Homogenization for dislocation based gradient visco-plasticity
In this work we study the homogenization for infinitesimal dislocation based gradient viscoplasticity with linear kinematic hardening and general non-associative monotone plastic flows. The constitutive equations in the models we study are assumed to be only of monotone type. Based on the generalized version of Korn’s inequality for incompatible tensor fields (the non-symmetric plastic distortion) due to Neff/Pauly/Witsch, we derive uniform estimates for the solutions of quasistatic initial-boundary value problems under consideration and then using a modified unfolding operator technique and a monotone operator method we obtain the homogenized system of equations. A new unfolding result for the -operator is presented in this work as well. The proof of the last result is based on the Helmholtz-Weyl decomposition for vector fields in general -spaces.
Communicated with Patrizio Neff
Key words: plasticity, gradient plasticity, viscoplasticity,
dislocations, plastic spin, homogenization, periodic unfolding, Korn’s inequality,
Rothe’s time-discretization method, rate-dependent models.
AMS 2000 subject classification: 35B65, 35D10, 74C10, 74D10, 35J25, 34G20, 34G25, 47H04, 47H05
We study the homogenization of quasistatic initial-boundary value problems arising in gradient viscoplasticity. The models we study use rate-dependent constitutive equations with internal variables to describe the deformation behaviour of metals at infinitesimally small strain.
Our focus is on a phenomenological model on the macroscale not including the case of single crystal plasticity. Our model has been first presented in . It is inspired by the early work of Menzel and Steinmann . Contrary to more classical strain gradient approaches, the model features from the outset a non-symmetric plastic distortion field , a dislocation based energy storage based solely on (and not ) and therefore second gradients of the plastic distortion in the form of acting as dislocation based kinematical backstresses. We only consider energetic length scale effects and not higher gradients in the dissipation.
Uniqueness of classical solutions in the subdifferential case (associated plasticity) for rate-independent and rate-dependent formulations is shown in . The existence question for the rate-independent model in terms of a weak reformulation is addressed in . The rate-independent model with isotropic hardening is treated in [21, 42]. The well-posedness of a rate-dependent variant without isotropic hardening is presented in [49, 50]. First numerical results for a simplified rate-independent irrotational formulation (no plastic spin, symmetric plastic distortion ) are presented in . In [26, 55] well-posedness for a rate-independent model of Gurtin and Anand  is shown under the decisive assumption that the plastic distortion is symmetric (the irrotational case), in which case one may really speak of a strain gradient plasticity model, since the full gradient acts on the symmetric plastic strain.
Let us shortly revisit the modeling ingredients of the gradient plasticity model under consideration. This part does not contain new results but is added for clarity of exposition. As usual in infinitesimal plasticity theory, the basic variables are the displacement and the plastic distortion . We split the total displacement gradient into non-symmetric elastic and non-symmetric plastic distortions
For invariance reasons, the elastic energy contribution may only depend on the symmetric elastic strains . For more on the basic invariance questions related to this issue dictating this type of behaviour, see [59, 40]. We assume as well plastic incompressibility , as is usual. The thermodynamic potential of our model is therefore written as
The positive definite elasticity tensor is able to represent the elastic anisotropy of the material. The plastic flow has the form
where is the elastic symmetric Cauchy stress of the material and is a multivalued monotone flow function which is not necessary the subdifferential of a convex plastic potential (associative plasticity). This ensures the validity of the second law of thermodynamics, see .
In this generality, our formulation comprises certain non-associative plastic flows in which the yield condition and the flow direction are independent and governed by distinct functions. Moreover, the flow function is supposed to induce a rate-dependent response as all materials are, in reality, rate-dependent.
Clearly, in the absence of energetic length scale effects (i.e. ), the -term is absent. In general we assume that maps symmetric tensors to symmetric tensors. Thus, for the plastic distortion remains always symmetric and the model reduces to a classical plasticity model. Therefore, the energetic length scale is solely responsible for the plastic spin (the non-symmetry of ) in the model.
Regarding the boundary conditions necessary for the formulation of the higher order theory we assume that the so-called micro-hard boundary condition (see ) is specified, namely
This is the correct boundary condition for tensor fields in spaces which admits tangential traces. We combine this with a new inequality extending Korn’s inequality to incompatible tensor fields, namely
Here, the domain needs to be sliceable, i.e. cuttable into finitely many simply connected subdomains with Lipschitz boundaries. This inequality has been derived in [43, 44, 45] and is precisely motivated by the well-posedness question for our model . The inequality (3) expresses the fact that controlling the plastic strain and the dislocation density in gives a control of the plastic distortion in provided the correct boundary conditions are specified: namely the micro-hard boundary condition. Since we assume that (plastic incompressibility) the quadratic terms in the thermodynamic potential provide a control of the right hand side in (3).
It is worthy to note that with only monotone and not necessarily a subdifferential the powerful energetic solution concept [37, 26, 35] cannot be applied. In our model we face the combined challenge of a gradient plasticity model based on the dislocation density tensor involving the plastic spin, a general non-associative monotone flow-rule and a rate-dependent response.
Setting of the homogenization problem.
Let be an open bounded set, the set of material points of the solid body, with a -boundary and be a set having the paving property with respect to a basis defining the periods, a reference cell. By we denote a positive number (time of existence), which can be chosen arbitrarily large, and for
The sets, and denote the sets of all –matrices and of all symmetric –matrices, respectively. Let be the set of all traceless –matrices, i.e.
Unknown in our small strain formulation are the displacement of the material point at time and the non-symmetric infinitesimal plastic distortion .
The model equations of the problem are
which must be satisfied in . Here, is a given material constant independent of and is the infinitesimal Eshelby stress tensor driving the evolution of the plastic distortion and is a scaling parameter of the microstructure. The homogeneous initial condition and Dirichlet boundary condition are
where is a normal vector on the boundary 111Here, with and denotes a row by column operation.. For simplicity we consider only homogeneous boundary condition and we assume that the cell of periodicity is given by . Then, we assume that , a given material function, is measurable, periodic with the periodicity cell and satisfies the inequality
for all and some positive constant . For every the elasticity tensor is linear symmetric and such that there exist two positive constants satisfying
We assume that the mapping is measurable and periodic with the same periodicity cell . Due to the above assumption (), the classical linear kinematic hardening is included in the model. Here, the nonlocal backstress contribution is given by the dislocation density motivated term together with corresponding Neumann conditions.
For the model we require that the nonlinear constitutive mapping is monotone for all , i.e. it satisfies
for all and all . We also require that
The mapping is periodic with the same periodicity cell . Given are the volume force and the initial datum .
It is well known that classical viscoplasticity (without gradient effects) gives rise to a well-posed problem. We extend this result to our formulation of rate-dependent gradient plasticity. The presence of the classical linear kinematic hardening in our model is related to whereas the presence of the nonlocal gradient term is always related to .
The development of the homogenization theory for the quasi-static initial boundary value problem of monotone type in the classical elasto/visco-plasticity introduced by Alber in  has started with the work , where the homogenized system of equations has been derived using the formal asymptotic ansatz. In the following work  Alber justified the formal asymptotic ansatz for the case of positive definite free energy222Positive definite energy corresponds to linear kinematic hardening behavior of materials., employing the energy method of Murat-Tartar, yet only for local smooth solutions of the homogenized problem. It is shown there that the solutions of elasto/visco-plasticity problems can be approximated in the norm by the smooth functions constructed from the solutions of the homogenized problem. Later in , under the assumption that the free energy is positive definite, it is proved that the difference of the solutions of the microscopic problem and the solutions constructed from the homogenized problem, which both need not be smooth, tends to zero in the norm, where is the periodicity cell. Based on the results obtained in , in  the convergence in is replaced by convergence in . In the meantime, for the rate-independent problems in plasticity similar results are obtained in  using the unfolding operator method (see Section 3) and methods of energetic solutions due to Mielke. For special rate-dependent models of monotone type, namely for rate-dependent generalized standard materials, the two-scale convergence of the solutions of the microscopic problem to the solutions of the homogenized problem has been shown in [61, 62]. The homogenization of the Prandtl-Reuss model is performed in [57, 62]. In  the author considered the rate-dependent problems of monotone type with constitutive functions , which need not be subdifferentials, but which belong to the class of functions introduced in Section 5. Using the unfolding operator method and in particular the homogenization methods developed in , for this class of functions the homogenized equations for the viscoplactic problems of monotone type are obtained in .
In the present work the construction of the homogenization theory for the initial boundary value problem (4) - (9) is based on the existence result derived in  (see Theorem 5.6) and on the homogenization techniques developed in  for classical viscoplasticity of monotone type. The existence result in  extends the well-posedness for infinitesimal dislocation based gradient viscoplasticity with linear kinematic hardening from the subdifferential case (see ) to general non-associative monotone plastic flows for sliceable domains. In this work we also assume that the domain is sliceable and that the monotone function belongs to the class . For sliceable domains , based on the inequality (3), we are able to derive then uniform estimates for the solutions of (4) - (9) in Lemma 5.8. Using the uniform estimates for the solutions of (4) - (9), the unfolding operator method and the homogenization techniques developed in [18, 48], for the class of functions we obtain easily the homogenized equations for the original problem under consideration (see Theorem 5.7). The distinguish feature of this work is that we use a variant of the unfolding operator due to Francu (see [24, 25]) and not the one defined in . The modified unfolding operator helps to resolve the problems connecting with the need of the careful treatment of the boundary layer in the definition of the unfolding operator in . To the best our knowledge this is the first homogenization result obtained for the problem (4) - (9). We note that similar homogenization results for the strain-gradient model of Fleck and Willis  are derived in [23, 27, 31] using the unfolding method together with the -convergence method in the rate-independent setting. In  the authors, based on the assumption that the model under consideration is of rate-independent type, are able to treat the case when is a -periodic function as well. In the rate-independent setting this is possible due to the fact that the whole system (4) - (9) can be rewritten as a standard variational inequality (see ) and then the subsequant usage of the techniques of the convex analysis enable the passage to the limit in the model equations. Contrary to this, in the rate-independent case this reduction to a single variational inequality is not possible and one is forced to use the monotonicity argument to study the asymptotic behavior of the third term in (6).
Suppose that is a bounded domain with a -boundary . Throughout the whole work we choose the numbers satisfying the following conditions
and denotes a norm in . Moreover, the following notations are used in this work. The space with consists of all functions in with weak derivatives in up to order . If is not integer, then denotes the corresponding Sobolev-Slobodecki space. We set . The norm in is denoted by (). The operator defined by
denotes the usual trace operator. The space with consists of all functions in with . One can define the bilinear form on the product space by
is a Banach space with respect to the norm
The well known result on the generalized trace operator (see [58, Section II.1.2]) can be easily adopted to the functions with values in . Then, according to this result, there is a bounded operator on
where denotes the dual of a Banach space . Next,
Let us define spaces and by
which are Banach spaces with respect to the norm
According to [34, Theorem 2]333This theorem has to be applied to each row of a function with values in to obtain the desired result. the spaces and are continuously imbedded into . We define and by
and denote by and the -spaces of harmonic functions on as
Then the spaces and for every fixed , , coincides with the spaces and given by
We also define the space by
which is a Banach space with respect to the norm
The space denotes the Banach space of -periodic functions in equipped with the -norm.
For functions defined on we denote by the mapping , which is defined on . The space denotes the Banach space of all Bochner-measurable functions such that is integrable on . Finally, we frequently use the spaces , which consist of Bochner measurable functions having -integrable weak derivatives up to order .
2 Maximal monotone operators
Let be a reflexive Banach space with the norm , be its dual space with the norm . The brackets denotes the dual pairing between and . Under we shall always mean a reflexive Banach space throughout this section. For a multivalued mapping the sets
are called the effective domain and the graph of , respectively.
A mapping is called monotone if and only if the inequality holds
A monotone mapping is called maximal monotone iff the inequality
A mapping is called generalized pseudomonotone iff the set is closed, convex and bounded for all and for every pair of sequences and such that , , and
we have that and .
A mapping is called strongly coercive iff either is bounded or is unbounded and the condition
is satisfied for each .
It is well known ([53, p. 105]) that if is a maximal monotone operator, then for any the image is a closed convex subset of and the graph is demi-closed.444A set is demi-closed if converges strongly to in and converges weakly to in (or converges weakly to in and converges strongly to in ) and , then A maximal monotone operator is also generalized pseudomonotone (see [9, 32, 53]).
We recall that the subdifferential of a lower semi-continuous and convex function is maximal monotone (see [54, Theorem 2.25]).
The duality mapping is defined by
for all .
Without loss of generality (due to Asplund’s theorem) we can assume that both and are strictly convex, i.e. that the unit ball in the corresponding space is strictly convex. In virtue of [9, Theorem II.1.2], the equation
has a solution for every and if is maximal monotone. The solution is unique (see [9, p. 41]).
we define two single valued operators: the Yosida approximation and the resolvent with .
By the definition, one immediately sees that . For the main properties of the Yosida approximation we refer to [9, 32, 53] and mention only that both are continuous operators and that is bounded and maximal monotone.
Convergence of maximal monotone graphs
In the presentation of the next subsections we follow the work , where the reader can also find the proofs of the results mentioned here.
The derivation of the homogenized equations for the initial boundary value problem (4) - (9) is based on the notion of the convergence of the graphs of maximal monotone operators. According to Brezis  and Attouch , the convergence of the graphs of maximal monotone operators is defined as follows.
Let , be maximal monotone operators. The sequence converges to as , (), if for every there exists a sequence such that strongly in as .
Obviously, if and are everywhere defined, continuous and monotone, then the pointwise convergence, i.e. if for every , , implies the convergence of the graphs. The converse is true in finite-dimensional spaces.
Let , be maximal monotone operators, and let and . If, as , , , and
See [18, Theorem 2.8]. ∎
The convergence of the graphs of multi-valued maximal monotone operators can be equivalently stated in term of the pointwise convergence of the corresponding single-valued Yosida approximations and resolvents.
Let , be maximal monotone operators and . The following statements are equivalent:
for every , as ;
for every , as ;
Moreover, the convergences and are uniform on strongly compact subsets of .
See [18, Theorem 2.9]. ∎
Canonical extensions of maximal monotone operators.
In this subsection we present briefly some facts about measurable multi-valued mappings. We assume that , and hence , is separable and denote the set of maximal monotone operators from to by . Further, let be a finite complete measurable space. The notion of measurability for maximal monotone mappings can be defined in terms of the measurability for appropriate single-valued mappings.
A function is measurable iff for every , is measurable
Given a mapping , one can define a monotone graph from to , where , as follows:
Let , the canonical extension of from to , where , is defined by:
Monotonicity of defined in Definition 2.10 is obvious, while its maximality follows from the next proposition.
Let be measurable. If , then is maximal monotone.
See [18, Proposition 2.13]. ∎
We have to point out here that the maximality of for almost every does not imply the maximality of as the latter can be empty (): and
For given mappings and their canonical extensions , one can ask whether the pointwise convergence implies the convergence of the graphs of the corresponding canonical extensions . The answer is given by the next theorem.
Let be measurable. Assume
for almost every , as ,
and are maximal monotone,
there exists and such that strongly in as ,
See [18, Proposition 2.16]. ∎
3 The periodic unfolding
The derivation of the homogenized problem for (4) - (9) is based on the periodic unfolding operator method. In 1990, Arbogast, Douglas and Hornung used a so-called dilation operator to study the homogenization of double-porosity periodic medium in  (see [12, 13] for further applications of the method). This idea has been extended and further developed in  for two-scale and multi-scale homogenization under the name of ”unfolding method”. Nowadays there exists an extensive literature concerning the applications and extensions of the unfolding operator method. We recommend an interested reader to have a look into the following survey papers [15, 17] and in the literature cited there. We recall briefly the definition of the unfolding operator due to Cioranescu, Damlamian and Griso ([16, 17]):
Let be an open set and . Let denote the standard basis in . For , denotes a linear combination with such that belongs to , and set
Then, for each , one has
We use the following notations:
The set is the largest union of cells () included in , while is the subset of containing the parts from cells intersecting the boundary .
Let be a reference cell, be a positive number and a map . The unfolding operator is defined by
From Definition 3.1 it easily follows that, for , the operator is linear and continuous from to and that or every in one has
Obviously, if satisfies
The fact, that we can not consider the integration on the righthand side in (16) over the whole domain and have to establish the validity of the unfolding criterion for integrals for a sequence of functions, can cause some difficulty due to the necessity of the careful treatment of the boundary layer in (17). In [24, 25] this problem has been resolved by extending the unfolding operator by the identity:
The unfolding operator in (18) conserves the integral, i.e. every in one has
which implies that it is an isometry between and . In case of a general bounded domain , i.e. when and , both definitions of the unfolding operator (15) and (18) are equivalent for the sequences, which are bounded in . For the sequences, which are unbounded in , the definitions differ (see [25, Section 4]). Since in this work we are dealing only with bounded sequence, we shall not introduce a new notation for the unfolding operator (18) and use the results in , which are proved for bounded sequences in and the unfolding operator defined by (15).
Let belong to .
For any , strongly in ,
Let be a bounded sequence in such that strongly in , then
For every relatively weakly compact sequence in , the corresponding is relatively weakly compact in . Furthermore, if
See [17, Proposition 2.9]. ∎
Next results present some properties of the restriction of the unfolding operator to the space .
Let belong to . Let converge weakly in to . Then
See [17, Corollary 3.2, Corollary 3.3]. ∎
Let belong to . Let converge weakly in to some . Then, up to a subsequence, there exists some such that
See [17, Theorem 3.5]. ∎
The last proposition can be generalized to -spaces with .
Let belong to and . Let converge weakly in to some . Then, up to a subsequence, there exists some