Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion
aa r X i v : . [ m a t h . A P ] J a n STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY WITH APRESSURE DEPENDENT YIELD CRITERION
JEAN-FRANC¸ OIS BABADJIAN AND MARIA GIOVANNA MORA
Abstract.
This work is devoted to establishing a regularity result for the stress tensor in quasi-static planar isotropic linearly elastic – perfectly plastic materials obeying a Drucker-Prageror Mohr-Coulomb yield criterion. Under suitable assumptions on the data, it is proved thatthe stress tensor has a spatial gradient that is locally squared integrable. As a corollary, theusual measure theoretical flow rule is expressed in a strong form using the quasi-continuousrepresentative of the stress. Introduction
Perfect plasticity is a class of models in continuum solid mechanics involving a fixed thresholdcriterion on the Cauchy stress. When the stress is below a critical value, the underlying mate-rial behaves elastically, while the saturation of the constraint leads to permanent deformationsafter unloading back to a stress-free configuration. Elasto-plasticity represents a typical inelasticbehavior, whose evolution is described by means of an internal variable, the plastic strain.To formulate more precisely the problem, let us consider a bounded open set Ω ⊂ R n (in thefollowing, only the dimension n = 2 will be considered), which stands for the reference configurationof an elasto-plastic body. In the framework of small strain elasto-plasticity the natural kinematicand static variables are the displacement field u : Ω × [0 , T ] → R n and the stress tensor σ :Ω × [0 , T ] → M n × n sym , where M n × n sym is the set of n × n symmetric matrices. In quasi-statics theequilibrium is described by the system of equations − div σ = f in Ω × [0 , T ] , for some given body loads f : Ω × [0 , T ] → R n . Perfect plasticity is characterized by the existenceof a yield zone in which the stress is constrained to remain. The stress tensor must indeed belongto a given closed and convex subset K of M n × n sym with non empty interior: σ ∈ K. If σ lies inside the interior of K , the material behaves elastically, so that unloading will bring thebody back to its initial configuration. On the other hand, if σ reaches the boundary of K (calledthe yield surface), a plastic flow may develop, so that, after unloading, a non-trivial permanentplastic strain will remain. The total linearized strain, denoted by Eu := ( Du + Du T ) /
2, is thusadditively decomposed as Eu = e + p. The elastic strain e : Ω × [0 , T ] → M n × n sym is related to the stress through the usual Hooke’s law σ := C e, Date : August 16, 2018.
Key words and phrases.
Elasto-plasticity, Convex analysis, Quasi-static evolution, Regularity, Functions ofbounded deformation, Capacity. where C is the symmetric fourth order elasticity tensor. The evolution of the plastic strain p :Ω × [0 , T ] → M n × n sym is described by means of the flow rule˙ p ∈ N K ( σ ) , (1.1)where N K ( σ ) is the normal cone to K at σ . From convex analysis, N K ( σ ) = ∂I K ( σ ), i.e. , itcoincides with the subdifferential of the indicator function I K of the set K (where I K ( σ ) = 0if σ ∈ K , while I K ( σ ) = + ∞ otherwise). Hence, from convex duality, the flow rule can beequivalently written as σ : ˙ p = max τ ∈ K τ : ˙ p =: H ( ˙ p ) , (1.2)where H : M n × n sym → [0 , + ∞ ] is the support function of K . This last formulation (1.2) is nothingbut Hill’s principle of maximum plastic work, and H ( ˙ p ) denotes the plastic dissipation.Standard models used for most of metals or alloys are those of Von Mises and Tresca. Thesekinds of materials are not sensitive to hydrostatic pressure, and plastic behavior is only generatedthrough critical shearing stresses. In these models, if σ D := σ − tr σn Id stands for the deviatoricstress, the elasticity set K is of the form { σ ∈ M n × n sym : κ ( σ D ) ≤ k } , where k >
0. The Von Mises yield criterion corresponds to κ ( σ D ) = | σ D | , while that of Trescato κ ( σ D ) = σ n − σ where σ ≤ · · · ≤ σ n are the ordered principal stresses. The mathematicalanalysis of such models has been performed in [28, 30, 5, 12].On the other hand, in the context of soil mechanics, materials such as sand or concrete turn outto develop permanent volumetric changes due to hydrostatic pressure. Typical models are thoseof Drucker-Prager and Mohr-Coulomb (see [18]), which can be seen as generalizations of the VonMises and Tresca models, respectively. In these cases, the elasticity set K takes the form { σ ∈ M n × n sym : κ ( σ D ) + α tr σ ≤ k } , where α, k >
0. The main difference between metals and soils, is that for the latter, there are ingeneral no directions along which the Cauchy stress is bounded. These models have been studiedin [9] (see also [25]).A common feature to all the models of perfect plasticity described so far is that they developstrain concentration leading to discontinuities of the displacement field. This has been a majordifficulty in defining a suitable functional framework for the study of such problems. It has beenovercome by the introduction of the space BD of functions of bounded deformation (see [29, 30])and through a suitable relaxation procedure (see [7, 24]). Solutions in the energy space must atleast satisfy the following regularity: for all t ∈ [0 , T ], u ( t ) ∈ BD (Ω) , e ( t ) , σ ( t ) ∈ L (Ω; M n × n sym ) , p ( t ) ∈ M (Ω; M n × n sym ) , where M (Ω; M n × n sym ) stands for the space of bounded Radon measures in Ω.Higher regularity of solutions appears therefore as a natural question. For dynamical problemsit has (only recently) been established in [23] that for any elasticity set K the solutions are smoothin short time, provided the data are smooth and compactly supported in space. Such a result doesnot hold in the static or quasi-static cases (see the examples in [28, Section 2] or [17, Section 10]).However, some partial regularity results are available for the stress in some particular situations.Indeed, it has been proved in [10, 26, 17, 14] that for a Von Mises elasticity set, the Cauchy stresssatisfies σ ∈ L ∞ (0 , T ; H (Ω; M n × n sym )) . Unfortunately, the proofs of these results are very rigid and do not easily extend to other types ofelasticity sets. A more general result has been obtained in [20], where it has been proved that the
TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 3 same regularity result holds if K = K D ⊕ R Id, where K D is a smooth compact convex subset ofdeviatoric symmetric matrices, with positive curvature.In the footsteps of [9], we address here the question of deriving similar regularity properties forthe stress tensor in the case of Drucker-Prager and Mohr-Coulomb elasticity sets. A result in thisdirection has been obtained in [27] in the static case. The object of this present work is to extendthis result to the quasi-static case. We indeed prove that in dimension n = 2, the stress tensor hasthe expected regularity σ ∈ L ∞ (0 , T ; H (Ω; M × )) (see Theorem 2.7).Our proof rests on a duality approach, analogous to that of [27] and [17]. Since the solutionsto the perfect plasticity model are singular, one needs first to regularize the problem. To thisaim we consider a visco-plastic approximation of Perzyna type (see [28]). In contrast with theKelvin-Voigt visco-elastic regularization used in [27, 17], it modifies the dissipation potential H (see (1.2)) in such a way that the regularized flow rule is not given by a differential inclusion as in(1.1), but is a differential equation. In other words, the regularized plastic strain rate is univocallydetermined by the stress (see (3.3)). To the best of our knowledge, this is the first time that thiskind of approximation is used to establish regularity of the stress.In the following we explain the strategy of our proof and the main difficulties. For simplicityof exposition we neglect the terms due to the visco-plastic approximation (which are those ε -dependent), we assume the solution to be smooth, and we show how a uniform estimate on the H (Ω; M n × n sym )-norm of σ ( t ) can be obtained.In the static case [27] and in the absence of external body loads, the equilibrium states ( u, e, p )minimize the energy functional( v, η, q ) ˆ Ω C η : η dx + ˆ Ω H ( q ) dx among all triples ( v, η, q ) satisfying the additive decomposition Ev = η + q and the boundarycondition. Minimizing first with respect to q shows that u is actually a minimizer of v ˆ Ω g ( Ev ) dx, where g is the convex conjugate of the auxiliary energy τ C − τ : τ + I K ( τ ), and the stress isthen given by σ = Dg ( Eu ) . (1.3)This formula for the stress is the starting point of the analysis. In order to get estimates on thespatial gradient of σ , it would be convenient to differentiate (1.3). Unfortunately, the function g being only of class C with Lipschitz continuous partial derivatives, the classical chain rule formulafor the composition of a Lipschitz function with a vector valued (Sobolev) function does not apply.To overcome this problem, we compute explicitely the expression of g (see (3.13)) and use a generalchain rule formula established in [3] to get a formula of the type ∂ k σ = D g ( Ev ) ∂ k Ev .The study of the quasi-static case introduces further difficulties. In the case of Von Mises plas-ticity [17], the previous method is applied to the incremental problem where ( u i , e i , p i ) minimizes( v, η, q ) ˆ Ω C η : η dx + ˆ Ω H ( q − p i − ) dx among all admissible triples ( v, η, q ) at time t i , or still u i minimizes v ˆ Ω g ( Ev − p i − ) dx. It is shown that the stress σ i := Dg ( Eu i − p i − ) satisfies an H (Ω; M n × n sym ) estimate that is uniformwith respect to the viscosity parameter, but may possibly depend on the time step. Afterwards,a uniform estimate with respect to the time step is established showing the desired regularity J.-F. BABADJIAN AND M.G. MORA result. The main drawback of this approach is that it necessitates to perform twice almost thesame computations.In contrast with [17], we directly work on a time continuous model, and use the underlyingvariational structure to establish a similar formula (see (3.15)) for the stress as σ = Dg ( ξ ) , ξ := e + ˙ p. The strategy consists then in differentiating the equilibrium equation, take ( ∂ k ˙ u ) ψ as test function(where ψ is a suitable cut-off function), and deduce an inequality that provides a bound on ∂ k σ (see Proposition 5.1). The main difficulty in doing so is to deal with a term of the form (see (5.13)) ˆ t ˆ Ω | ∂ k σ || ξ | ψ dx ds, (1.4)since ξ is only bounded in L (Ω × (0 , T ); M n × n sym ). We thus need to absorb this term by some of thecoercivity terms of the left hand side ˆ t ˆ Ω ∂ k σ : ∂ k σψ dx ds + ˆ t ˆ Ω ∂ k σ : ∂ k ξψ dx ds. To this end, we use the special structure of the function g , together with the formula ∂ k σ = D g ( ξ ) ∂ k ξ , to show that the integral in (1.4) can be controled by M (cid:18) ˆ t ˆ Ω ∂ k σ : ∂ k ξψ dx ds (cid:19) / , where M only depends on various norms of ( u, e, p ) in the energy space (see (5.23)–(5.24)).Note that the estimates performed in the proof of Proposition 5.1 should actually hold true inany space dimension. However, since the final estimate (5.27) involves the L (Ω × (0 , T ); R n )-normof ˙ u and this is controled only in dimension n = 2 owing to the continuous embedding of BD (Ω)into L (Ω; R ) , this estimate turns out to be uniform with respect to the viscosity parameter onlyin dimension n = 2. This special role played by the dimension n = 2 was already observed in thepapers [9, 25, 27], and is a recurrent feature in plasticity models where the elasticity set has nobounded directions.A direct consequence of this result is that, by means of the quasi-continuous representative ofthe stress with respect to the H -capacity, one can express the flow rule in a pointwise strong form.Indeed, since σ ( t ) ∈ L (Ω; M × ) and ˙ p ( t ) ∈ M (Ω; M n × n sym ), the product between σ ( t ) and ˙ p ( t ) is ingeneral not well defined. This issue is usually overcome by introducing a distributional notion ofduality σ ( t ) : ˙ p ( t ) as in [22]. In the present situation the precise representative, denoted by ˜ σ ( t ),is defined up to a set of zero capacity and thus it turns out to be | ˙ p ( t ) | -measurable. This enablesone to give a sense to the pointwise product of the (quasi-continuous) stress ˜ σ ( t ) with the measure˙ p ( t ), and, in particular, to express Hill’s principle of maximal plastic work (1.2) in a strong sense.The paper is organised as follows. In Section 2 we introduce the precise mathematical settingto formulate accurately the model of perfect plasticity, and state our main regularity result, The-orem 2.7. In Section 3 we approximate the perfect plasticity model by means of a visco-plasticregularization, and establish the convergence of the solutions, as well as some (non uniform) reg-ularity properties of the approximating solutions. Section 4 is devoted to establish a chain ruletype formula for the stress which, as explained above, is instrumental for the subsequent analysis.In Section 5 we prove an estimate on the (visco-plastic) stress in L ∞ (0 , T ; H (Ω; M × )), whichis uniform with respect to the viscosity parameter. Owing to this estimate we complete the proofof Theorem 2.7. The last section is devoted to show the validity of the flow rule by means of thequasi-continuous representative of the stress. In dimension n = 3, BD (Ω) only embeds into L / (Ω; R ) TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 5 Mathematical formulation of the problem
Notation.
Vectors and matrices. If a, b ∈ R n , we write a · b for the Euclidean scalar product, and wedenote by | a | = √ a · a the associated norm.We write M n × n for the set of real n × n matrices, and M n × n sym for that of all real symmetric n × n matrices. Given two matrices A and B ∈ M n × n , we use the Frobenius scalar product A : B = p tr( A T B ) (where A T is the transpose of A , and tr A is its trace), and we denoteby | A | = √ A : A the associated norm. If A ∈ M n × n , we denote by A D := A − n (tr A )Id thedeviatoric part of A , which is a trace free matrix. We recall that for any two vectors a, b ∈ R n , a ⊗ b := ab T ∈ M n × n stands for the tensor product, and a ⊙ b := ( a ⊗ b + b ⊗ a ) / ∈ M n × n sym denotesthe symmetric tensor product.2.1.2. Functional spaces.
We use standard notation for Lebesgue and Sobolev spaces.Let X ⊂ R n be a locally compact set and Y be an Euclidean space. We write M ( X ; Y ) (orsimply M ( X ) if Y = R ) for the space of bounded Radon measures in X with values in Y , endowedwith the norm | µ | ( X ), where | µ | ∈ M ( X ) is the variation of the measure µ . The Lebesgue measurein R n is denoted by L n , and the ( n − H n − .If U ⊂ R n is an open set, BD ( U ) stands for the space of functions of bounded deformation in U , i.e. , u ∈ BD ( U ) if u ∈ L ( U ; R n ) and Eu ∈ M ( U ; M n × n sym ), where Eu := ( Du + Du T ) / Du is the distributional derivative of u . We recall that, if U ⊂ R n is bounded and has a Lipschitzboundary, BD ( U ) can be embedded into L n/ ( n − ( U ; R n ). We refer to [30] for general propertiesof this space.2.1.3. Capacity.
We finally recall the definition and several facts about capacity (see [1]). LetΩ ⊂ R n be an open set. The capacity of a set A ⊂ Ω in Ω is defined byCap( A ) := inf (cid:26) ˆ Ω |∇ u | dx : u ∈ H (Ω) , u ≥ L n -a.e. in a neighborhood of A (cid:27) . One of the interests of capacity is that it enables one to give an accurate sense to the pointwisevalue of Sobolev functions (see [1, Section 6.1]). More precisely, every u ∈ H (Ω) has a quasi-continuous representative ˜ u , which is uniquely defined except on a set of capacity zero in Ω. Itmeans that ˜ u = u L n -a.e. in Ω, and that, for each ε >
0, there exists a closed set A ε ⊂ Ω such thatCap(Ω \ A ε ) < ε and ˜ u | A ε is continuous on A ε . In addition (see [1, Theorem 6.2.1]), there exists aBorel set Z ⊂ Ω with Cap( Z ) = 0 such thatlim r → + L n ( B r ( x ) ∩ Ω) ˆ B r ( x ) ∩ Ω u ( y ) dy = ˜ u ( x ) for all x ∈ Ω \ Z. Description of the model.
The reference configuration.
We denote byΩ ⊂ R a bounded connected open set with Lipschitz boundary ( A )the reference configuration of an elasto-plastic material.2.2.2. Boundary condition.
We assume that the body is subjected to a time-dependent boundarydisplacement, which is the trace on ∂ Ω of a function w ( t ) : Ω → R with w ∈ AC ([0 , T ]; H (Ω; R )) . ( A ) J.-F. BABADJIAN AND M.G. MORA
The elastic energy.
We consider an isotropic body whose fourth order elasticity tensor C isgiven by C e = λ (tr e )Id + 2 µe for all e ∈ M × , ( A )where λ and µ are the Lam´e coefficients satisfying µ > λ + µ >
0. Note that there exist twoconstants c , c > c | e | ≤ C e : e ≤ c | e | for all e ∈ M × . (2.1)Setting K := λ + µ , the inverse of C can be represented as C − σ = 14 K (tr σ )Id + 12 µ σ D for all σ ∈ M × . We define the elastic energy, for all e ∈ L (Ω; M × ), by Q ( e ) := 12 ˆ Ω C e ( x ) : e ( x ) dx. The elasticity set.
In this paper we are interested in the
Drucker-Prager and
Mohr-Coulomb models, where the elasticity domain is a closed and convex cone with vertex lying on the axis ofhydrostatic stresses given by K := { σ ∈ M × : | σ D | + α tr σ ≤ κ } . ( A )In the previous formula α > κ > External forces.
We consider a time-dependent body load f ( t ) : Ω → R satisfying f ∈ AC ([0 , T ]; L (Ω; R )) , ( A )which satisfies the usual safe-load condition: there exist χ ∈ AC ([0 , T ]; L (Ω; M × )) and a con-stant δ ∈ (0 , κ ) such that for every t ∈ [0 , T ] ( − div χ ( t ) = f ( t ) in Ω , | χ D ( t ) | + α tr χ ( t ) ≤ κ − δ in Ω . ( A )2.2.6. The dissipation energy.
We define the support function H : M × → [0 , + ∞ ] of K by H ( p ) := sup σ ∈ K σ : p for all p ∈ M × . Since K is closed and convex, H is convex, lower semicontinuous, and positively 1-homogeneous.In addition, since 0 belongs to the interior of K , the functions H enjoys the following coercivityproperty: there exists c > H ( p ) ≥ c | p | for all p ∈ M × . It is easy to establish the following explicit formula for the function H . Lemma 2.1.
For all p ∈ M × , H ( p ) = κ tr p α if | p D | ≤ tr p α , + ∞ if | p D | > tr p α . Note that in dimension n = 2 the two models are equivalent because of the algebraic identity √ | σ D | = σ max − σ min , where σ max (resp. σ min ) is the largest (resp. lowest) eigenvalue of σ D . TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 7
The dissipated energy functional is then defined, for all p ∈ L (Ω; M × ), by H ( p ) := ˆ Ω H ( p ( x )) dx. As a consequence of the previous properties of H , we infer that H is sequentially weakly lowersemicontinuous in L (Ω; M × ). Since L (Ω; M × ) is not reflexive (bounded sequences in thatspace are only weakly* sequentially compact in the space of measures), it will also be useful toextend the definition of H when p ∈ M (Ω; M × ). According to [21], we define the non-negativeBorel measure H ( p ) := H (cid:18) dpd | p | (cid:19) | p | , where dpd | p | is the Radon-Nikodym derivative of p with respect to its variation | p | . In general, themeasure H ( p ) is not even locally finite. However, if further H ( p ) has finite mass, i.e. , if H ( p ) is abounded Radon measure, we can define the dissipation functional H ( p ) := H ( p )(Ω) . In that case, the results of [15, 16] apply and H ( p ) can be expressed by means of a duality formula.If H ( p ) ∈ M (Ω), we get that (see [9]) ˆ Ω ϕ dH ( p ) = sup (cid:26) ˆ Ω ϕσ : dp : σ ∈ C ∞ (Ω; K ) (cid:27) , (2.2)for any ϕ ∈ C (Ω) with ϕ ≥
0, and in particular H ( p ) = sup (cid:26) ˆ Ω σ : dp : σ ∈ C ∞ (Ω; K ) (cid:27) . (2.3)Note also that the Reshetnyak Theorem (see [4, Theorem 2.38]) applies here, so that H is sequen-tially weakly* lower semicontinuous in M (Ω; M × ).2.2.7. Spaces of admissible fields.
Given a prescribed boundary displacement ˆ w ∈ H (Ω; R ), wewill consider the following spaces of kinematically admissible fields: A r ( ˆ w ) := (cid:8) ( v, η, q ) ∈ H (Ω; R ) × L (Ω; M × ) × L (Ω; M × ) : Ev = η + q a.e. in Ω , v = ˆ w H -a.e. on ∂ Ω (cid:9) , and A ( ˆ w ) := n ( v, η, q ) ∈ BD (Ω) × L (Ω; M × ) × M (Ω; M × ) : Ev = η + q in Ω , q = ( ˆ w − v ) ⊙ ν H on ∂ Ω o , where ν is the outer unit normal to ∂ Ω.The space of plastically admissible stresses is defined by K := { τ ∈ L (Ω; M × ) : τ ( x ) ∈ K for a.e. x ∈ Ω } , and the space of statically admissible stresses is given by S := { τ ∈ L (Ω; M × ) : div τ ∈ L (Ω; R ) } . J.-F. BABADJIAN AND M.G. MORA
Stress/strain duality.
The duality pairing between stresses and plastic strains is a priori notwell defined, since the former are only squared Lebesgue integrable, while the latter are possiblysingular measures. Following [22], we define the following distributional notion of duality.
Definition 2.2.
Let σ ∈ S and ( u, e, p ) ∈ A ( ˆ w ) with ˆ w ∈ H (Ω; R ). We define the distribution[ σ : p ] ∈ D ′ ( R ) supported in Ω by h [ σ : p ] , ϕ i = ˆ Ω ϕ ( ˆ w − u ) · div σ dx + ˆ Ω σ : [( ˆ w − u ) ⊙ ∇ ϕ ] dx + ˆ Ω σ : ( E ˆ w − e ) ϕ dx (2.4)for every ϕ ∈ C ∞ c ( R ). The duality product is then defined as h σ, p i := h [ σ : p ] , i = ˆ Ω ( ˆ w − u ) · div σ dx + ˆ Ω σ : ( E ˆ w − e ) dx. Remark 2.3.
Note that the first and second integrals in (2.4) are well defined since BD (Ω) isembedded into L (Ω; R ) for n = 2. Moreover, according to the integration by parts formula in BD (Ω) (see [8, Theorem 3.2]), if σ ∈ S ∩ C (Ω; M × ), we have h [ σ : p ] , ϕ i = ˆ Ω ϕσ : dp for all ϕ ∈ C ∞ c ( R ) . (2.5)A convolution argument shows that (2.5) remains true provided σ ∈ S∩C (Ω; M × ) and ϕ ∈ C ∞ (Ω).Using this notion of stress/strain duality, the duality formulas (2.2) and (2.3) can be nowextended to less regular statically and plastically admissible stresses. If p ∈ M (Ω; M × ) with H ( p ) < + ∞ , ˆ Ω ϕ dH ( p ) = sup n h [ σ : p ] , ϕ i : σ ∈ K ∩ S o , for all ϕ ∈ C ∞ (Ω) with ϕ ≥
0, and in particular, H ( p ) = sup n h σ, p i : σ ∈ K ∩ S o . The following result establishes a coercivity property of the functional p
7→ H ( p ) − h χ ( t ) , p i (see e.g. [9, Proposition 6.1]). Proposition 2.4.
Let ˆ w ∈ H (Ω; R ) and ( u, e, p ) ∈ A ( ˆ w ) . Then there exists a constant C δ,α ,depending on δ and α , such that the following coercivity estimate holds: H ( p ) − h χ ( t ) , p i ≥ C δ,α k p k M (Ω; M × ) for every t ∈ [0 , T ] . Initial condition.
We finally consider an initial datum ( u , e , p ) ∈ A ( w (0)) and σ := C e satisfying the stability conditions σ ∈ K , − div σ = f (0) in Ω . ( A )We are now in position to state the existence result obtained in [9]. Theorem 2.5.
Assume ( A ) – ( A ) . Then there exist u ∈ AC ([0 , T ]; BD (Ω)) ,e, σ ∈ AC ([0 , T ]; L (Ω; M × )) ,p ∈ AC ([0 , T ]; M (Ω; M × )) , with ( u (0) , e (0) , p (0)) = ( u , e , p ) that satisfy, for all t ∈ [0 , T ] : TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 9 (i) the kinematic compatibility ( Eu ( t ) = e ( t ) + p ( t ) in Ω ,p ( t ) = ( w ( t ) − u ( t )) ⊙ ν H on ∂ Ω , (ii) the static and plastic admissibility σ ( t ) = C e ( t ) , − div σ ( t ) = f ( t ) in Ω ,σ ( t ) ∈ K , (iii) the energy balance Q ( e ( t )) + ˆ t H ( ˙ p ( s )) ds = Q ( e ) + ˆ t ˆ Ω σ : E ˙ w dx ds + ˆ t ˆ Ω f · ( ˙ u − ˙ w ) dx ds. (2.6) Moreover, the stress σ is unique, and for a.e. t ∈ [0 , T ] the distribution [ σ ( t ) : ˙ p ( t )] is well defined,and it is a measure in M (Ω) satisfying Hill’s principle of maximum plastic work H ( ˙ p ( t )) = [ σ ( t ) : ˙ p ( t )] in M (Ω) . (2.7) Remark 2.6.
According to Lemma 2.1, the flow rule can be equivalently written as κ tr ˙ p ( t )2 α = [ σ ( t ) : ˙ p ( t )] , and | ˙ p D ( t ) | ≤ tr ˙ p ( t )2 α in M (Ω) . The main result of this work is the following regularity result.
Theorem 2.7.
Assume ( A ) – ( A ) and that α = 1 / √ in ( A ) . Under the additional hypothesesthat w ∈ H ([0 , T ]; H (Ω; R )) , χ ∈ W , ∞ ([0 , T ]; L ∞ (Ω; M × )) ∩ H ([0 , T ]; H (Ω; M × )) , f ∈ L ∞ (0 , T ; H (Ω; R )) ∩ L (0 , T ; H (Ω; R )) ∩ L ∞ (Ω × (0 , T ); R ) and e ∈ H (Ω; M × ) , the stresstensor satisfies σ ∈ L ∞ (0 , T ; H (Ω; M × )) . Perzyna visco-plastic approximations
In order to prove Theorem 2.7, we will need to consider a regularized problem. This will be doneby means of a so-called Perzyna visco-plastic approximation. The following result, formulated herein a modern language, has been established in [28].Since the initial data ( u , e , p ) given in ( A ) does not belong to the right energy space associ-ated to the visco-plastic model, we first need to regularize it. According to [13, Lemma 5.1], thereexists a sequence ( u ,ε ) ⊂ H (Ω; R ) such that u ,ε = w (0) H -a.e. on ∂ Ω, u ,ε → u stronglyin L (Ω; R ), and Eu ,ε ⇀ Eu weakly* in M (Ω; M × ). Setting p ,ε = Eu ,ε − e , we get that( u ,ε , e , p ,ε ) ∈ A r ( w (0)). Proposition 3.1.
Assume ( A ) – ( A ) . Let ε > and let ( u ,ε , e , p ,ε ) ∈ A r ( w (0)) be constructedas above. Then there exists a unique triple ( u ε , e ε , p ε ) ∈ AC ([0 , T ]; H (Ω; R )) × AC ([0 , T ]; L (Ω; M × )) × AC ([0 , T ]; L (Ω; M × )) such that ( u ε (0) , e ε (0) , p ε (0)) = ( u ,ε , e , p ,ε ) , for all t ∈ [0 , T ]( u ε ( t ) , e ε ( t ) , p ε ( t )) ∈ A r ( w ( t )) , σ ε ( t ) = C e ε ( t ) , − div σ ε ( t ) = f ( t ) in Ω , and for a.e. t ∈ [0 , T ] σ ε ( t ) − ε ˙ p ε ( t ) ∈ ∂H ( ˙ p ε ( t )) in Ω . (3.1) Remark 3.2.
For every q ∈ M × , we define the function H ε ( q ) := H ( q ) + ε | q | . The convex conjugate of H ε is given, for all τ ∈ M × , by H ∗ ε ( τ ) = | τ − P K ( τ ) | ε , where P K stands for the orthogonal projection onto the nonempty closed convex set K . Thefunction H ∗ ε turns out to be of class C and its differential is given by DH ∗ ε ( τ ) = τ − P K ( τ ) ε . (3.2)With these notation, the flow rule (3.1), can be equivalently written, for a.e. t ∈ [0 , T ], as σ ε ( t ) ∈ ∂H ε ( ˙ p ε ( t )) in Ω , or still, by convex analysis, ˙ p ε ( t ) = DH ∗ ε ( σ ε ( t )) in Ω . (3.3)We will show that the solution ( u ε , e ε , p ε ) of the visco-plastic model given by Proposition 3.1converges to a solution of the perfectly plastic model, in the sense of Theorem 2.5. Proposition 3.3.
Assume that ( A ) – ( A ) hold, and in addition that w ∈ H ([0 , T ]; H (Ω; R )) and χ ∈ W , ∞ ([0 , T ]; L ∞ (Ω; M × )) . Then, up to a subsequence (not relabeled), ( u ε , e ε , p ε ) ⇀ ( u, e, p ) weakly* in H ([0 , T ]; BD (Ω)) × H ([0 , T ]; L (Ω; M × )) × H ([0 , T ]; M (Ω; M × )) , where ( u, e, p ) is a solution of the perfectly plastic model as in Theorem 2.5. Note that this result was already proven in [28] for different type of elasticity sets that arebounded in the direction of deviatoric stresses (see also [12]). However we give below a slightlydifferent and simplified argument since some finer estimates established along the proof will beuseful in that of our regularity result Theorem 2.7.3.1.
A priori estimates.
We first establish some uniform a priori estimates which will enable oneto get weak compactness on the families ( u ε ) ε> , ( e ε ) ε> , and ( p ε ) ε> .3.1.1. First energy estimates.
Standard arguments show that the following energy balance holds:for all t ∈ [0 , T ], Q ( e ε ( t )) + ˆ t H ( ˙ p ε ( s )) ds + ε ˆ t ˆ Ω | ˙ p ε | dx ds = Q ( e ) + ˆ t ˆ Ω σ ε : E ˙ w dx ds + ˆ t ˆ Ω f · ( ˙ u ε − ˙ w ) dx ds, (3.4)or still, using the safe load condition ( A ), together with an integration by parts in time, Q ( e ε ( t )) + ˆ t H ( ˙ p ε ( s )) ds − ˆ t ˆ Ω χ : ˙ p ε dx ds + ε ˆ t ˆ Ω | ˙ p ε | dx ds = Q ( e ) + ˆ t ˆ Ω σ ε : E ˙ w dx ds − ˆ t ˆ Ω ˙ χ : ( e ε − Ew ) dx ds + ˆ Ω χ ( t ) : ( e ε ( t ) − Ew ( t )) dx − ˆ Ω χ (0) : ( e − Ew (0)) dx. (3.5)Therefore, an application of Proposition 2.4 leads to the following first energy estimates:sup ε> (cid:16) k e ε k L ∞ (0 ,T ; L (Ω; M × )) + k ˙ p ε k L (0 ,T ; L (Ω; M × )) + √ ε k ˙ p ε k L (0 ,T ; L (Ω; M × )) (cid:17) < + ∞ . (3.6) TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 11
Second energy estimates.
Writing the additive decomposition for the rates yields E ˙ u ε =˙ e ε + ˙ p ε in Ω × (0 , T ), taking the scalar product with ˙ σ ε − ˙ χ and integrating over Ω × (0 , T ) leads to2 ˆ t Q ( ˙ e ε ( s ) − C − ˙ χ ( s )) ds = ˆ t ˆ Ω ( ˙ σ ε − ˙ χ ) : E ˙ u ε dx ds − ˆ t ˆ Ω ( ˙ σ ε − ˙ χ ) : ˙ p ε dx ds − ˆ t ˆ Ω C − ˙ χ : ( ˙ σ ε − ˙ χ ) dx ds. (3.7)By integrating by parts in space and by using the equilibrium equation, together with the safe loadcondition ( A ), we can rewrite the first integral at the right hand side of (3.7) as ˆ t ˆ Ω ( ˙ σ ε − ˙ χ ) : E ˙ u ε dx ds = ˆ t ˆ Ω ( ˙ σ ε − ˙ χ ) : E ˙ w dx ds. Using the flow rule (3.3), we have ˙ p ε = DH ∗ ε ( σ ε ), so that, owing to the chain rule formula and aderivation under the integral sign, we get for a.e. t ∈ [0 , T ], ˆ Ω ˙ σ ε ( t ) : ˙ p ε ( t ) dx = ˆ Ω DH ∗ ε ( σ ε ) : ˙ σ ε ( t ) dx = ddt ˆ Ω H ∗ ε ( σ ε ( t )) dx. Thus, since σ ∈ K , − ˆ t ˆ Ω ˙ σ ε ( t ) : ˙ p ε ( t ) dx = − ˆ Ω H ∗ ε ( σ ε ( t )) dx + ˆ Ω H ∗ ε ( σ ) dx ≤ ˆ Ω H ∗ ε ( σ ) dx = 0 . Finally, using the safe load condition ( A ), together with (2.1) and H¨older’s inequality, we obtainfrom (3.7) the following estimate: k ˙ σ ε − ˙ χ k L (0 ,T ; L (Ω; M × )) ≤ k ˙ χ k L ∞ (Ω × (0 ,T ); M × ) k ˙ p ε k L (0 ,T ; L (Ω; M × )) + C (cid:16) k E ˙ w k L (0 ,T ; L (Ω; M × )) + k ˙ χ k L (0 ,T ; L (Ω; M × )) (cid:17) k ˙ σ ε − ˙ χ k L (0 ,T ; L (Ω; M × )) , where C > ε . By the first energy estimate (3.6) we infer thatsup ε> k ˙ σ ε k L (0 ,T ; L (Ω; M × )) < + ∞ . (3.8)3.1.3. Third energy estimates.
Writing next the energy balance (3.5) between two arbitrary times t and t , and integrating by parts in times yield Q ( e ε ( t )) + ˆ t t H ( ˙ p ε ( s )) ds − ˆ t t ˆ Ω χ : ˙ p ε dx ds + ε ˆ t t ˆ Ω | ˙ p ε | dx ds = Q ( e ε ( t )) + ˆ t t ˆ Ω σ ε : E ˙ w dx ds + ˆ t t ˆ Ω χ : ( ˙ e ε − E ˙ w ) dx ds. By applying Proposition 2.4, together with estimate (3.6), we deduce that C δ,α k p ε ( t ) − p ε ( t ) k L (Ω; M × ) ≤ C δ,α ˆ t t k ˙ p ε ( s ) k L (Ω; M × ) ds ≤ ˆ t t H ( ˙ p ε ( s )) ds − ˆ t t ˆ Ω χ : ˙ p ε dx ds ≤ ˆ Ω ( σ ε ( t ) + σ ε ( t )) : ( e ε ( t ) − e ε ( t )) dx + ˆ t t ˆ Ω σ ε : E ˙ w dx ds + ˆ t t ˆ Ω χ : ( ˙ e ε − E ˙ w ) dx ds ≤ ˆ t t h ε ( s ) ds, where h ε ∈ L (0 , T ) is defined, for a.e. s ∈ [0 , T ], by h ε ( s ) := M (cid:16) k ˙ e ε ( s ) k L (Ω; M × ) + k E ˙ w ( s ) k L (Ω; M × ) (cid:17) − ˆ Ω χ ( s ) : E ˙ w ( s ) dx, and M := sup ε> k σ ε k L ∞ (0 ,T ; L (Ω; M × )) + k χ k L ∞ (0 ,T ; L (Ω; M × )) . According to [11, Proposition A.3] and (3.8), this implies that p ε ∈ H ([0 , T ]; L (Ω; M × )),sup ε> k ˙ p ε k L (0 ,T ; L (Ω; M × )) ≤ sup ε> k h ε k L (0 ,T ) < + ∞ , (3.9)and owing to the Poincar´e-Korn inequality (see [30, Chap. 2, Rmk. 2.5(ii)]), thatsup ε> k ˙ u ε k L (0 ,T ; BD (Ω)) < + ∞ . (3.10)3.2. Passage to the limit in ε . According to (3.8), (3.9), and (3.10), there exists a subse-quence (not relabeled) and functions u ∈ H ([0 , T ]; BD (Ω)), e ∈ H ([0 , T ]; L (Ω; M × )), and p ∈ H ([0 , T ]; M (Ω; M × )) such that u ε ⇀ u weakly* in H ([0 , T ]; BD (Ω)) ,e ε ⇀ e weakly in H ([0 , T ]; L (Ω; M × )) ,p ε ⇀ p weakly* in H ([0 , T ]; M (Ω; M × )) . An application of the Ascoli-Arzel`a Theorem also shows that u ε ( t ) ⇀ u ( t ) weakly* in BD (Ω) ,e ε ( t ) ⇀ e ( t ) weakly in L (Ω; M × ) ,p ε ( t ) ⇀ p ( t ) weakly* in M (Ω; M × )for all t ∈ [0 , T ]. Since ( u ε (0) , e ε (0) , p ε (0)) = ( u ,ε , e , p ,ε ), by passing to the limit we deduce thatthe initial condition ( u (0) , e (0) , p (0)) = ( u , e , p ) is satisfied. Note also that, according to [12,Lemma 2.1], we have that ( u ( t ) , e ( t ) , p ( t )) ∈ A ( w ( t )) for all t ∈ [0 , T ]. Moreover, defining σ := C e ,we get that − div σ ( t ) = f ( t ) in Ω for all t ∈ [0 , T ].We then prove the validity of the stress constraint. Owing to (3.6) and (3.8), the fact that P K is 1-Lipschitz and that 0 ∈ K , up to another subsequence, we have ( P K ( σ ε ) ⇀ τ weakly in H ([0 , T ]; L (Ω; M × )) ,P K ( σ ε ( t )) ⇀ τ ( t ) weakly in L (Ω; M × ) for all t ∈ [0 , T ] , for some τ ∈ H ([0 , T ]; L (Ω; M × )) with τ ( t ) ∈ K for all t ∈ [0 , T ]. According to the flow rule(3.3), (3.2), and (3.6), we get σ ε − P K ( σ ε ) = ε ˙ p ε → L (0 , T ; L (Ω; M × )) , which implies that σ = τ and thus, that σ ( t ) ∈ K for all t ∈ [0 , T ].It remains to show the energy balance. According to the weak convergences established so far,it is possible to pass to the lower limit in the energy balance (3.4) to get, for all t ∈ [0 , T ], Q ( e ( t )) + ˆ t H ( ˙ p ( s )) ds ≤ Q ( e ) + ˆ t ˆ Ω σ : E ˙ w dx ds + ˆ t ˆ Ω f · ( ˙ u − ˙ w ) dx ds. The converse inequality, and thus equality (2.6), can be proved using a standard argument ofrate independent processes (see [12, Theorem 4.7]) by noticing that the conditions σ ( t ) ∈ K ∩ S , − div σ ( t ) = f ( t ) in Ωare equivalent to the minimality property Q ( e ( t )) − ˆ Ω f ( t ) · u ( t ) dx ≤ Q ( η ) + H ( q − p ( t )) − ˆ Ω f ( t ) · v dx TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 13 for all ( v, η, q ) ∈ A ( w ( t )). This can be seen by adapting the argument of [12, Theorem 3.6], whichonly requires the stress-strain duality pairing to be well defined.3.3. Higher regularity properties.
In the proof of our regularity result Theorem 2.7, we willneed some higher regularity properties for the solution of the visco-plastic model.
Proposition 3.4.
Assume that ( A ) – ( A ) hold, and, in addition, that w ∈ H ([0 , T ]; H (Ω; R )) , χ ∈ W , ∞ ([0 , T ]; L ∞ (Ω; M × )) ∩ H ([0 , T ]; H (Ω; M × )) , f ∈ L ∞ (0 , T ; H (Ω; R )) , and e ∈ H (Ω; M × ) . Then ˙ u ε ∈ L (0 , T ; H (Ω; R )) , e ε , σ ε ∈ H ([0 , T ]; H (Ω; M × )) , ˙ p ε ∈ L (0 , T ; H (Ω; M × )) . Let us introduce the following notation: given a generic function φ : R → R , we write ∂ hk φ ( x ) := φ ( x + he k ) − φ ( x ) h for k ∈ { , } and h > Fourth energy estimates.
Using the linearity of the equilibrium equations, we get that forall t ∈ [0 , T ], − div ( ∂ hk σ ε ( t )) = ∂ hk f ( t ) a.e. in { x ∈ Ω : dist( x, ∂ Ω) > h } . Let ϕ ∈ C ∞ c (Ω) be a cut-off function with h < dist(supp ϕ, ∂ Ω). We multiply the previous equationby ϕ ∂ hk ˙ u ε and integrate by parts in the space variables. In this way we get ˆ t ˆ Ω ϕ ∂ hk σ ε : ∂ hk E ˙ u ε dx ds + ˆ t ˆ Ω ∂ hk σ ε : ( ∇ ϕ ⊙ ∂ hk ˙ u ε ) dx ds = ˆ t ˆ Ω ϕ ∂ hk f · ∂ hk ˙ u ε dx ds. Using the additive decomposition ∂ hk E ˙ u ε = ∂ hk ˙ e ε + ∂ hk ˙ p ε , the regularized flow rule (3.1), and themonotonicity of the subdifferential of a convex function, we obtain Q ( ϕ∂ hk e ε ( t )) + ε ˆ t ˆ Ω ϕ | ∂ hk ˙ p ε | dx ds ≤ Q ( ϕ∂ hk e ) + ˆ t ˆ Ω ϕ ∂ hk f · ∂ hk ˙ u ε dx ds − ˆ t ˆ Ω ϕ∂ hk σ ε : ( ∇ ϕ ⊙ ∂ hk ˙ u ε ) dx ds ≤ C (cid:16) k e k H (Ω; M × ) + k f k L ∞ (0 ,T ; H (Ω; R )) k∇ ˙ u ε k L (0 ,T ; L (Ω; M × )) + k ϕ∂ hk σ ε k L ∞ (0 ,T ; L (Ω; M × )) k∇ ˙ u ε k L (0 ,T ; L (Ω; M × )) (cid:17) , where, here and in the following, C denotes a positive constant independent of h and ε , but possiblydepending on ϕ . Thanks to (2.1) and Young’s inequality, we infer that k ϕ∂ hk σ ε k L ∞ (0 ,T ; L (Ω; M × )) + ε k ϕ∂ hk ˙ p ε k L (0 ,T ; L (Ω; M × )) ≤ C (cid:16) k e k H (Ω; M × ) + k f k L ∞ (0 ,T ; H (Ω; R )) + k∇ ˙ u ε k L (0 ,T ; L (Ω; M × )) (cid:17) . According to Korn’s inequality in H (Ω; R ), we have ˆ Ω |∇ ˙ u ε ( t ) − ∇ ˙ w ( t ) | dx ≤ ˆ Ω | E ˙ u ε ( t ) − E ˙ w ( t ) | dx, hence k ϕ∂ hk σ ε k L ∞ (0 ,T ; L (Ω; M × )) + ε k ϕ∂ hk ˙ p ε k L (0 ,T ; L (Ω; M × )) ≤ C (cid:16) k e k H (Ω; M × ) + k f k L ∞ (0 ,T ; H (Ω; R )) k + k E ˙ u ε k L (0 ,T ; L (Ω; M × )) + k ˙ w k L (0 ,T ; H (Ω; R )) (cid:17) . Thus, using (3.6) and (3.8), we get that k ϕ∂ hk σ ε k L ∞ (0 ,T ; L (Ω; M × )) + ε k ϕ∂ hk ˙ p ε k L (0 ,T ; L (Ω; M × )) ≤ Cε .
Letting h ց
0, we then obtain that for every open set Ω ′ ⊂⊂ Ω, there exists a constant
C > ε ) such that k ∂ k σ ε k L ∞ (0 ,T ; L (Ω ′ ; M × )) + ε k ∂ k ˙ p ε k L (0 ,T ; L (Ω ′ ; M × )) ≤ Cε . (3.11)In particular, we get the following higher regularity properties: e ε , σ ε ∈ L ∞ (0 , T ; H (Ω; M × )) , ˙ p ε ∈ L (0 , T ; H (Ω; M × )) . (3.12)3.3.2. Fifth energy estimate.
Using the additive decomposition, we have E ( ∂ hk ˙ u ε ) = ∂ hk ˙ e ε + ∂ hk ˙ p ε in { x ∈ Ω : dist( x, ∂ Ω) > h } × (0 , T ) . Let ϕ ∈ C ∞ c (Ω) be a cut-off function with h < dist(supp ϕ, ∂ Ω). Taking the scalar product of theprevious relation with ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) and integrating over Ω × (0 , T ) yield2 ˆ T Q (cid:0) ϕ ( ∂ hk ˙ e ε ( s ) − C − ∂ hk ˙ χ ( s )) (cid:1) ds = ˆ T ˆ Ω ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) : E ( ∂ hk ˙ u ε ) dx ds − ˆ T ˆ Ω ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) : ∂ hk ˙ p ε dx ds − ˆ T ˆ Ω ϕ ( C − ∂ hk ˙ χ ) : ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) dx ds. We now look at the first integral on the right hand side. Integrating by parts in space and usingthe equilibrium equations, together with the safe load condition ( A ), yield ˆ T ˆ Ω ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) : E ( ∂ hk ˙ u ε ) dx ds = ˆ T ˆ Ω ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) : E ( ∂ hk ˙ w ) dx ds. According to (3.2), DH ∗ ε is Lipschitz continuous with Lipschitz constant less than 1 /ε . Therefore,the flow rule ˙ p ε = DH ∗ ε ( σ ε ) implies that | ∂ hk ˙ p ε | ≤ | ∂ hk σ ε | /ε in Ω × (0 , T ). Therefore, the Cauchy-Schwarz inequality, together with (2.1), yields k ϕ ( ∂ hk ˙ σ ε − ∂ hk ˙ χ ) k L (0 ,T ; L (Ω; M × )) ≤ C (cid:18) ε k ∂ hk σ ε k L (0 ,T ; L (Ω; M × )) + k w k H ([0 ,T ]; H (Ω; R )) + k χ k H ([0 ,T ]; H (Ω; M × )) (cid:19) , where C > h and ε . Therefore using (3.11), and passing to the limitas h →
0, we get the following estimate: for every open set Ω ′ ⊂⊂ Ω, there exists a constant
C > ε such that k ∂ k ˙ e ε k L (0 ,T ; L (Ω ′ ; M × )) ≤ Cε . As a consequence of this last estimate, together with (3.12), the additive decomposition Eu ε = e ε + p ε , and Korn’s inequality, we deduce the following higher regularity properties: e ε , σ ε ∈ H ([0 , T ]; H (Ω; M × )) , ˙ u ε ∈ L (0 , T ; H (Ω; M × )) . TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 15
A useful formula for the stress.
Following the strategy of [27], we introduce the function g : M × → R defined by g ( ξ ) := K ξ ) + µ | ξ D | if 2 µ | ξ D | + 2 αK tr ξ ≤ κ,κ tr ξ α − κ α K + β (cid:18) | ξ D | − tr ξ α + κ α K (cid:19) if 2 µ | ξ D | + 2 αK tr ξ > κ, (3.13)where β := (cid:18) µ + 12 α K (cid:19) − . According to the results of [25, 27], g is the convex conjugate of the function τ C − τ : τ + I K ( τ ).Consequently, we have g ∗ ( τ ) = 12 C − τ : τ + I K ( τ ) for all τ ∈ M × , and, in particular, g is convex, lower semicontinuous, and locally Lipschitz continuous. One cancheck that g is actually of class C , and that its differential is given by Dg ( ξ ) = K (tr ξ )Id + 2 µξ D if 2 µ | ξ D | + 2 αK tr ξ ≤ κ,κ α Id+2 β (cid:18) | ξ D | − tr ξ α + κ α K (cid:19) + (cid:18) ξ D | ξ D | − α Id (cid:19) if 2 µ | ξ D | + 2 αK tr ξ > κ for every ξ ∈ M × . In addition, there exists a constant a > α , κ , K ) suchthat g ( ξ ) ≥ a | ξ | , if 2 µ | ξ D | + 2 αK tr ξ ≤ κ. (3.14)According to convex analysis, the regularized flow rule (3.1) can be equivalently written, for a.e. t ∈ [0 , T ], as ˙ p ε ( t ) ∈ ∂I K (cid:0) σ ε ( t ) − ε ˙ p ε ( t ) (cid:1) in Ω , or still ˙ p ε ( t ) + e ε ( t ) − ε C − ˙ p ε ( t ) ∈ ∂g ∗ ( σ ε ( t ) − ε ˙ p ε ( t )) in Ω . Again by convex duality we can rewrite the regularized flow rule in terms of the following formula,which will be instrumental in the subsequent analysis: for a.e. t ∈ [0 , T ], σ ε ( t ) − ε ˙ p ε ( t ) = Dg (cid:0) ˙ p ε ( t ) + e ε ( t ) − ε C − ˙ p ε ( t ) (cid:1) in Ω . (3.15)4. A chain rule formula for the weak derivatives of the stress
In order to prove Theorem 2.7, we will show that σ ε ∈ L ∞ (0 , T ; H (Ω; M × )) with a boundon its norm that is uniform with respect to ε . To do that, it will be useful to differentiate formula(3.15) with respect to the space variables. Unfortunately, the function g is only of class C , withLipschitz continuous partial derivatives, so that the classical chain rule formula for vector-valuedSobolev functions does not apply. To overcome this issue, we will use the special form of g givenin (3.13), together with the generalized chain rule formula established in [3, Corollary 3.2].4.1. Some geometrical preliminaries.
We first establish several technical results which will beof use in the proof of that result. Let us introduce some notation: let G : M × → M × bedefined, for all ξ ∈ M × , by G ( ξ ) := Dg (cid:18) ξ + κ αK Id (cid:19) . We define the open sets A := { ξ ∈ M × : µ | ξ D | + αK tr ξ < } , A := (cid:26) ξ ∈ M × : | ξ D | < tr ξ α (cid:27) , A := M × \ ( A ∪ A ) . A A A A (tr ξ )Id Figure 1.
Representation of the sets A , A and A .The sets A and A are open cones in M × (which is a three-dimensional space) with same vertexat the origin, ∂ A ∩ ∂ A = { } , and axis given by the hydrostatic matrices (see Figure 1). Note,in particular, that if ξ ∈ A , then − µαK | ξ D | < tr ξ < α | ξ D | , so that | ξ | ≤ c | ξ D | for all ξ ∈ A , (4.1)for some constant c > µ , α , and K . It follows that G ( ξ ) = κ α Id + K (tr ξ )Id + 2 µξ D if ξ ∈ A , ξ ∈ A ∪ { } , β (cid:18) | ξ D | − tr ξ α (cid:19) (cid:18) ξ D | ξ D | − α Id (cid:19) if ξ ∈ A \ { } . (4.2)Note that G is Lipschitz continuous on M × , and its restriction to the open sets A , A , and A is smooth. The only non differentiability points lie at the interfaces ∂ A ∪ ∂ A . Let us also define,for all ξ and η ∈ M × , G ( ξ, η ) = K (tr η )Id + 2 µη D if ξ ∈ A , ξ ∈ A ∪ { } , β (cid:18) ξ D : η D | ξ D | − tr η α (cid:19) (cid:18) ξ D | ξ D | − α Id (cid:19) +2 β (cid:18) | ξ D | − tr ξ α (cid:19) | ξ D | (cid:18) η D − ξ D : η D | ξ D | ξ D (cid:19) if ξ ∈ A \ { } . Note that if ξ ∈ A ∪ A ∪ A , then G ( ξ, η ) = DG ( ξ ) η = D g (cid:18) ξ + κ αK Id (cid:19) η. TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 17
Remark 4.1.
Since, for each ξ ∈ M × , the bilinear form ( η , η )
7→ G ( ξ, η ) : η is symmetric andnon-negative, if follows from the Cauchy-Schwarz inequality (applied to this bilinear form) that G ( ξ, η ) : η ≤ p G ( ξ, η ) : η p G ( ξ, η ) : η for every ξ, η , η ∈ M × . In addition, there exists a constant C ∗ >
0, depending only on K , µ , α , and κ , such that G ( ξ, η ) : η ≤ C ∗ | η | for all ξ, η ∈ M × . Lemma 4.2.
Let σ ∈ ( ∂ A ∪ ∂ A ) \ { } , and τ , τ ∈ M × . We define the set T := (cid:8) σ + z τ + z τ : z ∈ R (cid:9) . Assume that G | T is differentiable at σ . Then, τ : ν ( σ ) = τ : ν ( σ ) = 0 , where ν ( σ ) := µ σ D | σ D | + αK Id if σ ∈ ∂ A ,σ D | σ D | − α Id if σ ∈ ∂ A is normal to ∂ A ∪ ∂ A at σ and oriented from A ∪ A to A .Proof. Suppose first that σ ∈ ∂ A , so that | σ D | = tr σ α . (4.3)Assume by contradiction that τ k : ν ( σ ) = 0 for some k ∈ { , } , i.e. , σ D : ( τ k ) D | σ D | − tr τ k α = 0 . (4.4)Then, for t ∈ R small enough so that σ + tτ k = 0, we can assume that σ + tτ k ∈ A if t < σ + tτ k ∈ A if t >
0. Since the restriction of G to A is constant, and { σ + tτ k : t ∈ R } isa direction in T along which (by assumption) G is differentiable at σ , we deduce that ∂∂τ k [ G | T ]( σ ) = lim t → − G ( σ + tτ k ) − G ( σ ) t = 0 . (4.5)Similarly, we use the differentiability of G in A to get that ∂∂τ k [ G | T ]( σ ) = lim t → + G ( σ + tτ k ) − G ( σ ) t = 2 β (cid:18) σ D : ( τ k ) D | σ D | − tr τ k α (cid:19) (cid:18) σ D | σ D | − α Id (cid:19) + 2 β (cid:18) | σ D | − tr σ α (cid:19) | σ D | (cid:18) ( τ k ) D − σ D : ( τ k ) D | σ D | σ D (cid:19) . (4.6)According to (4.3), the second term in the right hand side of the previous identity vanishes. Onthe other hand, the combination of (4.5), (4.6), and the assumption (4.4) yield σ D | σ D | = α Id, whichis impossible.Assume now that σ ∈ ∂ A , and, again by contradiction, that τ k : ν ( σ ) = 0 for some k ∈ { , } , i.e. , µ σ D : ( τ k ) D | σ D | + αK (tr τ k ) = 0 . (4.7) Then, for t ∈ R small enough so that σ + tτ k = 0, we can assume that σ + tτ k ∈ A if t < σ + tτ k ∈ A if t >
0. An analogous argument as before shows that ∂∂τ k [ G | T ]( σ ) = lim t → − G ( σ + tτ k ) − G ( σ ) t = K (tr τ k )Id + 2 µ ( τ k ) D , (4.8)and ∂∂τ k [ G | T ]( σ ) = lim t → + G ( σ + tτ k ) − G ( σ ) t = 2 β (cid:18) σ D : ( τ k ) D | σ D | − tr τ k α (cid:19) (cid:18) σ D | σ D | − α Id (cid:19) + 2 β (cid:18) | σ D | − tr σ α (cid:19) | σ D | (cid:18) ( τ k ) D − σ D : ( τ k ) D | σ D | σ D (cid:19) . (4.9)Equating (4.8) and (4.9), and taking the trace yield (cid:18) µ + 12 α K (cid:19) K tr τ k = − α (cid:18) σ D : ( τ k ) D | σ D | − tr τ k α (cid:19) , which is against (4.7). (cid:3) Let us denote by T σ ( ∂ A ) (resp. T σ ( ∂ A )) the tangent space to ∂ A (resp. ∂ A ) at σ . Lemma 4.3.
For every σ ∈ ∂ A i \ { } ( i = 1 , ), there exists some ρ > such that T σ ( ∂ A i ) ∩ B ( σ, ρ ) ⊂ A . Proof.
Let us check this property for i = 2. Let σ ∈ ∂ A \ { } . Since σ = 0, there exists some ρ > B ( σ, ρ ) ∩ A = ∅ . We define the convex function a ( τ ) := | τ D | − α tr τ for every τ ∈ M × . Since σ = 0, the function a is differentiable at σ with gradient equal to ν ( σ ),and for every τ ∈ T σ ( ∂ A ) ∩ B ( σ, ρ ), we have a ( τ ) ≥ a ( σ ) + ν ( σ ) : ( τ − σ ) . Since σ ∈ ∂ A , we have that a ( σ ) = 0, while the fact that τ − σ ∈ T σ ( ∂ A ) yields ν ( σ ) : ( τ − σ ) = 0.Therefore a ( τ ) ≥
0, that is, τ
6∈ A , and since τ
6∈ A neither, then τ ∈ A (see Figure 2). (cid:3) Proof of the chain rule formula.
As a consequence of the previous results, we get that achain rule type formula holds.
Corollary 4.4.
Let us set ξ ε := ˙ p ε + e ε − ε C − ˙ p ε − κ αK Id . Then for all k ∈ { , } , ∂ k σ ε − ε∂ k ˙ p ε = G ( ξ ε , ∂ k ξ ε ) = K tr( ∂ k ξ ε )Id + 2 µ∂ k ξ εD a.e. in A ξ ε , a.e. in A ξ ε , β (cid:18) ξ εD : ∂ k ξ εD | ξ εD | − tr( ∂ k ξ ε )2 α (cid:19) (cid:18) ξ εD | ξ εD | − α Id (cid:19) +2 β (cid:18) | ξ εD | − tr ξ ε α (cid:19) | ξ εD | (cid:18) ∂ k ξ εD − ξ εD : ∂ k ξ εD | ξ εD | ξ εD (cid:19) a.e. in A ξ ε , where A ξ ε := { ( x, t ) ∈ Ω × (0 , T ) : ξ ε ( x, t ) ∈ A } ,A ξ ε := { ( x, t ) ∈ Ω × (0 , T ) : ξ ε ( x, t ) ∈ A ∪ { }} ,A ξ ε := { ( x, t ) ∈ Ω × (0 , T ) : ξ ε ( x, t ) ∈ A \ { }} . TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 19 T σ ( ∂ A ) A A A σ • B ( σ, ρ ) Figure 2.
The tangent space T σ ( ∂ A ) of ∂ A at σ ∈ ∂ A \ { } might intersect A , but not in a small neighborhood B ( σ, ρ ) of σ . Proof.
In order not to overburden notation, we omit the subscript ε . Using the regularity properties(3.12), we deduce that ξ ∈ L (0 , T ; H (Ω; M × )), and by (3.15), we get σ − ε ˙ p = G ◦ ξ . Accordingto the generalized chain rule formula established in [3, Corollary 3.2], if we denote the affine spacepassing through ξ ( x, t ) by T ξ ( x,t ) := (cid:8) ξ ( x, t ) + ∂ ξ ( x, t ) z + ∂ ξ ( x, t ) z : z ∈ R (cid:9) , then we have that G | T ξ ( x,t ) is differentiable at ξ ( x, t ) for a.e. ( x, t ) ∈ Ω × (0 , T ), and for all 1 ≤ i, j ≤ ∇ σ ij − ε ∇ ˙ p ij = ∇ (cid:0) G ij | T ξ ( x,t ) (cid:1) ( ξ ) ∇ ξ. (4.10)First, since ∇ ξ = 0 a.e. in { ξ = 0 } , it results from formula (4.10) that ∇ σ − ε ∇ ˙ p = 0 a.e. in { ξ = 0 } , and as { ξ = 0 } ⊂ A ξ , we deduce that ∂ k σ − ε∂ k ˙ p = G ( ξ, ∂ k ξ ) a.e. in { ξ = 0 } . (4.11)Secondly, if ξ ( x, t ) ∈ A i for some i ∈ { , , } , then ξ ( x, t ) is a differentiability point of G , andusing again (4.10), ∂ k σ − ε∂ k ˙ p = DG ( ξ ) : ∂ k ξ = G ( ξ, ∂ k ξ ) a.e. in [ i =1 { ξ ∈ A i } . (4.12)It thus remains to check the formula when ξ ( x, t ) ∈ ∂ A \ { } = ( ∂ A ∪ ∂ A ) \ { } . Accordingto Lemma 4.2, we deduce that for a.e. ( x, t ) in that set, ∂ k ξ ( x, t ) : ν ( ξ ( x, t )) = 0 for all k ∈ { , } , and thus, T ξ ( x,t ) ⊂ T ξ ( x,t ) ( ∂ A ) ∪ T ξ ( x,t ) ( ∂ A ). Using next Lemma 4.3 yields that T ξ ( x,t ) ∩ B ( ξ ( x, t ) , ρ ( x, t )) ⊂ A for some ρ ( x, t ) >
0. Thus, according to the expression (4.2) of G in A together with (4.10), ∂ k σ − ε∂ k ˙ p = G ( ξ, ∂ k ξ ) a.e. in { ξ ∈ ∂ A \ { }} . (4.13)Gathering (4.11), (4.12), and (4.13) leads to the desired expression for the derivatives of σ . (cid:3) A uniform bound for the Sobolev norm of the stress
We next show that the L ∞ (0 , T ; H (Ω; M × )) norm of σ ε can be uniformly controled withrespect to ε . Proposition 5.1.
Under the same assumptions of Theorem 2.7, we have that, for every open set Ω ′ ⊂⊂ Ω , sup ε> k σ ε k L ∞ (0 ,T ; H (Ω ′ ; M × )) < + ∞ . Proof.
All constants appearing in this proof are independent of ε . In order to lighten the notation,we do not explicitely write the various dx , ds inside the integrals. As in Corollary 4.4, we alsoomit the subscript ε and we write ξ := ˙ p + e − ε C − ˙ p − κ αK Id ∈ L (0 , T ; H (Ω; M × )) . We start with the equilibrium equation − div σ = f in Ω × (0 , T ) . (5.1)Let Ω ′ ⊂⊂ Ω ′′ ⊂⊂ Ω and ϕ ∈ C ∞ c (Ω; [0 , ϕ ≡ ′ and ϕ ≡ \ Ω ′′ . By Proposition 3.4, if t ∈ [0 , T ] is arbitrary, we have ϕ ( ∂ k ˙ u ) χ [0 ,t ] ∈ L (0 , T ; H (Ω; R )).We then take the partial derivative of (5.1) with respect to x k ( k = 1 , Q t := Ω × (0 , t ) the space-time cylinder, we get ¨ Q t ∂ k σ : ( E∂ k ˙ u ) ϕ + ¨ Q t ∂ k σ : ( ∂ k ˙ u ⊗ ∇ ϕ ) = ¨ Q t ∂ k f · ( ∂ k ˙ u ) ϕ , (5.2)where, from now on, we use the summation convention over repeated indexes. Then, writing ∂ k ˙ u j = 2 E kj ˙ u − ∂ j ˙ u k , we find ¨ Q t ∂ k σ : ( ∂ k ˙ u ⊗ ∇ ϕ ) = 2 ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ − ¨ Q t ∂ k σ ij ∂ j ˙ u k ∂ i ϕ . An integration by parts in the last integral at the right hand side, together with (5.1), yields ¨ Q t ∂ k σ ij ∂ j ˙ u k ∂ i ϕ = ¨ Q t ∂ k f i ˙ u k ∂ i ϕ − ¨ Q t ˙ u k ∂ k σ ij ∂ j ∂ i ϕ , and thus, using the previous identities in (5.2), ¨ Q t ∂ k σ : ( E∂ k ˙ u ) ϕ = − ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ − ¨ Q t ˙ u k ∂ k σ ij ∂ j ∂ i ϕ + ¨ Q t (cid:0) − ∆ f · ˙ uϕ − ∂ k f · ˙ u∂ k ϕ + ∂ k f i ˙ u k ∂ i ϕ (cid:1) . (5.3)Using the additive decomposition and the definition of ξ , we have E ˙ u = ˙ e + ˙ p, ˙ p = ξ − e + ε C − ˙ p + κ αK Id . (5.4)Hence ¨ Q t ∂ k σ : ( E∂ k ˙ u ) ϕ = ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k ξ − ∂ k e + ε C − ∂ k ˙ p ) ϕ , TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 21 and inserting inside (5.3) yields ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ = − ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ − ¨ Q t ˙ u k ∂ k σ ij ∂ j ∂ i ϕ + ¨ Q t C − ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ + ¨ Q t (cid:0) − ∆ f · ˙ uϕ − ∂ k f · ˙ u∂ k ϕ + ∂ k f i ˙ u k ∂ i ϕ (cid:1) . (5.5) Step 1.
We first bound from below the left hand side of (5.5). First, by Corollary 4.4 we havethat ∂ k σ − ε∂ k ˙ p = G ( ξ, ∂ k ξ ), and thus( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ = X k =1 K (tr ∂ k ξ ) + 2 µ | ∂ k ξ D | a.e. in A ξ , A ξ , X k =1 β (cid:18) ξ D : ∂ k ξ D | ξ D | − tr ∂ k ξ α (cid:19) + X k =1 β (cid:18) | ξ D | − tr ξ α (cid:19) | ξ D | (cid:18) | ∂ k ξ D | − ( ξ D : ∂ k ξ D ) | ξ D | (cid:19) a.e. in A ξ . (5.6)On the other hand, according to Remark 4.1, we get that( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) = G ( ξ, ∂ k ξ ) : ( ∂ k σ − ε∂ k ˙ p ) ≤ p G ( ξ, ∂ k ξ ) : ∂ k ξ p G ( ξ, ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ≤ p ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ p C ∗ ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) , so that ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ≤ C ∗ ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ. Thus, applying this inequality in (5.5), we get that ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 12 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 12 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ − ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ − ¨ Q t ˙ u k ∂ k σ ij ∂ j ∂ i ϕ + ¨ Q t C − ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ + ¨ Q t (cid:0) − ∆ f · ˙ uϕ − ∂ k f · ˙ u∂ k ϕ + ∂ k f i ˙ u k ∂ i ϕ (cid:1) . (5.7) Step 2.
We now bound from above the right hand side of (5.7). Since ∂ i ϕ = ϕ (6 ϕ ∂ i ϕ ), and ∂ j ∂ i ϕ = ϕ (30 ϕ∂ j ϕ∂ i ϕ + 6 ϕ ∂ j ∂ i ϕ ), we deduce from the Cauchy-Schwarz and Young inequalities that ¨ Q t (cid:0) − ∆ f · ˙ uϕ − ∂ k f · ˙ u∂ k ϕ + ∂ k f i ˙ u k ∂ i ϕ (cid:1) ≤ C (cid:16) k f k L (0 ,T ; H (Ω; R )) + k ˙ u k L ( Q T ; R ) (cid:17) , − ¨ Q t ˙ u k ∂ k σ ij ∂ j ∂ i ϕ ≤ C (cid:18) ¨ Q t ∂ k σ : ∂ k e ϕ dx ds + k ˙ u k L ( Q T ; R ) (cid:19) , and ¨ Q t C − ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C (cid:18) ¨ Q t ∂ k σ : ∂ k e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ (cid:19) . If Cε < /
2, the last integral can be absorbed by the second one at the left hand side of (5.7). Wethen replace in (5.7) to get ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 12 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 12 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C (cid:18) ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k f k L (0 ,T ; H (Ω; R )) (cid:19) − ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ . (5.8)Using again (5.4), we have ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ = ¨ Q t ˙ e kj ∂ k σ ij ∂ i ϕ + ε ¨ Q t ˙ p kj ∂ k ˙ p ij ∂ i ϕ + ¨ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) (cid:18) ξ kj − e kj + ε ( C − ˙ p ) kj + κ αK δ kj (cid:19) ∂ i ϕ . Applying again the Cauchy-Schwarz and Young inequalities, we get that − ¨ Q t ∂ k σ ij ( E kj ˙ u ) ∂ i ϕ ≤ − ¨ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ + C ¨ Q t ∂ k σ : ∂ k e ϕ + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) ! + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 14 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ . Inserting inside (5.8) yields ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 12 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 14 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ − ¨ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ + C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) ! . (5.9) TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 23
The remaining of the proof consists in absorbing the integral ¨ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ by the left hand side of (5.9). Step 3.
We consider the following decomposition: ¨ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ = X r =1 ¨ A ξr ∩ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ . We first observe that, according to Corollary 4.4, ∂ k σ − ε∂ k ˙ p = 0 a.e. in A ξ , so that ¨ A ξ ∩ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ = 0 . (5.10)For r = 1, using that ∂ i ϕ = ϕ (6 ϕ ∂ i ϕ ), together with the Cauchy-Schwarz and Young inequali-ties, yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¨ A ξ ∩ Q t ( ∂ k σ ij − ε∂ k ˙ p ij ) ξ kj ∂ i ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ + C k ξ k L ( A ξ ; M × ) . (5.11)By formula (3.15) we have that σ − ε ˙ p = Dg (cid:18) ξ + κ αK Id (cid:19) . By convexity of g this implies that0 = g (0) ≥ g (cid:18) ξ + κ αK Id (cid:19) − ( σ − ε ˙ p ) : (cid:18) ξ + κ αK Id (cid:19) a.e. in Ω × (0 , T ). Integrating the previous inequality over A ξ and using the coercivity property(3.14) of g yield a ¨ A ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ + κ αK Id (cid:12)(cid:12)(cid:12)(cid:12) ≤ ¨ A ξ g (cid:18) ξ + κ αK Id (cid:19) ≤ ¨ A ξ ( σ − ε ˙ p ) : (cid:18) ξ + κ αK Id (cid:19) . Owing to the Cauchy-Schwarz inequality, we deduce that ¨ A ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ + κ αK Id (cid:12)(cid:12)(cid:12)(cid:12) ≤ a ¨ Q T | σ − ε ˙ p | , and therefore, k ξ k L ( A ξ ; M × ) ≤ C (cid:16) k e k L ( Q T ; M × ) + ε k ˙ p k L ( Q T ; M × ) (cid:17) . (5.12) Inserting (5.10), (5.11), and (5.12) inside (5.9), leads to ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 12 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + X i,k =1 ¨ A ξ ∩ Q t | ∂ k σ − ε∂ k ˙ p || ξ || ∂ i ϕ | ! . According to (4.1), and by definition of the set A ξ , we have | ξ | ≤ c | ξ D | a.e. in A ξ , hence ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 12 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + X i,k =1 ¨ A ξ ∩ Q t | ∂ k σ − ε∂ k ˙ p || ξ D || ∂ i ϕ | ! . (5.13) Step 4.
We next focus on the last term X i,k =1 ¨ A ξ ∩ Q t | ∂ k σ − ε∂ k ˙ p || ξ D || ∂ i ϕ | in the right hand side of (5.13). We will show that this term can be absorbed by the left hand sideof (5.13) by using the explicit expression (3.15) of σ − ε ˙ p . Let us consider the functions h := 2 β (cid:18) | ξ D | − tr ξ α (cid:19) , B := ξ D | ξ D | − α Id . According to (3.15) and (4.2), we have σ − ε ˙ p = κ α Id + hB a.e. in A ξ , (5.14)and by Corollary 4.4 we obtain ∂ k σ − ε∂ k ˙ p = ( ∂ k h ) B + h | ξ D | (cid:18) ∂ k ξ D − ξ D : ∂ k ξ D | ξ D | ξ D (cid:19) a.e. in A ξ . (5.15)As a consequence, since | B | ≤ α − , we deduce that ¨ A ξ ∩ Q t | ∂ k σ − ε∂ k ˙ p || ξ D ||∇ ϕ |≤ (1 + α − ) ¨ A ξ ∩ Q t |∇ h || ξ D ||∇ ϕ | + ¨ A ξ ∩ Q t h (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ξ D − ξ D : ∂ k ξ D | ξ D | ξ D (cid:12)(cid:12)(cid:12)(cid:12) |∇ ϕ | . (5.16) TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 25
We start by estimating the first integral in the right hand side of (5.16). Let us show that thereexists a constant c α > α ) such that |∇ h | ≤ c α ε ( ∂ k ˙ p : ∂ k ˙ p ) / + c α h | ξ D | " X l =1 (cid:18) | ∂ l ξ D | − ( ξ D : ∂ l ξ D ) | ξ D | (cid:19) / + c α | f | a.e. in A ξ . (5.17)Writing componentwise the equilibrium equation (5.1) yields − ∂ j σ ij = f i in Ω × (0 , T ), and using(5.15), ε∂ j ˙ p ij + ( ∂ j h ) B ij + h | ξ D | (cid:18) ∂ j ( ξ D ) ij − ξ D : ∂ j ξ D | ξ D | ( ξ D ) ij (cid:19) + f i = 0 a.e. in A ξ . According to [27, Lemma 3.1], the matrix B is invertible if and only if α = 1 / √
2, and in that case, | B − | ≤ c α for some constant c α >
0. Hence ∂ i h = − ε ( B − ) ik ∂ j ˙ p kj − h | ξ D | ( B − ) ik (cid:18) ∂ j ( ξ D ) kj − ξ D : ∂ j ξ D | ξ D | ( ξ D ) kj (cid:19) − ( B − ) ik f k a.e. in A ξ , and |∇ h | ≤ c α ε ( ∂ k ˙ p : ∂ k ˙ p ) / + c α h | ξ D | X j,k,l =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂ l ( ξ D ) kj − ξ D : ∂ l ξ D | ξ D | ( ξ D ) kj (cid:12)(cid:12)(cid:12)(cid:12) / + c α | f | = c α ε ( ∂ k ˙ p : ∂ k ˙ p ) / + c α h | ξ D | " X l =1 (cid:18) | ∂ l ξ D | − ( ξ D : ∂ l ξ D ) | ξ D | (cid:19) / + c α | f | a.e. in A ξ , which proves (5.17). According to this estimate, the fact that ∇ ϕ = ϕ (6 ϕ ∇ ϕ ), and the Cauchy-Schwarz inequality, we get that ¨ A ξ ∩ Q t |∇ h || ξ D ||∇ ϕ | ≤ c α ε ¨ A ξ ∩ Q t ( ∂ k ˙ p : ∂ k ˙ p ) / | ξ D ||∇ ϕ | + c α ¨ A ξ ∩ Q t h " X l =1 (cid:18) | ∂ l ξ D | − ( ξ D : ∂ l ξ D ) | ξ D | (cid:19) / |∇ ϕ | + c α ¨ Q t | f || ξ D ||∇ ϕ |≤ c α ε ¨ A ξ ∩ Q t ∂ k ˙ p : ∂ k ˙ p ϕ ! / ¨ A ξ ∩ Q t | ξ D | ϕ |∇ ϕ | ! / +6 c α ¨ A ξ ∩ Q t h | ξ D | " X l =1 (cid:18) | ∂ l ξ D | − ( ξ D : ∂ l ξ D ) | ξ D | (cid:19) ϕ ! / ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! / +6 c α k f k L ∞ ( Q T ; R ) k ξ D k L ( Q T ; M × ) ≤ c α (cid:18) ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ (cid:19) / (cid:18) ε ¨ Q t | ξ D | ϕ |∇ ϕ | (cid:19) / + 6 c α ¨ A ξ ∩ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ ! / ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! / + 6 c α k f k L ∞ ( Q T ; R ) k ξ D k L ( Q T ; M × ) , (5.18)where we used (5.6) in the last inequality. We next estimate the second integral in the right hand side of (5.16). Using again that ∇ ϕ = ϕ (6 ϕ ∇ ϕ ), together with the Cauchy-Schwarz inequality, we infer that X k =1 ¨ A ξ ∩ Q t h (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ξ D − ξ D : ∂ k ξ D | ξ D | ξ D (cid:12)(cid:12)(cid:12)(cid:12) |∇ ϕ |≤ ¨ A ξ ∩ Q t h | ξ D | " X k =1 (cid:18) | ∂ k ξ D | − ( ξ D : ∂ k ξ D ) | ξ D | (cid:19) ϕ ! / ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! / ≤ ¨ A ξ ∩ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ ! / ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! / , (5.19)where, once more, we used (5.6) in the last inequality. Gathering (5.16), (5.18), (5.19), insertinginside (5.13), and using Young’s inequality yield ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 14 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + ε k ξ D k L ( Q T ; M × ) + k ξ D k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) + ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! . (5.20)By the definition of ξ we obtain √ ε k ξ D k L ( Q T ; M × ) ≤ C √ ε k ˙ p k L ( Q T ; M × ) + k e k L ( Q T ; M × )) , (5.21) k ξ D k L ( Q T ; M × ) ≤ C (cid:16) k ˙ p k L ( Q T ; M × ) + k e k L ( Q T ; M × )) (cid:17) . Thus, equation (5.20) can be rewritten as ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 14 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) + ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | ! . (5.22) Step 5.
We now estimate the last term ¨ A ξ ∩ Q t h | ξ D | ϕ |∇ ϕ | in the right hand side of (5.22). To simplify notation, let us denote by ψ := ϕ |∇ ϕ | . Firstly,according to (5.14), we have that σ D − ε ˙ p D = hξ D / | ξ D | a.e. in A ξ , and thus h | ξ D | = ( σ D − ε ˙ p D ) : ξ D a.e. in A ξ . This implies that ¨ A ξ ∩ Q t h | ξ D | ψ = ¨ A ξ ∩ Q t ( σ D − ε ˙ p D ) : ξ D ψ. (5.23)According to (3.15) and (4.2), we have that σ D − ε ˙ p D = 0 a.e. in A ξ , and thus the Cauchy-Schwarzinequality yields ¨ A ξ ∩ Q t h | ξ D | ψ ≤ ¨ Q t ( σ D − ε ˙ p D ) : ξ D ψ + C k σ − ε ˙ p k L ( Q T ; M × ) k ξ k L ( A ξ ; M × ) . On the other hand, the definition of ξ and the additive decomposition (5.4) yield ¨ Q t ( σ D − ε ˙ p D ) : ξ D ψ ≤ ¨ Q t σ : ( E ˙ u ) ψ − ¨ Q t (tr σ )(div ˙ u ) ψ + k σ k L ( Q T : M × ) (cid:16) k e k H ([0 ,T ]; L (Ω: M × )) + ε k ˙ p k L ( Q T ; M × ) (cid:17) + ε k ˙ p k L ( Q T ; M × ) k ξ D k L ( Q T ; M × ) . (5.24)Inserting (5.23) and (5.24) into (5.22), and using (5.21) lead to ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 14 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) + ¨ Q t σ : ( E ˙ u ) ψ − ¨ Q t (tr σ )(div ˙ u ) ψ ! . (5.25) Step 6.
We finally estimate both last integrals in the right hand side of (5.25). Integrating byparts, and using again the Cauchy-Schwarz and Young inequalities yield ¨ Q t σ : ( E ˙ u ) ψ − ¨ Q t (tr σ )(div ˙ u ) ψ = − ¨ Q t (div σ ) · ˙ uψ − ¨ Q t σ : ( ˙ u ⊗ ∇ ψ )+ 12 ¨ Q t ∇ (tr σ ) · ˙ uψ + 12 ¨ Q t tr σ ( ˙ u · ∇ ψ ) ≤ C (cid:18) ¨ Q t ∂ k σ : ∂ k e ϕ + k e k L ( Q T ; M × ) + k ˙ u k L ( Q T ; M × ) (cid:19) . Inserting this ultimate relation inside (5.25) leads to ¨ Q t ∂ k σ : ∂ k ˙ e ϕ + ε ¨ Q t ∂ k ˙ p : ∂ k ˙ p ϕ + 14 ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ∂ k ξ ϕ + 18 C ∗ ¨ Q t ( ∂ k σ − ε∂ k ˙ p ) : ( ∂ k σ − ε∂ k ˙ p ) ϕ ≤ C (cid:18) ¨ Q t ∂ k σ : ∂ k e ϕ + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) (cid:19) . (5.26) Step 7.
The final step rests on an application of Gronwall’s Lemma. For all t ∈ [0 , T ] let usdenote q ( t ) := 12 ˆ Ω ∂ k σ ( t ) : ∂ k e ( t ) ϕ dx. Neglecting nonnegative terms at the left hand side of (5.26) (note that the third integral at theleft hand side of (5.26) is nonnegative by (5.6)), we can write (5.26) as q ( t ) ≤ q (0) + a (cid:18) ˆ t q ( s ) ds + b (cid:19) , where a := C and b := 1 + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) . By applying Gronwall’s inequality we deduce that q ( t ) ≤ ( q (0) + ab ) e at for all t ∈ [0 , T ] . Using (2.1), together with the fact that ϕ = 1 on Ω ′ , ϕ = 0 on Ω \ Ω ′′ and ϕ ≥
0, the previousinequality leads to ˆ Ω ′ ∂ k σ ( t ) : ∂ k σ ( t ) dx ≤ C (cid:16) k e k H (Ω ′′ ; M × ) + k ˙ u k L ( Q T ; R ) + k e k H ([0 ,T ]; L (Ω; M × )) + ε k ˙ p k L ( Q T ; M × ) + k ˙ p k L ( Q T ; M × ) + k f k L (0 ,T ; H (Ω; R )) + k f k L ∞ ( Q T ; R ) (cid:17) . (5.27)Using this inequality, the a priori estimates (3.6), (3.8), and (3.10), and the continuous embeddingof BD (Ω) into L (Ω; R ), we conclude that the right hand side of (5.27) is uniformly boundedwith respect to ε . (cid:3) We are now in position to complete the proof of our regularity result.
Proof of Theorem 2.7.
According to Propositions 3.1, 3.3, and 5.1, the solution σ ε to the vis-coplastic model is uniformly bounded in L ∞ (0 , T ; H (Ω; M × )) and converges to σ weakly in H ([0 , T ]; L (Ω; M × )), where σ is the solution of the perfectly plastic model. By uniqueness ofweak limits, we infer that σ ∈ L ∞ (0 , T ; H (Ω; M × )). (cid:3) TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 29 Strong form of the flow rule
In this section we use the Sobolev regularity property established in Theorem 2.7 to define asuitable intrisic representative of the stress, for which the flow rule holds in a pointwise form, andnot only in the measure theoretical sense of (2.7) (see [6, 12, 19] for analogous results). To thisaim we define the averages of the stress as follows: for all ( x, t ) ∈ Ω × [0 , T ], and all r >
0, let σ r ( x, t ) := 1 L ( B r ( x ) ∩ Ω) ˆ B r ( x ) ∩ Ω σ ( y, t ) dy. From Theorem 2.7, we know that σ ( t ) ∈ H (Ω; M × ) for a.e. t ∈ [0 , T ]. Therefore, for a.e. t ∈ [0 , T ] there exists a quasi-continuous representative of σ ( t ), denoted by ˜ σ ( t ), such that σ r ( t ) → ˜ σ ( t ) in Ω \ Z t , where Z t ⊂ Ω is a Borel set of zero capacity in Ω.
Theorem 6.1.
For a.e. t ∈ [0 , T ] the quasi-continuous representative of the stress ˜ σ ( t ) is | ˙ p ( t ) | -measurable and H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) = ˜ σ ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | | ˙ p ( t ) | -a.e. in Ω . Proof.
Since for a.e. t ∈ [0 , T ], σ ( t ) ∈ L (Ω; M × ) with div σ ( t ) ∈ L (Ω; R ), we have that σ r ( t ) → σ ( t ) in L (Ω; M × ) , div σ r ( t ) → div σ ( t ) in L (Ω; R ) , as r → + . As a consequence of the definition (2.4) of the stress/strain duality, we infer that,[ σ r ( t ) : ˙ p ( t )] ⇀ [ σ ( t ) : ˙ p ( t )] weakly* in D ′ ( R ) . (6.1)Since Cap( Z t ) = 0, we have H ( Z t ) = 0. Thus, since ˙ u ( t ) ∈ BD (Ω), by [2, Remark 3.3] we getthat | E ˙ u ( t ) | ( Z t ) = 0, and thus | ˙ p ( t ) | ( Z t ) = 0. It thus follows that for a.e. t ∈ [0 , T ], σ r ( t ) → ˜ σ ( t ) | ˙ p ( t ) | -a.e. in Ω , and since σ r ( t ) ∈ C (Ω; M × ), it results that ˜ σ ( t ) is | ˙ p ( t ) | -measurable. Using next that σ r ( x, t ) ∈ K for all t ∈ [0 , T ] and all x ∈ Ω, we get that H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | ( x ) (cid:19) ≥ σ r ( x, t ) : d ˙ p ( t ) d | ˙ p ( t ) | ( x ) for | ˙ p ( t ) | -a.e. x ∈ Ω,and, passing to the limit as r → + , H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | ( x ) (cid:19) ≥ ˜ σ ( x, t ) : d ˙ p ( t ) d | ˙ p ( t ) | ( x ) for | ˙ p ( t ) | -a.e. x ∈ Ω. (6.2)According to Fatou’s Lemma, we infer that for all ϕ ∈ C ∞ c (Ω) with ϕ ≥ ˆ Ω (cid:20) H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) − ˜ σ ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | (cid:21) ϕ d | ˙ p ( t ) | = ˆ Ω lim inf r → + (cid:20) H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) − σ r ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | (cid:21) ϕ d | ˙ p ( t ) |≤ lim inf r → + ˆ Ω (cid:20) H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) − σ r ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | (cid:21) ϕ d | ˙ p ( t ) | . Since σ r ( t ) ∈ C (Ω; M × ) ∩ S , we can use Remark 2.3 together with the convergence (6.1) to getthat ˆ Ω (cid:20) H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) − ˜ σ ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | (cid:21) ϕ d | ˙ p ( t ) |≤ ˆ Ω ϕ dH ( ˙ p ( t )) − lim sup r → + h [ σ r ( t ) : ˙ p ( t )] , ϕ i = ˆ Ω ϕ dH ( ˙ p ( t )) − h [ σ ( t ) : ˙ p ( t )] , ϕ i = 0 , (6.3)where we used the flow rule (2.7) in the last equality. Finally, gathering (6.2) and (6.3), we obtainthat for a.e. t ∈ [0 , T ], H (cid:18) d ˙ p ( t ) d | ˙ p ( t ) | (cid:19) = ˜ σ ( t ) : d ˙ p ( t ) d | ˙ p ( t ) | | ˙ p ( t ) | -a.e. in Ω , as required. (cid:3) Acknowledgements
The authors would like to thank the hospitality of SISSA, where a large part of this work has beencarried out. The research has been supported by the ERC under Grant No. 290888 “Quasistaticand Dynamic Evolution Problems in Plasticity and Fracture”, by GNAMPA under Project 2016“Multiscale analysis of complex systems with variational methods”, and by the International As-sociate Laboratory, Laboratorio Ypazia delle Scienze Matematiche (LYSM), between CNRS andINdAM.
References [1]
D.R. Adams, L.I. Hedberg : Function spaces and potential theory , Springer-Verlag, Berlin (1996).[2]
L. Ambrosio, A. Coscia, G. Dal Maso : Fine properties of functions with bounded deformation,
Arch.Rational Mech. Anal. (1997), 201–238.[3]
L. Ambrosio, G. Dal Maso : A general chain rule for distributional derivatives,
Proc. Amer. Math. Soc. (1990), 691–702.[4]
L. Ambrosio, N. Fusco, D. Pallara : Functions of bounded variation and free discontinuity problems , OxfordMathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000).[5]
G. Anzellotti : On the existence of the rates of stress and displacements for Prandtl-Reuss plasticity,
Quart.Appl. Math. (1984), 181–208.[6] G. Anzellotti : On the extremal stress and displacement in Hencky plasticity,
Duke Math. J. (1984),133–147.[7] G. Anzellotti, M. Giaquinta : Existence of the displacement field for an elasto-plastic body subject toHencky’s law and von Mises yield condition,
Manuscr. Math. (1980), 101–136.[8] J.-F. Babadjian : Traces of functions of bounded deformation,
Indiana Univ. Math. J. (2015), 1271–1290.[9] J.-F. Babadjian, M.G. Mora : Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models,
Quart. Appl. Math. (2015), 265–316.[10] A. Bensoussan, J. Frehse : Asymptotic behaviour of the time dependent Norton-Hoff law in plasticity theoryand H regularity, Comment. Math. Univ. Carolin. (1996), 285–304.[11] H. Br´ezis : Operateurs maximaux monotones et semi groupes de contractions dans les espaces de Hilbert ,North-Holland, Amsterdam-London; American Elsevier, New York (1973).[12]
G. Dal Maso, A. De Simone, M.G. Mora : Quasistatic evolution problems for linearly elastic–perfectly plasticmaterials,
Arch. Rational Mech. Anal. (2006), 237–291.[13]
G. Dal Maso, A. De Simone, F. Solombrino : Quasistatic evolution for Cam-Clay plasticity: a weak formu-lation via viscoplastic regularization and time rescaling,
Calc. Var. PDEs (2011), 125–181.[14] E. Davoli, M.G. Mora : Stress regularity for a new quasistatic evolution model of perfectly plastic plates,
Calc. Var. PDEs (2015), 2581–2614.[15] F. Demengel, R. Temam : Convex function of a measure,
Indiana Univ. Math. J. (1984), 673–709.[16] F. Demengel, R. Temam : Convex function of a measure: the unbounded case, FERMAT days 1985: mathe-matics for optimization, 103–134, North-Holland (1986).
TRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 31 [17]
A. Demyanov : Regularity of stresses in Prandtl-Reuss perfect plasticity,
Calc. Var. PDEs (2009), 23–72.[18] D.C. Drucker, W. Prager : Soil mechanics and plastic analysis of rock and concrete,
Quart. Appl. Math. (1952), 157–175.[19] G.A. Francfort, A. Giacomini, J.-J. Marigo : The taming of plastic slips in von Mises elasto-plasticity,
Interfaces Free Bound. (2015), 497–516.[20] A. Giacomini, M.G. Mora : In preparation.[21]
C. Goffman, J. Serrin : Sublinear functions of measures and variational integrals,
Duke Math. J. (1964),159–178.[22] R.V. Kohn, R. Temam : Dual spaces of stresses and strains, with applications to Hencky plasticity,
Appl. Math.Optim. (1983), 1–35.[23] C. Mifsud : Short-time regularity for dynamic evolution problems in perfect plasticity. 2016. hal-01370797[24]
M.G. Mora : Relaxation of the Hencky model in perfect plasticity,
J. Math. Pures Appl. (2016), 725–743.[25]
S. Repin, G. Seregin : Existence of weak solution of the minimax problem arising in Coulomb-Mohr plasticity,
Amer. Math. Soc. Transl. (1995), 189–220.[26]
G. Seregin : On the differentiability of extremals of variational problems of the mechanics of ideally elastoplasticmedia. (Russian)
Differentsial’nye Uravneniya (1987), 1981–1991.[27] G. Seregin : On the differentiability of the stress tensor in the Coulomb-Mohr theory of plasticity,
St. PetersburgMath. J. (1993), 1257–1271.[28] P. Suquet : Sur les ´equations de la plasticit´e: existence et r´egularit´e des solutions,
J. M´ecanique (1981),3–39.[29] P. Suquet : Un espace fonctionnel pour les ´equations de la plasticit´e,
Ann. Fac. Sci. Toulouse Math. (1979),77–87.[30] R. Temam : Mathematical problems in plasticity , Gauthier-Villars, Paris (1985). Translation of
Probl`emesmath´ematiques en plasticit´e , Gauthier-Villars, Paris (1983).(J.-F. Babadjian)
Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
E-mail address : [email protected] (M.G. Mora) Dipartimento di Matematica, Universit`a di Pavia, via Ferrata 1, 27100 Pavia, Italy
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