We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the “stiff” material, and a “soft” material that fills the remaining pores. We assume that the pores are of size and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate , , so that the contrast between the two materials becomes infinite as . We study the homogenisation limit in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements.
We consider a geometrically nonlinear elastic composite material that consists of a “stiff” matrix material and periodically distributed pores filled by a “soft” material: for and a fixed scaling parameter we consider the energy functional of non-linear elasticity
Summary and discussion of our result. To illustrate our result, here in the introduction we restrict ourselves to the special case . If we assume that the density of the body forces is small in magnitude, in the sense that for some , and has vanishing first moment, i.e. , then (1) can be expressed as
Let denote a metric space. A sequence of functionals Γ-converges to a functional , if
(a) (lower bound). For every and every in X we have .
(b) (recovery sequence). For every there exists a sequence in X such that .
A fundamental property of Γ-convergence is the following fact: If a sequence of functionals Γ-converges and the functionals are equicoercive, then the associated sequence of minima (resp. minimisers) converge (up to a subsequence) to the minimum (resp. a minimiser) of the Γ-limit, and any minimiser of the Γ-limit can be obtained as a limit of a minimizing sequence of the original functionals. Thanks to this property, Γ-convergence is especially useful for the study of the asymptotics of parametrised minimisation problems. The Γ-limit, if it exists, is unique; yet, the question whether a sequence of functionals Γ-converges or not, and the form of the Γ-limit depend on the topology of X. In particular, a sequence of functionals is more likely to Γ-converge in a stronger topology, while it is more likely to be equicoercive in a weaker topology. Therefore, it is natural to consider the strongest notion of convergence on X for which the functionals remain equicoercive. In the situation we are interested in, namely the asymptotics of the functionals defined on , it turns out that due to the presence of high-contrast, a natural and appropriate notion of convergence on is a variant of two-scale convergence that we discuss next. Let us first recall the standard notion of two-scale convergence from  and :
We say that a sequence weakly two-scale converges to if the sequence is bounded and
In the study of the asymptotics of we work with a variant of this definition that is tailor-made to capture the effects of high-contrast. For it can be summarised as follows (for details and the general case see Section 3.1): Given a sequence of displacements in we consider the unique decomposition into a function and a contribution that is harmonic in . We then write and say that converges to a pair with and , if weakly in , and , weakly two-scale. The component of a limit pair describes the (scaled) macroscopic displacement of the body, and is a two-scale function describing the (scaled) microscopic displacement on the pores relative to the deformed matrix material. As a main result (see Theorem 1 and Theorem 3) we prove (in fact in a slightly more general situation) that Γ-converges with respect to the type of convergence introduced above. It turns out that two different regimes emerge for and .
In the small strain regime (), the strain becomes infinitesimally small in the entire domain Ω, and the limit behaviour is expressed by a linearised, two-scale energy ,
(a) (lower bound). For every and every we have .
(b) (recovery sequence). For every there exists a sequence such that .
In the finite strain regime, which corresponds to , the displacement gradient becomes infinitesimally small only in the stiff component, while large strains still may occur in the soft pores. Therefore, the Γ-limit is a non-convex (partially linearised) functional of the form
Connection to acoustic wave propagation in high-contrast materials. The sequence of functionals , in either of the two regimes described above, occupies an intermediate position between a fully nonlinearly elastic composite and fully linearised models, as . Notably, linear models with high contrast, which are suitable for the description of small displacement fields (that often occur, say, in acoustic wave propagation) already exhibit a coupling between the macroscopic part and microscopic part of the minimiser of , which in our case is obtained as a limit in the small-strain regime . This can be seen by considering the time-harmonic solitons to the equations of elastodynamics with the elastic part of the energy given by (4), away from the sources of the elastic motion. In this case the function f in (5) (which in our analysis we assume to be independent of the fast variable for simplicity, an assumption that can be relaxed with no changes in the proofs needed) has to be replaced by the sum , with the integration in (5) carried over and Q at the same time, i.e. the work of the external forces (5) is replaced by the expression for the work of “self-forces”
Methods and previous results. In this paper we appeal to analytic methods that have been developed in the last two decades in the areas of nonlinear elasticity and homogenisation. Among these are the notion of two-scale convergence introduced in [1,32] and periodic unfolding (see  and references therein). The convergence statements of our main results are expressed in the language of Γ-convergence (see  and references therein). In order to treat the geometric nonlinearity of the considered functional, we make use of the geometric rigidity estimate (see ). Since we consider a low energy regime, linearisation and homogenisation take place at the same time. The simultaneous treatment of both effects is inspired by recent works [24,28–31] of the third author, where various problems involving simultaneous homogenisation, linearisation and dimension reduction are studied. The homogenisation of the kind of high-contrast composites that we study is related to the homogenisation for periodically perforated domains (e.g. see [8,33]). For instance, we make use of extensions across the pores. As a side result we prove a version of the geometric rigidity estimate for perforated domains (see Lemma 4 below). We would like to remark that while the present work is one of the few papers, along with [9,16], that treat the fully nonlinear high-contrast case, during the last decade there has been a significant amount of literature devoted to the mathematical analysis of phenomena associated with, or modelled by, a high degree of contrast between the properties of the materials constituting a composite, in the linearised setting. The first contributions in this direction are due to Zhikov , and Bouchitté and Felbacq , following an earlier paper by Allaire  and the collection of papers by Hornung  (see also the references therein), where the special role of high-contrast elliptic PDE was pointed out albeit not studied in detail. These works demonstrated that the behaviour of the field variable in such models is of a two-scale type in the homogenisation limit, i.e. the limit model cannot be reduced to a one-scale formulation and fields that depend on the fast variable remain in the effective model. They also noticed that the spectrum of such materials has a band-gap structure, as in (9), and indicated how this fact could be exploited for high-resolution imaging and cloaking. It has since been an adopted approach to the theoretical construction of “negative refraction” media, or more generally “metamaterials”, which is now a hugely popular area of research in physics (see e.g.  and references therein). On the analytical side, a number of further works followed, in particular [3,4,7,10,11,13–15,17,26,35], where various consequences of high contrast (or, mathematically speaking, the property of non-uniform ellipticity) in the underlying equations have been explored. Among these are the “non-locality” and “micro-torsion” effects in materials with high-contrast inclusions in the shape of fibres extending in one or more directions, the “partial band-gap” wave propagation due to the high degree of anisotropy of one of the constituent media, and the localisation of energy in high-contrast media with a defect (“photonic crystal fibres”), all of which can be thought of as examples of “non-standard”, or “non-classical”, behaviour in composites, which is not available in the usual moderate-contrast materials. In the present paper we aim to develop further a rigorous high-contrast theory in the context of finite elasticity, where the underlying model is nonlinear.
With this paper we continue the multiscale theme initiated in , where the regime of large deformation gradients in the soft component of the composite was considered. Let us emphasise two points that contrast our contribution to some earlier work within the related field. First, we note that, apart from [9,16], a number of other articles (e.g. [5,7,12]) have treated high-contrast periodic composites in the nonlinear context. However, the related results are of limited relevance to nonlinear elasticity, due to the convexity or monotonicity assumptions made in these works. In the present paper we study a class of functionals subject to the requirement of material fame indifference (see assumption (W1) in Section 2), which makes our analysis fit the fully nonlinear elasticity framework, as opposed to the works mentioned. Second, as was discussed above, the analysis of composites with “soft” inclusions within a “stiff” matrix cannot be reduced to a “decoupled” model where the perforated medium obtained by removing the inclusions is considered first and the displacement within the inclusions is found independently, which from the physics perspective can be viewed as a kind of resonance phenomenon; cf. (9) in the linearisation regime, for which an inherent energy coupling, in the limit as , between the soft and stiff components of the composite is essential. On a related note, the proof of the key compactness statement (Lemma 1) involves the simultaneous analysis of the displacements on the two components. We would also like to highlight the fact that in  the order of the relative scaling of the displacements on the soft and stiff components of the composite are assumed from the outset, while in the present work it is the result of the above compactness argument itself.
Organisation of the paper. In Section 2 we state the assumptions on the geometry of the composite and the material law. In Section 3 we present the main results, starting with results regarding two-scale compactness, convergence results in the small strain regime and finally the convergence result in the finite strain regime. All proofs are presented in Section 4.
Here we list some notation that we use throughout the text. Additional items will be introduced whenever they are first used in the text.
is the (integer) dimension of the space occupied by the material.
is the exponent in the notation for a Lebesgue space.
the reference period cell; is an open Lipschitz set whose closure is contained in Y, and .
Ω, and denote the reference domains of the composite, the set occupied by the pore material, and the domain occupied by the matrix material, respectively, see Section 2 for precise definition.
Unless stated otherwise, all function spaces , , , etc. consist of functions taking values in .
Function spaces whose notation contains subscript “c” consist of functions that vanish outside a compact set.
The function spaces , , and are introduced in Section 3.1.
We write · and : for the canonical inner products in and , respectively.
denotes the set of rotations in .
≲ stands for ⩽ up to a multiplicative constant that only depends on d, , Ω, and on p if applicable.
2.Geometric and constitutive setup
The pore geometry. The set defined above describes the “pores” contained within the cell Y. Note that is an open, bounded, connected set with Lipschitz boundary. Therefore, to each we can associate (see e.g. ) a unique harmonic extension characterised by
For a given domain and , we define the sets and as follows:
The composite. The two materials are described by energy densities , . Unless stated otherwise, we assume that for :
(W1) is frame-indifferent, i.e. for all and all ;
(W2) The identity matrix is a “natural state”, i.e. , and is non-degenerate:
(W3) has a quadratic expansion at I, i.e. there exists a non-negative quadratic form on and an increasing function with , such that
The scaling parameter γ. Throughout the paper denotes a fixed scaling parameter. It is a quantitative measure of the relative contrast between the two components of the composite.
Energy functional. We define the elastic energy as a functional of the displacement, as follows:
3.1.Compactness and two-scale convergence
We first present an a priori estimate and a two-scale compactness statement for sequences whose energy is equi-bounded in the sense that
We write a representation for in the spirit of an asymptotic decomposition as .
We study the convergence properties of the terms in this decomposition by appealing to two-scale convergence.
In the following lemma we address the first item above.
Let and .
(a) There exists a unique pair of functions and such that
(b) There exists a positive constant C that only depends on such that
As already explained in the introduction, for our purpose it is convenient to appeal to two-scale convergence, see Definition 2. We use the following shorthand notation:
is the space of -periodic functions in .
is the closed subspace of consisting of functions with on .
is the closed subspace of consisting of functions that satisfy the identity
Consider a sequence and let be associated with via (16). Suppose that there exists a sequence of positive numbers such that
Let , and . Let be an arbitrary sequence of positive numbers converging to zero. Then there exist function sequences , such that is related to as in (16), and
Our main result is formulated in terms of the notion of convergence described in the above lemmas. For convenience we use the following notation:
Given we write , if , , and both functions are related to as in (16).
We write , if and
We write , if and
3.2.Convergence in the small strain regime
Throughout this section we assume that the densities and satisfy the conditions (W1)–(W3). We show that in the small strain regime the limit functional
Let be a sequence of positive numbers and assume that as .
(a) (Compactness). Suppose that satisfy
(b) (Lower bound). Consider and suppose that for some and . Then the estimate
(c) (Recovery sequence). For all and there exists a sequence such that and
In the next result we consider a minimisation problem that involves the density of the “body forces” . We study the variational limit of the (scaled) total energy
(a) (Convergence of infima). One has
(b) (Convergence of minimisers). Let be a sequence of almost minimisers, i.e.
Next, we prove that almost minimisers satisfy the asymptotic relation
In addition to the properties of assumed in Section 2, we require the following assumption on the regularity of :
There exist an exponent and a constant C such that for all and related via (10) we have .
Note that Assumption 1 is satisfied if can be written as the disjoint union of a finite number of Lipschitz domains with for .
To illustrate the result of Theorem 2, consider the case with , where is smooth both in x and y and is periodic in y. If the domain Ω and the pore set are sufficiently regular, the minimisers and are smooth, by the classical elliptic regularity theory, see e.g. . In that case we may set
In the remainder of this section, we restrict to the special case of the introduction (in the small strain regime), i.e. we assume that , , and for some , so that the functionals and defined in (2) and (24) are identical. Hence, Theorem 1 and Proposition 1 prove two-scale Γ-convergence of and the convergence of the associated minimisation problems (as claimed in the Introduction), and Theorem 2 yields the two-scale expansion (6). We argue that the functionals Γ-converge to the (single scale) limit :
(a) (Compactness). Suppose that satisfy
(b) (Lower bound). For every with weakly in we have
(c) (Upper bound). For every we can find weakly in such that
(d) (Convergence of the minimisation problem). Let denote an infimizing sequence of , i.e.
3.3.Convergence in the finite strain regime
Throughout this section we assume that
satisfies the conditions (W1)–(W3).
is continuous and satisfies the growth condition
(a) (Lower bound). Consider a sequence and the associated decomposition . If and , then
(b) (Recovery sequence). For any and there exists a sequence such that and
Remark 2(Example in Section 1).
If we consider Theorem 3 and Proposition 1 in the case , and for some , then we recover the special case (in the finite strain regime) presented in the introduction. In particular, we deduce that the functionals two-scale Γ-converge (in the sense of Theorem 3) to . Arguing as in the small-strain regime, cf. Proposition 2, we deduce that Γ-converges (with respect to the weak topology in ) to , cf. (8). We leave the details to the readers.
Theorem 4(Geometric rigidity estimate, see ).
Let U be an open, bounded Lipschitz domain in , . There exists a constant with the following property: for each there is a rotation such that
In fact, we need the following modified version, which is adapted to perforated domains.
There exists a constant that only depends on Ω and such that for all and satisfying
Proof of Lemma 4.
Step 1. The proof of the inequality (36).
Let denote the union of ε-cells that are completely contained in Ω. Since , it suffices to prove (36) for Ω replaced by , respectively. In fact we shall prove the following stronger estimate: for all with we have
Step 2. The proof of the rigidity estimate (37).
Proof of Lemma 1.
In the following, the symbol ≲ stands for ⩽ up to a multiplicative constant that only depends on and Ω.
Step 1. Existence of the decomposition (16) and derivation of the estimate for .
Let denote the unique function in characterised by in and (16)(ii). Since is Lipschitz, we deduce that . This proves the existence of the decomposition. We claim that
Step 2. Derivation of the estimate for .
Since we have an improved Poincaré inequality (see e.g. [25, Lemma 1.6]):
Proof of Lemma 2.
Step 1. A priori estimate and basic compactness.
From Lemma 1 we deduce that
Step 2. The proof of the inclusion .
By a density argument, it suffices to show that
Step 3. The proof of the inclusion .
By a density argument, it suffices to show that
For any , there exists a mapping such that
Proof of Lemma 3.
Step 1. Characterisation of strong two-scale convergence via unfolding.
For and define
Step 2. Construction of .
We claim that there exists a sequence in whose elements satisfy (16)(ii) and
Step 3. Conclusion.
As can be shown by appealing to a combination of a density argument and a diagonal-sequence argument, similar to Step 1, there exists a sequence such that
As a preliminary remark, we note that two different effects play a role when passing to the limit in the small strain regime:
The non-convex energy functional is linearised at identity map (which is a stress-free state for ) – this corresponds to the passage from nonlinear to linearised elasticity.
The obtained linearised, still oscillating, convex-quadratic energy is homogenised.
Consider sequences that satisfy
(a) If and as , then
(b) If , as , and
In Lemma 6, following , the function is introduced, in order to truncate the peaks of . This is needed for exploiting the quadratic expansion (W3). Since both and are assumed to be bounded sequences in , we deduce, from the definition of , the fact that and the Chebyshev inequality, that
Let be sequences in and assume that satisfies (52), then the following implications are valid:
Suppose that be a sequence in and, as above, let denote the set indicator function of . Then the following implications hold:
Proof of Lemma 6.
Step 1. Linearisation.
We claim that the following statement holds for : Let denote a sequence in , and let be a sequence of positive numbers converging to zero, such that
Step 2. Proof of part (a).
Since the energy densities , are minimised at the identity, cf. (W2), we have
Step 3. Proof of part (b).
We claim that
We are now in a position to prove the Γ-convergence statement for the energies .
Proof of Theorem 1.
Step 1. Part (a) (Compactness).
Thanks to (W2) we have
Step 2. Part (b) (Lower bound).
Without loss of generality we assume that
Step 3. Part (c) (Upper bound).
Choose such that
Proof of Proposition 1.
Step 1. A priori estimate.
We claim that for every sequence in the following implication holds:
Step 2. The proof of parts (a) and (b).
The existence of a minimiser to follows by the direct method. The minimiser is unique, since the implication
The remaining claims of Proposition 1 follow from the standard Γ-convergence arguments (cf. [20, Corollary 7.20]), provided the functionals , , are equi-coercive and Γ-converge to . Indeed, thanks to (25c), it is easy to check that implies
For the proof of Theorem 2 we make use of the following lemma:
Let and let be a sequence with in . Then there exists a sequence such that the following properties hold for a subsequence of (not relabelled):
(a) in ;
(b) in a neighbourhood of ;
(c) is equi-integrable;
(d) as .
Proof of Theorem 2.
It suffices to prove the theorem for a subsequence. Throughout the proof we write
Step 1. Convergence of and of the corresponding energy values.
We claim that, as , one has
Step 2. Equi-integrable decomposition.
We claim that for a subsequence (not relabelled) there exist sequences such that satisfies
and are equi-integrable,
the indicator function defined by
Step 3. Error estimate.
We claim that
The difference of the two quadratic terms on the right-hand side converges to zero, since is associated with a recovery sequence, and thanks to (66). On the other hand, since strongly two-scale converges, and by (65), we deduce that
Step 4. Conclusion (Proof of (31)).
We split the estimate into
Thanks to the definition of , the argument for (73) can be reduced to the following statement: For all we have
Proof of Proposition 2.
Step 1. Proof of (a).
Arguing as in Step 1 in the proof of Proposition 1 we find that , and thus the compactness part of Theorem 1 (and the fact that two-scale convergence implies weak convergence) yields (for a subsequence, which we do not relabel):
Step 2. Proof of (b).
We may restrict to the case
Step 3. Proof of (c).
Let . It suffices to argue that there exist and with and
Step 4. Proof of (d).
By Proposition 1 we have and thus . By definition of we have
4.3.Proof of Theorem 3: Finite strain regime
We define, for , , and , the following functionals:
Suppose that . There exists a constant such that for all and , we have
(a) Consider a sequence . If and , then
(b) For all there exists a sequence such that , , and
For the stiff part one can prove (similar to Lemma 6) the following lemma:
(a) Consider . If weakly in , then
(b) For all there exists a sequence such that
We proceed to the proof of Theorem 3.
Proof of Theorem 3.
Step 1. The proof of parts (a) and (b).
Step 2. Proof of (33): convergence of the minima.
For brevity set