Existence, numerical convergence and evolutionary relaxation for a rate-independent phase-transformation model

Abstract

We revisit the model for a two-well phase transformation in a linearly elastic body that was introduced and studied in Mielke et al. (2002 Arch. Ration. Mech. Anal. 162, 137–177). This energetic rate-independent system is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in Mielke et al. (2002), (ii) to establish the convergence of suitable numerical approximations via space–time discretization and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed-mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.

1. Introduction

Microstructures occur in many material models and are important for macroscopic effects such as elastoplasticity or the hysteresis in shape-memory materials. On typical macroscopic and mesoscopic length scales, such materials are usually modelled by a strain tensor and some internal variables such as phase indicators, magnetization, plastic tensor or hardening variables. In most cases, the stored-energy density depends only on the point values of these variables and thus defines a material model without any length scale. Thus, even steady states, which occur as minimizers of the energy, may develop microstructures on arbitrary fine scales. For static problems, a rich theory was developed based on the seminal work [1], which introduced Young measures as an essential tool.

For evolutionary problems, the situation is much less developed, as the temporal behaviour of such microstructures is significantly more difficult. For rate-independent systems, which do not have an intrinsic time scale and hence are sufficiently close to static problems, a major step forward was made using incremental minimization problems, namely for finite-strain elastoplasticity in [2–4], for brittle fracture in [5,6] and for shape-memory materials in [7–9].

All these approaches have in common that they are based on incremental minimization problems for an energetic rate-independent system (ERIS) , where is a (possibly nonlinear) state space, is the energy potential and is the dissipation distance, which measures the minimal energy needed to change the state from q to . Given an initial state , the approximate incremental minimization problem then reads:

1.1

where τ=T/J>0 is the time step. Here, the error level ε=0 is allowed if there exist minimizers of . However, in many cases, one has to take ε>0, since no minimizer exists because of the formation of microstructures. Instead, for every ε>0, there exists a solution .

Using a fixed initial condition q₀, the static theory can be employed to study the microstructure that arises in for ε→0 (e.g. [10]). However, if one wants to study the microstructure in , there will be a strong dependence on the microstructure of , and similarly strongly depends on . This problem gets even more involved if we define the piecewise-constant interpolants via

Then, the major mathematical task in evolutionary relaxation is to establish the convergence of a suitable subsequence for τ_n, ε_n→0 to a limit and to determine an evolution equation for all such limits.

For nonlinear material models without an internal length scale, this programme is largely open. There are particular results for brittle fracture (e.g. [11,12]), in damage modelling [13] and for a very particular plasticity model [14]. This work is a continuation of the two-phase model introduced in [7,8], where (i) we generalize the existence result for the separately relaxed problem postulated there, (ii) we provide a numerical convergence result for space–time discretizations and (iii) finally, we show that the above-mentioned evolutionary relaxation holds true, i.e. that all accumulation points of approximations q_τ,ε are indeed solutions of the separately relaxed model.

All these works lead to so-called energetic solutions (also called quasi-static evolutions in [6,15,12]) for ERIS (see definition 2.1). This notion of solutions is formulated in terms of a global stability condition (S) and an energy balance (E). The former simply means that the solution satisfies

where the (approximate) stability sets for an ERIS are defined via

A crucial step in the existence and Γ-convergence theory for ERIS (see §3b) is the so-called closedness of the stability sets, in the following sense:

where refers to the limit system . We say that the latter is the separately relaxed ERIS for the family if and in a suitable topology on . Yet, in general, one cannot conclude that accumulation points q of energetic solutions for are energetic solutions for the limit system .

In this work, we want to highlight that the method of mutual recovery sequences (MRS) (originally called ‘joint recovery sequences’ in [16]) is an ideal tool for existence and convergence theory for ERIS. This is a general abstract version of the jump-transfer or crack-transfer lemmas used in [11,12,17]. It can be seen as an evolutionary counterpart to the classical limsup condition, or condition on the existence of recovery sequences, for static Γ-convergence. However, here the condition is for a sequence of ERIS , its supposed limiting system , a sequence of states and an arbitrary test state . Throughout this work, we assume that is a weakly or strongly closed subset of a reflexive Banach space Q and use → and to denote strong and weak convergence, respectively.

Definition 1.1 (Mutual recovery sequences (MRS)) —

Given the ERIS for , a sequence with t_k→t_* and in , and , a sequence is called an MRS if and

The importance here is that we have to recover mutual information on the energy increment and the dissipation with the help of one sequence . This is clearly distinct from separate relaxation, where there is no interaction between the two quantities. In particular, this relates to the obvious fact that, for an evolutionary theory, we need a recovery condition that couples properties of the energy storage and the dissipation. Another instance of an explicit coupling occurs in EDP-convergence (EDP = energy-dissipation principle) for generalized gradient systems defined in [18].

To highlight the major advantages of MRS, it is sufficient to look at the case and for , since, even for showing the existence of energetic solutions for one ERIS, the concept of MRS is relevant and non-trivial. The simplest case occurs if is weakly continuous and is weakly lower semi-continuous; then we can always choose the constant MRS , since and . There is a huge literature for non-local material models, where the energy is regularized by gradient terms or some non-local terms, while the dissipation remains local like [19–24]. Indeed, if Q=U×W^s,q(Ω) for some s>0 and q>1, then weak continuity of holds for Caratheodory functions D (because D has at most linear growth by the triangle inequality). However, in this case, the theory of MRS is not really needed.

To see the cancellation effect in the definition of the MRS, we consider a Hilbert space , a quadratic energy and a translation-invariant dissipation distance , which includes the case of classical linearized elasticity. Here, the MRS can be chosen as

Moreover, using the quadratic structure of we find

1.2

Note that and are false in general. Thus, the appropriate choice of leads to a cancellation, and we conclude that is indeed an MRS.

The full strength of the tool of MRS is seen in material modelling without internal length scale. There we are able to adjust the microstructure in suitably to recover the dissipation as well as the energy increment. Indeed, often (including this work) it is possible to find such that

1.3a

and

1.3b

After we recall some of the modelling for N-phase materials in §2, we concentrate on the special two-phase model of Mielke et al. [8], which relies on the relaxed two-well energy derived in [25]. Here, θ:Ω→[0,1] denotes the mesoscopic volume fraction of phase 2, and is the displacement. Thus the states are with

The particular case has the special structure that is quadratic, namely

where is the linearized strain tensor, and is a symmetric linear operator. The dissipation distance has the form

where κ_1→2 and κ_2→1 are positive material constants.

Because of the constraint θ∈[0,1], the quadratic trick in (1.2) cannot be used to construct an MRS. However, it is shown in proposition 3.3 that

1.4

for a suitable depending nonlinearly on θ_* and defines an MRS satisfying (1.3). Indeed, the choice of g gives , and (1.3a) follows by the affine structure in (1.4).

To control the energy difference, we exploit the quadratic structure of the energy and the property that the material model is scale-invariant. As a consequence, the reduced energy

is defined by a symmetric bounded linear operator that is a pseudo-differential operator with non-negative symbol Λ satisfying Λ(rξ)=Λ(ξ) for all r>0 and . Thus, as was already done in [8,26], the H-measure theory can be employed. In particular, if θ_n generates the H-measure μ, then generates the H-measure g²μ, and we find

Using g²≤1 and μ≥0 gives the desired estimate (1.3b), and is an MRS. This provides the major step in the existence of energetic solutions for the two-phase model (theorem 3.1).

In §4, we generalize the theory by approximating the spaces U and by suitable finite-element spaces U_k⊂U_k+1 and . We provide conditions that all accumulation points of the corresponding approximate minimizers are indeed solutions for the limiting ERIS . The MRS is obtained by suitably projecting the sequence defined in (1.4).

The final section (§5) solves the question of evolutionary relaxation. We start from the microscopic pure-phase model where θ is restricted to be either 0 or 1, i.e.

In terms of the above theory, we set on and otherwise. In [26], it was shown that the ‘separately relaxed’ ERIS is a lower relaxation of in the sense of Mielke [27]. This means that each energetic solution of can be approximated by solutions of the approximate incremental minimization problem (1.1), but now using the state space .

Our theorem 5.1 shows that all accumulation points q of approximate solutions q_τ,ε are indeed energetic solutions for the ERIS . Thus, we conclude that the lower relaxation is also an upper relaxation in the sense of Mielke [27]. This reveals that the two-phase model under consideration is very special. In general, one should not expect that the separate relaxation is also an upper or a lower relaxation. This can only happen if the macroscopic information kept in the relaxation (here the phase fraction θ) is enough to characterize all relevant macroscopic quantities. In [8,26], it was shown that simple laminates are sufficient to study the separate and the lower relaxation. Interestingly, our method solves the question of upper relaxation even in cases where there are microstructures that are not laminates.

The difficulty in the construction of MRS lies in the fact that , while the weak limit in general. Similarly, for general test functions , we have to find with . This will be done by constructing hierarchical microstructures based on θ_n and much finer laminates with normal direction ω_* such that Λ(ω_*)=0 (see proposition 5.2).

2. Pure and relaxed N-phase models

We start with general N-phase models and then restrict to the two-phase model as discussed in [8], where also a detailed physical motivation in terms of separate relaxation is given. We also refer to [28,26].

(a). A microscopic model with pure phases

We consider a bounded Lipschitz domain , where Γ_Dir⊂∂Ω with is the part of the boundary on which displacement (Dirichlet) boundary conditions are applied. The displacement will be of the form g_Dir(t)+u(t), where u lies in the fixed space

In the case of N pure phases, we consider N different stored-energy densities

where denotes the linearized elastic strain, is the elastic tensor of the ith phase, A_i is the transformation strain and β_i is the height of the ith well. All these quantities may depend on temperature, but we consider an isothermal setting.

For later purposes, we associate the ith phase with the ith unit vector and call the functions a phase-indicator field, where

For characterizing a simple evolutionary model, we add a dissipation distance , where κ_i→j:=d_N(e_i,e_j) denotes the energy per unit volume that is dissipated when a phase transformation from i to j takes place. The induced dissipation distance on is defined via

The associated ERIS for the pure N-phase model is given via the state space , the dissipation distance from above, and the energy-storage functional

where for i=1,…,N, and ℓ:[0,T]→U* includes possible time-dependent volume or surface loadings. In particular, we assume

2.1

(b). Incremental minimization and energetic solutions

Following the seminal work [2,3], it was suggested in [7,8] to consider incremental minimization problems for ERIS for a given time discretization which we take equidistant for simplicity, i.e. τ=T/J with . For an initial state , we consider approximate minimizers satisfying the following:

2.2

For positive ε such approximate minimizers always exist, and we can define piecewise-constant interpolants via

2.3

For ε=0, one asks for existence of true minimizers, which in the present, non-relaxed case is not to be expected in general.

The major task is now the characterization of all possible limits, i.e. accumulation points, of q_τ,ε for (τ,ε)→(0,0) and to derive a suitable evolutionary model (e.g. in the sense of Mielke [27]) having these limits as solutions. In general, this task is still much too difficult; however, we will see in §5 that it is solvable for the two-phase model (i.e. N=2) with .

The main achievement in [7,8] was the observation of the general fact that all possible limits of the above approximate incremental minimization problem lead to so-called energetic solutions for rate-independent systems.

Definition 2.1 (Energetic solutions) —

A function is called an energetic solution of the ERIS if lies in L¹([0,T]) and if for all t∈[0,T] the stability (S) and the energy balance (E) hold:

2.4

where the dissipation is defined as the supremum over all partitions 0≤t₀<t₁<⋯<t_N−1<t_N≤t and all of the sums .

We will see in §3b how under natural conditions the stability and energy balance arise naturally from the incremental minimization problem. However, in the present pure-phase model, this does not work, since we have to pass to limits in q_τn,εn(t) without any compactness. Thus, we have to work on the weak completion of . This leads to so-called relaxed models.

(c). A separately relaxed N-phase model

Instead of treating the microscopic phase indicators z with z(x)∈P_N, we may consider a mixture theory on the mesoscopic level, where z takes values in the Gibbs simplex

Here, z_i(x)∈[0,1] denotes the volume fractions of the ith phase at a mesoscopic material point x∈Ω. With this, we introduce the relaxed state space

Extending to by outside of , we can define the lower semi-continuous envelope , which is called the (static) relaxation of . It has the form

where the relaxed stored-energy density is given in terms of the cross-quasi-convexification [8], eqn (4.5)

For see also [29,9].

Similarly, by an optimal transport problem based on the weight , one can define a dissipation distance which takes the form for a one-homogeneous function (i.e. Ψ_N(γv)=γΨ_N(v) for all γ≥0 and ) [8], §4.3. This leads to the relaxed dissipation distance defined via

and the so-called separately relaxed ERIS .

So far, the existence of energetic solutions for such relaxed systems is still open. However, we gained already that the incremental problem (2.2) has solutions for ε=0. Indeed, since is convex and continuous, it is weakly lower semi-continuous. Moreover, is weakly lower semi-continuous, because of its construction as a cross-quasi-convexification (cf. [30]). We refer to [31–33] for such static relaxations in the context of material modelling.

However, for passing to the limit of time steps τ→0, it remains open how to show the closedness of the stability sets in the weak topology. Nevertheless, there is some hope that energetic solutions for exist. Unfortunately, we are only able to show that this is true for the case N=2 if (theorem 3.1). We emphasize that showing the existence of an energetic solution for the ERIS is a first step only.

The more important step is to show that accumulation points of approximate solutions of the microscopic pure-state system are indeed solutions of the mesoscopic relaxed model . We expect that this is typically not the case. The point is that the relaxed model only takes into account the mesoscopic volume fractions z_i(t,x)∈[0,1] of the phases i=1,…,N. However, in general situations, it is necessary to take into account the type of the microstructures. For instance, a rotating laminate may have constant volume fraction, but must dissipate microscopically; see the discussions in [13,15,24,34]. It is surprising that we are able to prove the evolutionary relaxation property in the two-phase case with (see §5). The main idea here follows an observation in [26], where it is shown that looking at suitable laminates with a fixed normal is sufficient, even though other microstructures may occur.

3. Existence for the relaxed two-phase model

In this section, we provide the first existence result for the two-phase problem. Moreover, we introduce the general theory of [35,36] for establishing convergence of approximations obtained from incremental minimization procedures. The major step is the proof of the weak closedness of the stability sets, which will be treated afterwards. For this, we employ H-measures, which are well adapted for the treatment of the quadratic energies occurring in the two-phase problem.

(a). Set-up and existence result

For the rest of this work, we restrict to the case of N=2 phases and use the scalar θ∈[0,1] as the volume fraction of phase i=2, i.e.

Moreover, we assume , where is the symmetric and positive definite elasticity tensor. We also write

equipped with the weak topology. The relaxed energy is defined on and reads

3.1

Here, , and A_i is the transformation strain of the ith phase. According to [8,9,29], the constant γ is determined by Kohn's relaxation result [25] for the elastic double-well problem, that is

3.2

where A:=A₂−A₁, and the acoustic tensor is defined via

Using the two positive thresholds κ_1→2 and κ_2→1, the dissipation distance reads

3.3

Thus, the ERIS is specified, and we can define the stability sets

Note that the relaxed energy and dissipation defined above correspond to and for N=2 from §2c.

Theorem 3.1 (Energetic solutions) —

Under the above assumptions the two-phase model has an energetic solution for all initial conditions . Moreover, every accumulation point of the approximations for (τ,ε)→(0,0) obtained from the approximate incremental minimization problem (2.2) is an energetic solution.

We mention that the theory in [37] also shows that all energetic solutions are accumulation points of approximations obtained via a slight variant of (2.2); see also [38], §4.2.

(b). General strategy for the convergence proof

Here, we give the general strategy of constructing energetic solutions that was developed in [8,11]. We follow the six steps as introduced in [39] and [36], §2.1.6; but in the present model many features are much simpler, since we can use the quadratic structure of the energy and the weak sequential compactness of the space . Step 3 will rely on the existence of MRS, which is established in §3d.

Step 0: Construction of approximate solutions. For every time step τ=T/J and any ε≥0 and the given initial value q₀=q(0), the approximate incremental problem (2.2) has solutions , j=1,…,J. For ε>0, this is indeed trivial; while for ε=0, we can use the weak lower semi-continuity of . Thus, the piecewise-constant interpolants are well defined.

Step 1: A priori estimates. Since lies in a bounded ball of radius R=|Ω|^1/2 in Z:=L²(Ω), we always have

Owing to for j=0,…,J, the quadratic structure of together with Korn's inequality show that there is a constant C₁>0 such that

Finally, we may insert into (2.2) and sum over j=1,…,J to find

3.4

independently of ε∈[0,1] and τ=T/J. This estimate does not give any information on u_τ;ε, but with , we find

Step 2: Selection of convergent subsequences. Because of the uniform total variation bound for θ_τ,ε, we can apply the abstract version of Helly's selection principle. Hence, for every sequence with τ_k,ε_k→0 for , there exists a subsequence (τ_kn,ε_kn) with and a function such that

3.5

Define the function u:[0,T]→U to be the unique minimizer of ; then it is easy to show that for all t. Thus, we conclude the convergence along the whole subsequence, namely

Step 3: Stability of the limit. The most difficult step in the proof is to show that the accumulation point is stable in the sense of (S) in definition 2.1, i.e. . For this we first show that is approximately stable for time t=jτ, which follows by the triangle inequality for as follows. Indeed for all , we have

3.6

which also will be abbreviated by .

In order to establish the stability , we want to pass to the limit along the sequence (τ_kn,ε_kn)→(0,0) by choosing suitable test functions in the above estimate. The crucial point is to find an MRS such that and

This step will be discussed explicitly in the three results of the sections ‘Mutual recovery sequences I to III’. Using such that and inserting the MRS into (3.6) yields

where we used and (2.1). This is the desired stability .

Step 4: Upper energy estimate. We return to the dissipation estimate (3.4) in Step 1, which can be written as

Since is affine in q, it is weakly continuous, and using implies the convergence of the last term. Together with the lower semi-continuities and (3.5) we find

which is the desired upper energy estimate.

Step 5: The lower energy estimate

holds for all 0≤s<t≤T generally for all measurable functions that are stable for all r∈[s,t], which was established in Step 3 (see [8], Thm. 2.5 or [35], Prop. 3.11). Combining this with Step 4 provides the energy balance (E) in definition 2.1 for energetic solutions, and the proof of theorem 3.1 is finished, except for the construction of the MRS.

The remaining part in the above proof is the difficult Step 3, where the stability of the accumulation point is established. In [8], §5, this step was done under the restrictive assumption of convexity of . Here, we show that the proof via the construction of MRS is more flexible. Of course, we still need a fine tool from weak convergence theory, namely H-measure or microlocal defect measures; see [40–42] for the more general microlocal compactness forms.

(c). Pseudo-differential operators and H-measures

To understand the set of stable states a little better, we can use the facts that is quadratic, that is uniformly convex (by Korn's inequality and ), and that depends on θ only. Thus,

where the minimizer u_elast(t) of satisfies u_elast(⋅)∈g_Dir+C¹([0,T];U). The linear operator satisfies . Defining we arrive at the quadratic functional

where β∈C¹([0,T];L²(Ω)) and α∈C¹([0,T]). While the energetic shift α is irrelevant, the function β can be seen as a time-dependent driving force that depends linearly on g_Dir(t) and ℓ(t) via u_elast(t).

The important feature here is that the quadratic functional is given in terms of the linear operator , which is a symmetric pseudo-differential operator of order 0, which means that

where denotes the extension by 0 outside of Ω, and is a compact operator in L²(Ω). The more important first part consists of the Fourier transform and the Fourier multiplier Λ, which is also called a symbol. The order 0 of the pseudo-differential operator relates to the homogeneity of Λ, namely Λ(rξ)=r⁰Λ(ξ) for r>0 and ξ≠0. For our two-phase problem, Λ takes the specific form

see [8] and (3.2) for the definition of γ and Σ. Thus, the continuous spectrum of equals {Λ(ω)| , lies in and contains 0, because of the definition of γ. In particular, a possible negative part of must be compact, and is indeed lower semi-continuous.

For pseudo-differential operators, we can use H-measures (cf. [40,41]) to calculate the limits of quadratic functionals under weak convergence in L²(Ω). To formulate our results shortly, we simply write , if and the sequence θ_n−θ_* generates the H-measure μ. The latter means that for all ϕ∈C_c(Ω) and we have

The following results will be central for our construction of MRS.

Proposition 3.2 (H-measures) —

For p>4 assume that in L^p(Ω) and b_n→b_* and w_m→0 in L^p(Ω). Then, we have

3.7a

and

3.7b

Proof. —

Relation (3.7a) is a well-known standard result [40], Cor. 1.2 and 1.12.

The same reference contains result (3.7b) under the stronger assumption b_n=b_* and . Using the a priori bounds , we can extend the result since , and there exists a subsequence such that for . We want to show that .

We approximate b∈L^p(Ω) by with B_δ→b_* in L^p(Ω) and write b_nv_n=z_n+y_n with z_n=B_δv_n and y_n=(b_n−B_δ)v_n. The vector-valued H-measure for the vector (z_n,y_n)^⊤ has components with , where we exploit B_δ∈C_c(Ω). Using b_nv_n=z_n+y_n, we have . Moreover, for the total variations of the measures μ_ij, we have

Using , we obtain the estimate

Thus, we conclude that as desired, and even without taking a subsequence. ▪

More results on H-measures involving fine laminates are given in proposition 5.2, which is proved in §5c.

(d). Mutual recovery sequences I

Fix t∈[0,T] and consider a stable sequence , i.e. with . To show the stability , we have to find an MRS for every test function . This will be done with the help of the function

3.8

Proposition 3.3 (Mutual recovery sequence I) —

Assume that and and that is arbitrary. Then, the sequence with

is a recovery sequence satisfying

3.9

Proof. —

We first discuss the dissipation, which only depends on θ. The construction of g via F is such that

This follows immediately from the explicit representations

Thus, we can calculate the dissipation by using the domains , namely

Note that the weak convergence and the linearity of the integrals over Ω_± allow us to pass to the limit . Thus, the first relation in (3.9) is established.

To establish the second relation, we use that for we have , which is equivalent to . Thus, it suffices to show

where can be arbitrary.

We now use that in L^p(Ω) for all p>1. By the construction of , we also have in L^p(Ω) for all p>1. Choosing a subsequence (not relabelled), we can assume that θ_n and generate H-measures μ and , respectively. Applying proposition 3.2 on H-measures with we obtain from (3.7b) and arrive via (3.7a) at

due to Λ≥0 and g(x)∈[0,1]. Because this holds along any subsequence, the second relation in (3.9) is established. ▪

4. Numerical approximation

We now exploit the flexibility and robustness of the method of MRS, which allow us to go much further than the theory in [8]. Indeed, we can numerically approximate the problem, e.g. by standard finite-element methods as used in [28].

For this we consider finite-dimensional subspaces U_k and Z_k of and Z = L²(Ω) that are asymptotically dense, i.e.

4.1

Moreover, assume that the discretization of is compatible with the constraint θ(x)∈[0,1]. We set and assume and that is dense in .

(a). An abstract convergence result

Based on the above general assumptions, we add two major conditions. For each k, we need a (maybe nonlinear) mapping such that the following holds:

4.2

To formulate the conditions between the compatibility of the discretization of u through the spaces U_k and the discretization of θ via , we again use the quadratic structure of . For , we define the reduced functionals

By (4.1), we have and for fixed (t,θ). The second major condition is that the convergence is uniform with respect to (t,θ), namely

4.3

To formulate the existence and convergence result, we again use that we are able to restrict to the variable θ. We consider the sequence of ERIS given by

and . We use the discretized stability sets

The numerical incremental minimization problem for τ=T/J with reads

As in (2.3) we define the piecewise-constant interpolants .

Theorem 4.1 (Convergence of numerical approximation) —

Let conditions (4.1), (4.2) and (4.3) hold. Moreover, consider stable initial conditions such that

Then all accumulation points for and τ→0 (in the sense of (3.5)) of the numerical approximations are energetic solutions of .

The proof is identical to the one in §3b, where now the crucial construction of MRS for the numerical approximation is given in §4b. We refer to §4c for possible ways to fulfil the assumptions (4.2) and (4.3) by concrete numerical discretizations.

(b). Mutual recovery sequences II

The construction follows closely the one for the existence result. However, we have to take care that the MRS lies in the discrete finite-dimensional space .

Proposition 4.2 (MRS for the discretized system) —

Let the conditions (4.1), (4.2) and (4.3) be satisfied. Then, for any sequence (θ_k) with , t_k→t_* and and any , the sequence

is an MRS satisfying

In particular, we conclude that .

Proof. —

We first observe that and , which follows from the definition of g via the specific form of F. Setting , we have

where the first term converges to 0 as in the proof of proposition 3.3. Since , the second term is bounded by Cα_k(g,h), which converges to 0 by condition (4.2).

For the energy difference, we use and σ_k as in (4.3) to obtain

By taking a subsequence, we may assume that the limsup is achieved, , and . Using (3.7b) yields , since with and ∥w_n∥_L²≤α_k(g,h)→0 by condition (4.2). Thus, using σ_k→0 (i.e. condition (4.3)) and t_k→t_*, we conclude via (3.7a), namely

since Λ≥0 and 0≤g≤1. This proves the proposition. ▪

(c). Conditions for numerical approximations

We now show that the two major conditions (4.2) and (4.3) can be easily satisfied by suitable discretizations. For this, we assume that for each , there is a triangulation of Ω, such that Ω decomposes into d-dimensional tetrahedra T (convex hull of d+1 points) plus some intersections of tetrahedra with Ω along the boundary. By

we denote the fineness of the triangulation . For any , we denote by Z_k the space of functions that are constant on each of the subsets . To satisfy the condition Z_k⊂Z_k+1, we need to choose a nested triangulation where new tetrahedra are constructed by inserting a point in the interior of T and generating smaller tetrahedra by connecting this point with all the faces of T.

We denote by the L² orthogonal projection from Z to Z_k which reads

Given the above construction, the following three conditions are equivalent:

• (i)  ;

• (ii)  ;

• (iii)  .

Lemma 4.3 —

The operator constructed above satisfies (4.2).

Proof. —

We consider arbitrary with . For , we use

where we used that θ_k is constant on each tetrahedron. Using 0≤θ_k≤1 yields

Thus, we conclude that and (iii) from above implies the desired result (4.2). ▪

We show that the second condition (4.3) can always be satisfied by choosing a suitably fine discretization for the displacements u∈U. Considering the same family of nested triangulations as above, we set

Here, the crucial point is that m_k has to be chosen sufficiently large, i.e. the fineness of the finite-element space U_k for the displacements is much higher than that for the phase indicator . In particular, this implies that the dimension of U_k may be much higher than that of Z_k. It is well known (cf. e.g. [43]) that is dense in if and only if .

Lemma 4.4 —

Under the above assumptions, there exists a sequence m_k such that condition (4.3) holds for Z_k and U_k given above.

Proof. —

For each , we set

such that σ_k in (4.3) has the form

where it is essential that ς has the larger index m_k while .

If N_k is the number of tetrahedra in , then is the convex hull of the J_k:=2^N_k extremal points in which are given by functions taking the values 0 or 1 on each tetrahedron. Because every has the convex representation with λ_j≥0 and , we can use the convexity of (which follows from U_k⊂U) to obtain

Since m↦ς_m(θ) decays monotonically to 0 for each k, there is a minimal M(θ,k) such that ς_m(θ)≤1/k for m≥M(θ,k). We now set

then ς_m(θ)≤1/k for all and all m≥m_k. Thus, we conclude that σ_k≤1/k, which implies the desired condition (4.3). ▪

While the above construction shows that it is in principle possible to find converging discretizations, the method is not satisfactory. It would be desirable to show that the discrete spaces U_k can be formulated on the same triangulation instead of the much finer triangulation . It is not clear that this can be achieved with some kind of conforming discretization (i.e. U_k⊂U) as used in [28]. However, it might be easier to construct a non-conforming scheme like discontinuous Galerkin schemes to satisfy condition (4.3). Moreover, the latter condition turned out to be sufficient for our convergence result in theorem 4.1, but there might be substantially weaker abstract conditions that would allow for a larger class of discretization schemes.

5. Evolutionary relaxation

The original microscopic problem was described by pure phases with z(t,x)∈{e₁,e₂}, i.e. the phase indicator θ should only take the values θ=0 for phase 1 or θ=1 for phase 2. Thus, we define the pure, or unrelaxed, state space

Obviously, is a subset of , but it is not weakly closed. In fact, is the convex hull of , while contains all extremal points of .

We may consider the full ERIS or the equivalent reduced ERIS , but it is not clear whether this system has any energetic solutions for general loadings via g_Dir and ℓ. However, following the ideas in [8,16,27] (see also [15] for a similar relaxation of a RIS related to fracture), one can define upper and lower incremental relaxations [27], def. 4.1. Indeed, for a special case of our two-phase problem, the lower relaxation was established in [26].

(a). The relaxation result

Here, we want to address the time-continuous relaxation as introduced in [16], §4. For this, we consider approximate incremental minimization problems for defined via (2.2) with a fixed initial state . Now for every ε_n>0, we choose an approximate solution for the time-discretized problem. As before we denote by the piecewise-constant interpolants.

Since θ_τ,ε satisfies an a priori dissipation bound independently of τ=T/N and ε∈]0,1], we can extract subsequences (τ_k,ε_k)→(0,0) such that

5.1

In the spirit of [27,16], we call an (upper) time-continuous relaxation of if all accumulation points θ obtained via (5.1) are energetic solutions for the ERIS .

The following result, which should be seen as a specific non-trivial instance of the general theory in [16], §4, provides the mathematically rigorous relaxation result that all accumulation points of the pure-phase approximation solutions are indeed solutions of the relaxed model. In particular, it justifies the model derived in [8] via separate relaxation as a true upper relaxation of the evolutionary problem. The property of lower relaxation was already established in [26].

Theorem 5.1 (Evolutionary relaxation) —

Consider the functions with ε>0 and τ = T/J with obtained via the approximate incremental minimization problem (2.2). Furthermore, assume that is stable in , i.e. for all . Then, every accumulation point satisfying (5.1) is an energetic solution of the ERIS as discussed in §3.

As before, the only non-trivial part of the proof is Step 3, where we have to establish the stability of the accumulation points , i.e. . As before, we will deduce this from the stability of the approximations θ_τ,ε. However, the non-relaxed (approximate) stability sets are defined via

where the test functions are in the much smaller set of pure phases only. Thus, the desired closedness condition, which reads

5.2

is more difficult, because we not only have to pass to the limit, but also have to enlarge the space of test functions from to .

Hence, for a construction of MRS, we must approximate functions by suitable functions . In particular, for values , we need to introduce new oscillations between the values 0 and 1, which implies that the oscillations captured in the H-measure generated by cannot always be bounded by the H-measure μ which is generated by θ_k. However, we may introduce the necessary oscillations in such a way that they do not increase the energy too much. For this, we essentially use that by the very definition of as the relaxation of there is at least one direction such that Λ(ω_*)=0, i.e. laminates with normal ω_* do only contribute to the energy as much as their weak limit. This is also the essential point in the lower relaxation result established in [26].

(b). Mutual recovery sequences III

We use the following construction for MRS. For , , and α_k as in (5.2) and arbitrary test functions , we have to find an MRS with . We employ the function with

which satisfies . For an arbitrary function and , we define the piecewise-constant approximations

i.e. is a semi-open cube of side length 1/m containing x. As in [26], Thm. 3.5, we set

which is locally near x∈Ω a laminate with normal ω_* and volume fraction η_k(x)≈η(x). Clearly, and with (e.g. [42], Lem. 12).

The following theorem 5.3 gives a construction for MRS, which relies on the fact that we can introduce oscillations via which are much faster than oscillations in θ_n. The enforcement of a decoupling of spatial scales via k_n≫n of the microstructures generated by and θ_n, respectively, allows us to calculate the generated H-measure of the maximum function explicitly. The proof of this result will be postponed to §5c, and figure 1 gives a sketch of the construction, where very fine laminates generated by are combined with the microstructure of θ_n.

Figure 1.

Themicrostructure of θ_n (green circles) is combined with the much finer laminates (violet stripes) generated by for k≫n. The function equals 0 in white regions and equals 1 in all coloured regions. (Online version in colour.)

Proposition 5.2 —

Assume , the convergence and . Then, there exists a sequence such that for all sequences with for all n, the functions satisfy

5.3

We expect that the microlocal compactness forms developed in [42] are the optimal tools to give a clearer proof of the following result and to provide a stronger characterization of the possible limiting objects. Fortunately, for our two-phase problem, the H-measure is already sufficient.

The above construction allows us to define a suitable MRS for a test function . As before, we will be able to guarantee the sign condition

5.4

Thus for α∈{+,0,−}, we define the indicator functions for the domains

Now we are ready to choose the sequence

5.5

and formulate the final result on the MRS for the relaxation problem.

Theorem 5.3 (MRS for evolutionary relaxation) —

Let , , and α_k be given as in (5.2) and . Then, there exist such that the sequence defined in (5.5) with is an MRS satisfying the relations

5.6

Moreover, if , then

5.7

Proof. —

Step 1: For the weak convergence, we first use proposition 5.2 to see that . Similarly, ; see Step 1 in the proof of proposition 5.2. Since obviously , we conclude that as desired, if k_n is sufficiently large.

Step 2: The convergence of the dissipation distances is an easy consequence of Step 1, if we observe the obviously true sign condition (5.4).

Step 3: Next, we derive the H-measure relation (5.7) for which we assume . By decomposing into , we can treat the three parts separately, since . Clearly, we obtain , i.e. , which means and .

For the part , we can directly apply proposition 5.2 with η=η⁺, which provides and as given in (5.7). The result on Ω₋ follows similarly, e.g. by substituting θ by 1−θ. Hence, (5.7) is established.

Step 4: To show the limsup estimate in (5.6), we first choose a subsequence realizing the limsup. Choosing a further subsequence (not relabelled), we may assume . Thus, owing to proposition 3.2 and (5.7) we find

where we used because of Λ(±ω_*)=0, Λ≥0, and 0≤b≤1 in (5.7). ▪

(c). Proof of proposition 5.2

We consider a sequence with . For a given , we have to construct a sequence such that for all the functions satisfies

5.8

Proof of proposition 5.2 —

Step 1: As in [26], we use that for a,b∈{0,1} we have the simple relation . Hence, using , we have

We first consider the weak limit, where fixing n and considering we find due to . Since the weak L²-convergence in is metrizable by some metric d_w, we can choose such that for all . Now, implies and

whenever , i.e. we have as desired.

Step 2: For the H-measure, we use proposition 3.2 to conclude that the term (1−η)θ_n generates the H-measure , while the third term η is constant and hence does not contribute to .

Step 3: We next show that the first term generates the measure if k_n grows sufficiently fast. The Fourier transform (where the extension by 0 on is suppressed) satisfies the convolution formula

We choose a radius R_n such

Since , the Fourier transform converges to 0 in the balls for and n fixed. Moreover, the fast oscillations in x↦H(η_k(x),k²ω_*⋅x)−η(x) lead to a spreading of the Fourier transform in the directions ξ≈±λω_* with λ≥2k²π. Indeed, recalling |ω_*|=1 and setting

we find the relation

Choosing such that ρ(k)≤1/n for all , we see that the convolution f_n*g_n has most of its mass inside the set , i.e.

Since the radial projection of on converges to {ω_*,−ω_*}, the H-measure generated by is , where α is the weak limit of . As in Step 1, we obtain α=(1−θ_*)²η(1−η), where we may increase if necessary.

Step 4: We still have to show that the sum

generates the H-measure . For this, it suffices to show that and have their masses well separated. By Step 3, we know that the essential part of the mass of is contained in Ξ_kn+{ξ||ξ|≤R_n}, while the essential part of the mass of is concentrated in . Increasing if necessary, for every test function φ∈C_c(Ω), we find . Thus, we conclude that

Thus, proposition 5.2 is proved. ▪

Authors' contributions

S.H. developed the main idea of the construction of the mutual recovery sequence and wrote §§2 and 3 as well as a first draft of §4. A.M. refined and simplified §4, developed the proof for the evolutionary relaxation in §5, and finalized the full manuscript. Both authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

Research partially supported by DFG via FOR 797 MicroPlast under Mie 459/5-2 and by ERC via AdG267802 AnaMultiScale.