Existence and regularity of solutions for an evolution model of perfectly plastic plates

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.


Introduction
This paper studies the dynamic evolution of a thin plate in Prandtl-Reuss plasticity, continuing the investigation undertaken in [18], where the authors rigorously derived a reduced model describing such situation. Here, we pursue the analysis of that model by extending the existence result in [18] to the case of nonzero external forces, and proving a regularity result for the stress field and for the vertical component of the displacement. Indeed, we point out that in [18] the stress is proved to be square integrable with respect to the space variables, without addressing any higher regularity; here we show H 1 loc -regularity. Moreover, we also prove that the vertical component of the displacement field satisfies H 1 loc -regularity in space uniformly in time.
We remark that an analogous investigation has been previously carried out in the simpler case of quasistatic evolution, namely neglecting inertial effects and assuming that the system is in dynamic equilibrium at every time. Indeed, this paper may be seen as the dynamic counterpart of [9], where the authors analyse the stress regularity for the reduced model derived in [8], describing the quasistatic evolution of a plastic plate. The regularity of the elastic stress is a classical issue in perfect plasticity, investigated for instance also in [4], [11], and [13] (see the related results in [2]). For general treatment of quasistatic evolutions of rate-independent systems we refer to [17,19].
Formulation of the model. Before to discuss our regularity result, we briefly describe the mechanical model studied in this paper; more details are provided in Sections 3 and 4. This model was proposed and rigorously justified in [18], as the limit of a dynamic evolution for a three-dimensional plate, when the thickness tends to zero. We emphasize however that the limit model is purely three-dimensional, because the dependence of the stress on the variable x 3 is in general not trivial; a similar situation appears also in the quasistatic setting (cf. [9,Section 5] for an explicit example).
An elastoplastic plate is described in the reference configuration by the set Ω := ω × − 1 2 , 1 2 , where ω ⊂ R 2 is an open, bounded set corresponding to the base of the plate. The state of the plate at time t is described by the triplet (u(t), e(t), p(t)), where u(t) is the displacement, e(t) is the elastic strain and p(t) is the plastic strain. We assume that the displacement u(t) is of Kirchhoff-Love type: namely that u 3 (t) is independent of the transverse variable x 3 and there existsū(t) : ω → R 2 with u α (t) =ū α (t) − x 3 ∂ α u 3 (t) in Ω, for α = 1, 2. 1 As observed in [6], the physical interpretation of this condition is that straight lines normal to the mid-surface remain straight and normal after the deformation, within the first order.
The evolution of the system is guided by two main elements: a time-dependent Dirichlet boundary datum w(t), prescribing the displacement field on a portion Γ d = γ d × − 1 2 , 1 2 of the lateral boundary ∂ω × − 1 2 , 1 2 , and an external force L, which in turn acts on the body, an will be decomposed as the sum of vertical and horizontal loads g and f .
Here Eu(t) := Du(t)+Du T (t) 2 is the symmetrized gradient of u, ⊙ is the symmetrized tensor product, ν ∂Ω is the outer unit normal to ∂Ω, and H 2 denotes the two-dimensional Hausdorff measure. The third equation expresses the Kirchhoff-Love structure of the displacement. In view of such property, in the following we identify e and p with their nontrivial components taking values in M 2×2 sym . Regarding the second equation, one may expect instead the usual boundary condition u(t) = w(t) on Γ d ; indeed, this will be our requirement on the approximating evolution problems. Yet, as happens already in the quasistatic setting, the existence of the solution (u(t), e(t), p(t)) to the limit evolution problem is provided only in the product space , where BD(Ω) is the space of functions with bounded deformation in Ω, and M b (Ω∪Γ d ; M 2×2 sym ) denotes the space of bounded Borel measures on Ω ∪ Γ d . For this reason the boundary condition has to be formulated in a weak sense, as above. The mechanical interpretation of the second equation in (cf1) is that u(t) may not attain the boundary condition and, if this is the case, a plastic slip of strength proportional to w(t) − u(t) develops along Γ d .
(cf2) Constitutive law: where C r is the elasticity tensor. (cf3) Evolution equations: givenσ andσ be respectively the stretching and the bending components of σ (cf. Definition 3.1), we have together with corresponding homogeneous Neumann boundary conditions on ∂ω\γ d , corresponding to absent boundary tractions. (cf4) Stress constraint: sym : |ξ| r ≤ α 0 }, where |ξ| r := |ξ| − 1 3 (tr ξ) 2 for every ξ ∈ M 2×2 sym , so that K r is a convex, compact ellipsoid in M 2×2 sym containing the origin (cf. Section 3). This assumption, adopted also in the quasistatic setting in [9], follows from the restriction, in the original three dimensional problem, to the case of von Mises yield criterion (see, e.g., [16]), namely assuming that the set of admissible stresses is a cylinder B α0 +RI 3×3 , with B α0 being the ball of radius α 0 centered at the origin of the space of trace-free M 3×3 sym matrices and I 3×3 being the identity matrix in M 3×3 sym . (cf5) Flow rule:ṗ(t) belongs to the normal cone to K r at σ(t). It is easy to see that condition (cf5) can be written in an equivalent way as (cf5') Maximum dissipation principle: where H r is the support function of K r (for a precise definition see (3.9)). We remark that, for the lack of spatial regularity in p(t) mentioned above, this expression has to be read with the following precise sense. The left-hand side is defined using the theory of convex functions of measures as where dp(t)/d|p(t)| is the Radon-Nikodym derivative of p(t) with respect to its total variation |p(t)|. The right-hand side of (cf5') requires an ad-hoc elastic stress-plastic strain duality notion, that was originally given in [8], and is summarized here in Section 3.
Main result. The main result of the paper is Theorem 5.1, stating the existence of a solution to the dynamic evolution model discussed above, and additional regularity for the stress field and third component of the displacement; both these variables are shown to be locally H 1 with respect to the space variables. More precisely, we prove that for every open set ω ′ compactly contained in ω there exists a positive constant C 1 (ω ′ ) such that, for α = 1, 2, and for every open set Ω ′ compactly contained in Ω there exists a positive constant C 2 (Ω ′ ) such that sup Analogous estimates hold for the vertical component u 3 of the displacement u. We observe that the estimate (1.1) is stronger than (1.2), since it is global in the direction x 3 . We remark that all the arguments used are purely local, thus cannot be used to study the behaviour of stress up to the boundary ∂Ω. Finally, we recall that also the existence result presents some novelty, due to the presence of general external forces in the bulk, not considered in [18].
Strategy of the proof. The proof is developed in several steps, which can be resumed into two partial existence results of approximating solutions to (u, e, p), which are obtained considering progressive approximations of the flow rule (cf5).
The key point is the study of the so-called approximating Norton-Hoff problem, see Theorems 4.4 and 4.5. In these theorems we show that for any natural numbers N ≥ 4 there exists a triplet (u N , e N , p N ) solving (cf1)-(cf3) and the conditioṅ where φ N is a convex potential approximating the indicator function of K r as N → +∞. These results, together with a priori estimates on the norms of the solutions independent of N , can be considered of individual interest and are one of the main novelties of this work. In turn, these Norton-Hoff problems are studied by considering a second approximation of flow rule, obtained by truncating the potential φ N at a certain threshold. The corresponding approximating solutions, obtained in Lemma 4.1, depend on a truncation parameter λ > 0, which will be sent to +∞ at the limit. The proof of Lemma 4.1 is carried out by standard time discretization and using an implicit Euler scheme.
In Proposition 4.7 we recover additional regularity in the Norton-Hoff problems, specifically for the stress and the third component of the displacement. The main result is obtained by letting N → +∞, showing that the approximating solutions converge to the solution of the original problem (cf1)-(cf5), and that the additional regularity is preserved at the limit.
Comparison with the quasistatic case. As mentioned above, Theorem 5.1 can be seen as the dynamic correspondent of the regularity result of [9] for the quasistatic case, and the two approaches present several analogies and a similar overall structure. However, the introduction of inertial effects produces new difficulties, requiring a different proof for the existence of the approximating solutions, and the use of several ad-hoc techniques and part integration formulas. In particular we remark that, due to the inertial terms, the kinematic admissibility and the evolution equations are now coupled, so that the argument adopted in the quasistatic context cannot be repeated.
Differences between the quasistatic and dynamic models may be found also in the requirements on the external data. In [9] the boundary Dirichlet datum w is taken in the space H 1 ([0, T ]; H 1 (Ω; R 3 ) ∩ KL(Ω)). In the dynamic framework such regularity is no longer sufficient. Indeed, we need a control also on the second derivative in time of w and the third derivative of w 3 . Moreover, in order to treat the spatial regularity of the stress we need a local control of the H 2 norm of w (see (4.1) for the precise assumption). Moreover, we have to assume an uniform safe-load condition on the external forces. Again, due to the presence of inertia, we need a control on the velocity and acceleration of the external forces via the safe-load variable. These are stronger hypotheses than the ones in the quasistatic case, but at the presence stage seem necessary in the dynamic one.
The paper is organized as follows. In Section 2 we recall some mathematical preliminaries. In Section 3 we give some mechanical preliminaries and describe the setting of the problem. In Section 4 we present the existence and regularity results for the approximating problems. By Lemma 4.1 and then Theorem 4.4 we prove the existence of solutions to the Norton-Hoff problem. In Proposition 4.7 we prove the a priori estimates for the space derivatives of the stress. Finally we prove that these estimates hold true for the original problem in Theorem 5.1, where the existence of a solution to (cf1)-(cf5) is stated.

Mathematical preliminaries and notations
Measures. We denote with L n the Lebesgue measure on R n , and with H n−1 the (n − 1)dimensional Hausdorff measure. For any given Borel set B ⊂ R n , we denote the space of bounded Borel measures on B with values in a finite dimensional Hilbert space X as M b (B; X); such space is endowed with the norm µ M b := |µ|(B), where |µ| ∈ M b (B; R) denotes the variation of the measure µ. For every measure µ ∈ M b (B; X), by Lebesgue decomposition Theorem there exist a measure µ a , absolutely continuous with respect to the Lebesgue measure L n , and a measure µ s singular with respect to L n , such that µ = µ a + µ s . When µ s = 0, we identify µ with its density with respect to L n .
Moreover, if B is locally compact in the relative topology, by Riesz-Markov Theorem we can identify M b (B; X) with the topological dual space of C 0 (B; X). We recall that C 0 (B; X) is the space of continuous functions f : B → X such that, for every ǫ > 0, the set {|f | ≥ ǫ} is compact. Using this property, we define the weak * topology of M b (B; X). We will denote the dual pairings between measures and continuous functions, as well as between other couples of spaces, as usual, by ·, · .
Functions with bounded deformation. Let us denote with M n×n sym the space of n × n real symmetric matrices. Given an open set U ⊂ R n , we define the space BD(U ) of functions with bounded deformation as the space whose elements are the functions u ∈ L 1 (U ; R n ) with symmetric gradient (in the sense of distributions) Eu := 1 2 (Du + Du T ) belonging to M b (U ; M n×n sym ). The space BD(U ), endowed with the norm is a Banach space. We refer to [20] for the properties of the space BD(U ); we recall here the most relevant for our purposes.
If the set U is bounded with Lipschitz boundary, then the space BD(U ) is continuously embedded in L n/(n−1) (U ; R n ) and compactly embedded in L p (U ; R n ), for every p < n/(n − 1).
A sequence (u k ) k is said to converge weakly * in BD(U ) to u if u k ⇀ u weakly in L 1 (U ; R n ) and Eu k ⇀ Eu weakly * in M b (U ; M n×n sym ). Every bounded sequence in BD(U ) admits a weakly * converging subsequence.
Furthermore, to every function u ∈ BD(U ) we can associate a trace, still denoted by u, belonging to L 1 (∂U ; R n ). Given a subset Γ of ∂U with positive H n−1 -measure, we have where the constant C > 0, depends only on U and Γ Functions with bounded Hessian. Given an open set U ⊂ R n , we define the space BH(U ) of functions with bounded Hessian as the space whose elements are the functions u ∈ W 1,1 (U ), with Hessian D 2 u (in the sense of distributions) belonging to M b (U ; M n×n sym ). The space BH(U ), endowed with the norm is a Banach space. We refer to [10] for the main properties of BH(U ).
According to the regularity of U , we can deduce several properties of BH(U ) If U has the cone property, then the space BH(U ) coincides with the space of functions in L 1 (U ) whose Hessian belongs to M b (U ; M n×n sym ). If U is bounded with Lipschitz boundary, the space BH(U ) can be embedded into W 1,n/(n−1) (U ). Moreover, if U is bounded with C 2 boundary, then the traces of u and ∇u, still denoted by u and ∇u are well defined for every u ∈ BH(U ); moreover we have u ∈ W 1,1 (∂U ), ∇u ∈ L 1 (∂U ; R n ), and ∂u ∂τ = ∇u · τ ∈ L 1 (∂U ) for every τ tangent vector to ∂U . Finally, if n = 2, then BH(U ) embeds into the space C(U ), of continuous functions on U .
Maximal monotone operators. Let X be a Banach space and let X ′ be its dual. Let A : X → X ′ be an operator, possibly multivalued, and let D(A) := {x ∈ X : Ax = Ø} be its domain. A is monotone if y 1 − y 2 , x 1 − x 2 ≥ 0 for every x 1 , x 2 ∈ D(A) and y 1 ∈ Ax 1 , y 2 ∈ Ax 2 .
A monotone operator is said to be maximal if it satisfies the following property: if (x, y) ∈ X × X ′ are such that y − η, x − ξ ≥ 0 for every ξ ∈ D(A) and η ∈ Aξ, then y ∈ Ax. We also recall the following useful property of maximal monotone operators (see [3, Chapter II, Lemma 1.3]): Proposition 2.1. Let A : X → X ′ be a maximal monotone operator, possibly multivalued. We assume that Then y ∈ Ax.
For the general properties of maximal monotone operators we refer, e.g., to [3] and [5].
A direct computation shows that the dual norm | · | * of | · | r is Moreover, it is easy to check that for every ξ ∈ M 2×2 sym |ξ| * = |ξ + (tr ξ)I| r , The elasticity tensor and its inverse. We denote by C r the elasticity tensor, which we recall to be a symmetric positive definite linear operator C r : M 2×2 sym → M 2×2 sym . Morever we denote by A r : M 2×2 sym → M 2×2 sym its inverse A r := C −1 r . It follows that there exist two constants α A and β A , with 0 < α A ≤ β A , such that In particular it holds |A r ξ| ≤ 2β A |ξ| for every ξ ∈ M 2×2 sym .
(3.6) Dissipation potential. Let K be a closed convex set of M 3×3 sym such that there exist two constants r H and R H , with 0 < r K ≤ R K , such that sym represents the set of admissible stresses in the reduced problem and can be characterised as follows: (see [8,Section 3.2]). In particular we note that if for some α 0 > 0, then (3.7) gives We define the set K r (Ω) := {σ ∈ L 2 (Ω; M 2×2 sym ) : σ(x) ∈ K r for a.e. x ∈ Ω}. The plastic dissipation potential is given by the support function of K r , namely H r : M 2×2 sym → [0, +∞) defined as H r (ξ) := sup σ∈Kr σ : ξ for every ξ ∈ M 2×2 sym . (3.9) It follows that H r is convex and positively one-homogeneous and there are two constants 0 < r H < R H such that Therefore H r satisfies the triangle inequality With this definition it is easy to deduce the important property where ∂H r (0) is the classical subdifferential of H r at the origin.
Kirchhoff-Love admissible triples. We now focus on the admissible configurations of the plate. Regarding the displacement u, we require it to be in the set KL(Ω) of Kirchhoff-Love displacements: We have the following alternative characterization of Kirchhoff-Love displacements: u ∈ KL(Ω) if and only if u 3 ∈ BH(ω) and there existsū ∈ BD(ω) such that The termsū, u 3 are called the Kirchhoff-Love components of u. For any given w ∈ H 1 (Ω; R 3 )∩KL(Ω), used to prescribe the Dirichlet boundary condition, we define the set A KL (w) of Kirchhoff-Love admissible triplets, as the class of all triplets In view of this latter property, in the following, given (u, e, p) ∈ A KL (w), we will always identify e with a function in L 2 (Ω; M 2×2 sym ) and p with a measure in M b (Ω ∪ Γ d ; M 2×2 sym ). To provide a useful characterisation of admissible triplets in A KL (w), let us first recall the definitions of zeroth and first order moments.
We definef ,f ∈ L 2 (ω; M 2×2 sym ) and f ⊥ ∈ L 2 (Ω; M 2×2 sym ) as the following orthogonal components (with respect to the scalar product of for a.e. x ∈ Ω. We namef the zero-th order moment of f andf the first order moment of f .
where ⊗ is the usual product of measures, and L 1 is the Lebesgue measure restricted to the third component of R 3 . We nameq the zero-th order moment of q andq the first order moment of q.
We are now ready to state the following characterisation of A KL (w).
As discussed in the introduction, since the plastic strain exists only as a measure, to properly use the stress-strain duality between Σ(Ω) and Π Γ d (Ω) we need first to consider a suitable notion of duality pairing (cf. [8]).
For every σ ∈ Σ(Ω) and ξ ∈ BD(ω), we define the distribution [σ : Eξ] on ω as For every σ ∈ Σ(Ω) and For every σ ∈ Σ(Ω) and v ∈ BH(ω), we set the distribution [σ : Finally, we combine the notions above for zeroth and first order moments to define, for every σ ∈ Σ(Ω) and We are now ready to define the stress-strain duality pairings as (3.14) To conclude, we recall the following integration by parts formula.
Since we shall consider homogeneous Neumann boundary condition for the stress, it is convenient to state the previous result in this setting.
The functions Φ N and Ψ λ . Let N ∈ N, N ≥ 4 be a fixed parameter and let α 0 > 0. We The function φ N is convex and of class C 1 with differential where (z) + := z ∧ 0 denotes the positive part of z. Therefore Hence, it is easy to see that Note that (3.15) implies that Dψ λ is Lipschitz continuous. Let us also introduce the functions The conjugate of ψ λ . In this subsection we compute the conjugate function of ψ λ . First of all we recall this definition.
Definition 3.7. Let X be a Banach space and let f : for every x ∈ X * .
We need the following three lemmas.
sym and assume that We will prove that ξ 1 = ξ 2 . Let us distinguish three cases: Then taking the norm | · | * of both members of the equality (3.20) and owing to (3.4), we infer and then, from (3.20), we obtain From this it easily follows that In such a case we have λ N −2 |ξ 1 | r = λ N −2 |ξ 2 | r from which |ξ 1 | r = |ξ 2 | r , and then we conclude as in case (i). Lemma 3.9. Let X be a reflexive Banach space. Let Ψ : X → R be convex and of class C 1 . If DΨ : X → X ′ is injective, then Ψ * is univalued and in particular differentiable, namely ∂Ψ * = DΨ * .
Proof. It is well-known from convex analysis that y = DΨ(x) if and only if x ∈ ∂Ψ * (y). Suppose that there are two elements x 1 , x 2 ∈ ∂Ψ * (y), then y = DΨ(x 1 ) = DΨ(x 2 ) implies x 1 = x 2 . This shows that ∂Ψ * is univalued, then Ψ * is differentiable and DΨ * = ∂Ψ * . Lemma 3.10. Let X be a reflexive Banach space and let Ψ : X → R be convex, of class C 1 , with DΨ : X → X ′ injective, and satisfying the condition Let F : X ′ → R be a convex function of class C 1 satisfying the following condition: for all x ∈ X, y ∈ X ′ we have that y = DΨ(x) if and only if x = DF (y). Then F = Ψ * + C for some constant C ∈ R.
Proof. Since DΨ * is univalued by Lemma 3.9, it holds y = DΨ(x) if and only if x = DΨ * (y). Therefore, if y = DΨ(x) we also have x = DF (y) and thus DF (y) = DΨ * (y). In particular we obtain that DF and DΨ * coincide on the range of DΨ. We then conclude if we show that DΨ is surjective. But this follows from hypothesis (3.21) and, e.g., [3, Proposition 2.6, Chapter II].
We are now ready to compute ψ * λ .
As a consequence of Proposition 3.11 we deduce that the conjugate function of Ψ λ in (3.19) is

The dynamic evolution problem: regularity
In this section we introduce the approximate problem to the dynamic model. We prove some preliminary lemmas before Theorem 4.4 which states the existence of a solution to the Norton-Hoff approximation. Then in Proposition 4.7 we study the regularity of the obtained solutions.
The Dirichlet datum of the problem is realized by a prescribed boundary displacement w with the following regularity . The total external load is a function L : [0, T ] → L 2 (Ω; R 3 ) which we decompose as horizontal and vertical forces f and g, namely We assume that the external forces satisfy the following uniform safe-load condition. Namely, we suppose the existence of ̺ : and satisfying, for all t ∈ [0, T ], , for some γ > 0 fixed and independent of t. The space Finally we make the following technical assumption on the external vertical force g, namely Notice that such condition does not follow from the safe-load condition. We assume for simplicity that there are no external loads on the Neumann boundary Γ n . Finally let us suppose that the initial data u 0 , σ 0 , and (v 0 ) 3 , satisfy the following properties We will denotev 0 simply by v 0 . Moreover we set p 0 := Eu 0 − A r σ 0 ∈ L 2 (Ω; M 2×2 sym ). Now we can state and prove the following result.
Lemma 4.1. Let T > 0, K r be of the form (3.7), assume (4.1), and that f and g satisfy (4.2)-(4.4). Assume that the initial conditions satisfy (4.5). Then for every integer N ≥ 4 and λ > 0 the problem for a constant C > 0 independent of λ and N .
Moreover we can prove the following additional regularity of solution of the problem (4.6).
Lemma 4.2. Let (u λ , σ λ , p λ ) be the solution of Lemma 4.1. Then there exists a constant C > 0 independent of λ and N such that Moreover, if we assume that the function f is independent of time t, then ) and there exists a constant C > 0 independent of λ and N such that for a constant C > 0 independent of λ and N . Coupling this with the estimate forσ λ in (4.8) we infer that for a constant C > 0 independent of λ and N . Using the fact that u λ is a Kirchhoff-Love function we conclude We will prove the two lemmas in a unique proof.
Proof. We proceed in six steps and we use a standard time discretization technique, with the aid of an implicit Euler scheme.
Step 1: Time discretization. For every integer k > 0 we consider a partition of the time interval [0, T ] into k subintervals of equal length δ k := T k , i.e., t i k := iδ k for every i = 0, . . . , k. We define For i = 1, . . . , k the triplet (u i k , σ i k , p i k ) is defined recursively as the solution of the minimum problem min where In (4.13) we have defined w i k := w(t i k ) for i = 0, . . . , k, and where f i k := f (t i k ), g i k := g(t i k ). We also set w −1 k := 2w 0 k − w 1 k and w −2 k := 2w −1 k − w 0 k . It follows from Proposition 3.11 that Ψ * λ is coercive with respect to the L 2 −norm and has more than linear growth. This fact, together with (3.5), implies that F i is coercive, lower semicontinuous and strictly convex. Then Korn inequality in H 1 ensures the existence and uniqueness of a solution for the problem (4.13).
Step 2: Euler-Lagrange equations. For all i = 0, . . . , k we claim that the solution (u i k , σ i k , p i k ) ∈ A reg (w i k ) of (4.13) satisfies the following system: where e i k := A r σ i k . The system (4.15) is satisfied by definition for the index i = 0. Moreover, by admissibility, and the first line in (4.15) follows. Let ǫ ∈ (−1, 1) and let ϕ ∈ H 1 (Ω; This, by arbitrariness of ϕ, entails , where the Neumann boundary conditions derive from (3.12) and (3.13). It remains to prove the last condition in (4.15). Let ǫ ∈ (−1, 1) and let η ∈ L 2 (Ω; M 2×2 sym ). We have , so that we can compare the values of F i at this point and at the minimum point (u i k , σ i k , p i k ), obtaining the last inequality following by convexity. From this, dividing by ǫ and letting ǫ → 0 + , we get where we also used that A r σ, C r η = σ, η for every σ, η ∈ M 2×2 sym . By arbitrariness of η expression (4.17) leads to This fact together with Lemma 3.8 and Lemma 3.9 yields in Ω, (4.18) which is equivalent to the last condition in (4.15).
For every i = −2, . . . , k we set We define two types of interpolations. The piecewise constant interpolations (u k , e k , p k ) : , and similarly are defined ω k and (v 3 ) k . In the same way we denote by g k and f k the piecewise constant function defined as and ̺ −1 k , we see that the expression above makes sense also for i = −1.
The piecewise affine interpolations are instead and analogous expressions forw k ,σ k ,p k , (ṽ 3 ) k , and (ω 3 ) k ,g k ,f k , and̺ k . Notice that, thanks to (4.12), setting t −1 k = −δ k , all the functions introduced so far are naturally defined on the interval [−δ k , T ] and it holds (v 3 Let us show that there exists a constant C > 0 independent of k, λ, and N , such that and To prove (4.19) we test (4.16) by ϕ = u i k − u i−1 k − w i k + w i−1 k . Then, owing to the first and last conditions in (4.15), we infer . For j ≥ 2, summing expression (4.21) on i = 1, . . . , j and rearranging terms we get Consequently, we rewrite the previous expression as Standard applications of Hölder and Young inequalities, together with the Gronwall Lemma gives the following estimates where the constant C > 0 depends on the following quantities and is independent of N , k, and λ. To obtain the previous estimate we have treated the term t j k 0 ̺ k , Dψ λ (σ k ) dt as follows. We have used the fact, ensured by the safe-load condition Now let us prove (4.20). Subtracting the first expression in (4.15) at step i with the one at step i − 1, dividing by δ k , and then testing the result with σ i k − σ i−1 k , we infer The first term in the left-hand side of (4.24) can be estimated from below by Using the last condition in (4.15), we note that the second term in the left-hand side of (4.24) is equal to which is nonnegative, since the differential of a convex function is a monotone operator. From (4.16) we can write the right-hand side of (4.24) as Let us first consider the case that f is independent of t. This implies that the second term of the right-hand side is null. Using this expression, let j ≥ 2 and sum (4.24) on i = 1, . . . , j. We get where the constant is independent of k, λ, and N , and depends on Let us now consider the general case in which f might depend on time t. This means that in the right-hand side of (4.25) there is the additional term Here the constant C depends on the norms (4.29) Coupling this last inequality with (4.25) we arrive at for all t ∈ [0, T ], where a 1 and a 2 belong to L 1 ([0, T ]) and are independent of k, λ, and N . Integrating in the variable t ∈ [0, T ] and using Young inequality we conclude (4.20), thanks to (4.19). Finally, writing Eũ k =ẽ k +p k , from the fact that p 0 := Eu 0 − e 0 ∈ L 2 (Ω; M 2×2 sym ), (4.15), (4.19), (4.20), and the Korn inequality, it follows that where C is a constant that does not depend on k, λ, and N .
Step 4: Existence. From (4.19), (4.20), and (4.30) we deduce that there exist such that, up to subsequences, This proves that σ λ =σ λ and v λ 3 =u λ 3 . In particular u λ 3 ∈ H 2 ([0, T ]; L 2 (ω)). Using that p 0 ∈ L 2 (Ω; M 2×2 sym ), estimate (3.15), and the first condition in (4.15), we infer Now it is convenient to introduce the notation implies, thanks to (4.34), that Let us show that (u, σ) satisfy (4.6). The definitions of σ k and ξ k allow us to rewrite (4.16) as First of all, as a consequence of the first condition in (4.15), we have that Let us consider the first term on the right hand-side. From the second equation in (4.15) it follows that Thanks to (4.42), we deduce that while, as a consequence of (4.38) and (4.42), the last term in the right-hand side of (4.48) tends to 0 as k → +∞. Finally, from (4.36) and (4.39), we get where in the inequality we have used (4.39). All these considerations lead to Note that the last equality in the previous formula derives from an integration by parts and from (4.43).
It remains to check that the initial conditions hold. The definition of the interpolate approximating functions gives Thus the initial conditions follow from the previous equalities together with (4.32),(4.36) and (4.42). This proves the existence of solutions for (4.6) and concludes Step 4.
Step 6: Conclusion. The estimates (4.7) and (4.8) are obtained by simply using (4.32)-(4.37) to pass to the limit as k → +∞ in (4.19), (4.20). Similarly, starting from (4.26) we infer (4.9). Now we are ready to prove the existence and uniqueness of solutions for an approximating problem of Norton-Hoff type.
Since (4.7) and (4.8) are uniform estimates with respect to λ, we can pass to the limit as λ → +∞ and deduce the existence of ; L 2 (ω)) such that, up to subsequences, In particular (4.50) and (4.52) hold true thanks to (4.8). By (4.7) there exists ), satisfying, up to a subsequence, (4.57) Passing to the limit in the first equation of (4.6) we get for a.e. t ∈ [0, T ]. By (4.54) and (4.55) we can pass to the limit in (4.43) as λ tends to +∞ and infer that Eu λ , σ λ dt and exploiting (4.7) and the estimates on w and the external forces, we arrive at where C is a positive constant independent of λ and N . Set τ λ := χ {|σ λ |r≤λ} σ λ . From (4.60) it follows that Using (4.60) again, we infer where the equality derives from the definition of τ λ . Hence  Morever, by (4.61), the inequality in (4.51) is achieved.
Step 4: uniqueness. The proof of uniqueness is very similar to the one in Step 5 of Lemma 4.1 that can be easily adapted.
Theorem 4.5. In the hypotheses of Theorem 4.4, assume in addition that the force f is independent of time t. Then the solution (σ N , u N ) also satisfies . and the following estimates: where C > 0 is a constant independent of N .
Remark 4.6. Since u N is a Kirchhoff-Love displacement, the first estimate in (4.69) together with the continuous immersion of BD(ω) into L 2 (ω; R 2 ) and of BH( where C is a positive constant independent of N .
Proof. The proof follows the same lines of that of Theorem 4.4, with the following changes.
In step one we observe that, thanks to estimate (4.9) we conclude estimate (4.67) and the second one in (4.69), which in particular imply the additional regularity σ N ∈ W 1,∞ ([0, T ]; L 2 (Ω; M 2×2 sym )), and u N 3 ∈ W 2,∞ ([0, T ]; L 2 (ω)). We now prove (4.68). Let us multiply the second equation in (4.15) by σ i k − σ i−1 k . This yields It is clear that the first term on the left-hand side is positive, while ) as a consequence of the convexity of ψ λ . Therefore, integrating by parts, we get Let j > 1 and sum on i = 1, . . . , j. Rearranging terms we infer Hence, since f is independent of time the last term is null and we can use (4.9) to estimate the right-hand side as By (4.19), (4.20), and (4.26), we can pass to the limit as k → +∞ and we get for a constant C independent of N . Consequently where τ λ has been first used in (4.61). Therefore convergence (4.62) also takes place with respect to the weak star topology of the space L ∞ ([0, T ]; L N (Ω; M 2×2 sym )), so that σ ∈ L ∞ ([0, T ]; L N (Ω; M 2×2 sym )) and (4.68) holds true.
Let us prove the first estimate in (4.69). We test the kinematic compatibility by σ N . We arrive at We now observe that the estimates (4.67) and the second in (4.69) imply that the right-hand side is uniformly bounded. Hence we conclude

74)
• for every open set Ω ′ compactly contained in Ω there exists a constant C 2 > 0 depending on Ω ′ but independent of N such that (4.75) • for every open set ω ′ compactly contained in ω there exists a constant C 3 > 0 depending on ω ′ but independent of N such that Before proving the result let us introduce the difference quotient operator, that for h ∈ R is defined as for a given function v : R 3 → R. We recall some important properties of the difference quotient. If f ∈ L p (Ω; R) and g ∈ L q (Ω; R), ψ ∈ C 0 (Ω; R) with either f or g with compact support in Ω, then for h sufficiently small we have Step 1. Let us first prove higher regularity for σ λ , the approximating solution of σ N found in Lemma 4.1. Let ϕ ∈ C ∞ c (ω), let α ∈ {1, 2}, and h ∈ (−1, 1) be sufficiently small. We multiply the first equation in (4.6) by The second term on the left-hand side of (4.78) can be rewritten as In particular it is positive, because for all ξ, η ∈ M 2×2 sym . Let us focus on the right-hand side of (4.78). Since u λ (t) ∈ KL(Ω) we get Integration by parts yields where we implicitely assume sum over the indices β and γ, whereas α is kept fixed. This last expression can be rewritten as where in the second equality we have used that −∂ α u γ = −2(Eu) αγ + ∂ γ u α , while in the last equality we have used (4.77).
As for the third term on the right-hand side of (4.82), we write Combining (4.83) and (4.84) and using that u ∈ KL(Ω) we obtain .
Using this, (4.78), (4.79), (4.80), (4.82), the computations carried on so far lead to for all t ∈ [0, T ]. From (3.15) it follows that Thanks to (4.7) and (4.8) we have (σ λ ,u λ ) ∈ H 1 ([0, T ]; L 2 (Ω; M 2×2 sym ))×L 2 ([0, T ]; L 2 (Ω; R 3 )). Using the regularity of the external data (4.1) and (4.2), the previous inequality implies for all t ∈ [0, T ] because u(t) ∈ KL(Ω) and BH(ω) embeds into H 1 (ω). Notice that these estimates are uniform in t ∈ [0, T ]. Therefore we can pass to the limit in (4.86) as h tends to 0, and thanks to (4.81) we get (4.87) The first, sixth, and seventh term in the right-hand side of (4.87) can be estimated using Cauchy-Schwartz and Hölder inequality. Similarly the second term is estimated by Now we consider the third term on the right-hand side of (4.87). Using the expression of Dψ λ , we get where we have used (4.7), (4.8), and (4.60). In such a way the last term is absorbed by the corresponding one in the left-hand side of (4.87). The part of the third term in the right-hand side of (4.87) containing D α ̺ is treated together with the last two terms. The estimate for the fifth term is analogous and is based on the boundedness (4.60). Let us treat the fourth term. To this aim we recall thatu λ is uniformly bounded with respect to λ and N in L 1 ([0, T ]; L 2 (Ω)) by (4.11). Therefore where C is independent of λ and N . We finally study the last two terms in (4.87). Since Dψ λ (σ λ ) is uniformly bounded in L 1 ([0, T ]; L 1 (Ω; M 2×2 sym )), it suffices to observe that ̺ be- ). Combining all these estimates and the hypotheses we conclude where the constant C, as well as r L 1 ([0,T ]) , are independent of λ and N . Therefore, by applying the Gronwall Lemma we infer that there is a constant C > 0 independent of λ and N such that Step 2. By (4.54) and (4.56) we know that ϕσ λ (t) ⇀ ϕσ N (t) weakly in L 2 (Ω), (4.90) for all t ∈ [0, T ]. Estimate (4.89) entails that the previous convergences take place in the H 1 (Ω) and H 1 (ω) weak topologies, respectively. Therefore by lower-semicontinuity we conclude for all t ∈ [0, T ], with the constant C > 0 independent of N . We have proved that (D α σ N ) N and (D αu Step 3. Now we prove higher regularity with respect to the third coordinate x 3 , namely (4.75). Given ϕ ∈ C ∞ c (Ω), we test the first equation in (4.49) by . The right-hand side can be integrated by parts as Integrating this equation with respect to time, using (3.5), (4.52), and the regularity of w and ̺, we infer Using the estimate obtained in Step 2, the Gronwall Lemma implies that the right hand-side of (4.94) is uniformly bounded with respect to N . Hence we have L 2 + C, with C independent of h and N . Passing to the limit as h tends to 0 and using the assumptions on σ 0 we get α A ϕD 3 σ N (t) 2 L 2 ≤ C, with C independent of N . This concludes the proof.
We conclude with two observations. Remark 5.2. As pointed out in [18,Remark 5.3], one cannot expect in general the uniqueness of the horizontal components of the displacement u and of the plastic strain p.