follows r∗ = a(t) + O(t) · (r − r0 ),

(10)

where O(t) is an arbitrary orthogonal tensor, a(t) is an arbitrary vector function and the constant vector r0 represents a fixed point position (a pole). We can treat equivalent motions as the one body motion considered in different reference frames. In micropolar mechanics, body deformations are described by the position vector r and the trihedron dk or the orthogonal tensor H. After Kafadar and Eringen (1971); Le and Stumpf (1998), we assume that in the equivalent motion the directors dk transform similarly to r: d∗k = O(t) · dk . It follows that in the equivalent motion the microrotation tensors relate by the equation: H∗ = O(t) · H.

(11)

We underline that two deformations of the micropolar medium are equivalent if the position vectors and microrotation tensors relate through Eqs. (10) and (11). Denoting by superscript “*” the stress tensors in the equivalent motions we formulate the property of objectivity for T and M as follows T∗ = O(t) · T · O(t)T , M∗ = det O(t) O(t) · M · O(t)T , (12) 5

for any orthogonal tensor O(t). The difference in transformations of T and M relates with the fact that M is a secondorder axial tensor (pseudotensor) while T is a polar second-order tensor or true tensor, see Eremeyev et al. (2013) for details. For proper orthogonal tensors that is rotation tensor there is not any difference between tensors and pseudotensors. Pseudotensors change its sign during mirror reflection transformation. An interesting discussion of the physical nature of axial vectors and mirror reflection can be found in Feynman et al. (1977). Thus, operators A1 and A2 satisfy the relations A1 [Ot (s) · Ft (s), Ot (s) · Ht (s), Ot (s) · Grad Ht (s)] = O(t) · A1 [Ft (s), Ht (s), Grad Ht (s)] · OT (t), A2 [O (s) · F (s), O (s) · H (s), O (s) · Grad H (s)] = det O(t) O(t) · A2 [F (s), H (s), Grad H (s)] · O (t), t

t

t

t

t

t

t

t

t

T

(13) (14)

Using (13) and (14) we can prove that (9) can be represented as follows T(t) = H(t) · B1 [Et (s), Kt (s)] · HT (t),

M(t) = H(t) · B2 [Et (s), Kt (s)] · HT (t),

(15)

where B1 and B2 are operators depending on histories of two Lagrangian strain measures E and K defined by formulas E = HT · F − I,

1 K = −  : (HT · Grad H). 2

(16)

Here I is the 3D identity tensor and  = −I×I is the third-order permutation tensor, : stands for the double dot product. Indeed, substituting O = HT into (13) and (14) we obtain that T

T

T

T

A1 [Ft (s), Ht (s), Grad Ht (s)] = H(t) · A1 [Ht (s) · Ft (s), I, Ht (s) · Grad Ht (s)] · HT (t), A2 [Ft (s), Ht (s), Grad Ht (s)] = H(t) · A2 [Ht (s) · Ft (s), I, Ht (s) · Grad Ht (s)] · HT (t). The third-order tensor HT · Grad H can be represented as follows HT · Grad H = I × K, where K is the second-order pseudotensor given by (16)2 , see Pietraszkiewicz and Eremeyev (2009a). As a result the constitutive equations take the form (15). For the wryness tensor K other representations can be established, for example the following formula holds true K = HT · B · F − β,

(17)

where where B and β are the microstructure curvature tensors in the actual and reference configurations, respectively, given by formulaes 1 1 B = dk × grad dk , β = Dk × Grad Dk . 2 2 For the micropolar continuum these tensors are used by Yeremeyev and Zubov (1999); Eremeyev and Pietraszkiewicz (2012), while within the framework the six-parameter shells theory the microstructure curvature tensors are introduced by Chr´os´cielewski et al. (2004); Eremeyev and Pietraszkiewicz (2006), see also Eremeyev et al. (2013). Instead of Lagrangian strain measures E and K one can use the following Lagrangian strain tensors U = HT · F,

Y = HT · B · F.

(18)

Unlike E and K which vanish in the absence of deformation, U and Y take the values I and β, respectively. In the case of elastic behaviour Eqs. (15) reduce to T(t) = H(t) · f1 [E(t), K(t)] · HT (t),

M(t) = H(t) · f2 [E(t), K(t)] · HT (t),

(19)

where vector functions f1 and f2 can be expressed with use of the strain energy function W = W(E, K). Various representations of W for non-linear elastic micropolar solids are given by Kafadar and Eringen (1971); Eringen and 6

Kafadar (1976); Eringen (1999); Ramezani et al. (2009b); Eremeyev and Pietraszkiewicz (2012). For isotropic micropolar elastic solids W considered as a function of two strain measures can be represented as a scalar function depending on 15 joint invariants of E and K given by Eqs. (A.2) in Appendix A as follows W = W(I1 , . . . I15 ). In particular, assuming W in the form W = W1 (E) + W2 (K) one obtains that functions W1 and W2 depend on 6 invariants of E and K given by Eqs. (A.1), respectively. In addition to the Lagrangian strain measures one can also use the Eulerian strain measures defined by Eremeyev and Pietraszkiewicz (2012) as follows e = I − H · F−1 ,

k = H · K · F−1 ,

u = H · F−1 ,

y = H · β · F−1 .

(20)

4. Relative strain measures For using the fading memory concept let us introduce relative strain measures. Within the framework of so-called relative description of deformation of continuum we consider the actual configuration χ at instant t as the reference one while the actual configuration χ at instant τ is considered as actual one, see Truesdell (1966, 1991); Truesdell and Noll (2004) and Astarita and Marrucci (1974). Following Eringen (1999) the relative deformation gradient and the relative microrotation tensor are introduced by formulas Ft (τ) = F(τ) · F−1 (t),

Ht (τ) = dk (τ) × dk (t) = H(τ) · H−1 (t).

(21)

Obviously, Ft (t) = I, Ht (t) = I. Using Ft (τ) and Ht (τ) we introduce the relative strain measures Et (τ), Kt (τ) by the relations 1 (22) Et (τ) = Ht (τ)T · Ft (τ) − I, Kt (τ) = −  : (Ht (τ)T · Grad Ht (τ)). 2 In what follows we denote the histories of relative tensors Ft (τ), Ht (τ), etc., as follows Ftt (s) = Ft (t − s),

Htt (s) = Ht (t − s).

From (21) and (22) it follows the following relations: Ut (s) = HT (t) · Utt (s) · H(t) · U(t), Ett (s)

=

Utt (s)

Yt (s) = HT (t) · Ytt (s) · H(t) · U(t), Ktt (s)

− I,

=

Ytt (s)

− B(t)

(23) (24)

and Utt (0) = I,

Ytt (0) = B(t),

Ett (0) = 0,

Ktt (0) = 0,

where we introduced the histories Utt (s) = Ut (t − s), Ytt (s) = Yt (t − s), Ett (s) = Et (t − s) and Ktt (s) = Kt (t − s). Substituting Eqs. (23) and (24) into (15) we obtain h     i T(t) = H(t) · B1 HT (t) · Utt (s) − I · H(t) · U(t), HT (t) · Ytt (s) − B(t) · H(t) · U(t) · HT (t), h     i M(t) = H(t) · B2 HT (t) · Utt (s) − I · H(t) · U(t), HT (t) · Ytt (s) − B(t) · H(t) · U(t) · HT (t). The latter relations transform to h i T(t) = H(t) · C1 U(t), Y(t), HT (t) · Utt (s) · H(t), HT (t) · Ytt (s) · H(t) · HT (t), h i M(t) = H(t) · C2 U(t), Y(t), HT (t) · Utt (s) · H(t), HT (t) · Ytt (s) · H(t) · HT (t) with new operators C1 and C2 .

7

(25) (26)

Further reduction of constitutive equation is possible assuming some type of anisotropy as was done by Eremeyev and Pietraszkiewicz (2012). In what follows we are restricted ourselves by isotropic behavior. In this case C1 and C2 should satisfy the restrictions, h i OT · C1 U(t), Y(t), HT (t) · Utt (s) · H(t), HT (t) · Ytt (s) · H(t) · O (27) h i = C1 OT · U(t) · O, (det O) OT · Y(t) · O, OT · HT (t) · Utt (s) · H(t) · O, (det O) OT · HT (t) · Ytt (s) · H(t) · O , h i (det O)OT · C2 U(t), Y(t), HT (t) · Utt (s) · H(t), HT (t) · Ytt (s) · H(t) · O (28) h i T T T T t T T t = C2 O · U(t) · O, (det O) O · Y(t) · O, O · H (t) · Ut (s) · H(t) · O, (det O) O · H (t) · Yt (s) · H(t) · O , for all orthogonal tensors O, O−1 = OT , see Yeremeyev and Zubov (1999); Eremeyev and Pietraszkiewicz (2012) for details. Let us substitute O = HT into (27) and (28). As a result we obtain h i T(t) = C1 H · U(t) · HT (t), H · Y(t) · HT (t), Utt (s), Ytt (s) , h i M(t) = C2 H · U(t) · HT (t), H · Y(t) · HT (t), Utt (s), Ytt (s) . Thus, the constitutive equations of any isotropic micropolar medium with memory take the following form T(t) = D1 [e(t), k(t), Utt (s), Ytt (s)],

M(t) = D2 [e(t), k(t), Utt (s), Ytt (s)],

(29)

where D1 and D2 are isotropic operators. The history of Ut (τ) and Yt (τ) can be represented as series with respect of two families of tensors Ak and Bk as follows ∞ ∞ X X (−1)k k (−1)k k Utt (s) = s Ak (t), Ytt (s) = s Bk (t). (30) k k k=0 k=0 In the micropolar continuum tensors Ak and Bk play a role of the Rivlin–Ericksen tensors used in the nonlinear viscoelasticity of simple materials. They are given by the recurrent relations Ak+1 =

d Ak + Ak · gradv − ω × Ak , dt

A0 = I,

d Bk + Bk · gradv − ω × Bk , B0 = B. dt Tensors Ak and Bk can be also represented using the derivative of U and Y as follows Bk+1 =

Ak = H ·

dk U −1 ·F , dtk

Bk = H ·

dk Y −1 ·F dtk

(31)

or by formulae Ak+1 = A◦k ,

Bk+1 = B◦k ,

(32)

where the corotational derivative is defined by the relations (. . .)◦ = H ·

i d d h T H · (. . .) · F · F−1 ≡ (. . .) + (. . .) · gradv − ω × (. . .). dt dt

(33)

Let us note that A1 and B1 coincide with the strain rates used in the theory of micropolar continuum A1 = ε ≡ grad v − I × ω,

B1 = κ ≡ grad ω.

(34)

For example, stress power in the micropolar continuum is given by w = T : ε + M : κ. In the micropolar continuum one can introduce other strain rate tensors which, for example, are analogues of the White–Metzner tensors in mechanics of simple materials, see Astarita and Marrucci (1974); Drozdov (1998). Such tensors are used as arguments of constitutive equations of inelastic micropolar materials such as viscoelastic 8

micropolar fluids and solids and in the micropolar plasticity. In particular, applying any objective time derivative to the Eulerian strain measures (20) one can obtain another set of strain rate tensors. Let us only note that within the framework of micropolar materials there are more possibilities to introduce objective time derivatives than in the Cauchy continuum, see for example Ramezani and Naghdabadi (2010); Grammenoudis and Tsakmakis (2001, 2005b). In particular, the objective derivative introduced by (33) is different from others including the gyration rate introduced by Ramezani and Naghdabadi (2010). Considered finite approximation of series (30) we obtain the model of micropolar material of order (m, n) T(t) = F1 [e(t), k(t), B(t), A1 (t) . . . Am (t), B1 (t) . . . Bn (t)],

M(t) = F2 [e(t), k(t), B(t), A1 (t) . . . Am (t), B1 (t) . . . Bn (t)], (35)

where F1 and F2 are isotropic operators. Finally, in order to consider various form of rate-type constitutive equations of micropolar materials we introduce implicit constitutive equations of differential type in the following form g1 [T◦{M} . . . T◦ , M◦{N} . . . M◦ , M(t), e(t), k(t), B, A1 . . . Am , B1 . . . Bn ] = 0, ◦{M}

g2 [T

...T ,M ◦

◦{N}

(36)

. . . M , M(t), e(t), k(t), B, A1 . . . Am , B1 . . . Bn ] = 0, ◦

where g1 and g2 are isotropic tensor-valued functions of M + N + m + n + 3 tensorial arguments, and ◦{k} stands for kth objective derivative. The latter equations generalize the idea of implicit constitutive equations originally introduced by Rajagopal (2003, 2006), see also Rajagopal and Srinivasa (2009); Prˆusˇa and Rajagopal (2012) and the reference therein. Implicit model of material has some advantages, such as, in particular, the possibility to describe limiting small strains (Rajagopal, 2011). In what follows we present some linear cases of (36) related with viscoelastic and hypo-elastic micropolar materials. 5. Examples of constitutive equations In this section we discuss some constitutive equations for inelastic fluid and solid micropolar materials which exhibit the kinematically independent rotational degrees of freedom. As was mention in Introduction these constitutive equations may be applied for description of large deformations of rate-dependent materials such as viscoelastic and viscoelastoplastic media. 5.1. Linear viscous micropolar fluid The simplest example of an inelastic micropolar material is the micropolar viscous fluid with the constitutive equations T = −p(ρ)I + α1 ε + α2 εT + α3 I tr ε, M = β1 κ + β2 κT + β3 I tr κ, (37) where p is the pressure, ρ is the density, α1 , α2 , α3 and β1 , β2 , β2 are viscosities, see Aero et al. (1965); Eringen (1966, 2001). Indeed, Eqs. (37) follow from (29) and (30) with certain assumptions. First we keep in series (30) terms with A1 = ε and B1 = κ only. We neglect dependence on k(t) in (29) while dependence on e(t) is assumed to be dependence on det F only. As a result, the stress and couple stress tensors are tensor-valued functions of strain rates ε, κ and density ρ. Finally, for the micropolar viscous fluid we assume that T and M depend on ε and κ linearly. Starting with pioneering papers by Aero et al. (1965); Eringen (1966) the model of viscous micropolar fluid is applied to model magnetic liquids, polymer suspensions, liquid crystals, and other types of fluids with microstructure, and in the case of flows in micro- and nano-sized channels, see Migoun and Prokhorenko (1984); Łukaszewicz (1999) and Eringen (2001). For example, action of a magnetic field on a magnetic fluid called also ferrofluid leads to appearance of volumen couples as in micropolar hydromechanics, see Rosensweig (1987).

9

5.2. Non-linear viscous micropolar fluid The further generalization of (37) is non-linear viscous micropolar fluid with the following constitutive equations: T = −p(ρ)I + Tv (ε, κ),

M = Mv (ε, κ),

(38)

where Tv (ε, κ) and M = Mv (ε, κ) are non-linear isotropic functions of two non-symmetric second order tensors. Such model may be applied for highly viscous suspensions or ferrofluids. Assuming existence of dissipative potential that is a scalar isotropic function Φ(ε, κ) ≥ 0 such that ∂Φ ∂Φ , Mv = , ∂ε ∂κ we can apply the theory of invariants for representation of Φ, see Spencer (1971); Zheng (1994) and Eqs. (A.2) in Appendix A. As a result Φ depends on 15 invariants of I j (ε, κ), j = 1 . . . 15 and should satisfy the requirement Tv =

Φ(I1 , I2 , I3 , I4 , I5 , I6 , I7 , I8 , I9 , I10 , I11 , I12 , I13 , I14 , I15 )

(39)

= Φ(I1 , I2 , I3 , I4 , I5 , I6 , −I7 , −I8 , I9 , −I10 , I11 , −I12 , I13 , −I14 , I15 ), since I7 , I8 , I10 , I12 , I14 are the relative invariants and change sign during the non-proper transformations, see Eremeyev and Pietraszkiewicz (2012). For linear viscous fluid Φ takes the form of a quadratic potential 2 Φ = α1 I12 + α2 I2 + α3 I4 + β1 I10 + β2 I11 + β3 I13

and Eqs. (38) reduce to the linear case (37). 5.3. Viscoelastic micropolar fluids Following Yeremeyev and Zubov (1999) the model of viscous micropolar fluid can be generalized to the case of viscoelastic behaviour. The viscoelastic micropolar fluid has the constitutive relations following from (35): T = H1 [ρ(t), B(t), Ett (s), Ktt (s)],

M = H2 [ρ(t), B(t), Ett (s), Ktt (s)],

(40)

where H1 and H2 are isotropic operators. In particular, using (30) we define the viscoelastic micropolar fluid of differential type of order (m, n) as a micropolar fluid with following constitutive dependencies: T = h1 (ρ, B, A1 . . . Am , B1 . . . Bn ),

M = h2 (ρ, B, A1 . . . Am , B1 . . . Bn ),

(41)

where h1 and h2 are tensor-valued isotropic functions of m + n + 1 tensorial arguments. Yeremeyev and Zubov (1999) proved that for such viscometer flows as the plane shear flow, flow in channel, Poiseuille flow any viscoelastic micropolar fluid of differential type of order (m, n) can not be distinguished from the fluid of order (1, 0). For the Couette flow any viscoelastic micropolar fluid of differential type of order (m, n) can not be distinguished from the fluid of order (1, 1). Since M, Bk , k = 0, 1, . . . n are pseudo-tensors and change sign during the mirror reflection transformation, the principle of material frame-indifference requires that h1 and h2 satisfy the relations h1 (ρ, B, A1 . . . Am , B1 . . . Bn ) = h1 (ρ, −B, A1 . . . Am , −B1 . . . − Bn ),

(42)

−h2 (ρ, B, A1 . . . Am , B1 . . . Bn ) = h2 (ρ, −B, A1 . . . Am , −B1 . . . − Bn ). Let us note that the mathematical correctness of the initial-boundary-value problems for fluids of differential type of order (m, n) requires additional initial data for m > 2, n > 2, see, for example, Lebedev (1975). Using the theory of representations of isotropic tensor-valued functions presented for example by Zheng (1994) further reduction of constitutive equations (41) is also possible, in general. On the other hand such reduction may lead to complicated and awkward relations, especially in the case of many arguments. In particular, the number of invariants of an isotropic scalar-valued function of several tensors may be greater than the number of components, see Spencer (1971); Zheng (1994). As a result, for more explicit reduction one needs to assume other properties of (41) in addition to isotropy and constraints (42) such as linearity with respect to some arguments, etc. This new model combines elastic properties related with rotational degrees of freedom as ones observed in the case of liquid crystals and viscoelastic properties which natural for concentrated polymeric fluids. Thus, the model may be applied in hydrodynamics of thermotropic and lyotropic liquid crystals, especially of biaxial liquid crystals, polymer melts, suspensions and other microstructured (oriented) fluids, see Eringen (1999); Lhuillier and Rey (2004a,b); Madsen et al. (2004); Lin et al. (2012); Gay-Balmaz et al. (2013). 10

5.4. Micropolar hypo-elasticity Ramezani and Naghdabadi (2010); Ramezani et al. (2008) generalized the concept of hypo-elasticity presented by Truesdell and Noll (2004); Xiao et al. (1997a); Besson et al. (2010) to the micropolar continuum model. Hypo-elastic constitutive relations are special rate type constitutive equations. The model was extensively discussed in the literature including difference between nonlinear elasticity, see Truesdell (1963); Truesdell and Noll (2004). Hypo-elasticity and the classical hyper-elasticity are different models , neither includes the other, in general. A hypo-elastic material may have permanent memory of all its past configurations. The model describes inelastic deformations possessing no potential, in which the stress-strain relation may defined incrementally. It is able to predict yield-like phenomenon at large simple shear deformation. For example, the micropolar hypo-elasticity is used for description of granular media, see Tejchman and Bauer (2005); Tejchman et al. (2012). In addition, since the constitutive equations are given in an incremental form they may be useful in numerics. Within the framework of the hypo-elastic micropolar solids the constitutive equations for T and M are formulated as follows T◦ = η1 (T, ε, κ),

M◦ = η2 (M, ε, κ),

(43)

where ◦ denotes an objective time derivative, and η1 and η2 are isotropic functions of their arguments and linear with respect to strain rates ε and κ. As a result, Eqs. (43) take the form T◦ = C1 (T) : ε + C2 (T) : κ,

M◦ = C3 (M) : ε + C4 (M) : κ,

(44)

where C1 , C2 , C3 , C4 are 4th-order tensors, which depend on stress and couple stress tensors, in general. Using the representations theorems of the theory of invariants (Spencer, 1971; Zheng, 1994), Ramezani and Naghdabadi (2010) derived polynomial-type expressions for these tensors. In addition to results of Ramezani and Naghdabadi (2010) let us note that the principle of material frame-indifference leads to the following restrictions for η1 and η2 and Ck : η1 (T, ε, κ) = η1 (T, ε, −κ),

−η2 (M, ε, κ) = η2 (M, ε, −κ),

(45)

which lead to constraints C2 = 0, C3 (M) = −C3 (−M), C4 (M) = C4 (−M). As a result, the constitutive equations by Ramezani and Naghdabadi (2010) are significantly reduced. Constitutive equations (43) or (44) can be extended as follows T◦ = η1 (T, M, ε, κ),

M◦ = η2 (T, M, ε, κ),

(46)

M◦ = C3 (T, M) : ε + C4 (T, M) : κ

(47)

−η2 (T, M, ε, κ) = η2 (T, −M, ε, −κ),

(48)

T◦ = C1 (T, M) : ε + C2 (T, M) : κ, with the following constraints η1 (T, M, ε, κ) = η1 (T, −M, ε, −κ), C1 (T, M) = C1 (T, −M),

−C2 (T, M) = C2 (T, −M),

−C3 (T, M) = C3 (T, −M),

C4 (T, M) = C4 (T, −M).

Let us note that constitutive equations (46) and (47) contain huge number of material parameters even for isotropic materials. On the other hand, unlike to more simple hypo-elastic model (44) new model (47) with constraints (48) may describe coupling between rotational and translational motion of material particles during finite deformations. For the hypo-elastic micropolar materials and for micropolar fluids it is natural to use the Eulerian static boundary conditions (4) and/or the kinematic boundary conditions expressed in terms of velocities, that is conditions (7). 5.5. Micropolar viscoelasticity The constitutive equations (29) include various forms of micropolar viscoelastic behaviour under finite deformations. Here we present few examples of constitutive equations of differential type. The constitutive equations of the form T = Φ1 (e, k, ε, κ), τ1 T + T = Ψ1 (ε, κ), ◦

τ1 T◦ + T = Ω1 (e, k, ε, κ),

M = Φ2 (e, k, ε, κ),

(49)

τ2 M + M = Ψ2 (ε, κ),

(50)

τ2 M◦ + M = Ω2 (e, k, ε, κ)

(51)

◦

11

play a role of Kelvin-Voigt, Maxwell and standard models in micropolar viscoelasticity, respectively. Here ◦ denotes an objective time derivative, τ1 and τ2 are relaxation time parameters and Φ1 , Φ2 , Ψ1 , Ψ2 , Ω1 and Ω2 are constitutive tensor-valued functions. Using higher order objective time derivatives and tensors Ak , Bk one can present constitutive equations of differential type of any order. Linear models of micropolar viscoelasticity are discussed by Eringen (1967, 1999). Equations (44), (47), (50) and (51) are examples on general implicit constitutive equations (36). 5.6. Isotropic yield criteria and other rate-type constitutive equations Description of plastic behaviour of such non-classical materials as nanostructured metals and alloys, granular media within the framework of micropolar plasticity was highlighted by McDowell (2008, 2010); Luscher et al. (2010) among others. Application of micropolar plasticity is motivated by the possibility of modelling of kinematically independent rotations of grains and subgrains and observed strain localization during plastic deformations de Borst (1991); Dietsche et al. (1993). The foundations of the micropolar elastoplasticity were developed by Vardoulakis (1989); de Borst (1993); Steinmann (1994a,b); Forest et al. (2000); Forest and Sievert (2006); Grammenoudis and Tsakmakis (2001, 2005a,b, 2009); Neff (2006). Lippmann (1969) generalized the Huber–von Mises yield criterion for the case of micropolar material considering action of couple stresses, see recent results by Steinmann (1994a,b); Grammenoudis and Tsakmakis (2001, 2005a,b, 2007, 2009). ! p 3 1 Y(T, M) ≡ J2 − Y p = 0, J2 = devT : devT + M : M , (52) 2 `p where devT = T − 1/3Itr T is the deviatoric part of T, ` p is the characteristic length, and Y p is the yield strength in uniaxial tension. Obviously, the constraint is Y(T, M) = Y(T, −M) is fulfilled. For small deformations the governing equations of micropolar elastoplasticity is formulated on the basis of an additive decomposition of infinitesimal strain rates into a sum of elastic and plastic parts ε = εe + ε p ,

κ = κe + κ p ,

(53)

where indices “e” and “p” relate with elastic and plastic parts, respectively. In the case of small deformations ε and κ are defined through the time derivative of the linear strain measures ε=

d , dt

κ=

d κ, dt

 = (Grad u − I × θ),

κ = Grad θ,

where u and θ are vectors of infinitesimal translations and rotations, see Pietraszkiewicz and Eremeyev (2009a,b). The constitutive equations are given by T = λItr  + µ1 ( −  p ) + µ2 ( −  p )T ,

κ − κ p ) + γ(κ κ − κ p )T , M = αItr κ + β(κ

where λ, µ1 , µ2 , α , β and γ are elastic moduli and the constraints tr  p = 0 and tr κ p = 0 are assumed. For plastic strains the following flow rule d ∂Y d ∂Y ε p ≡  p = λ˙ , κ p ≡ κ p = λ˙ , dt ∂T dt ∂M where λ˙ is a plastic multiplier. In the case of finite deformations the micropolar elastoplasticity is based on the multiplicative decomposition of the deformation gradient and microrotation tensor into elastic and plastic parts as follows, F = Fe · F p ,

H = He · H p ,

(54)

and the corresponding representation of strain measures Ee , E p , Ke , K p , and strain rates, elastic constitutive equations and the flow rule ∂Y ∂Y ε p ≡ E◦p = λ˙ , κ p ≡ K◦p = λ˙ , ∂T ∂M see Steinmann (1994a); Grammenoudis and Tsakmakis (2001, 2007) for details. The version of the micropolar plasticity under finite deformations based on additive decomposition is formulated by Ramezani et al. (2009a). 12

The crucial point in the micropolar elastoplasticity is formulation of an yield criterion as a function of two nonsymmetric stress measures, that is stress and couple stress tensors. As a result, an yield criterion can be represented as a scalar-value function of stress and couple stress tensors as follows Y = Y(T, M)

(55)

with the property Y(T, M) = Y(T, −M). Function Y can be represented as a function of 15 joint invariants of T and M listed in (A.2). The more general yield criteria may be useful for description of behaviour of micro- and nanostructured materials, such as for example foams which are pressure sensitive, granular and porous media, etc. For the classical Cauchy continuum models there are many rate-dependent and rate-independent constitutive models proposed for inelastic behavior, see, for example, Naumenko and Altenbach (2007); Besson et al. (2010); Clayton (2011); Bertram (2012) and the references therein. A constitutive model for the inelastic deformation tensor and three state variables to capture hardening, softening and damage is proposed by Naumenko et al. (2011a). In Altenbach et al. (2008) a stress-range dependent model of creep is presented. That is, for low stress levels diffusion type creep is described, while for moderate stress levels power law type creep is considered. A constitutive model for inelastic processes at high temperature including creep, visco-plasticity, and thermo-mechanical fatigue is presented by Naumenko et al. (2011b). To capture inelastic behavior at high temperature and for a wide range of deformation rates a special non-linear fluid model is developed by Schmicker et al. (2013). A constitutive model with the damage tensor is developed and applied for the analysis of thin-walled structures by Altenbach et al. (2002). Naumenko and Altenbach (2005) proposed a model of anisotropic creep is developed and identified for a multi-pass weld metal. The constitutive laws of elasto-plasticity with internal variables including various hardening models and damage are considered by Cuomo and Contrafatto (2000); Contrafatto and Cuomo (2002, 2006). For the micropolar inelastic continua similar models can be also suggested. The essential feature of these models is the following form of constitutive equations ε p = Υ1 (T, M),

κ p = Υ2 (T, M),

where ε p and κ p are the inelastic parts of strain rates while Υ1 and Υ2 are tensor-valued function of stress tensors which can also depend on temperature, other internal variables and structure tensors. The detailed description and further development of such constitutive models lie outside the scope of this paper and may be presented in forthcoming papers. Here we underline that the objectivity and the theory of isotropic tensor-valued functions give one a powerful tool for highly nonlinear inelastic behaviour of micropolar materials. 6. Conclusions We introduced a new family of strain rate tensors for micropolar materials which is used in construction of constitutive equations of inelastic materials. With the help of introduced strain rates we discussed the possible forms of constitutive equations of the nonlinear inelastic micropolar continuum, that is micropolar viscous and viscoelastic fluids and solids, hypo-elastic and viscoelastoplastic materials. Considering the fact that some of strain rates are not true tensors but pseudotensors we obtain some constitutive restrictions following from the material frame indifference principle. For example, with these restrictions the general form of hypoelastic micropolar material is more simple than it was presented by (Ramezani and Naghdabadi, 2010). Using the theory of tensorial invariants we present the general form of constitutive equations of some types of inelastic isotropic micropolar materials including several new constitutive equations. Let us note one practically important difference between models of Cauchy and Cosserat continua. For some cases ideas of constitutive modelling used in the Cauchy mechanics can be straightforward extended for the micropolar continuum maybe with some complications. Among such cases are micropolar elasticity (Ramezani et al., 2009b; Pietraszkiewicz and Eremeyev, 2009a; Eremeyev and Pietraszkiewicz, 2012) and hypo-elasticity (Ramezani and Naghdabadi, 2010), mechanics of viscous micropolar fluids (Aero et al., 1965; Eringen, 1966). On the other hand, for the description of elastic and inelastic behaviour some specific strain measures and strain rates can be applied, that is logarithmic Hencky’s strain measure and related logarithmic strain rate, see Xiao et al. (1997a,b, 2000); Bruhns et al. (1999) and recent review by Bruhns (2014). For the micropolar media strain measures and related strain rates are non-symmetric second-order tensors. This makes the straightforward transfer of such models of classical continuum mechanics based on logarithmic objective derivative to the case of the micropolar continuum more complex or impossible, in general. 13

Appendix A. System of invariants of two non-symmetric tensors For modelling of constitutive relations the theory of invariants and tensor functions is widely applied, see Spencer (1971); Smith (1994); Boehler (1987); Zheng (1994) and the references therein. This theory gives the possibility for further simplification of tensor functions especially in the case of functions of two and many tensorial arguments. Following general representations of isotropic and hemitropic scalar-valued functions of one non-symmetric tensor E given by Spencer and Rivlin (1962); Smith (1965, 1994); Spencer (1965, 1971); Smith and Smith (1971) with the help of algebraic theory of the invariants, an isotropic scalar-valued function of one non-symmetric tensors E can be constructed as a function of six invariants In , n = 1, . . . , 6, where I1 = tr E,

I2 = tr E2 ,

I3 = tr E3 ,

I4 = tr (E · ET ),

I5 = tr (E2 · ET ),

I6 = tr (E2 · ET 2 ).

(A.1)

According to Zheng (1994), a isotropic scalar-valued function of two non-symmetric tensors E and K depends on following 15 invariants: I1 (E, K) I4 (E, K) I7 (E, K) I10 (E, K) I13 (E, K)

= = = = =

tr E, tr (E · ET ), tr (E · K), tr K, tr (K · KT ),

I2 (E, K) I5 (E, K) I8 (E, K) I11 (E, K) I14 (E, K)

= = = = =

tr E2 , tr (E2 · ET ), tr (E2 · K), tr K2 , tr (K2 · KT ),

I3 (E, K) I6 (E, K) I9 (E, K) I12 (E, K) I15 (E, K)

= tr E3 , = tr (E2 · ET 2 ), = tr (E · K2 ), = tr K3 , = tr (K2 · KT 2 )

(A.2)

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