RATE-INDEPENDENT PHASE TRANSITIONS IN ELASTIC MATERIALS: A YOUNG-MEASURE APPROACH.

A quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the usage of suitable regularity arguments in order to prove an existence result for a notion of evolution presenting some improvements with respect to the one defined in [13], for infinitely many phases.

Assuming that the reference configuration of the crystalline material is a bounded region D ⊂ R d , the state of the system is determined by two functions: the deformation v : D → R N and the internal variable z : D → Z ⊂ R m , which takes into account the phase transformations of the material.
In our framework, Z is a finite set {θ α : α = 1, . . ., q} , representing the different phases (or phase variants) of the crystal, and z represents the phase distribution of the material.Then the stored energy of the system can be written as: From a physical point of view, the energy functional should also depend on the temperature, but we omit this dependence since we are dealing with isothermal transformations.We assume that changes of the phase distribution of the material lead to an energy dissipation, which is represented by where H is a metric distance on Z , z old is the old phase distribution and z new the new one.Moreover, we require that the admissible deformations satisfy a prescribed time-dependent boundary condition ϕ(t) , which we impose on the whole boundary ∂D to avoid some technical difficulties; for the same reason, we neglect any contribution due to external forces.
The natural form for the stored-energy density W is a multiple-well potential form (see [23], [22], [14], [11], [20], [21]), more in general we deal with a density which does not satisfies any convexity assumption with respect to z .As in [9], this lack of convexity gives rise to many technical difficulties, making unsolvable in usual functional spaces the incremental minimum problems used in the construction of approximate solutions (see [16] and references therein); it is also responsible for the formation of microstructures (see, e.g., [14], [25]).To overcome these difficulties, many authors have proposed to introduce suitable regularizing terms in the energy functional (see [2], [11], [15]).To avoid any artificial regularization, in this paper we follow the same approach of [9], and set the problem in a suitable space of Young measures, where the incremental minimum problems can be solved.
Since we are assuming that the internal variable takes only a finite number of values, we are now able to give a more explicit description of the Young measure ν which is going to substitute the pair (z, ∇v) in our extended setting: ν can be written as for suitable families (λ α ) α of Young measures on D with values in R N ×d , and (b α ) α in L ∞ (D; [0, 1]) , with α b α = 1 a.e. in D .
To find a correct extension to the Young measures setting of the dissipation functional and of the corresponding notion of total dissipation on a time interval, we need to use the tool of compatible systems of Young measures, introduced in [4].Our discrete setting again allows to deal with a more explicit expression for these objects: every compatible system of Young measures µ on D , with time set A and values in Z can be written as In this language, when a Young measure with values in Z is representable by a function z , the corresponding b is defined by b α = 1 {x∈D : z(x)=θα} for every α.
In this case the Young measure representation can be interpreted in the following way: the material assumes a pure phase distribution, i.e., to every point x is associated a pure phase θ ∈ Z .While in the general case we say that the material has a mixed phase distribution meaning that at each point x we have a mixture of phases θ α with volume fractions b α (x) .
If we consider two time instants s < t, b st αβ (x) represents the volume fraction at x undergoing the phase transition from θ α at time s to θ β at time t .
The aim of the paper is to prove an existence result for the quasistatic evolution in a time interval [0, T ] , defined as a pair (b, λ) = (b, (λ t ) t∈[0,T ] ) satisfying an admissibility condition and suitably reformulated stability condition and energy inequality.
The admissibility condition requires suitable approximation properties by functions which satisfy the boundary condition.
The stability condition will be a global minimality condition, but the set of competitors is a proper subset of the admissible pairs; for this reason we call this condition partial-global stability.Specifically, a quasistatic evolution (b, λ) satisfies for every t ∈ [0, T ] the following property for every ũ ∈ H 1 0 (D; R N ) , and every measurable function M on D with values in a special set of q × q real matrices.The elements of this set are the matrices with nonnegative entries such that the sum of the entries of each column is 1 ; in probabilistic language they are called stochastic matrices (see, e.g., [1,Part 2]), and their entries M βα represent the probability of a transition from phase θ α to phase θ β .In our model, M βα (x) is the proportion of the volume fraction at x originally in phase θ α undergoing a phase transition to θ β .According to the picture described so far, the quantity ) can be interpreted as the energy density dissipated at the point x by the phase transition from θ α to θ β .Therefore, the following expression represents the energy which would be dissipated on the whole domain D , if we performed the microscopic phase transition determined by M .We observe that any other phase distribution ( bα ) α can be obtained by the action of a suitable stochastic matrix: indeed, it is enough to choose M βα (x) := bβ (x) for every α, β .
From the stability property we can deduce a pointwise condition.If we call active at x the phases θ α for which b t α (x) > 0 , then the Euler equation for the internal variable can be written as follows: for a.e.x with active phase θ α , we have for every β .According to the above physical picture, this condition can be interpreted as an optimality condition of the active phases.Clearly, an Euler equation for the deformation can be derived as well: it is the classical equilibrium condition on the stress σ (see Remark 5.5 for the definition of σ ).
The energy inequality expressed in terms of (b, λ) takes the following form: where the supremum is taken over all partitions . The above definition justifies the use of compatible systems of Young measures: the energetic effect of the phase transitions occurring from the instant s i−1 to the instant s i can only be described by an object b si−1si αβ , representing the volume fraction at x undergoing the phase transformation from θ α to θ β .The knowledge of b si−1 α and b si β separately does not keep the complete information about the energy spent in the transition.Indeed, if we consider the case of a homogeneous phase distribution b si−1 α = 1/q for every α , and we suppose that the material undergoes a transition from s i−1 to s i just permuting the phases and leaving the volume fractions unchanged, we have b si = b si−1 ; hence the dissipation computed using only b si−1 and b si is 0 , while the dissipation energy computed using b si−1si depends on the permutation and it is different from zero.Therefore, the previous description seems to give a more realistic picture of the dissipation phenomenon, if compared with that proposed in [14, Section 5], which only takes into account the contribution of single time instants.
The proof of the existence theorem (Theorem 6.2) follows the classical scheme of time-discretization, resolution of incremental minimum problems, and passage to the limit of the approximate solutions.
The new feature concerns the choice of the solutions to the discretized minimum problems.In the spirit of [7], we use the Ekeland Principle to choose minimizers satisfying special approximability properties.Then the regularity results for quasiminima of integral functionals (see [12]) are used to prove a uniform bound over the moments of order 2r > 2 of the selected minimizers, and consequently of the approximate solutions (b t n , λ t n ) .As a by-product of this selection, we get the continuity of the functional Thanks to this continuity property, we are able to obtain in the limit the stability condition and the energy inequality written above, which improve the notion of quasistatic evolution proposed in [9].Due to technical difficulties, we cannot still have a complete energy balance; nevertheless the energy inequality holds for any pair of time instants, while the quasistatic evolution considered in [9] satisfies it only on intervals of the form (0, t) .Moreover, under weaker assumptions on W than in [9], we can obtain a better notion of stability, since the minimality property is now satisfied with a quite large set of competitors including all possible rearrangements of the phase distribution.One technical point in the proof of the stability condition is the approximation of the right hand-side of (1.1) by integrals corresponding to functions satisfying the prescribed boundary condition.This is done by adapting to our problem the classical Riemann Lebesgue Lemma.
The outline of this paper is as follows.In Section 2 and 3 we provide some mathematical preliminaries and technical tools.In Section 4 we fix the setting of the problem.In Section 5 we describe the admissible set where we look for the quasistatic evolution, which is defined in Section 6. Section 7 is devoted to the proof of the existence theorem, and finally in Section 8 we derive the Euler equations for the partial-stability condition.

Mathematical preliminaries
The symbol 1 B indicates the characteristic function of a subset B of R d .The Lebesgue measure on R d , d ≥ 1 , is denoted by L d ; we sometimes use the notation |E| for the Lebesgue measure of a measurable subset E of R d .The Borel σ -algebra on D is denoted by B(D) .For 1 ≤ p ≤ +∞ , • p is the usual norm on L p , while W 1,p (D; R N ) denotes the usual Sobolev space of all L p functions from an open We recall the well-known following lemma.
The symbol M q×q St denotes the set of all stochastic matrices of size q × q , i.e. the set of all matrices (M βα ) β,α with • 0 ≤ M βα ≤ 1 for every α , β , For the notion of quasi-minimum and the related results, we refer to the Appendix.

Young measures and discrete sets of values
For the mathematical preliminaries about measures and Young measures we refer to [9, Section 2 and 4].Here we just fix some notation.Let (Ω, F) be a measure space, Ξ a finite dimensional Hilbert space, and µ ∈ Y (D; Ξ) ; for every B(D) -F -measurable function f : D × Ξ → Ω , the image measure, defined by µ(f −1 (B)) for every measurable set B ⊆ Ω , will be denoted by f (µ) ; for every bounded measurable function g : D → R, the product gµ is defined by for every bounded Borel function g : D × Ξ → R. In particular δ ξ0 is the Young measure associated to the constant function u(x) ≡ ξ 0 , which should not be confused with the measure δ ξ0 .
We recall the statement of a lemma which will be useful in the regularization of the approximate solutions.We will use the statement of Fonseca, Müller, and Pedregal (see [10,Lemma 1.2]).Lemma 3.1.(Decomposition Lemma) Let (v j ) j be a bounded sequence in H 1 (D; Ξ).Then there exist a subsequence (v j k ) k of (v j ) j , and a bounded sequence as k → ∞.
In the whole paper D is a bounded connected open subset of R d with Lipschitz boundary; Z denotes a nonempty finite subset {θ 1 , . . ., θ q } of R m , and H is a metric on Z ; A ⊆ R denotes a set of indices.
The space of Young measures on D with values in Z is indicated with Y (D; Z) and the space of compatible systems on D with time set A and values in Z is denoted by SY (A, D; Z) .It is easy to see that µ ∈ Y (D; Z) if and only if its disintegration (µ x ) x∈D can be written as where b α are functions in L ∞ (D; [0, 1]) satisfying the condition In disintegrated form, formula (3.2) can be written as Therefore Y (D; Z) can be identified with the set of all families b = (b α ) q α=1 in L ∞ (D; [0, 1]) satisfying condition (3.3).
as k → ∞, for every finite sequence Now we state a lemma to describe the canonical form of the space Y p (D; Z × R N ×d ) of all Young measures on D with values in Z ×R N ×d and finite p -moments, for 1 < p < +∞ . where 3) and, for every α = 1, . . ., q , λ α is a Young measure on D with values in R N ×d such that for every α = 1, . . ., q .
On the other hand, if ν belongs to ∈ Y p (D; Z × R N ×d ) and (ν x ) x∈D is its disintegration, for a.e.x ∈ D and every α = 1, . . ., q we define b α (x) := ν x ({θ α } × R N ×d ); (3.12) let us fix a probability measure ω on R N ×d ; for α = 1, . . ., q and for a.e.x ∈ D let us define a probability measure λ x α on R N ×d by setting for every By construction b α is measurable with nonnegative values for every α , and (λ x α ) x is a measurable family of probability measures satisfying (3.11), for every α .It is now immediate to see that the measure ν whose disintegration is given by νx is exactly ν .Indeed every Borel subset B of Z ×R N ×d can be written as the union of disjoint sets of the form {θ α }×B α , for suitable B α ∈ B(R N ×d ) , for α = 1, . . ., q ; hence we have Remark 3.5.The functions b α and the measures b α λ α , α = 1, . . ., q , satisfying the properties described in the previous lemma are uniquely determined by ν .In particular if we consider the disintegration of λ α , (λ x α ) x∈D , we obtain that λ x α is uniquely determined for a.e.
3) for every h, and

The mechanical model
The reference configuration will be the set D introduced in the previous section.
We indicate the deformation with v and the internal variable with z .We denote the stored energy density by W : R m × R N ×d → [0, +∞) and the dissipation rate density by H : Z 2 → [0, +∞) .For every θ, θ ∈ R m and F ∈ R N ×d , we make the following assumptions: (W.1) there exist positive constants c, C such that Let W be the functional for every z ∈ L ∞ (D; Z) and every v ∈ H 1 (D; R N ) , and H the functional where the supremum will be taken among all finite partitions s = τ The prescribed boundary datum on ∂D at time t is denoted by ϕ(t) ; we assume ϕ ∈ AC([0, T ]; W 1,p (D; R N )) , with 2 < p < +∞ .
The kinematically admissible values at time t for v are those which make the total energy finite and satisfy the boundary condition, i.e., v = ϕ(t) on ∂D H d−1a.e.(in the sense of traces).From the previous assumption it follows that the kinematically admissible values at time t are contained in A(t) , where A(t) := ϕ(t) + H 1 0 (D; R N ) .

Admissible set in terms of Young measures
Definition 5.1.Given A ⊂ R and w : A → H 1 (D; R N ) , we define the admissible set for the time set A and the boundary datum w , Ad(A, q, w) , as the set of all pairs (b, λ) ∈ S(A, D, q)×(Y (D; R N ×d ) q ) A such that property (3.11) (for p = 2 ) is satisfied by b t α λ t α , for every α and t , and the following condition holds: for every finite sequence t 1 < • • • < t n in A, for every i = 1, . . ., n , and every k ∈ N, there exist a measurable partition (D i,k α ) q α=1 of D and a function (2) for every i = 1, . . ., n there exists a subsequence (k i j ) j , possibly depending on i , such that for every α = 1, . . ., q .
The following remark compares the notion of Ad(A, q, w) with the notion of admissible set in terms of Young measures AY (A, Z, w) , as defined in [9, Section 6.2].
Then (b, λ) ∈ Ad(A, q, w) if and only if (ν, µ) ∈ AY (A, Z, w) , i.e. for every finite sequence as k → ∞; (app2) Z for every i = 1, . . ., n , there exists a sequence of integers (k i j ) j , possibly depending on i, such that ν ti 2-weakly*, ( as j → ∞.Indeed, given (D i,k α ) α satisfying the approximation property for b ti α we define z k i by z k i (x) = θ α whenever x ∈ D i,k α , or equivalently, given z k i satisfying the approximation property for ν ti , we consider D i,k α := {x ∈ D : The closure properties of Ad(A, q, w) are described by the following lemma, which is the formulation in our discrete setting of [9, Lemma 6.7].
Lemma 5.3.Let (w j ) j be a sequence of functions from A into H 1 (D, R m ) , such that w j (t) → w(t) strongly in H 1 , for every t ∈ A and let (b, λ) ∈ S(A, D, q) × (Y (D; R N ×d ) q ) A with (b t , λ t ) satisfying (3.11) for p = 2, for every t ∈ A. Assume that for every finite sequence as j → ∞ for every (α 1 , . . ., α n ) ∈ A q n , and such that for every i there exists a sequence of integers (j i h ) h , possibly depending on i , satisfying as h → ∞ for every α = 1, . . ., q .Then (b, λ) ∈ Ad(A, q, w).
The following lemma will be used in order to provide a class of competitors for the discretized minimum problem in Section 7.1.
Consider now the case of M β,α in C 1 (D) .Fixed a positive parameter ε , consider a finite family (Q i ε ) i=1 of disjoint cubes in R d , with diameter ε , covering D , and set for every i = 1, . . ., I(ε) , and every α , β .For a fixed α , we can define a measure Let us fix i = 1, . . ., I(ε) and reproduce the arguments used in the constant case: consider a measurable partition (( ε , for every β (it is possible to find such a partition since , by the hypotheses on M ), and define (z α ) i ε : R d → Z as the 1 -periodic measurable function satisfying for every β = 1, . . ., q .For every δ > 0 , consider the function (z α ) i ε,δ : R d → Z defined by (z α ) i ε,δ (x) := (z α ) i ε ( x δ ) , for a.e.x ∈ R d .Fixed ε , we obtain as before that as δ → 0 , where zα ε,δ : R d → Z is the function defined by zα ε,δ := Since for every x ∈ D m α there exists a unique i x = 1, . . ., I(ε) with x ∈ Q ix ε , we have for every x ∈ D m α and every β = 1, . . ., q .Therefore we have which tends to 0 as ε → 0 .Since Y (D; Z ×R N ×d ) is contained in a bounded subset of the dual of a separable Banach space, it is metrizable with respect to the weak* topology.Let us denote by d a metric inducing on Y (D; Z × R N ×d ) the weak* topology, so that we have Applying as before the same argument to µ t1...tm , we deduce, using a diagonalization argument, that there exist sequences δ k → 0 and ε k → 0 such that (1) for every α , 1 ν α tm 2 -weakly* as k → ∞; (2) for every α , we have weakly*, as k → ∞.Now it is enough to define zε,δ : D → Z , by zε,δ := α 1 D m α zα ε,δ , to prove the thesis.
It remains only to treat the general case of M βα ∈ L ∞ (D) .We can reproduce the same construction proposed in the C 1 -case; the only difference is that we have to use an approximation argument to show that ν α ε ν α tm .Indeed it is enough to consider, for every β , a sequence (M n βα ) n in C 1 (D) , with M n αβ → M βα strongly in L 1 (D) , as n → ∞ , and let (M n βα We know that for every x ∈ D m α and every β = 1, . . ., q .On the other hand, using Lemma 2.1, we can deduce that Let us now fix η > 0 ; choosing n such that β M βα − M n βα 1 ψ ∞ ≤ η/2 , we have ; therefore we obtain that ν α ε ν α tm as ε → 0 and we can prove the thesis as in the previous case. Remark 5.5.If (b, λ) ∈ Ad(A, q, w) , for every t ∈ A there exists a unique function v(t) ∈ w(t) + H 1 0 (D; R N ) such that ∇v(t) = α b t α bar(λ t α ) ; moreover, for every t ∈ A , the function σ(t) representing the stress and defined by 6. Definition of quasistatic evolution and main result.

Proof of the main theorem
The proof is obtained via time-discretization, resolution of incremental minimum problems, and passing to the limit.

7.1.
The incremental minimum problem.The first step of the proof consists in the definition of an approximate solution via an inductive minimization process.

From the definition of
as k → ∞ .Thanks to Lemma 3.1, we can assume, without loss of generality, that (|∇v i n,k | 2 ) k are equiintegrable; hence by the Fundamental Theorem for Young measures (see, e.g., [3]) we may assume that sup Denote by I i n the minimum value of (7.1) over A i n (b i−1 , λ i−1 ) .Thanks to (7.10), we can deduce that lim Now we want to consider the following auxiliary minimum problem, for every k : For every k , we choose vi Using v i n,k as competitor in (7.12), we can easily deduce, from (7.13) and the growth hypothesis on W , that 2 ), for a suitable positive constant Ĉ , independent of n .Hence, thanks to (7.9), sup k ∇v i n,k 2 2 is bounded; in particular there exists νi n ∈ Y 2 (D; Z × R N ×d ) such that, up to a subsequence, δ (z i n,k ,∇v i n,k ) νi n 2 -weakly* as k → ∞.Thanks to Lemma 3.1 we can assume, up to a subsequence, that α δ θα , by Remark 3.5 there exists a family of Young measures λi n = (( λi We have The construction of νi n implies that the pair (b, λ) , with λ := (λ t 0 n , . . ., is an element of Ad({t 0 n , . . ., t i n }, q, ϕ) ; moreover it satisfies the"memory properties" (7.2) and (7.3) required to be in A i n (b i−1 , λ i−1 ) .Hence we can deduce from (7.15) and (7.16) that (b, λ) is a minimizer of (7.1) over ) .Now we want to apply Ekeland Principle in order to construct a more regular sequence (v i n,k ) k which, together with z i n,k , generates νi n .We define ûi n,k as the function vi Consider the functional E defined on the Banach space W 1,1 0 (D; R N ) by This functional is strongly lower semicontinuous with respect to the W 1,1 0 topology, it is positive and not infinite everywhere: hence we apply Ekeland's Principle (see [8,Corollary 6.1, p.30]) to W 1,1 0 (D; R N ) endowed with the norm u W 1,1 0 := ∇u 1 , and we deduce that there exists ūi n,k ∈ H 1 0 (D; R N ) with the following properties: for every u ∈ H 1 0 (D; R N ) , where c is the Poincaré constant of the domain D .In particular these properties imply that sup for a suitable positive constant C independent of k , n , and i, and Using the growth hypotheses on W , it is easy to deduce from (7.19) that, for k sufficiently large, vi and i.
We can now apply Theorem 8.7, and conclude that there exist two constants γ > 0 and r > 1 , both independent of k , n , and i, such that for every k .In particular, thanks to (7.20), we have for a suitable constant γ > 0 independent of k , n , and i .Thanks to the equiintegrability of |∇v i n,k | 2 , using the Fundamental Theorem for Young measures we can deduce that This concludes the proof.
Using the minimization process described so far, it is possible to construct inductively (b i n , λ i n ) , for every i = 1, . . ., k(n) and every n .Set τ n (s) := t i n , whenever t i n ≤ s < t i+1 n , where we set t k(n)+1 n := T + 1 n .For every i and n we set and define for a.e.x ∈ D , whenever t i n ≤ t < t i+1 n .For every α = 1, . . ., q , we define for every i = 1, . . ., k(n) , and for every t ∈ [0, T ] and every As in [9, Section 7.2], we want to deduce a discrete version of the energy inequality for (b n , λ n ) .We briefly recall the argument for the reader's convenience.
(since ν n are piecewise constant interpolations of Young measures with finite second moments), we can deduce from (7.25) that, for n sufficiently large, , for suitable positive constants C and c independent of t and n .Since this can be repeated for every t ∈ [0, T ] , we deduce  (7.30)where ρ n → 0 as n → ∞.Let T be a dense countable subset of T containing 0 .Thanks to (7.26) and Remark 3.7, we can find with a diagonalization process a subsequence of (λ n ) n , still indicated by (λ n ) n , and λ t = (λ t α ) α ∈ Y (D; R N ×d ) q for every t ∈ T , such that and for every t ∈ T .Note that the family of coefficients b appearing here is the same as in (7.31), because π D×Z ((ν n ) t ) = (µ n ) t for every t ∈ [0, T ] and thanks to Remark 3.5; moreover, by construction of (ν n , µ n ) we have for every t ∈ T .This implies that the map (6.1) is measurable on [0, T ] ; moreover for every t ∈ T we have lim sup The family ν will denote the element of Y 2r (D; Z × R N ×d ) T corresponding to (bλ) .Let t ∈ [0, T ] \ T , and fix a sequence s j in T converging to t with s j < t; by (7.32), and (7.38), we have for every j ; again by Remark 3.7, we can find a subsequence, not relabelled, and Note that, since π D×Z (ν t ) = µ t for every t ∈ T , the left continuity of b defined in (7.31) ensures that the family of coefficients appearing in (7.42) is the same as in (7.31).
If we deal with quasi-minima satifying a prescribed boundary condition, the following result can be proved with similar arguments (see [13,Section 6.5]).
Then there exist a positive constant C > 0 , depending only on Q, such that