Abstract
We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary Γ-convergence.
1 Introduction
The main goal of this paper is to complete the study of limiting models stemming from the interplay of homogenization and dimension reduction in perfect plasticity which we have initiated in [7], as well as to show how the stress-strain approach introduced in [26] for the homogenization of elasto-perfect plasticity can be used to identify effective theories for composite plates. In our previous contribution, we considered a composite thin plate whose thickness h and microstructure-width
In this work, instead, we analyze the two limiting regimes corresponding to the settings
To the authors’ knowledge, apart from [7] there has been no other study of simultaneous homogenization and dimension reduction for inelastic materials. In the purview of elasticity, we single out the works [8, 14] (see also the book [44]) where first results were obtained in the case of linearized elasticity and under isotropy or additional material symmetry assumptions, as well as [5] for the study of the general case without further constitutive restrictions and for an extension to some nonlinear models. A Γ-convergence analysis in the nonlinear case has been provided in [10, 43, 36, 4, 48], whereas the case of high-contrast elastic plates is the subject of [6].
We briefly review below the literature on dimension reduction in plasticity and that on the study of composite elastoplastic materials. Reduced models for homogeneous perfectly plastic plates have been characterized in [21, 22, 40, 31] in the quasistatic and dynamic settings, respectively, whereas the case of shallow shells is the focus of [41]. In the presence of hardening, an analogous study has been undertaken in [38, 39]. Further results in finite plasticity are the subject of [15, 16].
Homogenization of the elastoplastic equations in the small strain regime has been studied in [45, 35, 34]. We also refer to [28, 30] for a study of the Fleck and Willis model, and to [33] for the case of gradient plasticity. Static and partial evolutionary results for large-strain stratified composites in crystal plasticity have been obtained in [12, 11, 17, 20], whereas static results in finite plasticity are the subject of [18, 19]. Inhomogeneous perfectly plastic materials have been fully characterized in [27], an associated study of periodic homogenization is the focus of [26].
The main result of the paper, Theorem 6.2, is rooted in the theory of evolutionary Γ-convergence (see [42]) and consists in showing that rescaled three-dimensional quasistatic evolutions associated to the original composite plates converge, as the thickness and periodicity simultaneously go to zero, to the quasistatic evolution corresponding to suitable reduced effective elastic energies (identified by static Γ-convergence) and dissipation potentials, cf. Section 5.4. As one might expect, for
Essential ingredients to identify the limiting models are to establish a characterization of two-scale limits of rescaled linearized strains, as well as to prove variants of the principle of maximal work in each of the two regimes. These are the content of Theorem 4.14, as well as Theorem 5.31 for the case
We finally mention that, for the regimes analyzed in this contribution, we obtain more restrictive results than in [7], for an additional assumption on the ordering of the phases on the interface, cf. Section 3.1 needs to be imposed in order to ensure lower semicontinuity of the dissipation potential, cf. Remark 3.3.
We briefly outline the structure of the paper.
In Section 2 we introduce our notation and recall some preliminary results. Section 3 is devoted to the mathematical formulation of the problem, whereas Section 4 tackles compactness properties of sequences with equibounded energy and dissipation. In Section 5 we characterize the limiting model, we introduce the set of limiting deformations and stresses, and we discuss the duality between stress and strain. Eventually, in Section 6 we prove the main result of the paper, i.e., Theorem 6.2, where we show convergence of the quasistatic evolution of
2 Notation and preliminary results
Points
for
Given two vectors
In analogy with the notation used for points
In the same way, for every
The natural embedding of
We will adopt standard notation for the Lebesgue and Hausdorff measure, as well as for Lebesgue and Sobolev spaces, and for spaces of continuously differentiable functions.
Given a set
Let E be an Euclidean space.
We will distinguish between the spaces
With a slight abuse of notation,
We will frequently make use of the standard mollifier
where the constant
Throughout the text, the letter C stands for generic positive constants whose value may vary from line to line.
A collection of all preliminary results which will be used throughout the paper can be found in [7, Section 2]. For an overview on basic notions in measure theory, functions of bounded variation (BV), as well as functions of bounded deformation (BD) and bounded Hessian (BH), we refer the reader to, e.g., [25, 1, 2], to the monograph [46], as well as to [23].
2.1 Convex functions of measures
Let U be an open set of
where r and R are two constants, with
Using the theory of convex functions of measures (see [32] and [24]) it is possible to define a nonnegative Radon measure
for every Borel set
and being lower semicontinuous on
Let
Analogously, the
From (2.2) it follows that
2.2 Generalized products
Let S and T be measurable spaces and let μ be a measure on S.
Given a measurable function
In particular, for any measurable function
Note that in the previous formula
Let
Given a η-measurable function
for every bounded Borel function
2.3 Traces of stress tensors
In this last subsection we collect some properties of classes of maps which will include our elastoplastic stress tensors.
We suppose here that U is an open bounded set of class
for every
whenever
We will consider the normal and tangential parts of
Since
More generally, if U has Lipschitz boundary and is such that there exists a compact set
3 Setting of the problem
We describe here our modeling assumptions and recall a few associated instrumental results.
Unless otherwise stated,
to be the reference configuration of a linearly elastic and perfectly plastic plate.
We consider a nonzero Dirichlet boundary condition on the whole lateral surface, i.e., the Dirichlet boundary of
We work under the assumption that the body is only submitted to a hard device on
3.1 Phase decomposition
We recall here some basic notation and assumptions from [26].
Recall that
and to every function
With a slight abuse of notation we will also write
The torus
We will write
where
Correspondingly, ω is composed of finitely many phases
We are interested in the situation when the period ε is a function of the thickness h, i.e.,
exists in
We say that a multi-phase torus
Remark 3.1.
Notice that under the above assumptions,
Remark 3.2.
We point out that we assume greater regularity than that in [26], where the interface
The set of admissible stresses.
We assume that there exist convex compact sets
Finally, we define
We will require an ordering between the phases at the interface.
Namely, we assume that
at the point
We will call this requirement the assumption on the ordering of the phases.
Remark 3.3.
The restrictive assumption on the ordering between the phases will allow us to use Reshetnyak’s lower semicontinuity theorem to obtain lower semicontinuity of the dissipation functional, cf. the proof of Theorem 6.2. Notice that in the regime
The elasticity tensor.
For every i, let
Let
where
Let
It follows that Q satisfies
The dissipation potential.
For each i, let
It follows that
The dissipation potential
For every
For a point
For all other points we take
Remark 3.4.
We point out that H is a Borel, lower semicontinuous function on
Admissible triples and energy.
On
where
The function v represents the displacement of the plate, while f and q are called the elastic and plastic strain, respectively.
For every admissible triple
The first term represents the elastic energy, while the second term accounts for plastic dissipation.
3.2 The rescaled problem
As usual in dimension reduction problems, it is convenient to perform a change of variables in such a way to rewrite the system on a fixed domain independent of h.
To this purpose,
we consider the open interval
We consider the change of variables
and the linear operator
To any triple
Here the measure
According to this change of variable we have
where
and
We also introduce the scaled Dirichlet boundary datum
By the definition of the class
We are thus led to introduce the class
for every
Notice that if
In the following, with a slight abuse of notation, we will still denote this extension by
Kirchhoff–Love admissible triples and limit energy.
We consider the set of Kirchhoff–Love displacements, defined as
We note that
In particular, if
If, in addition,
For every
Note that the space
is canonically isomorphic to
To provide a useful characterization of admissible triplets in
Definition 3.5.
For
for a.e.
for a.e.
The coefficient in the definition of
Analogously, we have the following definition of zero-th and first order moments of measures.
Definition 3.6.
For
for every
where
We are now ready to recall the following characterization of
Proposition 3.7.
Let
3.3 The reduced problem
For a fixed
where for every
We observe that for every
This implies, in particular, for every
Let
By the properties of Q, we have that
We also define the tensor
We remark that by (3.20) it holds
and
The reduced dissipation potential.
The set
where
The plastic dissipation potential
It follows that
Therefore
Finally, for a fixed
i.e.,
3.4 Definition of quasistatic evolutions
For every
Definition 3.8.
Let
from
for every
for every
the function
The following existence result of a quasistatic evolution for a general multi-phase material can be found in [27, Theorem 2.7].
Theorem 3.9.
Assume (3.2), (3.3), and (3.5).
Let
Our goal is to study the asymptotics of the quasistatic evolution when h goes to zero. The main result is given by Theorem 6.2.
3.5 Two-scale convergence adapted to dimension reduction
We briefly recall some results and definitions from [26].
Definition 3.10.
Let
if for every
The convergence above is called two-scale weak* convergence.
Remark 3.11.
Notice that the family
for every
We collect some basic properties of two-scale convergence in the proposition below (the first one is a direct consequence of Remark 3.11 and the second one follows from the definition). Before stating the proposition recall (3.1).
Proposition 3.12.
The following statements hold:
Any sequence that is bounded in
Let
4 Compactness results
In this section, we provide a characterization of two-scale limits of symmetrized scaled gradients.
We will consider sequences of deformations
The boundedness of
We first recall a compactness result for sequences of non-oscillating fields (see [21]).
Proposition 4.1.
Let
Then, there exist functions
Now we turn to identifying the two-scale limits of the sequence
4.1 Corrector properties and duality results
In order to define and analyze the space of measures which arise as two-scale limits of scaled symmetrized gradients of
Let V and W be finite-dimensional Euclidean spaces of dimensions N and M, respectively. We will consider kth order linear homogeneous partial differential operators with constant coefficients
More precisely, the operator
where the coefficients
We define the space
of functions with bounded
Here, the distributional
where
The total
Let
In order to characterize the two-scale weak* limit of scaled symmetrized gradients, we will generally consider two domains
We will assume that
Assumption 1.
If
Furthermore, there exists a countable collection
The following theorem is our main disintegration result for measures in
Proposition 4.2.
Let Assumption 1 be satisfied.
Let
and
Moreover, the map
Lastly, we give a necessary and sufficient condition with which we can characterize the
Assumption 2.
For every
Assumption 3.
The following Poincaré–Korn-type inequality holds in
The proof of the following result is given in [7, Proposition 4.3].
Proposition 4.3.
Let Assumptions 1, 2 and 3 be satisfied.
Let
For every
There exists
Next we will apply these results to obtain auxiliary claims which we will use to characterize two-scale limits of scaled symmetrized gradients.
4.1.1 Case
γ
=
0
We consider
Remark 4.4.
We note that
Analogously, let
Remark 4.5.
It is known that Assumption 1 and Assumption 2 are satisfied in
where
However, since integrating first derivatives of periodic functions over the periodicity cell provides a zero contribution, we precisely obtain the desired Poincaré–Korn-type inequality.
As a consequence of Proposition 4.2 and Proposition 4.3, we infer the following results.
Proposition 4.6.
Let
and
Moreover, the maps
Proposition 4.7.
Let
For every
There exists
Proposition 4.8.
Let
For every
There exists
4.1.2 Case
γ
=
+
∞
In this scaling regime, we consider
Further, we choose
Clearly Assumption 1, Assumption 2 and Assumption 3 are satisfied in
Proposition 4.9.
Let
and
Moreover, the maps
Proposition 4.10.
Let
For every
There exists
Proposition 4.11.
Let
For every
There exists
4.2 Additional auxiliary results
4.2.1 Case
γ
=
0
In order to simplify the proof of the structure result for the two-scale limits of symmetrized scaled gradients, we will use the following lemma.
Lemma 4.12.
Let
for some
for every
for some
there exists an open set
Then, there exists
Proof.
Every
for every Borel set
Let ν be the measure such that
We first observe that, from assumption (i) and (iii), it follows that
Let
then we have
Suppose now that
By a density argument, we infer that
for every
Since
4.2.2 Case
γ
=
+
∞
The following result will be used in the proof of the structure result for the two-scale limits of symmetrized scaled gradients. We note, however, that this result is independent of the limit value γ.
Proposition 4.13.
Let
for some
Proof.
The proof follows closely that of [26, Proposition 4.10].
By compactness, the exists
Since
Consider
By a density argument, we infer that
for every
This yields the claim. ∎
4.3 Two-scale limits of scaled symmetrized gradients
We are now ready to prove the main result of this section.
Theorem 4.14.
Let
Then there exist
and a (not relabeled) subsequence of
If
If
Proof.
Owing to [46, Chapter II, Remark 3.3], we can assume without loss of generality that the maps
In the following, we will consider
Step 1: We consider the case
for a.e.
Set
We have that
where we choose
Then
since
From the above constructions, we infer
For the
for every
Similarly, we compute
Thus, considering an open set
Since
From Proposition 4.1 there holds
thus we infer that
Further, multiplying (4.8) with
Using similar calculations as in (4.3) and (4.3), we obtain that only the first term is not negligible in the limit, from which we conclude that, for any
Consider now
where in the last equality we used Green’s theorem. Passing to the limit, by (4.14) and (4.15), we have
From (4.12), (4.13), (4.3) and Lemma 4.12, we conclude that
where
for a suitable
Step 2: Consider the case
Let
We consider a test function
such that
For each
Then, observing that
we find
Recalling (4.4) and (4.5), we deduce
Thus, recalling that
and since for arbitrary test function we can subtract their mean value over
Similarly, from the observation that
we deduce
Suppose now that
Then we have
Furthermore,
From (4.3), (4.3) and (4.3), and Proposition 4.11, and recalling that
This concludes the proof of the theorem. ∎
5 Two-scale statics and duality
In this section we define a notion of stress-strain duality and analyze the two-scale behavior of our functionals.
The main goal is to prove the principle of maximum plastic work in Section 5.4, which we will use in Section 6 to prove the global stability of the limiting model.
In Section 5.1 we characterize the duality between stress and strain on the torus
5.1 Stress-plastic strain duality on the cell
5.1.1 Case
γ
=
0
Definition 5.1.
The set
where
Recalling (3.21), by conditions i and ii we may identify
Definition 5.2.
The family
such that
Recalling the definitions of zero-th and first order moments of functions and measures (see Definition 3.5 and Definition 3.6), we introduce the following analogue of the duality between moments of stresses and plastic strains.
Definition 5.3.
Let
for every
Remark 5.4.
Note that the second integral in (5.2) is well defined since
From this it follows that for
Through the approximation by convolution (5.4) then extends to arbitrary continuous
Remark 5.5.
If α is a simple
where
From (2.4) it follows that if U is an open set in
Remark 5.6.
It can be shown that if
where
From [23, Théoreme 2 and Appendice, Théoreme 1] it follows that if U is an open set in
We are now in a position to introduce a duality pairing between admissible stresses and plastic strains.
Definition 5.7.
Let
so that
for every
Remark 5.8.
Notice that
The following proposition will be used in Section 5.4 to prove the main result of this section.
Proposition 5.9.
Let
Proof.
The proof is divided into two steps.
Step 1. In this step we consider a phase
We also have that for every
where
As a consequence, for
for every
for n large enough.
Since
Passing to the limit, we have
This proves (5.9) on every phase.
Step 2.
In this step we consider a curve α that is of class
and
where
Since, for each i,
where
From these and by using Remark 5.4, Remark 5.5 and (5.12) we conclude for every k,
By summing over k we infer (5.9) on α.
The final claim goes by combining Step 1 and Step 2 and using the fact that both measures in (5.9) are zero on
5.1.2 Case
γ
=
+
∞
We first define the set of admissible stresses and configurations on the torus.
Definition 5.10.
The set
Notice that in (i) we neglect the third column of Σ.
Definition 5.11.
The family
such that
We also define a notion of stress-strain duality on the torus.
Definition 5.12.
Let
for every
Remark 5.13.
Note that the integrals in (5.15) are well defined since
The following proposition provides an estimate on the total variation of
Proposition 5.14.
Let
Proof.
Using a convolution argument we construct a sequence
According to the integration by parts formulas for
From these two equalities, together with the above convergence and the expression in (5.15), we compute
In view of the
from which the claims follow. ∎
The following proposition characterizes
Proposition 5.15.
Let
Furthermore, if
where
Proof.
To prove (5.17), let
Note that
Since
we compute using (5.14)
Owing to the assumption on
Now
and
By defining
the
and we infer from (5.21) that
for a suitable
and
Since (5.22) implies
The following proposition is analogous to Proposition 5.9 and will also be used in Section 5.4 to prove the main result of this section.
Proposition 5.16.
Let
Proof.
To establish the stated inequality, we consider the behavior of the measures on each phase
Furthermore,
Thus for every
Since
Passing to the limit we have
The inequality on the phase
where
5.2 Disintegration of admissible configurations
Let
5.2.1 Case
γ
=
0
Definition 5.17.
Let
such that
and also such that there exist
The following lemma gives the disintegration result that will be used in the proof of Proposition 5.30.
Lemma 5.18.
Let
Above,
where
Proof.
The proof is a consequence of Proposition 4.6 and follows along the lines of [7, Lemma 5.8]. ∎
Remark 5.19.
From the above disintegration, we have that, for η-a.e.
Thus, the quadruplet
is an element of
5.2.2 Case
γ
=
+
∞
Definition 5.20.
Let
such that
and also such that there exist
The following lemma provides a disintegration result in this regime and will be instrumental for Proposition 5.32.
Lemma 5.21.
Let
Above,
where
Remark 5.22.
From the above disintegration, we have that, for η-a.e.
Thus, the quintuplet
is an element of
5.3 Admissible stress configurations and approximations
For every
We recall some properties of the limiting stress that can be found in [21].
If we consider the weak limit
In the following, we further characterize the sets of two-scale limits of sequences of elastic stresses
5.3.1 Case
γ
=
0
We first introduce the set of limiting two-scale stress.
Definition 5.23.
The set
where
The following proposition motivates the above definition.
Proposition 5.24.
Let
Then
Proof.
Properties (v) and (vi) follow from Section 5.3.
To prove (i), let
converges strongly in
which is sufficient to conclude that
To prove (ii), we define
and consider the set
The construction of
Since compactness of
Finally, to prove (iii) and (iv), let
By a direct computation we infer
Hence, taking such a test function in
Suppose now that
from which we deduce that, for a.e.
Thus,
The following lemma approximates the limiting stresses with respect to the macroscopic variable and will be used in Proposition 5.30. It is proved under the assumption that the domain is star-shaped.
Lemma 5.25.
Let
where
and
Proof.
The approximation is done by dilation and convolution and is analogous to [7, Lemma 5.13]. ∎
5.3.2 Case
γ
=
+
∞
In this regime, the set of limiting two-scale stresses is defined as follows.
Definition 5.26.
The set
where
The previous definition is motivated by the following.
Proposition 5.27.
Let
Then
Proof.
Properties (iii), (iv) and (v) follow in view of Section 5.3.
To prove (i), we consider the test function
converges strongly in
Suppose now that
from which we can deduce that
To conclude the proof, it remains to show the stress constraint
The following lemma is analogous to Lemma 5.25.
Lemma 5.28.
Let
Further, if we set
and
Proof.
The proof is again analogous to [7, Lemma 5.13].
The only difference is that the convolution and dilation used to define
5.4 The principle of maximum plastic work
We introduce the following functionals: Let
and
The aim of this subsection is to prove the following inequality between two-scale dissipation and plastic work, which in turn will be essential to prove the global stability condition of two-scale quasistatic evolutions. It is used in Step 2 of the proof of Theorem 6.2 and
its proof is a direct consequence of Theorem 5.31 for the case
Corollary 5.29.
Let
for every
The proof relies on the approximation argument given in Lemmas 5.25 and 5.28 and on two-scale duality, which can be established only for smooth stresses by disintegration and duality pairings between admissible stresses and plastic strains (given by (5.8) and (5.15)).
The problem is that the measure η defined in Lemmas 5.18 and 5.21 can concentrate on the points where the stress (which is only in
5.4.1 Case
γ
=
0
The following proposition defines the measure λ through two-scale stress-strain duality based on the approximation argument.
Proposition 5.30.
Let
Furthermore, the mass of λ is given by
Proof.
The proof is divided into two steps.
Step 1.
Suppose that ω is star-shaped with respect to one of its points.
Let
where η is given by Lemma 5.18 and the duality
for every
where the last inequality stems from item c in Lemma 5.25. This in turn implies that
from which we conclude that is
Let now
Since
where
where in the last equality we used that
Furthermore, since
Taking into account that
Likewise taking into account that
Let now
By items a and (f) in Lemma 5.25, we have in the limit
Taking
Step 2.
If ω is not star-shaped, then since ω is a bounded
Let
For each i, let
Since
such that
with
for every
and
Then we can see that
The following theorem provides a two-scale Hill’s principle (cf. [26, Theorem 5.12]).
Theorem 5.31.
Let
where
Proof.
Take
By Proposition 5.9, we have for η-a.e.
Since
Consequently,
By passing to the limit, we infer the desired inequality. ∎
5.4.2 Case
γ
=
+
∞
The following proposition is the analogue of Proposition 5.30.
Proposition 5.32.
Let
Furthermore, the mass of λ is given by
Proof.
Suppose that ω is star-shaped with respect to one of its points.
Let
where η is given by Lemma 5.21 and the duality
for every
where the last inequality stems from item c in Lemma 5.28. This in turn implies that
from which we conclude that is
Let now
From this point on, the proof is exactly the same as the proof of Proposition 5.30 by defining in the analogous way
The following theorem is analogous to Theorem 5.31.
Theorem 5.33.
Let
where
Proof.
Let
By Proposition 5.16, we have for η-a.e.
Since
By passing to the limit, we have the desired inequality. ∎
6 Two-scale quasistatic evolutions
The associated
In this section we prescribe for every
We now give the notion of the limiting quasistatic elastoplastic evolution.
Definition 6.1.
A two-scale quasistatic evolution for the boundary datum
for every
for every
the function
for
for
Recalling the definition of a h-quasistatic evolution for the boundary datum
Theorem 6.2.
Let
for
be a h-quasistatic evolution in the sense of Definition 3.8 for the boundary datum w such that
for the boundary datum
in case
in case
Proof.
The proof is divided into several steps, in the spirit of evolutionary Γ-convergence and it follows the lines of [7, Theorem 6.2].
We present the proof in the case
Step 1: Compactness. First, we prove that there exists a constant C, depending only on the initial and boundary data, such that
for every
where the last integral is well defined as
Second, from the latter inequality in (6.10) and (3.5), we infer that
for every
Next, we note that
In view of the assumption
Owing to (2.3), we infer that
for every
for every
for every
Since
Lastly, we consider for every
Then we can choose a (not relabeled) subsequence such that
where
Step 2: Global stability.
From Step 1 we have
Furthermore, since from the first step of the proof
where the last equality is a straightforward computation. From the above, we immediately deduce
hence the global stability of the two-scale quasistatic evolution (qs1)${}^{\rmhom}_{\gamma}$.
We proceed by proving that the limit functions
Identifying
Integrating over
Using the variant of Poincaré–Korn’s inequality as in Step 1, we can infer that
This implies that the whole sequences converge without depending on t, i.e.,
Step 3: Energy balance. In order to prove the energy balance of the two-scale quasistatic evolution (qs2)${}^{\rmhom}_{\gamma}$, it is enough (by arguing as in, e.g., [13, Theorem 4.7] and [27, Theorem 2.7]) to prove the energy inequality
For a fixed
In view of the strong convergence assumed in (6.2) and (6.13), by the Lebesgue’s dominated convergence theorem we infer
Hence, we have
Taking the supremum over all partitions of
Remark 6.3.
The prevailing effect of dimension reduction for the case
The prevailing effect of homogenization in the regime
We emphasize the fact that, to the best of our knowledge, neither a homogenization of the plate model in [21] (the model derived there is for homogeneous material) nor a dimension reduction of the homogenized model obtained in [26] have been studied in the literature.
Funding source: Austrian Science Fund
Award Identifier / Grant number: F65
Award Identifier / Grant number: V 662
Award Identifier / Grant number: Y1292
Award Identifier / Grant number: I 4052
Funding source: OeAD-GmbH
Award Identifier / Grant number: HR08/2020
Funding statement: M. Bužančić and I. Velčić were supported by the Croatian Science Foundation under Grant Agreement no. IP-2018-01-8904 (Homdirestroptcm). The research of E. Davoli was supported by the Austrian Science Fund (FWF) projects F65, V 662, Y1292, and I 4052. All authors are thankful for the support from the OeAD-WTZ project HR 08/2020.
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