Abstract
We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material’s degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016).
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1 Introduction
Plasticity and damage describe the inelastic behavior of materials in response to applied forces, respectively, accounting for permanent deformations and for discontinuities on microscales, both of surface type (microcracks) and of volume type (microvoids). In spite of their different macroscopical implications, the initial causes of the two phenomena are identical, and in particular in metals they are originated by movement and accumulation of dislocations (cf. [26, Chapter 7]). Several strain gradient plasticity models have been proposed (see e.g., [1, 6, 16, 20, 21, 23]) in order to provide a description of the interaction among dislocations, and to capture size effects, such as strengthening and strain hardening, caused by these defects in the range 500 nm–50 \(\upmu \)m.
In this paper, we present a mathematical model coupling scalar damage with the Gurtin–Anand gradient plasticity in small strain assumptions (for Gurtin–Anand plasticity see the original paper [21], and the mathematical treatment of the model in [17, 18, 35]). The aim of the present formulation is to account for different phenomena occurring in solid mechanics: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material’s degradation, and on the other hand, the dislocation density affects the damage growth.
The coupling between plasticity and damage is also investigated for instance in [2] and in [8, 9]. In [2], a model is proposed that combines perfect plasticity and damage, and the corresponding one-dimensional response is studied; in [8, 9] are proved existence results for this model in general dimension, basing on global and local minimization, respectively. In these works are considered suitable regularizations for the damage variable, but they do not include plastic strain derivatives.
We prove the existence of quasistatic evolutions for the present model in the framework of the energetic approach to rate-independent processes (see e.g., [29] for the abstract formulation, [10, 38] for applications to perfect plasticity, [32, 41] for damage, [8] , and [18]). Moreover, we study the asymptotics of these evolutions as the strain gradient terms tend to zero: Precisely we show, under stronger regularity assumptions, the convergence to evolutions for the coupled elastoplastic damage model studied in [8].
We now present the strong formulation and next our existence result for the corresponding energetic solutions. Let the reference configuration of a given elastoplastic body be a Lipschitz set \(\Omega \subset {{\mathbb {R}}}^n\), \(n\ge 2\), with \(\partial \Omega \) partitioned into \(\partial _D \Omega \) and \(\partial _N \Omega \). According to the classical theory for isotropic plastic materials in small strain assumptions, the variables
respectively, denoting the displacement, the elastic strain and the plastic strain, satisfy for every \(t\in {[0, T]}\) the additive decomposition (we denote the total strain as \(\mathrm {E}u=\tfrac{\nabla u +\nabla u^T}{2}\))
Moreover, assuming isotropic damage, we employ the variable \(\alpha :{[0, T]}\times \Omega \rightarrow [0,1]\) for the damage state of the body: here \(\alpha (t,x)=1\) stands for no damage and \(\alpha (t,x)=0\) for maximal damage in the vicinity of a point \(x\in \Omega \) at time t. In this presentation of the strong formulation, we consider smooth variables, both in space and in time.
We study the evolution for u, e, p, and \(\alpha \) in a time interval [0, T] when the body undergoes an imposed boundary displacement \(w(t):\partial _D \Omega \rightarrow {{\mathbb {R}}}^n\) on \(\partial _D \Omega \), namely
and volume and surface forces (on \(\partial _N \Omega \)), whose densities are denoted by \( f(t):\Omega \rightarrow {{\mathbb {R}}}^n\) and \(g(t):\partial _N \Omega \rightarrow {{\mathbb {R}}}^n\).
The starting point, as in the approach of Gurtin and Anand [21], is to consider \(\dot{e}(t)\), \(\dot{p}(t)\), and \(\nabla \dot{p}(t)\) as independent rate-like kinematical descriptors with conjugated internal forces \(\sigma (t)\), \(\sigma ^p(t)\), and \({\mathbb {K}}^p(t)\) such that the (internal) power expenditure within a subdomain \({\mathcal {B}}\subset \Omega \) at a time t is expressed by
Then, the stress configuration of the system is described by \(\sigma (t)\), which is the usual Cauchy stress, by a second-order tensor \(\sigma ^p(t)\) and by a third-order tensor \({\mathbb {K}}^p(t)\). (We denote by “\(\,{\cdot }\,\)” the scalar product between tensors of the same order, independently of the order.) In [21], a balance between the power of the internal forces (0.1) and the one of the external forces usually considered in gradient plasticity is imposed for every subdomain and every virtual velocity of the fields u, e, p; then, the following macroforce and microforce balance conditions are deduced:
and
where \(\nu \) is the outward normal to \(\Omega \) and we denote the deviatoric part of a matrix A by \(A_D\). Moreover, we have that for every subbody \({\mathcal {B}}\) with outward normal \(\nu \), the deviatoric matrix \({\mathbb {K}}^p \nu \) represents the surface density of microtractions associated to the plastic strain (cf. [20, Sections 9 and 11] for the connection between microtractions and thermodynamic force between dislocations). As in [21, Section 8], we assume null microscopic power expenditure at the boundary, namely
This condition corresponds to require that the material in contact with the body does not expend microscopic power on \(\partial \Omega \). In [21], it is observed that such an assumption is consistent at least with the macroscopic condition \(\sigma (t)\nu =0\), namely with a null macroscopic surface force g(t); following [17, 18], we include (sf2c) in a model where g(t) may be not null.
The total energy density for our model is
where \(\mu \) and k are nonincreasing and positive functions giving, respectively, the elastic shear and the bulk modulus, d is a continuous nonnegative function, and L, \(\ell \) are length scales. We can look at \(\psi \) as the sum of two parts: The first three terms correspond to the free energy density proposed by Gurtin and Anand, with the elastic moduli depending on the damage; the part of \(\psi \) depending only on \(\alpha \) and \(\nabla \alpha \) is taken as in [2, 34] and comes from mechanical considerations in [34]. In particular, the density of energy dissipated by the material in the process of damage growth is included in the free energy and the dependence on the damage gradient is quadratic.
The assumptions on \(\mu \) and k imply that the elastic tensor \({\mathbb {C}}(\alpha )\), defined by
is equicoercive and nonincreasing with respect to \(\alpha \); then, we consider incomplete damage with softening. The term \(\tfrac{L^2}{2}\mu (\alpha )|{\mathrm {curl}}\,p|^2\) is the density of energy stored by the geometrically necessary dislocations.
Dislocations are line defects within a crystal structure that are characterized by two vectors: the Burgers vector, b, that measures the slip displacement associated with the line defect, and a unit vector t, that points in the direction of the dislocation line. There are two main types of dislocations: the edge dislocations, where b and t are perpendicular, and the screw dislocations, where the two vectors are parallel. In the most general case, the dislocation line lies at an arbitrary angle to its Burgers vector and the dislocation has a mixed edge and screw character. The energy stored per unit length by a dislocation is proportional to \(\mu |b|^2\), see e.g., in [24, Section 4.4] and [26, Section 1]. The macroscopic Burgers tensor \({\mathrm {curl}}\,p\) measures the incompatibility of the field p and, for every unit vector m, \(({\mathrm {curl}}\,p)\, m\) is the Burgers vector, measured per unit area, associated with small loops orthogonal to m, namely with those dislocation whose lines pierce the plane with normal m (see [21, Section 3]); then, \({\mathrm {curl}}\,p\) provides a measure of the dislocation density.
Therefore, in order to minimize \(\mu (\alpha )|{\mathrm {curl}}\,p|^2\) it is convenient to damage portions of the material with high dislocation density (recall that \(\mu \) is nondecreasing). Actually this type of interplay between damage and dislocations complies with various models of microcrack formation and coalescence by dislocation pileup (see e.g., [7, 37, 39, 43]); moreover, one can use the length scale L as a parameter tuning the relevance of this term in the process of damage growth.
By the standard assumption that \(\sigma :=\frac{\partial \psi }{\partial e}\), the constitutive equation for the effective Cauchy stress
is derived.
In analogy to [21], we define the energetic higher-order stress \({\mathbb {K}}^p_{{\mathrm {en}}\,}\) as the symmetric deviatoric part in the first two components [cf. (1.1)] of the partial derivative of \(\psi \) with respect to \({\mathrm {curl}}\,p\), namely
for every \({\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\)-valued function A, the dissipative higher-order stress \({\mathbb {K}}^p_{{\mathrm {diss}}\,}\) by
Moreover, we impose a maximum plastic dissipation principle requiring the constraint condition
with \(K(\alpha (t,x))\) an ellipsoid (called stability domain) in the product space of the deviatoric matrices with the third-order tensors symmetric deviatoric in the first two components, and the flow rule
where \(N_E(\xi )\) denotes the normal cone to a convex set E at \(\xi \in E\). In other words, if \((\dot{p}(t,x),\nabla \dot{p}(t,x))\ne (0,0)\) then \((\sigma ^p(t,x), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t,x))\) belongs to the boundary of \(K(\alpha (t,x))\) and
for a suitable \(\lambda (t,x)>0\). Here \(l>0\) is a dissipative length scale and \(S_1\), \(S_2\) are nondecreasing positive functions of damage. Notice that we can deal with two different softening-type behaviors corresponding to different directions of the generalized constraint sets, as proposed in [21, Remark in Subsection 6.3] for a generalization of the model with a further internal variable. The three length scales l, \(\ell \), and L are constitutive parameters of the material.
In order to derive the equation governing the evolution of the damage variable, we introduce the total energy \({\mathcal {E}}\), that is obtained by integrating \(\psi \), and then it reads as
with
Following, e.g., [34], the strong formulation of damage evolution is provided by the Kuhn–Tucker conditions
where \(\partial _\alpha {\mathcal {E}}\) is the Gâteaux derivative of \({\mathcal {E}}\) with respect to \(\alpha \). The expression above makes sense for \(\beta \) sufficiently regular, see Proposition 5.3 for details.
Notice that, under regularity assumptions, (sf6b) implies that
where \(\partial _\alpha \psi = \mu '(\alpha )|e_D|^2+\tfrac{1}{2} k'(\alpha )|{\mathrm {tr}}\,e|^2+\tfrac{L^2}{2}\mu '(\alpha )|{\mathrm {curl}}\,p|^2+ d'(\alpha )\).
The conditions (sf1)–(sf6) constitute the strong formulation of the present model of Gurtin–Anand gradient plasticity coupled with damage. We now give the weak formulation of this model in the sense of [29]: The existence of a corresponding evolution is the main result of the paper.
Recalling (0.1) and (0.2), we get that the energy dissipated on a subbody \({\mathcal {B}}\), namely the difference between the power expended and the rate of the free energy, is
We have only a plastic term, since the density of the energy dissipated by damage growth is comprised in \(\psi \). By (sf5), the expression above is nonnegative (as expected from thermodynamical considerations) and we are led to define the plastic potential as the relaxation of the functional
We therefore consider for every \(\alpha \in H^1(\Omega )\) and \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
Here, \(\nabla p\) and \({\mathrm {D}}^sp\) are the absolutely continuous and the singular part of \({\mathrm {D}}p\) with respect to the Lebesgue measure \({\mathcal {L}}^n\), and \({\widetilde{\alpha }}\) is the precise representative of \(\alpha \), which is well defined at \({\mathcal {H}}^{n-1}\)–a.e. \(x\in \Omega \). Notice that if \(S_1\) and \(S_2\) have constant value \(S_Y\), then \({\mathcal {H}}\) depends only on p and coincides with the plastic potential in [17] and [18]. Moreover, when \(l=0\), we recover the potential in [2], where the stability domain is a convex set in the space of deviatoric matrices. We remark that the \(H^1\) damage regularization employed here is the one used in engineering (see for instance [2, 27]). This is an improvement with respect to the elastoplastic damage models in [8, 9]: The strong damage regularizations therein (respectively, \(W^{1,\gamma },\,\gamma >n\), and \(H^m,\,m>\frac{n}{2}\)) permitted us to work with a continuous field \(\alpha \) (see also e.g., [25], with \(H^m\) regularization), and therefore to use Reshetnyak’s Theorem for the plastic dissipation. Here, in contrast, in order to get the lower semicontinuity of \({\mathcal {H}}\) we prove an abstract Reshetnyak-type lower semicontinuity theorem (Theorem 3.1) tailored to the discontinuous functions and to the special measures considered. The proof exploits also tools from the theory of capacity.
As in [8] and in [9], the plastic dissipation corresponding to an evolution of \(\alpha \) and p in a time interval [s, t] is the \({\mathcal {H}}\)- variation of p with respect to \(\alpha \) on [s, t], namely
Thus, if w, f, g are absolutely continuous from \({[0, T]}\) into \(H^1(\Omega ;{{\mathbb {R}}}^n)\), \(L^n(\Omega ;{{\mathbb {R}}}^n)\), \(L^n(\partial _N \Omega ;{{\mathbb {R}}}^n)\), respectively, we define quasistatic evolution for the Gurtin–Anand model coupled with damage any function
that satisfies the following conditions:
-
(qs0)
irreversibility for every \(x \in \Omega \) the function\( {[0, T]}\ni t \mapsto \alpha (t, x)\) is nonincreasing;
-
(qs1)
global stability (u(t), e(t), p(t)) is admissible for the boundary condition w(t) (i.e., its energy is finite and (sf1) hold) for every \(t\in {[0, T]}\) and
$$\begin{aligned} {\mathcal {E}}(\alpha (t),e(t), {\mathrm {curl}}\,p(t)) -\langle {\mathcal {L}}(t),u(t)\rangle \le {\mathcal {E}}(\beta ,\eta , {\mathrm {curl}}\,q) -\langle {\mathcal {L}}(t),v\rangle +{\mathcal {H}}(\beta , q- p(t)) \end{aligned}$$for every \(\beta \le \alpha (t)\) and every triple \((v, \eta , q)\) admissible for w(t), where
$$\begin{aligned} \langle {\mathcal {L}}(t),u\rangle :=\int _\Omega f(t)\,{\cdot }\,u \,\mathrm {d}x+\int _{\partial _N \Omega } g(t)\,{\cdot }\,u\, \mathrm {d}{\mathcal {H}}^{n-1}\,; \end{aligned}$$ -
(qs2)
energy balance the function \(t\mapsto p(t)\) from [0, T] into \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) has bounded variation and for every \(t\in [0,T]\)
$$\begin{aligned} \begin{aligned}&{\mathcal {E}}(\alpha (t),e(t), {\mathrm {curl}}\,p(t)) -\langle {\mathcal {L}}(t),u(t)\rangle + {\mathcal {V}}_{\mathcal {H}}(\alpha , p; 0,t) \\&\quad = {\mathcal {E}}(\alpha (0), e(0), {\mathrm {curl}}\,p(0)) -\langle {\mathcal {L}}(0),u(0)\rangle + \int _0^t {\langle \sigma (s), \mathrm {E} \dot{w}(s)\rangle \,\mathrm {d}s}\\&\qquad -\int _0^t\langle \dot{{\mathcal {L}}}(s),u(s)\rangle \,\mathrm {d}s-\int _0^t \langle {\mathcal {L}}(s),\dot{w}(s)\rangle \,\mathrm {d}s. \end{aligned} \end{aligned}$$
Our existence result (Theorem 2.5) for quasistatic evolutions is based on time discretization and on approximation by means of solutions to incremental minimization problems, as common for globally stable quasistatic evolutions. The condition (qs0) corresponds to (sf6b), while (qs1) provides the desired balance equations (sf2) and the constraint condition (sf4). In order to deduce the plastic flow rule (sf5) and the activation condition for damage (sf6b), we have to assume more regularity on \(\alpha \) and p to differentiate in time the energy balance (qs2). In particular, to recover the strong formulation from the weak one we have to work with a continuous damage field. Let us also mention that the mathematical treatment of the evolution problem for a model with a damage regularization of the type \(W^{1,\gamma }\), \(\gamma >1\), instead of \(H^1\), is analogous to the one developed here.
Let us make some comments about the energetic formulation of the evolution, presented above. It is known (see e.g., [28, Ex. 6.1]) that the request of global stability may lead to jumps in the system response that could happen both too early and too far; in correspondence with these jumps, the system may overtake potential barriers. However, the description of the process is meaningful at least up to the first jump time. In order to avoid such unphysical phenomena in rate-independent evolutions, one may follow an approach, proposed and developed in recent years (see e.g., [30, 31] for an abstract treatment and [11, 12, 25] for some applications), based on vanishing viscosity approximation: the evolution is seen as limit of solutions to some rate-dependent systems containing a viscous dissipation that tends to zero. In the context of the coupling between perfect plasticity and damage, the first step for the viscous approximation in [9] was to write the Euler equation associated to the incremental minimum problem. In the present setting, it is not clear for instance how to differentiate the plastic dissipation with respect to \(\alpha \), if we desire an \(H^1\) regularization for the damage variable.
In the last part of the paper, we study the limit evolutions as the length scales l and L tend to zero. In [18], it is proven that, in this case, evolutions for the classical Gurtin–Anand formulation converge weakly for every time to evolutions for von Mises perfect plasticity model. We show an analogous convergence of the quasistatic evolutions for the present model to evolutions for the coupled elastoplastic damage model proposed in [8], which corresponds to the perfect plasticity for heterogeneous materials studied in [38] when the damage is constant in time. However, we have to consider a stronger (gradient) damage regularization for the Gurtin–Anand model with damage, since in [8] (and [9]) the space continuity of \(\alpha \) is needed.
An important difference with respect to the analysis in [18] relies on the fact that we cannot still characterize the global stability in the limit model by the equilibrium conditions for the Cauchy stress and the plastic constraint (cf. [10, Theorem 3.6]). Therefore, our proof is different from that in [18]. Indeed, we exploit the approximation in a strong sense of every admissibile triple for perfect plasticity with more regular ones that assume the boundary datum in a classical sense; in a forthcoming work, M.G. Mora proves such approximation for every Lipschitz domain; here, we show it in dimension two, and in higher dimension for a star-shaped domain. (See also Remark 6.2.)
The structure of the paper is the following: in Sect. 1, we fix the notation and recall some basic facts about the theory of capacity; in Sect. 2, we introduce the model starting from the mathematical formulation of the classical Gurtin–Anand model provided in [17]; and we give the definition of quasistatic evolutions, and state the existence result, which is proved in Sects. 3 and 4. The connection between strong and energetic formulation of the evolution is studied in Sect. 5, while Sect. 6 is devoted to the asymptotic analysis for vanishing strain gradient terms.
2 Notation and preliminaries
We recall in this section the definitions and the main properties of the mathematical objects employed in the paper.
2.1 Measures and function spaces
We denote by \({\mathcal {L}}^n\) the Lebesgue measure on \({{\mathbb {R}}}^n\) and by \({\mathcal {H}}^{s}\) the s-dimensional Hausdorff measure, for every \(s>0\). Given a locally compact subset B of \({{\mathbb {R}}}^n\) and a finite dimensional Hilbert space X, we use the symbol \(M_b(B; X)\) for the space of bounded X-valued Radon measures on B, the indication of X being omitted when \(X{=}{\mathbb {R}}\). This space is endowed with the norm \({\Vert \mu \Vert }_1 := \vert \mu \vert (B)\), where \(\vert \mu \vert \in M_b(B)\) is the total variation of the measure \(\mu \). For every \(\mu \in M_b(B; X)\), we denote by \(\mu ^a\) and \(\mu ^s\) the absolutely continuous and the singular part of \(\mu \) with respect to \({\mathcal {L}}^n\). By the Riesz Representation Theorem, \(M_b(B; X)\) can be regarded as the dual of \(C_0(B; X)\), the space of continuous functions \( \varphi :B \rightarrow X\) such that \(\{\vert \varphi \vert \ge \varepsilon \}\) is compact for every \(\varepsilon >0\) (see, e.g., [36, Theorem 6.19]). The weak\(^*\) topology of \(M_b(B; X)\) is defined using this duality. Moreover, we say that a sequence \((\mu _k)_k\subset M_b(B; X)\) converges strictly to a bounded Radon measure \(\mu \) if and only if it converges in the weak\(^*\) topology and \(|\mu _k|(B)\rightarrow |\mu |(B)\). We use the symbol \(\Vert \cdot \Vert _p\) for the \(L^p\) norm and \(\Vert \cdot \Vert _{1,q}\) for the norm of the Sobolev spaces \(W^{1,q}\). Notice that if \(L^1(B;X)\) is identified with the space of bounded measures \(\mu \) with \(\mu ^s=0\) (considering the density of \(\mu ^a\) with respect to \({\mathcal {L}}^n\)), then \({\Vert \cdot \Vert }_1\) coincides with the induced norm, so that the notation is consistent. Throughout the paper, we adopt the brackets \(\langle \cdot , \cdot \rangle \) to denote the product between dual spaces, the arrows \(\rightarrow \), \(\rightharpoonup \), and \(\mathrel {\mathop {\rightharpoonup }\limits ^{*}}\) for the strong, weak, and weak\(^*\) convergences, respectively, and \(\mathrel {\mathop {\rightarrow }\limits ^s}\) for the strict convergence of measures.
Given an open subset U of \({{\mathbb {R}}}^n\), the space BV(U; X) is the set of the functions \(u\in L^1(U;X)\) whose distributional derivative \({\mathrm {D}} u\) is a vector-valued bounded Radon measure. This is a Banach space with respect to the norm
A sequence \((u_k)_k\) converges to u weakly\(^*\) in BV if and only if \(u_k \rightarrow u\) in \(L^1\) and \({\mathrm {D}}u_k \mathrel {\mathop {\rightharpoonup }\limits ^{*}}{\mathrm {D}}u\) in \({M_b}\). We recall that if U is bounded and has Lipschitz boundary, then every bounded sequence in BV(U; X) has a weakly\(^*\) convergent subsequence and BV(U; X) is continuously embedded into \(L^q(U;X)\) for every \(1\le q\le \tfrac{n}{n-1}\), the embedding being compact for \(1\le q < \tfrac{n}{n-1}\). For the general theory of BV functions, we refer to [3].
2.2 Capacity
We recall some facts about the theory of capacity, referring to [22] for a complete treatment of the subject. Given an open subset U of \({{\mathbb {R}}}^n\) and \(1\le q <+\infty \), for every \(E\subset U\) the q-capacity of E in U is defined by
We shall use the shorter notation \({C}_q(E)\) when there is no ambiguity on the domain. The q-capacity is indeed a Carathéodory outer measure such that if \(1<q< n\) and \({C}_q(E)=0\), then the Hausdorff dimension of E is at most \(n-q\). We say that a real-valued function u is \({C}_q\)-quasicontinuous in U if for every \(\varepsilon >0\) there is an open set G such that \({C}_q(G)<\varepsilon \) and the restriction of u to \(U\setminus G\) is continuous. A sequence of real-valued functions \(u_k\) converges \({C}_q\)-quasiuniformly in U to u if for every \(\varepsilon >0\) there is an open set G such that \((G)<\varepsilon \) and \(u_k\rightarrow u\) uniformly in \(U\setminus G\). For every \((u_k)_k\subset C(U)\cap W^{1,q}(U)\) that is a Cauchy sequence in \(W^{1,q}(U)\), there exist a function \(u\in W^{1,q}(U)\) and a subsequence converging locally \({C}_q\)-quasiuniformly (namely quasiuniformly in the compact subsets of U) to u. It follows that such a limit u is \({C}_q\)-quasicontinuous, that \(u_k\rightarrow u\) pointwise \({C}_q\)-quasieverywhere in U (that is, pointwise except on a set of \({C}_q\)-capacity zero), and that every \(W^{1,q}\) function admits a quasicontinuous representative uniquely defined up to a \({C}_q\)-negligible set. For every \(u\in W^{1,q}(U)\), its precise representative \({{\widetilde{u}}}\), that is defined as the approximate limit of u in the Lebesgue points and takes value zero elsewhere, is a \({C}_q\)-quasicontinuous representative of u. When \(u_k\rightharpoonup u\) in \(W^{1,q}(U)\) there exists a subsequence \((u_j)_j\) such that \({\widetilde{u_j}}\rightarrow {{\widetilde{u}}}\) in \(\mu \)-measure, for every \(\mu \) nonnegative bounded Radon measure that vanishes on all \({C}_q\)-negligible Borel sets (cf. [5, Proposition 3.5 and Remark 3.4]). These results hold also for vector-valued functions, as one can see considering each component.
2.3 Matrices
We denote by \({\mathbb {M}}^{n\times n}\) (respectively by \({\mathbb {M}}^{n\times n\times n}\)) the space of \(n \times n\) real matrices (resp. third-order tensors) endowed with the Euclidean scalar product \(\xi \cdot \eta := \sum _{i,j} \xi _{ij} \eta _{ij}\) (resp. \({\mathbb {A}} \cdot {\mathbb {B}}:= \sum _{i,j,k} {\mathbb {A}}_{ijk} {\mathbb {B}}_{ijk}\)) and with the corresponding Euclidean norm \(\vert \xi \vert :=(\xi \cdot \xi )^{1/2}\). Moreover, \({\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\) denotes the subspace of symmetric matrices and \({\mathbb {M}}^{n{\times }n}_D\) the subspace of trace free matrices in \({\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\). Given \(\xi \in {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\), its orthogonal projection on \({\mathbb {M}}^{n{\times }n}_D\) is the deviator \(\xi _D:=\xi -\frac{1}{n}({\mathrm {tr}}\,\xi ) I\).
The symmetrized gradient of an \({{\mathbb {R}}}^n\)-valued function u(x) is the \({\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\)-valued function \(\mathrm {E}u(x)\) with components \(E_{ij}u:=\tfrac{1}{2}({\mathrm {D}}_ju_i+{\mathrm {D}}_iu_j)\), where \({\mathrm {D}}_i\) denotes the derivative \(\frac{\partial }{\partial x_i}\) for \(1\le i\le n\).
The gradient, the divergence, and the curl of a \({\mathbb {M}}^{n\times n}\)-valued function \(\xi (x)=(\xi _{ij}(x))\) are defined as
where \(\epsilon _{ipq}\) is the standard permutation symbol.
We say that a third-order tensor \({\mathbb {A}}=(a_{ijk})\) is symmetric deviatoric in its first two components, and we write \({\mathbb {A}}\in {\mathbb {M}^{n{\times } n{\times } n}_{D}}\), if
The divergence of a \({\mathbb {M}}^{n\times n \times n}\)-valued function \({\mathbb {A}}(x)=(a_{ijk}(x))\) is the \({\mathbb {M}}^{n\times n}\)-valued function given by
3 Quasistatic evolutions for the Gurtin–Anand model coupled with damage
In this section, we introduce the weak formulation of our model, corresponding to the strong formulation described in the Introduction, and we specify the mathematical framework adopted.
3.1 The reference configuration
The reference configuration of the elastoplastic body considered is a bounded, open, and Lipschitz set \(\Omega \subset {{\mathbb {R}}}^n\), \(n\ge 2\), whose boundary is decomposed as
\(\partial _D \Omega \) being the part of \(\partial \Omega \) where the displacement is prescribed, while traction forces are applied on \(\partial _N \Omega \). Here, \(\partial _D \Omega \) and \(\partial _N \Omega \) are open (in the relative topology), with the same boundary \(\Gamma \) such that
3.2 The external loading
We consider an evolution up to a time \(T>0\), driven by an absolutely continuous loading: This is given by an imposed boundary displacement (in the sense of trace on \(\partial _D \Omega \))
and by volume and surface forces (on \(\partial _N \Omega \)) with densites
For every \(t\in {[0, T]}\) we define \({\mathcal {L}}(t):W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\rightarrow {\mathbb {R}}\) as
It is easily seen that \({\mathcal {L}}(t)\) is linear and continuous on \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\).
3.3 Admissible configurations
As usual in linearized plasticity, the variables
denoting the displacement and the elastic and plastic strains, respectively, satisfy for every \(t\in {[0, T]}\) the additive strain decomposition
that corresponds to small strain assumptions (\(\mathrm {E}u=\tfrac{\nabla u +\nabla u^T}{2}\) is the linearized strain).
Given \(\overline{w} \in H^1(\Omega ;{{\mathbb {R}}}^n)\), an admissible configuration relative to \(\overline{w}\) is a triple (u, e, p) such that
the second equality in (2.1b) being in the sense of traces. The set of admissible configurations is then
Notice that if \(u:\Omega \rightarrow {{\mathbb {R}}}^n\) measurable, \(e\in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) satisfy (2.1b), then \(u \in W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) by properties of BV functions and Korn’s inequality.
3.4 The damage variable
The damage state of the body is described by a scalar internal variable
We shall see that during the evolution \(\alpha (t)\in H^1(\Omega ;[0,1])\) for every \(t\in {[0, T]}\), by the expression of our total energy. At a given \(x\in \Omega \), as \(\alpha (\cdot ,x)\) decreases from 1 to 0, the material point x passes from a sound state to a fully damaged one.
3.5 The elastic energy
In our formulation, the elastic shear and bulk moduli of the body, denoted, respectively, by \(\mu \) and k, depend on the damage state \(\alpha \). We assume that they are Lipschitz and nondecreasing functions defined on \({\mathbb {R}}\) and constant in \({\mathbb {R}}^-\) with
This corresponds to say that the stiffness decreases as the damage grows and that an elastic response is present even in the most damaged state. Defining for every \(\alpha \in {\mathbb {R}}\) the elastic tensor \({\mathbb {C}}(\alpha )\) by
the assumptions above imply that
for suitable positive constants \(\gamma _1\) and \(\gamma _2\). The elastic energy is
3.6 The energy stored by the dislocations
As explained in [21, Section 3], the macroscopic Burgers tensor \({\mathrm {curl}}\,p\) measures the incompatibility of the field p and it provides a measure of the dislocation density. Following the approach of Gurtin–Anand, the energy stored by the dislocations is given by
with \(L>0\) a length scale and \(\mu \) the shear modulus. Notice that, since \(\mu \) is nondecreasing, in order to minimize \(\mu (\alpha )|{\mathrm {curl}}\,p|^2\) it is convenient to damage portions of the material with high dislocation density.
Remark 2.1
Let us consider the functionals \({\mathcal {Q}}_1\) and \({\mathcal {Q}}_2\): their densities are the functions \((\alpha ,\xi )\mapsto \tfrac{1}{2}{\mathbb {C}}(\alpha )\,\xi \,{\cdot }\,\xi \) and \((\alpha , \xi )\mapsto \tfrac{L^2}{2} \mu (\alpha )|\xi |^2\), convex in \(\xi \) and continuous. Then, the Ioffe-Olach Semicontinuity Theorem (cf. [4, Theorem 2.3.1]) gives that \({\mathcal {Q}}_1\) and \({\mathcal {Q}}_2\) are lower semicontinuous with respect to the strong convergence of the first variable in \(L^1(\Omega )\) and the weak convergence of the second variable in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), namely for \(i\in \{1,2\}\)
3.7 The total energy
The total energy of a quadruple \((\alpha , u,e,p)\) such that \(\alpha \in {H^1(\Omega )}\) and \((u,e,p)\in A(\overline{w})\) for some \(\overline{w}\) is given by:
where \(\ell >0\) is an internal length and
with
We include in the total energy the function D and a quadratic gradient damage term. This choice is motivated by [34], where an analogous expression of (elastic) strain work is derived for an isotropic material in absence of prestress, under the assumption that the strain work depends also on \(\nabla \alpha \), by an expansion up to the second order in the strain and in \(\nabla \alpha \). The term \(D(\alpha )\) is related to the energy dissipated during the damage growth up to \(\alpha \).
3.8 The plastic dissipation
We now introduce a term which accounts for the energy dissipated in the evolution of plasticity. Let us first define the plastic potential \({\mathcal {H}}\) for every \((\alpha ,p)\in {H^1(\Omega )}\times {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) as
with \({\widetilde{\alpha }}\) the precise representative of \(\alpha \), which is well defined at \({\mathcal {H}}^{n-1}\)–a.e. \(x\in \Omega \) (indeed it is a \(C_2\)-quasicontinuous representative of \(\alpha \)), and \(\nabla p\) and \({\mathrm {D}}^sp\) the absolutely continuous and the singular part of \({\mathrm {D}}p\) with respect to the Lebesgue measure \({\mathcal {L}}^n\). We recall that
where \(J_p\) is the jump set of p, the functions \(p^+\) and \(p^-\) are the approximate upper and lower limit of p, respectively, and \({\mathrm {D}}^cp\) is the Cantor part of \({\mathrm {D}}p\) (see [3, Section 3.9]). We assume for \(i\in \{1,2\}\)
This definition of \({\mathcal {H}}\) is a generalization of the one in [18], where \({S_1}(\alpha )={S_2}(\alpha )={S^0_Y}>0\). Notice that for every \(\alpha \) in \({H^1(\Omega )}\) and \(p_1\), \(p_2\) in \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
and \({\mathcal {H}}\) is positively 1-homogeneous in p. Moreover, for every \(\alpha \) in \({H^1(\Omega )}\) and \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
where \(r:={S_1}(0)\wedge (l{S_2}(0))\) and \(R:=\sup _{\mathbb {R}}{S_1}\vee (l\sup _{\mathbb {R}}{S_2})\).
Given \(\alpha :[s,t]\rightarrow {H^1(\Omega )}\) and \(p:[s,t] \rightarrow {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) evolutions of damage and plastic strain in a time interval [s, t], the plastic dissipation corresponding is defined as the \({\mathcal {H}}\)- variation of p with respect to \(\alpha \) on [s, t], namely
We denote the variation of p on [s, t] by
3.9 The safe load conditions
Besides the assumptions (H2), we require that the forces satisfy the following strong safe load condition: for every \(t\in {[0, T]}\) there exists \(\varrho (t)\in L^n(\Omega ;{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\) such that
and there exists \(c_0>0\) such that for every \(A\in {\mathbb {M}}^{n{\times }n}_D\) with \(|A|\le c_0\) we have
We also assume that the functions \(t\mapsto \varrho (t)\) and \(t\mapsto \varrho _D(t)\) are absolutely continuous from \({[0, T]}\) into \(L^2(\Omega ;{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\) and \(L^\infty (\Omega ;{\mathbb {M}}^{n{\times }n}_D)\), respectively. Notice that the second equality in (H11.1) is well defined in the dual of the space of traces on \(\partial _N \Omega \) of \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) since \(\varrho (t)\) and \({\mathrm {div}}\,\varrho (t)\) are in \(L^n\) for every t, and that for every \((u,e,p)\in A(w)\) the representation formula
holds, where \(\langle \cdot ,\cdot \rangle \) denotes the pairing between \(H^{-1/2}(\partial _D \Omega ;{{\mathbb {R}}}^n)\) and \(H^{1/2}(\partial _D \Omega ;{{\mathbb {R}}}^n)\) (here we use \({\mathcal {H}}^{n-2}(\Gamma )<\infty \)).
Remark 2.2
The conditions (H11) are standard assumptions in the study of evolutions in perfect plasticity and strain gradient plasticity, when there are not null external forces (see e.g., [10, Equations (2.17) and (2.18)] and [18, Equations (4.13) and (4.14)]). However, as observed in [15, Remark 2.9], it is not a trivial issue the feasibility, for a given pair (f(t), g(t)) of loads, of finding a stress tensor \(\varrho (t)\) satisfying (H11). The safe load conditions are important in order to provide the following coercivity estimate for the plastic dissipation:
for every \(t\in {[0, T]}\), \(\alpha \in {H^1(\Omega )}\), and \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\). This can be obtained by adapting the proof of [18, Lemma 4.3], and it is based on the fact that \(\varrho _D(t)\) belongs to the ball centered in the origin of \({\mathbb {M}}^{n{\times }n}_D\) with radius \((S_1(0)\wedge S_2(0))-c_0\). From (2.6), it is immediate to deduce that
3.10 Quasistatic evolutions
We are now ready to give the definition of quasistatic evolution for the present model. We define, for given \(\alpha \in {H^1(\Omega )}\) and \(\overline{w}\in H^1(\Omega ;{{\mathbb {R}}}^n)\),
Definition 2.3
A quasistatic evolution for the Gurtin–Anand model coupled with damage is a function \((\alpha (t), u(t),e(t),p(t))\) from \([0,\, T]\) into \({H^1(\Omega ; [0,1])}\times W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n){\times }L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}){\times }{BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) that satisfies the following conditions:
-
(qs0)
irreversibility for every \(x \in \Omega \) the function\( {[0, T]}\ni t \mapsto \alpha (t, x)\) is nonincreasing;
-
(qs1)
global stability for every \(t\in [0,T]\) we have \((u(t),e(t),p(t))\in A(w(t))\) and
$$\begin{aligned} {\mathcal {E}}(\alpha (t),e(t), {\mathrm {curl}}\,p(t)) -\langle {\mathcal {L}}(t),u(t)\rangle \le {\mathcal {E}}(\beta ,\eta , {\mathrm {curl}}\,q) -\langle {\mathcal {L}}(t),v\rangle +{\mathcal {H}}(\beta , q- p(t)) \end{aligned}$$for every \((\beta ,v, \eta , q)\in {\mathcal {A}}(\alpha (t), w(t))\);
-
(qs2)
energy balance the function \(t\mapsto p(t)\) from [0, T] into \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) has bounded variation and for every \(t\in [0,T]\)
$$\begin{aligned} \begin{aligned}&{\mathcal {E}}(\alpha (t),e(t), {\mathrm {curl}}\,p(t)) -\langle {\mathcal {L}}(t),u(t)\rangle + {\mathcal {V}}_{\mathcal {H}}(\alpha , p; 0,t) \\&\quad = {\mathcal {E}}(\alpha (0), e(0), {\mathrm {curl}}\,p(0)) -\langle {\mathcal {L}}(0),u(0)\rangle + \int _0^t {\langle \sigma (s), \mathrm {E} \dot{w}(s)\rangle \,\mathrm {d}s}\\ {}&\qquad -\int _0^t\langle \dot{{\mathcal {L}}}(s),u(s)\rangle \,\mathrm {d}s-\int _0^t \langle {\mathcal {L}}(s),\dot{w}(s)\rangle \,\mathrm {d}s , \end{aligned} \end{aligned}$$where \(\sigma (s):= {\mathbb {C}}(\alpha (s)) e(s)\).
Remark 2.4
We shall prove in Lemma 4.1 that such an evolution is measurable and the integrals in (qs2) are well defined.
We now state the main result of the paper, that will be proved in Sects. 3 and 4.
Theorem 2.5
(Existence of quasistatic evolutions) Assume (H1), (H2), (H3)–(H6), (H8)–(H10) and (H11), and let \((\alpha _0, (u_0, e_0, p_0)) \in {H^1(\Omega ; [0,1])}\times A(w(0))\) satisfy the stability condition
for every \((\beta ,v, \eta , q)\in {\mathcal {A}}(\alpha _0, w(0))\). Then, there exists a quasistatic evolution for the Gurtin–Anand model coupled with damage \(t\mapsto (\alpha (t),u(t),e(t),p(t))\) such that \(\alpha (0)=\alpha _0\), \(u(0)=u_0\), \(e(0)=e_0\), \(p(0)=p_0\).
4 The minimization problem
This section is focused on the minimization problem employed in the construction of time discrete approximations for a quasistatic evolution. If \(\overline{\alpha }\in {H^1(\Omega ; [0,1])}\) and \(\overline{p} \in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) are the current values of the damage variable and of the plastic strain, and \(w\in H^1(\Omega ;{{\mathbb {R}}}^n)\), \(f\in L^n(\Omega ;{{\mathbb {R}}}^n)\), and \(g\in L^n(\partial _N \Omega ;{{\mathbb {R}}}^n)\) are the updated values of the boundary displacement and of the body and surface loads, the updated values of the internal variables \(\alpha ,u,e,p\) are obtained by solving the problem
where
First, we show the existence of solutions to this problem and their main properties, and afterward a stability property of the solutions with respect to variations of the data.
The following semicontinuity theorem will be used several times in the following, for instance to prove the existence of solutions to (3.1). Notice that in the case when the energy includes a gradient damage term \(\Vert \nabla \alpha \Vert _\gamma ^\gamma \), with \(\gamma >n\) the result follows easily from Reshetnyak’s Lower Semicontinuity Theorem (see [8, 9]). Instead, for the current regularization \(\Vert \nabla \alpha \Vert _2^2\), the proof relies on the specific form of \({\mathcal {H}}\); in particular, we use the fact that \({\mathrm {D}}p\) is the gradient of a BV function and then it vanishes on sets with dimension lower than \(n-1\).
Theorem 3.1
The plastic potential \({\mathcal {H}}\) defined in (H9) is lower semicontinuous with respect to the weak–\({H^1(\Omega )}\) convergence of \(\alpha _k\) and the weak\(^*\)–\({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) convergence of \(p_k\), namely
for every \(\alpha _k\rightharpoonup \alpha \) in \({H^1(\Omega )}\) and \(p_k\mathrel {\mathop {\rightharpoonup }\limits ^{*}}p\) in \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\).
Proof
Let \((\alpha _k)_k\) and \((p_k)_k\) be two sequences in \({H^1(\Omega )}\) and \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) such that \(\alpha _k\rightharpoonup \alpha \) in \({H^1(\Omega )}\) and \(p_k\mathrel {\mathop {\rightharpoonup }\limits ^{*}}p\) in \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\). We divide the proof into two steps, starting from the case when the functions \(p_k\) are uniformly bounded, that is \(\Vert p_k\Vert _\infty <M\), for a suitable \(M>0\).
Step 1 (\(p_k\) uniformly bounded) Notice that for \(\beta \in {H^1(\Omega )}\cap L^\infty (\Omega )\) and \(q \in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\cap L^\infty (\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) we have that \(\beta \, q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) and
where \({\widetilde{\beta }}\) is the precise representative of \(\beta \). Indeed, it is well known that this formula holds for \(\beta \in C^1(\Omega )\); thus, we can argue by approximation, considering a sequence \((\beta _k)_k\subset C^1(\Omega )\) uniformly bounded in \(L^\infty (\Omega )\) such that \(\beta _k\rightarrow \beta \) in \({H^1(\Omega )}\). Therefore, the total variations \(\Vert {\mathrm {D}}(\beta _k\,q)\Vert _1\) are uniformly bounded and then up to a subsequence
Moreover, up to a further subsequence, \(\beta _k \rightarrow \beta \) pointwise \(C_2\)-quasieverywhere (see Sect. 1), which implies \(\beta _k(x)\rightarrow {\widetilde{\beta }}(x)\) for \(|{\mathrm {D}}q|\)–a.e. \(x\in \Omega \); then, we recover (3.4) by using the fact that \(q\in L^\infty (\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) and the Dominated Convergence Theorem for the convergence of the right-hand side.
We now take \(q=p_k\), \(\beta =S_i(\alpha _k)\), and recall that \(S_i\) are bounded and Lipschitz maps (cf. (H10)). Since \(S_i(\alpha _k)\rightarrow S_i(\alpha )\) in \(L^2(\Omega )\) and the sequences \((S_i(\alpha _k))_k\) are equibounded in \(L^\infty (\Omega )\) and in \({H^1(\Omega )}\), we get that \(S_i(\alpha _k)\rightarrow S_i(\alpha )\) in \(L^r(\Omega )\) for every \(r\in [1,+\infty )\) and \(S_i(\alpha _k)\rightharpoonup S_i(\alpha )\) in \({H^1(\Omega )}\), for \(i=1,2\). In particular
Evaluating (3.4) with \(q=p_k\) and \(\beta ={S_2}(\alpha _k)\) we get
Hence, the measures \({\mathrm {D}}({S_2}(\alpha _k) p_k)\) have uniformly bounded total variations, and (3.5) implies that
On the other hand, since \(p_k\rightarrow p\) in \(L^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) and we are assuming the \(p_k\) uniformly bounded, then \(p_k\rightarrow p\) in \(L^r(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) for every \(r\in [1,+\infty )\) and
By difference (and (3.4) with \(q=p\) and \(\beta =\alpha \)) we obtain that
In order to prove (3.3), we observe that by definition \({\mathcal {H}}\) is the total variation of a convex function of a measure, defined in the sense of [19]; precisely for every \(\beta \in {H^1(\Omega )}\) and \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
where
and \(({S_1}(\beta )q,{S_2}({\widetilde{\beta }}){\mathrm {D}}q)\in M_b(\Omega ;{\mathbb {M}}^{n{\times }n}_D\times {\mathbb {M}^{n{\times } n{\times } n}_{D}})\) is the product measure of \({S_1}(\beta )q\) and \({S_2}({\widetilde{\beta }}){\mathrm {D}}q\). From (3.5) and (3.6) it follows that
In view of (3.7), by Reshetnyak’s Lower Semicontinuity Theorem (cf. [3, Theorem 2.38]) applied to the convex function \(\overline{f}\) and to the measures above, we get (3.3).
Step 2 (General case) We now approximate the functions \(p_k\) with bounded functions, without increasing the total variation of the gradient. For every \(x\in \Omega \), \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), and \(R>1\) we define
where \(\omega _R\in C^1({\mathbb {R}}^+\cup \{0\}; [0,1])\) is a nonincreasing map such that
and \({\widehat{R}}(R)\) is some radius bigger than R. We can take for instance
The resulting function \(\omega _R\) has a \(C^1\) discontinuity at \((R+1)e^{\frac{4(R+1)^2-1}{2(R+1)}}\), where \(g_R\) vanishes; however, we can modify it near the corner to obtain a \(C^1\) function by using a smooth cutoff \(h_R\) such that \(|h_R'(\varrho )|\le |g_R'(\varrho )|\) and \(h_R(\varrho )+\varrho ^2(h_R'(\varrho ))^2\le 1\) (Fig. 1).
The cutoff function \(\omega _R\)
By construction \(|\varphi _R(q)|\le {\widehat{R}}\) a.e. in \(\Omega \), and we can see that \(\varphi _R(q)\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) with
Let us prove (3.8) first in the case \(q\in C^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\). Here, we see every matrix \(\xi \) as a vector in \({\mathbb {R}}^{n^2}\); then
which gives
by the Cauchy inequality and the fact that \(\omega _R\) is nonnegative and nondecreasing. Therefore, the inequality (3.8) is proved when \(q\in C^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\). We now show the general case: since these measures are regular, it is sufficient to prove (3.8) on open sets.
Given \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) and U an open subset of \(\Omega \), by the Anzellotti-Giaquinta Approximation Theorem (cf. [3, Theorem 3.9]) there exists \((q_k)_k\subset C^1(U;{\mathbb {M}}^{n{\times }n}_D)\) such that \(q_k \mathrel {\mathop {\rightharpoonup }\limits ^{*}}q\upharpoonright _U\) in \(BV(U;{\mathbb {M}}^{n{\times }n}_D)\) and
by regularity of \(\omega _R\) we get that
as \(k\rightarrow \infty \). By semicontinuity of the total variation with respect to weak\(^*\) convergence, the inequality (3.8) is proved for open sets, and this concludes the proof of (3.8).
By (3.9) we have that \(\varphi _R(p_k)\mathrel {\mathop {\rightharpoonup }\limits ^{*}}\varphi _R(p)\) in \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) as \(k\rightarrow \infty \); then, from the Step 1 (recall that \(|\varphi _R(p_k)|\le {\widehat{R}}\) a.e. in \(\Omega \)) it follows that
and we want to pass to the limit as \(R\rightarrow \infty \). First, we prove that for every k
To this end, it is useful to rewrite \({\mathcal {H}}\) as
where \(\left| \left( {S_1}(\beta ){S_2}(\beta )^{-1} q,l\,{\mathrm {D}}q\right) \right| \) is the variation of the product measure
Since by construction \(|\varphi _R(q)|\le |q|\) a.e. in \(\Omega \) for every \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) we get by (3.8) that
for every \(\beta \in {H^1(\Omega )}\), \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), and \(R>1\). Taking \(\beta =\alpha _k\), \(q=p_k\), and integrating the positive function \({S_2}({\widetilde{\alpha }}_k)\), we obtain (3.10).
Therefore, the proof is completed if we show that
The chain rule for BV functions proved in [42] gives in our case
where \({\widetilde{p}}(x)\) is the approximate limit of p at any Lebesgue point x, and then
It is known from the theory of BV functions that \(p^+(x),\,p^-(x)\in {\mathbb {R}}\) for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Omega \) and hence \({\mathcal {H}}^{n-1}\)-a.e. \(x\in \Omega \setminus J_p\) is a Lebesgue point for p. Since \(\omega _R(|x|)=1\) if \(|x|\le R\), it follows that
and
By (3.8) we have that
Then, we can pass to the limit in (3.12) using the Dominated Convergence Theorem and obtain (3.11). Therefore, the proof is concluded. \(\square \)
Using Theorem 3.1, we can prove the existence of solutions to the minimization problem (3.1) by applying the direct method of the Calculus of Variations.
Lemma 3.2
Problem (3.1) admits a solution, and for every \((\alpha , u,e,p)\) solution of (3.1), it holds that \(\alpha \in {H^1(\Omega ; [0,1])}\).
Proof
Let
be a minimizing sequence for (3.1); by (H8.2), (H5.1), and (H10) we can assume \(\alpha _k\in {H^1(\Omega ; [0,1])}\) for every k. Since \((0, w, \mathrm {E}w,0)\in {\mathcal {A}}(\overline{\alpha }, w)\) and
we get that \({\mathcal {E}}(\alpha _k,e_k,{\mathrm {curl}}\,p_k)-\langle {\mathcal {L}},u_k\rangle +{\mathcal {H}}(\alpha _k,p_k-\overline{p})\) is uniformly bounded in k and
by the representation formula (2.5). By definition of \({\mathcal {E}}\) and (2.7), we obtain that
and hence, there exist \(\alpha \in {H^1(\Omega ; [0,1])}\), \(e\in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) such that up to a subsequence
Moreover, \({\mathrm {curl}}\,p \in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) and
Using the embedding \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\hookrightarrow L^{\frac{n}{n-1}}(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) and Korn’s inequality, it follows easily from \((u_k,e_k,p_k)\in A(w)\) that \(u_k\) are uniformly bounded in \(W^{\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\): then, up to a further subsequence
for a suitable u such that \((u,e,p)\in A(w)\). Collecting the semicontinuity properties (2.2) and (3.3), we get that \((\alpha ,u,e,p)\) is a minimizer, and the proof is completed. \(\square \)
In the very same way of [8, Lemma 3.2], we deduce the remark below from the properties of \({\mathcal {H}}\).
Remark 3.3
If \((\alpha , u,e,p)\) solves (3.1) then
for every \(({\widetilde{\alpha }},{\widetilde{u}},{\widetilde{e}},{\widetilde{p}})\in {\mathcal {A}}(\alpha ,w)\).
The following lemma states some differential conditions for a triple (u, e, p) such that \((\alpha ,u,e,p)\) satisfies (3.13). We shall make use of these conditions to recover the classical formulation of the model.
Lemma 3.4
Let \((\alpha ,u,e,p)\) satisfy (3.13). Then
for every \((v,\eta ,q)\in A(0)\), where \(\sigma :={\mathbb {C}}(\alpha )e\). Moreover
Proof
Let us fix \((v,\eta ,q)\in A(0)\). Since for every \(\varepsilon \in {\mathbb {R}}\)
from the remark above we have that for every \(\varepsilon \in {\mathbb {R}}\)
Then, the positive homogeneity of \({\mathcal {H}}\) gives that for every \(\varepsilon \in {\mathbb {R}}\)
Dividing by \(\varepsilon \) and passing to the limit as \(\varepsilon \rightarrow 0\), we recover (3.14).
Choosing in (3.14) (v, Ev, 0) for every \(v\in C^\infty (\overline{\Omega };{{\mathbb {R}}}^n)\) with \(v=0\) on \(\partial _D \Omega \), we get (3.15). Notice that the normal trace of \(\sigma \) on \(\partial \Omega \) is well defined in \(H^{-1/2}(\partial \Omega ;{{\mathbb {R}}}^n)\) since \(\sigma \in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) with divergence in \(L^2(\Omega ;{{\mathbb {R}}}^n)\). \(\square \)
The lemma below will permit us to say that when both \(\alpha \) and p are continuous at a given time then all the evolution is continuous there. In contrast with [8, Lemma 3.4], here it is not useful to write \(\omega _{12}\) in terms of \(\Vert \alpha _1 - \alpha _2\Vert _{\infty }\); indeed, we will consider the case when a sequence of functions \(\alpha _1\) tends to a function \(\alpha _2\) weakly in \(H^1(\Omega )\), and this does not provide uniform convergence in \(\Omega \).
Lemma 3.5
For \(i=1,2\) let \(w_i \in H^1(\Omega ; {{\mathbb {R}}}^n)\), \(f_i\in L^n(\Omega ;{{\mathbb {R}}}^n)\), \(g_i\in L^\infty (\partial _N \Omega ;{{\mathbb {R}}}^n)\), and let \({\mathcal {L}}_i\) be defined by (3.2) with \(f=f_i\) and \(g=g_i\). Suppose that \((\alpha _i,u_i,e_i,p_i)\) satisfies (3.13) with data \(w=w_i\), \({\mathcal {L}}={\mathcal {L}}_i\), and let
Then, there exists a positive constant C depending on L, \(\mu (0)\), \(\gamma _1\), \(\gamma _2\), R, \(\Omega \), \(\partial _N \Omega \) such that
Proof
Let
Since \((v,\eta ,q)\in A(0)\), by (3.14) we have that
Gathering the inequalities above and using (2.3) we obtain that
and then, by the definition of \(\eta \),
Arguing as in the proof of [10, Theorem 3.8] we see that there exists a constant \(\widehat{C}\) depending only on \(\Omega \) and \(\partial _N \Omega \) such that
Since
we conclude the former of (3.16) from (3.17) using the Cauchy inequality. The latter estimate is easily shown using the compatibility conditions (2.1b) and Korn’s Inequality. \(\square \)
We now prove a stability result for the solutions of (3.13) with respect to the weak convergence of the data.
Theorem 3.6
(Stability of solutions to (3.13)) Let \(w_k \in H^1(\Omega ; {{\mathbb {R}}}^n)\), \({\mathcal {L}}_k\in \big (W^{\frac{n}{n-1}}({\Omega }; {{\mathbb {R}}}^n)\big )'\), \(\alpha _k\in {H^1(\Omega ; [0,1])}\), and \((u_k, e_k, p_k) \in A(w_k)\) for every k. Assume that these sequences of functions converge weakly\(^*\) (weakly for reflexive spaces) in their target spaces to functions \(w_\infty \), \({\mathcal {L}}\), \(\alpha _\infty \), \(u_\infty \), \(e_\infty \), and \(p_\infty \), respectively.
Then \((u_\infty ,e_\infty ,p_\infty )\in A(w_\infty )\). If, in addition,
for every k and every \((\widehat{\alpha }_k,\widehat{u}_k, \widehat{e}_k,\widehat{p}_k)\in {\mathcal {A}}(\alpha _k, w_k)\), then
for every \((\alpha , u, e, p) \in {\mathcal {A}}(\alpha _\infty ,w_\infty )\).
Proof
The fact that \((u_\infty ,e_\infty ,p_\infty )\in A(w_\infty )\) is immediate by the definition of admissible triple and the weak convergences assumed.
Let us now fix \((\alpha , u, e, p) \in {\mathcal {A}}(\alpha _\infty ,w_\infty )\) and test (3.18) by
Indeed by assumption \((\widehat{\alpha }_k,\widehat{u}_k,\widehat{e}_k,\widehat{p}_k)\in {\mathcal {A}}(\alpha _k,w_k)\), and moreover \(\widehat{\alpha }_k \rightharpoonup \alpha \) and \( \alpha \vee \alpha _k \rightharpoonup \alpha _\infty \) in \({H^1(\Omega )}\), \(\widehat{u}_k\rightharpoonup u\) in \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\), \(\widehat{e}_k\rightharpoonup e\) in \(L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\), \(\widehat{p}_k\mathrel {\mathop {\rightharpoonup }\limits ^{*}}p\) in \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\).
Since for every \(\alpha \in {H^1(\Omega )}\) and every \(\eta _1, \eta _2 \in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) we have that
and for every \(\alpha , \beta \in {H^1(\Omega )}\)
then the inequality (3.18) can be rewritten, adding to both sides \(-{\mathcal {Q}}_1(\widehat{\alpha }_k,e_k) - {\mathcal {Q}}_2(\widehat{\alpha }_k,{{\mathrm {curl}}\,} p_k)\), thus obtaining
Notice that for every \(\eta \in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\)
Moreover, \((x,\beta ,\xi )\mapsto [{\mathbb {C}}(\beta )-{\mathbb {C}}(\beta \wedge \alpha (x))]\xi \,{\cdot }\,\xi \) and \((x,\beta ,\xi )\mapsto (\mu (\beta )-\mu (\beta \wedge \alpha (x)))|\xi |^2\) are measurable functions from \(\Omega \times {\mathbb {R}}\times {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}\) into \({\mathbb {R}}^+\cup \{0\}\), continuous in the variable \(\beta \) and convex in \(\xi \). Therefore, the Ioffe-Olach Semicontinuity Theorem (cf. [4, Theorem 2.3.1]) implies that
for every \(i\in \{1,2\}\) and \(\eta _k\rightharpoonup \eta _\infty \) in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\). Then, it follows that
On the other hand
Indeed, since \(\widehat{\alpha }_k\rightharpoonup \alpha \) in \({H^1(\Omega )}\), up to a subsequence \(k_j\) we have that \(\widetilde{\widehat{\alpha }}_{k_j}(x)\rightarrow \widetilde{\alpha }(x)\) for \(|{\mathrm {D}}(p-p_\infty )|\)–a.e. \(x\in \Omega \); therefore, by the Dominated Convergence Theorem
and, since the limit is independent of the subsequence, the convergence above holds for the whole sequence. The convergence of \({\mathcal {D}}(\widehat{\alpha }_k)\) to \({\mathcal {D}}(\alpha )\) follows easily from (H8). Let us consider the first term in \(\delta _k\): the symmetry of \({\mathbb {C}}(\beta )\) for every \(\beta \in {\mathbb {R}}\) gives that
Since \({\mathbb {C}}(\beta )\) is bounded uniformly with respect to \(\beta \in {\mathbb {R}}\) and \(\widehat{\alpha }_k\rightharpoonup \alpha \) in \({H^1(\Omega )}\), by the Dominated Convergence Theorem we get that
From the fact that \(e_k\rightharpoonup e_\infty \) in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) we conclude that
recalling (3.20). In the same way, we get that
and then we conclude (3.23). Gathering (3.22) and (3.23) we get (3.19) and the proof is completed. \(\square \)
5 Existence of quasistatic evolutions
This section is devoted to the proof of Theorem 2.5, basing on discrete time approximation. First, we construct a sequence of approximate evolutions by solving, for the k-th approximant, k incremental problems of the type (3.1) which we have studied in Sect. 3; then, we show that this sequence converges in a suitable sense to a quasistatic evolution for the Gurtin–Anand model coupled with damage. Henceforth, we assume the hypotheses of Theorem 2.5, in particular the stability condition on the initial datum \((\alpha _0, u_0,e_0,p_0)\).
Before starting the proof of the existence result, we prove that the integrals in the energy balance (qs2) of Definition 2.3 are well defined. This follows immediately by the following lemma.
Lemma 4.1
Let \((\alpha ,u,e,p)\) be a quasistatic evolution and \(\sigma (t):={\mathbb {C}}(\alpha (t))e(t)\), according to Definition 2.3. Let \(r\in [1,\infty )\). Then, the functions \(t\mapsto \alpha (t)\in L^r(\Omega )\), \(t\mapsto u(t)\in W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\), \(t\mapsto e(t)\in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), and \(t\mapsto \sigma (t)\in {L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) are strongly continuous except at most for a countable subset of \({[0, T]}\), and
Proof
By the irreversibility condition and [8, Lemma A.2],s it follows that there exists a countable set \(E_1\subset {[0, T]}\) such that \(\alpha \) is continuous at every \(t\in {[0, T]}\setminus E_1\) with respect to the \(L^r\) norm, for every \(r\in [1,\infty )\). The condition (qs2) gives that \(p\in L^\infty (0,T;{BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)})\); then by (qs1), taking \((\beta ,v, \eta , q)=(0,w(t),\mathrm {E}w(t),0)\) for every t, we deduce that \(\alpha (t)\), u(t), e(t) are uniformly bounded in \({H^1(\Omega )}\), \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\), \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), respectively. Thus for every \(t\in {[0, T]}\setminus E_1\)
Since p has bounded variation into the space \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), the set \(E_2\) of its discontinuity points is at most countable. Moreover, by the uniform bound for \(\mu (\alpha )\) and \({\mathbb {C}}(\alpha )\), (4.1), and the Dominated Convergence Theorem, it follows that for every \(t\in {[0, T]}\setminus E_1\)
Then, using Lemma 3.5 (recall that the loading is continuous in time) we obtain that e and u are strongly continuous in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) and \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) at every \(t\in {[0, T]}\setminus E\), with \(E=E_1\cup E_2\).
Hence, by (4.1), \(\sigma (s)\rightarrow \sigma (t) \) in \(L^1(\Omega ;{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\) as \(s\rightarrow t\) for every \(t\in {[0, T]}\setminus E\). Since \({\mathbb {C}}(\alpha )\) is uniformly bounded, and then \(|\sigma (s)|\le C |e(s)|\) in \(\Omega \), we deduce that this convergence is indeed strong in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), applying the Dominated Convergence Theorem. Finally, \(\alpha \) is measurable into \({H^1(\Omega )}\) by the former of (4.1) and the fact that \({H^1(\Omega )}\) is separable. This concludes the proof. \(\square \)
For every \(k\in {\mathbb {N}}\) we define approximate evolutions \((\alpha _k, u_k, e_k, p_k)\) by induction. Let us set \(t_k^i:=T\tfrac{i}{k}\) for \(i=0,\dots ,k\) and
For \(i=1,\dots ,k\) let \((\alpha _k^i, u_k^i,e_k^i, p_k^i)\) be a solution to the incremental problem
where \( w_k^i:=w(t_k^i)\) and \({\mathcal {L}}_k^i:={\mathcal {L}}(t_k^i)\). Notice that Lemma 3.2 ensures the existence of solutions. Then we define for \(i=0,\dots ,k-1\) and \(t\in [t_k^{i}, t_k^{i+1})\)
while \((\alpha _k(T),e_k(T ),u_k(T),p_k(T)):=(\alpha _k^k,u_k^k,e_k^k,p_k^k)\).
The proposition below gives that these piecewise constant approximants satisfy a discretized version of the stability condition (qs1), a discretized energy inequality, and some a priori estimates. The proof follows the line of [18, Proposition 6.2], with some modifications due to the presence of the damage variable.
Proposition 4.2
For every \(k\in {\mathbb {N}}\) the evolution \((\alpha _k, u_k, e_k, p_k)\) defined in (4.3) satisfies the following conditions:
-
(qs0)\(_k\) for every \(x \in \Omega \) the function\( t \in [0,T] \mapsto \alpha _k(t, x)\) is nonincreasing;
-
(qs1)\(_k\) for every \(t\in [0,T]\) we have \((u_k(t),e_k(t),p_k(t))\in A(w_k(t))\) and
$$\begin{aligned} {\mathcal {E}}(\alpha _k(t),e_k(t), {\mathrm {curl}}\,p_k(t)) -\langle {\mathcal {L}}_k(t),u_k(t)\rangle \le {\mathcal {E}}(\beta ,\eta , {\mathrm {curl}}\,q) \\ \quad -\,\langle {\mathcal {L}}_k(t),v\rangle +{\mathcal {H}}(\beta , q- p_k(t)) \end{aligned}$$for every \((\beta ,v, \eta , q)\in {\mathcal {A}}(\alpha _k(t), w_k(t))\);
-
(qs2)\(_k\) for every \(t\in [t_k^i,t_k^{i+1})\)
$$\begin{aligned} \begin{aligned}&{\mathcal {E}}(\alpha _k(t),e_k(t), {\mathrm {curl}}\,p_k(t))-\langle {\mathcal {L}}_k(t),u_k(t)\rangle +{\mathcal {V}}_{\mathcal {H}}(\alpha _k, p_k;0,t)\\&\quad \le {\mathcal {E}}(\alpha _0, e_0, {\mathrm {curl}}\,p_0) - \langle {\mathcal {L}}(0),u_0\rangle \\&\quad + \int _0^{t_k^i} {\langle \sigma _k(s), E \dot{w}(s)\rangle \,\mathrm {d}s}-\int _0^{t_k^i}\langle \dot{{\mathcal {L}}}(s),u_k(s)\rangle \,\mathrm {d}s-\int _0^{t_k^i} \langle {\mathcal {L}}_k(s),\dot{w}(s)\rangle \,\mathrm {d}s +\delta _k, \end{aligned} \end{aligned}$$where \(\delta _k\rightarrow 0\) as \(k\rightarrow \infty \).
Moreover there exists a positive constant C independent of k and \(t\in {[0, T]}\) such that
Proof
The condition (qs0)\(_k\) holds since \(\alpha _k^i\le \alpha _k^{i-1}\). Moreover \(( u_k(t),e_k(t),p_k(t))\in A(w_k(t))\) for every \(t\in {[0, T]}\), by definition of the approximate evolutions. By (4.2) and Remark 3.3 we get
for every k, \(i=1,\dots , k\), and \((\beta , u,e,p)\in {\mathcal {A}}(\alpha _k^i,w_k^i)\), which gives (qs1)\(_k\).
In order to prove (qs2)\(_k\) let us fix \(i \in \{1,\dots ,k\}\), \(t\in [t_k^{i-1},t_k^i)\), \(u:=u_k^{h-1}-w_k^{h-1}+w_k^h\), and \(e:=e_k^{h-1}-\mathrm {E}w_k^{h-1}+\mathrm {E}w_k^h\) for a given integer h with \(1\le h\le i\). Testing (4.2) for \(i=h\) by \(( \alpha _k^{h-1}, (u,e,p_k^{h-1}))\in {\mathcal {A}}(\alpha _k^{h-1},w_k^h)\) we get
where
Iterating for \(1\le h\le i\) we deduce (qs2)\(_k\), with \(\delta _k=\sum _{h=1}^i \delta _{k,h}\). Indeed, since \(p_k\) is piecewise constant and continuous from the right, and \(\alpha _k\) is nonincreasing, the supremum in the definition of \({\mathcal {V}}_{\mathcal {H}}\) is attained by the subdivision \((t_k^h)_h\), namely (cf. [8, Lemma A.1])
Moreover, the absolute continuity of the loading (H2) implies that \(\delta _k\rightarrow 0\) as \(k\rightarrow \infty \).
Let us now prove (4.4). By (2.5) we can rewrite the inequality in (4.5) as
By the absolute continuity of w and \(\varrho \)
with \(\omega _{k,h}:= {\mathcal {Q}}_1(\alpha _k^{h-1}, \mathrm {E}w_k^h -\mathrm {E}w_k^{h-1}) \rightarrow 0\) as \(k\rightarrow \infty \). Let \(t\in [t_k^i,t_k^{i+1})\); summing up for \(h=1,\dots ,i\) we get
with \(\bar{\varrho }_k(s)=\varrho (t_k^j)\) if \(s\in (t_k^{j-1},t_k^j]\) and \(\omega _k=\sum _{h=1}^i \omega _{k,h} \rightarrow 0\) as \(k\rightarrow \infty \). By (2.7) we obtain the estimate
Therefore, \(\Vert e_k(t)\Vert _2\) is uniformly bounded in k and t by the hypotheses on \({\mathcal {Q}}_1\) and the regularity assumptions on the external loading; hence, \(\alpha _k(t)\), \({\mathcal {V}}(p_k;0,t)\), and \({{\mathrm {curl}}\,}\,p_k(t)\) are bounded as well. Finally, also \(u_k(t)\) is bounded by Korn’s inequality. This concludes the proof. \(\square \)
The following lemma shows (in the spirit of [10, Theorem 4.7]) that in order to prove that an evolution satisfies Definition 2.3, it is sufficient to verify the irreversibility and the global stability condition (qs0), (qs1), and (qs2) as an inequality.
Lemma 4.3
Let \((\alpha , u,e,p):{[0, T]}\rightarrow {H^1(\Omega ; [0,1])}\times W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\times L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\times {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) be such that the conditions (qs0) and (qs1) of Definition 2.3 hold. Moreover, assume that p is a function with bounded variation from [0, T] into \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) and that for every \(t\in [0,T]\)
where \(\sigma (s):= {\mathbb {C}}(\alpha (s)) e(s)\). Then \((\alpha , u,e,p)\) is a quasistatic evolution for the Gurtin–Anand model coupled with damage.
Proof
Let us fix \(t\in {[0, T]}\) and let us define \(s_k^i:=\tfrac{i}{k}t\) for every \(k\in {\mathbb {N}}\) and \(i=0,1,\dots , k\). For given k and i we set \(u:=u(s_k^i)-w(s_k^i)+ w(s_k^{i-1})\) and \(e:=e(s_k^i)-\mathrm {E}w(s_k^i)+ \mathrm {E}w(s_k^{i-1})\); from the fact that \((\alpha (s_k^i), u, e, p(s_k^i))\in {\mathcal {A}}(\alpha (s_k^{i-1}),w(s_k^{i-1}))\), the global stability condition (qs1) implies
This inequality can be rewritten as
where for \(s\in (s_k^{i-1}, s_k^i]\) we define
and
Since \(\sum _i{\mathcal {H}}(\alpha (s_k^i), p(s_k^i)-p(s_k^{i-1}))\le {\mathcal {V}}_{\mathcal {H}}(\alpha , p;0,t)\), iterating the last inequality for \(1\le i\le k\) we obtain
where \(\overline{\delta }_k:=\sum _{i=1}^k\overline{\delta }_{k,i}\rightarrow 0\) as \(k\rightarrow \infty \). Lemma 4.1 implies that \(\overline{\sigma }_k(s)\rightarrow \sigma (s)\) in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) and \(\overline{u}_k(s)\rightarrow u(s)\) in \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) for a.e. \(s\in (0,t)\). Taking into account the continuity in time of the external loading and using the Dominated Convergence Theorem, the inequality (4.7) passes to the limit as \(k\rightarrow \infty \) and we deduce that
Then the energy balance (qs2) is proved. \(\square \)
In the following theorem, we prove that the piecewise constant interpolants defined in (4.3) converge in a suitable sense, up to subsequences, to a quasistatic evolution for the Gurtin–Anand model coupled with damage.
Theorem 4.4
In the hypotheses of Theorem 2.5, for every \(k\in {\mathbb {N}}\) let \((\alpha _k, u_k, e_k, p_k)\) be the evolution defined in (4.3). Then, there exist a subsequence (not relabeled) and a quasistatic evolution \((\alpha ,u,e,p)\) for the Gurtin–Anand model coupled with damage such that \((\alpha (0),u(0),e(0),p(0))=(\alpha _0, u_0,e_0,p_0) \) and for every \(t\in {[0, T]}\)
Proof
Since the functions \(\alpha _k\) are nonincreasing in time and \(\alpha _k(t,x)\in [0,1]\), we get that the \(\alpha _k\) are uniformly bounded in \(BV(0,T;L^1(\Omega ))\). Moreover, by the a priori estimates (4.4), \(\Vert \alpha _k(t)\Vert _{1,2}\le C\) for every k and t. Therefore, we can apply the generalized version of the classical Helly Theorem given in [13, Helly Theorem] to conclude that there exist a subsequence (not relabeled) and a function \(\alpha :{[0, T]}\rightarrow {H^1(\Omega ; [0,1])}\) nonincreasing in time such that \(\alpha _k(t) \rightharpoonup \alpha (t)\) in \({H^1(\Omega )}\) for every \(t \in {[0, T]}\). By (4.4), it also follows that \({\mathcal {V}}(p_k;0,T)\le C\) for every k; then, [10, Lemma 7.2] implies the existence of \(p \in BV(0,T;{BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)})\) such that the convergence (4.8d) holds up to a subsequence. The uniform bound in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\) for the \({{\mathrm {curl}}\,} p_k\) gives also that \({\mathrm {curl}}\,p_k(t)\rightharpoonup {\mathrm {curl}}\,p(t)\) in \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\).
Let us fix \(t\in {[0, T]}\). The a priori estimates on \(u_k\) and \(e_k\) imply that there exist two functions \({\hat{u}}\in W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) and \({\hat{e}}\in L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\), and an increasing sequence \((k_j)_j\) (possibly depending on t) such that \(u_{k_j}(t)\rightharpoonup {\hat{u}}\) in \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\) and \(e_{k_j}(t)\rightharpoonup {\hat{e}}\) in \(L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\). By Theorem 3.6, the global stability condition (qs1)\(_k\) proved in Proposition 4.2 for the approximate evolutions passes to the limit, so the quadruple \((\alpha (t),{\hat{u}},{\hat{e}},p(t))\) is a solution to the minimization problem
In particular, \(({\hat{u}},{\hat{e}})\) minimizes the functional \((u,e)\mapsto {\mathcal {Q}}_1(\alpha (t),e) -\langle {\mathcal {L}}(t),u\rangle \), which is strictly convex in e, on the convex set \(K:=\{(u,e)\,{:}\, (u,e,p(t))\in A(w(t))\}\). Then, \(({\hat{u}},{\hat{e}})\) is uniquely determined, using also Korn’s inequality; if we define \((u(t),e(t)):=({\hat{u}},{\hat{e}})\), we obtain that (4.8b) holds and that \(e_{k}(t)\rightharpoonup e(t)\) in \(L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\), without passing to further subsequences.
By construction, the quadruple \((\alpha ,u,e,p)\) satisfies the conditions (qs0), (qs1) in Definition 2.3, and \(p \in BV(0,T;{BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)})\). By Lemma 4.3, it is enough to show the inequality (4.6) for every \(t\in {[0, T]}\) in order to conclude that \((\alpha ,u,e,p)\) is a quasistatic evolution for the Gurtin–Anand model coupled with damage.
Let us then fix \(t\in {[0, T]}\) and consider the discrete inequality (qs2)\(_k\) in Proposition 4.2 given by
By the approximation properties already shown, the fact that \({\mathcal {L}}_k(t)\rightarrow {\mathcal {L}}(t)\) strongly in \((W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n))'\), and the Dominated Convergence Theorem, the right-hand side converges to the right-hand side of (qs2) and
as \(k \rightarrow \infty \). On the other hand, from the lower semicontinuity of \({\mathcal {H}}\) proved in Lemma 3.3 and the definition of plastic dissipation (2.4) it follows that
Moreover, the weak lower semicontinuity of the energetic terms implies that
By (4.9), (4.10), and (4.11), we can pass to the limit in (qs2)\(_k\) and deduce (4.6) and the existence result. Furthermore, we obtain the convergence of the total energy, and thus, again by lower semicontinuity,
and then (4.8a), (4.8c), (4.8e), by strict convexity. This concludes the proof. \(\square \)
The main existence result, Theorem 2.5, is now a consequence of the previous theorem.
6 Properties of quasistatic evolutions and classical formulation
In this section, we study the connection between the energetic formulation for the Gurtin–Anand model coupled with damage, given in Definition 2.3, and the strong formulation of the model, described in the Introduction. We shall prove that, without any further regularity assumption with respect to the hypotheses of Theorem 2.5, the classical balance equations (sf2) and the constraint condition (sf4) are satisfied during every evolution. Moreover, under additional regularity assumptions, also the flow rule (sf5) holds almost everywhere in space and time, and the evolution of damage is governed by the Kuhn–Tucker-type conditions (sf6). Notice the improved regularity is required in order to differentiate the energy balance condition, while the classical balance equations (sf2) and the constraint condition are obtained without any differentiation.
In the following, we assume that \((\alpha ,u,e,p)\) is a quasistatic evolution for the Gurtin–Anand model coupled with damage, according to Definition 2.3. For every \(t\in {[0, T]}\) let \({\mathbb {K}}^p_{{\mathrm {en}}\,}(t)\in {\mathbb {M}^{n{\times } n{\times } n}_{D}}\) be given by
and let \(\sigma (t):={\mathbb {C}}(\alpha (t))e(t)\).
As in perfect plasticity [10], the balance equations for the Cauchy stress \(\sigma \) easily follow from the global stability condition (qs1), computing the corresponding Euler equation. By Lemma 3.4 we get that for every \(t\in {[0, T]}\) and every \((v,\eta ,q)\in A(0)\)
and then
Following [18], we now characterize the plastic potential \({\mathcal {H}}\) as the supremum of certain duality products. A similar type of characterization for the plastic potential is given also in perfect plasticity (cf. [38, Corollary 3.8] and [8, equation (2.23)]). In view of the dependence of \({\mathcal {H}}\) on the damage \(\alpha \), we have to introduce the closed space of measures that vanishes on sets with 2-capacity zero, which was not useful in [18].
Lemma 5.1
Let us define the closed linear subspace of \({M_b(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\)
where we recall that \({C}_2 (E)\) is the 2-capacity of the set E, and let us set for every \(\alpha \in {H^1(\Omega )}\)
Then for every \(\alpha \in {H^1(\Omega )}\) and \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
where \( \langle (A,{\mathbb {B}},{\mathbb {L}}), (p, \nabla p, {\mathrm {D}}^s p)\rangle :=\langle A, p \rangle + \langle {\mathbb {B}}, \nabla p\rangle + \langle {\mathbb {L}}, {\mathrm {D}}^s p\rangle \, \) is the duality pairing between \(L^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\times L^1(\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\times {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\) and its dual space.
Proof
Let us fix \(\alpha \in {H^1(\Omega )}\) and consider the function
defined by
This definition is well posed because \({\widetilde{\alpha }}\in L^\infty (\Omega ;{\mathbb {L}})\) for every \({\mathbb {L}}\in {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\), and
for every \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\).
Since \({\mathcal {F}}(\alpha ;\cdot ,\cdot ,\cdot )\) is strongly continuous and convex we have \({\mathcal {F}}(\alpha ;\cdot ,\cdot ,\cdot )={\mathcal {F}}(\alpha ;\cdot ,\cdot ,\cdot )^{**}\), where \(^*\) is the symbol for the Fenchel transformation. Moreover, using the fact that
for every \(\varepsilon , \delta >0\), \(\xi _1,\, \xi _2\in {\mathbb {R}}^d\), \(\zeta _1,\,\zeta _2\in {\mathbb {R}}^m\), with the equality if and only if \(\xi _1=C \delta ^2 \xi _2\) and \(\zeta _1=C \varepsilon ^2 \zeta _2\) for any \(C>0\), it is not difficult to show that \({\mathcal {F}}^*(\alpha ;\cdot ,\cdot ,\cdot )\) is the indicator function of the set \({\mathcal {K}}_\alpha (\Omega )\). Therefore, we deduce that \({\mathcal {F}}(\alpha ;\cdot ,\cdot ,\cdot )\) is the Fenchel transform of the indicator of \({\mathcal {K}}_\alpha (\Omega )\), that gives (5.4). \(\square \)
We now derive the existence of three higher-order stresses conjugated to p(t), \(\nabla p(t)\), \({\mathrm {D}}^s p(t)\) for every t, and prove that they satisfy the constitutive relations and the constraint condition (sf4) in the classical formulation.
Proposition 5.2
For every \(t\in {[0, T]}\) there exists a triple \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t))\in {\mathcal {K}}_{\alpha (t)}(\Omega )\) such that, setting \({\mathbb {K}}^p(t):={\mathbb {K}}^p_{{\mathrm {en}}\,}(t)+{\mathbb {K}}^p_{{\mathrm {diss}}\,}(t)\), it holds the following
which implies the balance equations
Proof
Let us fix \(t\in {[0, T]}\). From the inequality (5.2), we can deduce that the linear functional
depends only on q. Indeed, since A(0) is a linear space, if both \((v_1,\eta _1,q)\) and \((v_2,\eta _2,q)\) belong to A(0) we have \((v_1-v_2,\eta _1-\eta _2,0)\in A(0)\) and then \(\langle \sigma (t), \eta _1-\eta _2 \rangle -\langle {\mathcal {L}}(t),v_1-v_2\rangle =0\). We can thus consider the linear functional
defined on the linear subspace of \(L^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\times L^1(\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\times {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\)
By the Hahn-Banach Theorem for seminorms (see [14, Theorem 5.7]), we can extend in a continuous way \(\varphi \) to the whole \(L^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\times L^1(\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\times {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\) keeping the constraint condition in (5.2):
for every \((A,{\mathbb {B}}, {\mathbb {L}})\in L^1(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\times L^1(\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\times {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\). Since \(\varphi \) is linear and bounded there exist \(\sigma ^p(t)\in L^\infty (\Omega ;{\mathbb {M}}^{n{\times }n}_D)\), \({\mathbb {K}}^p_{{\mathrm {diss}}\,}(t)\in L^\infty (\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\), and \(\mathbb {S}^p(t)\in ({M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})})'\) such that
Therefore, choosing \((A,{\mathbb {B}}, 0)\) and \((0,0, {\mathbb {L}})\) in (5.9) we get that \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t))\in {\mathcal {K}}_{\alpha (t)}(\Omega )\) (recall (5.5) and the definition (5.3)).
Moreover, by (5.8) it follows that
for every \((v,\eta ,q)\in A(0)\). Hence (5.6) follows recalling the definition of \({\mathbb {K}}_{{\mathrm {en}}\,}^p(t)\).
In order to show (5.7) let us consider \(q\in C^\infty (\overline{\Omega };{\mathbb {M}}^{n{\times }n}_D)\) and choose \((0,-q,q)\in A(0)\) in (5.6). We obtain that
Since \(q(x)\in {\mathbb {M}}^{n{\times }n}_D\) for every x, we can replace \(\sigma (t)\) by \(\sigma _D(t)\) and rewrite the inequality above as
The former equation in (5.7) follows immediately; as for the latter, it is enough to integrate by parts, taking into account that the normal trace of \({\mathbb {K}}^p(t)\) on \(\partial \Omega \) is in \(H^{-1/2}(\partial \Omega ;{\mathbb {R}}^{n\times n})\) since \({\mathbb {K}}^p(t)\in L^2(\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\) with divergence in \(L^2(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\) by (5.1), (5.11), and the fact that \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t))\in {\mathcal {K}}_{\alpha (t)}(\Omega )\). Hence, (5.7) is proved and the proof is concluded. \(\square \)
The classical flow rule (sf5) and the Kuhn–Tucker condition for the evolution of the damage can be derived by differentiating the energy balance equation (qs2); therefore, some regularity assumptions are needed both on the constitutive coefficients and on the evolution. For instance, if \(\alpha \) and p are absolutely continuous from \({[0, T]}\), respectively, into \(C(\overline{\Omega })\) and \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), then, adapting the argument of [8, Lemma A.4], we have that for every \(t\in {[0, T]}\)
Proposition 5.3
(Kuhn–Tucker conditions and maximum plastic work principle) Assume that the elastic moduli \(\mu \), k in (H4), and the constitutive functions d, \({S_1}\), \({S_2}\) are of class \(C^1\). Moreover, let \(\alpha \), u, e, p be absolutely continuous from \({[0, T]}\) into \(C(\overline{\Omega })\cap H^1(\Omega )\), \(W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n)\), \({L^2(\Omega ; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), respectively. Then, for every \(t\in {[0, T]}\) the functional \(C(\overline{\Omega })\cap H^1(\Omega )\ni \beta \mapsto {\mathcal {E}}(\beta , e(t), {\mathrm {curl}}\,p(t))\) is differentiable at \(\alpha (t)\) with Gâteaux derivative in the direction \(\beta \in C(\overline{\Omega })\cap H^1(\Omega )\) given by
Moreover
for every \(t\in {[0, T]}\) and every \(\beta \in C(\overline{\Omega })\cap H^1(\Omega )\), \(\beta \le 0\) in \(\Omega \),
for a.e. \(t\in (0,T)\). Finally, for a.e. \(t\in (0,T)\)
where \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t))\in {\mathcal {K}}_{\alpha (t)}(\Omega )\) is given by Proposition 5.2.
Proof
The differentiability of \(\beta \mapsto {\mathcal {E}}(\beta , e(t), {\mathrm {curl}}\,p(t))\) and the expression of its derivative follow from the regularity assumptions on the constitutive functions and on the evolution. Let \(t\in {[0, T]}\) and \(\beta \in C(\overline{\Omega })\cap H^1(\Omega )\), \(\beta \le 0\) in \(\Omega \). For every \(h>0\), considering \((\alpha (t)+h \beta , u(t),e(t),p(t))\in {\mathcal {A}}(\alpha (t),w(t))\) as a test pair in (qs1), we get
Letting \(h \rightarrow 0\) we obtain (5.13).
Since the evolution is assumed to be absolutely continuous, we can differentiate with respect to t the energy balance (qs2). Recalling (5.12) we get that for a.e. t
It is easy to see that \((\dot{u}(t)-\dot{w}(t), \dot{e}(t)-E\dot{w}(t),\dot{p}(t))\in A(0)\), when it exists; thus, using (5.6) (cf. also (5.10)), the previous inequality gives that for a.e. t
Since \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t)) \in {\mathcal {K}}_{\alpha (t)}(\Omega )\), by (5.4) and 5.13 (recall that \(\dot{\alpha }(t)\le 0\) in \(\Omega \)) the equality (5.16) implies (5.14) and (5.15) for a.e. t. \(\square \)
Remark 5.4
Notice that we can interpret the equalities (5.14) and (5.15) as two threshold criteria. Indeed, by (5.13) and (5.4), we have that \(\dot{\alpha }(t)\) minimizes \(\left\langle \partial _\alpha {\mathcal {E}}(\alpha (t),e(t),{\mathrm {curl}}\,p(t)), \beta \right\rangle \) among every \(\beta \in C(\overline{\Omega })\cap H^1(\Omega )\), \(\beta \le 0\) in \(\Omega \), while the supremum in (5.4) is attained on \((\sigma ^p(t), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t), \mathbb {S}^p(t))\). In other words, (5.14) and (5.15) may be regarded as a principle of minimal loss of elastic stiffness and a maximum plastic work principle, respectively. The two conditions (5.13) and (5.14) are called Kuhn–Tucker conditions.
By (5.15), we deduce a weak form of the flow rule, expressed by the following conditions.
Corollary 5.5
Gathering (5.4) and (5.15), we get that
for every \((A,{\mathbb {B}})\in L^\infty (\Omega ;{\mathbb {M}}^{n{\times }n}_D)\times L^\infty (\Omega ;{\mathbb {M}^{n{\times } n{\times } n}_{D}})\) with \(\frac{|A(x)|^2}{{S_1}(\alpha (t,x))^2}+\frac{|{\mathbb {B}}(x)|^2}{l^2 {S_2}(\alpha (t,x))^2}\le 1\) a.e. in \(\Omega \), and
for every \({\mathbb {L}}\in ({M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})})'\) such that \(|\langle {\mathbb {L}}, \mu \rangle |\le l \int _\Omega {S_2}(\widetilde{\alpha }(t))\,\mathrm {d}|\mu |\) for \(\mu \in {M_b^2(\Omega ; {\mathbb {M}^{n{\times } n{\times } n}_{D}})}\).
Indeed, both (5.4) and (5.15) hold if and only if
for every \((A,{\mathbb {B}},{\mathbb {L}})\in {\mathcal {K}}_{\alpha (t)}(\Omega )\).
We are now ready to prove that the classical flow rule (sf5) holds for a.e. (t, x).
Proposition 5.6
(Flow rule) In the hypotheses of Proposition 5.3, let \(t\in {[0, T]}\) such that \(\dot{p}(t)\) and \(\nabla \dot{p}(t)\) exist and let \(x\in \Omega \) be a Lebesgue point for \(\sigma ^p(t)\), \({\mathbb {K}}^p_{{\mathrm {diss}}\,}(t)\), \(\dot{p}(t)\) and \(\nabla \dot{p}(t)\). Then the condition
implies that
while if
we have
with
Proof
Let us fix t and x satisfying the assumption in the statement, and let us define the convex set
By assumption \((\sigma ^p(t,x), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t,x))\in C_{t,x}\). Given \((F_0,{\mathbb {G}}_0)\in C_{t,x}\) we set
Since \(\alpha (t)\in C(\overline{\Omega })\) we get \((F,{\mathbb {G}})\in C(\overline{\Omega };{\mathbb {M}}^{n{\times }n}_D)\times C(\overline{\Omega };{\mathbb {M}^{n{\times } n{\times } n}_{D}})\); by construction \((F(x),{\mathbb {G}}(x))= (F_0, {\mathbb {G}}_0)\) and \(\frac{|F(z)|^2}{{S_1}(\alpha (t,z))^2}+\frac{|{\mathbb {G}}(z)|^2}{l^2{S_2}(\alpha (t,z))^2}\le 1\) in \(\Omega \). We now fix \(r>0\) and test (5.17a) by
which is an admissible test function by convexity of the constraint set. Hence, we obtain that for every \(r>0\)
As \(r\rightarrow 0\) we get (recall that x is a Lebesgue point for the functions involved)
Since \((F_0,{\mathbb {G}}_0)\) is arbitrary in \(C_{t,x}\), it follows that \((\dot{p}(t,x), \nabla \dot{p}(t,x))\) is in the normal cone to \(C_{t,x}\) at \((\sigma ^p(t,x), {\mathbb {K}}^p_{{\mathrm {diss}}\,}(t,x))\) and this proves the result. \(\square \)
7 Asymptotic analysis for vanishing strain gradient effects
In this section, we study the relation between the Gurtin–Anand model coupled with damage and the coupled elastoplastic damage model proposed in [8].
In [18], it is proven that quasistatic evolutions for the Gurtin–Anand model converge in a suitable sense, as the strain gradient terms vanish, to evolutions for perfectly plastic bodies in the formulation of [10]. Then we expect, when l, L tend to zero, the convergence of quasistatic evolutions in Definition 2.3 to evolutions for perfectly plastic bodies with damage studied in [8]. Indeed, the latter model corresponds, when the damage is constant in time, to the perfect plasticity model for heterogeneous materials in [38]. However, while the classical Gurtin–Anand formulation reduces to von Mises perfect plasticity model by setting l and L equal to zero (recall that l is related to the thickness of the plastic shear bands and L to the energy stored by the geometrically necessary dislocations), in the presence of damage the models have two different gradient damage regularizations, because in [8, 9] the space continuity of \(\alpha \) is needed. Thus, we start from a coupled gradient plasticity-damage model with a regularizing term \(\Vert \nabla \alpha \Vert _\gamma ^\gamma \), \(\gamma >n\), instead of \(\Vert \nabla \alpha \Vert _2^2\). Moreover, in the model in [8] there is a term related to a fatigue phenomenon, which depends on a parameter \(\lambda \). For simplicity, we do not consider here the fatigue and thus we take \(\lambda =0\).
For technical reasons (see Remark 6.2), we also require that the only loading is the displacement field w applied to the whole of \(\partial \Omega \).
Under this assumptions, Theorem 6.1 shows that evolutions for the Gurtin–Anand model coupled with damage converge weakly for every time to evolutions in [8].
For \(l_k\rightarrow 0\) and \(L_k\rightarrow 0\), let
be the total energy and the plastic dissipation of the Gurtin–Anand model coupled with damage for the length scales \(l=l_k\), \(L=L_k\), \(\ell =\sqrt{2}\). Moreover, let \((\alpha _k(t), u_k(t),e_k(t),p_k(t))\) from \([0,\, T]\) into \({W^{1, \gamma }(\Omega ; [0,1])}\times W^{1,\frac{n}{n-1}}(\Omega ;{{\mathbb {R}}}^n){\times }L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}}){\times }{BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
be a corresponding quasistatic evolution with the prescribed displacement w. Namely the following conditions hold:
-
(qs0)
irreversibility for every \(x \in \Omega \) the function \( {[0, T]}\ni t \mapsto \alpha _k(t, x)\) is nonincreasing;
-
(qs1)
global stability for every \(t\in [0,T]\) we have \((u_k(t),e_k(t),p_k(t))\in A(w(t))\) and
$$\begin{aligned} {\mathcal {E}}_k(\alpha _k(t),e_k(t), {\mathrm {curl}}\,p_k(t)) \le {\mathcal {E}}_k(\beta ,\eta , {\mathrm {curl}}\,q) +{\mathcal {H}}_k(\beta , q- p_k(t)) \end{aligned}$$for every \((\beta ,v, \eta , q)\in {\mathcal {A}}(\alpha (t), w(t))\);
-
(qs2)
energy balance the function \(t\mapsto p_k(t)\) from [0, T] into \({BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) has bounded variation and for every \(t\in [0,T]\)
$$\begin{aligned} \begin{aligned}&{\mathcal {E}}_k(\alpha _k(t),e_k(t), {\mathrm {curl}}\,p_k(t)) + {\mathcal {V}}_{{\mathcal {H}}_k}(\alpha _k, p_k; 0,t) \\&\quad = {\mathcal {E}}_k(\alpha _k(0), e_k(0), {\mathrm {curl}}\,p_k(0)) + \int _0^t {\langle \sigma _k(s), E \dot{w}(s)\rangle \,\mathrm {d}s} , \end{aligned} \end{aligned}$$where \(\sigma _k(s):= {\mathbb {C}}(\alpha _k(s)) e_k(s)\).
We now recall the notion of globally stable evolution for the coupled elastoplastic damage model considered in [8], when the parameter \(\lambda \) therein is zero.
The class of admissible configurations for a given boundary datum \(w\in H^1(\Omega ;{{\mathbb {R}}}^n)\) in perfect plasticity is the set
and we define in analogy to (2.8)
Here
endowed with the norm
is the Banach space of functions with bounded deformation on \(\Omega \); for its general properties we refer to [40]. Notice that we use the subscripts “\(\mathrm {pp}\)” (perfect plasticity with damage) to distinguish objects with analogous meaning in the two models and that the term \(w-u\) appearing in the definition of \(A_{{\mathrm {pp}}}\) is intended in the sense of traces on \(\partial \Omega \).
For every \(\beta \in C(\overline{\Omega })\) and \(q\in {M_b(\overline{\Omega }; {\mathbb {M}}^{n{\times }n}_D)}\) we set
in analogy to \({\mathcal {H}}\). Here, we adopt a multiplicative formulation for the constraint sets (indeed we are in von Mises setting). The plastic dissipation \({\mathcal {V}}_{{\mathcal {H}}_{{\mathrm {pp}}}}(\beta ,q)\) is defined in the same way of \({\mathcal {V}}_{\mathcal {H}}\), starting from \({\mathcal {H}}_{{\mathrm {pp}}}\), and the total energy is
with \({\mathcal {Q}}_1\) and D as in (H6) and (H8.1).
A quasistatic evolution for the coupled perfect plasticity-damage model is a function \({[0, T]}\ni t\mapsto (\alpha (t), u(t),e(t),p(t))\) from \([0,\,T]\) into \(\in {W^{1, \gamma }(\Omega ; [0,1])}{\times } BD(\Omega ){\times }L^2({\Omega };{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\)
\({\times }{M_b(\overline{\Omega }; {\mathbb {M}}^{n{\times }n}_D)}\) satisfying the following conditions:
-
(qs0)\(_{\mathrm {pp}}\) irreversibility for every \(x \in \Omega \) the function\( {[0, T]}\ni t \mapsto \alpha (t, x)\) is nonincreasing;
-
(qs1)\(_{\mathrm {pp}}\) global stability for every \(t\in [0,T]\) we have \((u(t),e(t),p(t))\in A_{{\mathrm {pp}}}(w(t))\) and
$$\begin{aligned} {\mathcal {E}}_{{\mathrm {pp}}}(\alpha (t),e(t)) \le {\mathcal {E}}_{{\mathrm {pp}}}(\beta ,\eta ) +{\mathcal {H}}_{{\mathrm {pp}}}(\beta , q- p(t)) \end{aligned}$$for every \((\beta , v, \eta , q)\in {\mathcal {A}}_{\mathrm {pp}}(\alpha (t),w(t))\);
-
(qs2)\(_{\mathrm {pp}}\) energy balance the function \(t\mapsto p(t)\) from [0, T] into \({M_b(\overline{\Omega }; {\mathbb {M}}^{n{\times }n}_D)}\) has bounded variation and for every \(t\in [0,T]\)
$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{{\mathrm {pp}}}(\alpha (t),e(t)) + {\mathcal {V}}_{{\mathcal {H}}_{{\mathrm {pp}}}}(\alpha , p; 0,t) = {\mathcal {E}}_{{\mathrm {pp}}}(\alpha (0), e(0)) + \int _0^t {\langle \sigma (s), \mathrm {E} \dot{w}(s)\rangle \,\mathrm {d}s} , \end{aligned} \end{aligned}$$where \(\sigma (s):= {\mathbb {C}}(\alpha (s)) e(s)\).
Assuming \(\Omega \) Lipschitz and (H2.1), (H3), (H5), (H8.2), and (H10), it is proven in [8] that for every initial data \((\alpha _0, u_0, e_0, p_0) \in {\mathcal {A}}_{\mathrm {pp}}(1,w(0))\) such that
for every \((\beta , v, \eta , q)\in {\mathcal {A}}_{\mathrm {pp}}(\alpha _0,w(0))\), there exists a quasistatic evolution for the coupled perfect plasticity-damage model \((\alpha ,u,e,p)\) such that \((\alpha (0),u(0),e(0),p(0))= (\alpha _0,u_0, e_0, p_0)\).
Now, we consider the limit as \(k\rightarrow \infty \), assuming for the initial conditions that
for suitable \(\alpha _0\), \(e_0\), \(u_0\), \(p_0\), and
Under this assumption, we can prove the convergence result below.
Theorem 6.1
Let \(\Omega \subset {{\mathbb {R}}}^n\) be open, bounded, and Lipschitz; if \(n\ge 3\), let \(\Omega \) be also star-shaped. Assume \(\partial _D \Omega =\partial \Omega \), (H2.1), (H3), (H5), (H8.2), and (H10). Moreover, for \(l_k\rightarrow 0\) and \(L_k\rightarrow 0\), let \((\alpha _{k},u_{k},e_{k},p_{k})\) be a quasistatic evolution for the Gurtin–Anand model coupled with damage associated with \(l_k\) and \(L_k\) such that the conditions 6.1 hold. Then, there exists a quasistatic evolution for the perfect plasticity model coupled with damage \((\alpha ,u,e,p)\) with \(\alpha (0)=\alpha _0\), \(u(0)=u_0\), \(e(0)=e_0\), \(p(0)=p_0\) such that, up to a subsequence,
for every \(t\in {[0, T]}\).
Remark 6.2
An important difference with respect to the analysis in [18] relies on the fact that we cannot still characterize the global stability in the limit evolution by the equilibrium conditions for the Cauchy stress and the plastic constraint (see [10, Theorem 3.6]). This calls for the approximation in a strong sense of admissible triples for perfect plasticity with ones that are admissible for the Gurtin–Anand model. We show this relaxation result in the lemmas below both in the case of dimension two and in dimension three under the additional assumption that the domain is star shaped. Actually, in the paper [33], M.G. Mora proves the approximation property for every Lipschitz domain; then, Theorem 6.1 can be proved for this domains.
Lemma 6.3
(Approximation, \(n\ge 3\)) Let \(\Omega \subset {{\mathbb {R}}}^n\), \(n\ge 3\), be open, bounded, star-shaped and Lipschitz. Then, for every \((u,e,p)\in A_{{\mathrm {pp}}}(0)\) there exists a sequence of triples \((u_k,e_k,p_k)\in A(0)\) such that
Proof
Without loss of generality, we can assume that \(\Omega \) is star-shaped with respect to 0. For an open set \({\widetilde{\Omega }}\) such that \(\overline{\Omega }\subset {\widetilde{\Omega }}\) let us define
For k large enough we set
and
Then, it is not difficult to see that
and that, taking the restriction of \({\widehat{u}}_k\), \({\widehat{e}}_k\), \({\widehat{p}}_k\) to \(\overline{\Omega }\), we have
Moreover, if we regularize by convolution for every k with the sequence of mollifiers \((\varrho _{\frac{1}{h}})_{{\mathbb {N}}\ni h>k}\), we get (taking the restrictions to \(\overline{\Omega }\)) a sequence of functions
such that
as \(h\rightarrow \infty \). Indeed, by 6.3 it is enough to show that \({\widehat{p}}_k^h\mathrel {\mathop {\rightarrow }\limits ^s}{\widehat{p}}_k\quad \text {in }M_b(\Omega ;{\mathbb {M}}^{n{\times }n}_D)\), and this holds again by 6.3 since the regularization by convolution of a measure entails strict convergence on open subsets whose boundaries are not charged by the measure itself (see [3, Theorem 2.2]).
By a diagonal argument, we obtain \((u_k,e_k,p_k)\) as \(({\widehat{u}}_k^{h_k},{\widehat{e}}_k^{h_k}, {\widehat{p}}_k^{h_k})\) with \(h=h_k\) sufficiently large. \(\square \)
We now show the relaxation property for perfect plasticity triples in a bidimensional domain. The construction of the approximants is similar to the one made in [15, Theorem 6.2, Step 1].
Lemma 6.4
(Approximation, \(n= 2\)) Let \(\Omega \subset {\mathbb {R}}^2\) be open, bounded, and Lipschitz. Then for every \((u,e,p)\in A_{{\mathrm {pp}}}(0)\), there exists a sequence of triples \((u_k,e_k,p_k)\in A(0)\) such that
Proof
Let us define
Since \((u ,e,p)\in A_{{\mathrm {pp}}}(0)\), we get that
Let \(\{Q_{\nu _k}(x_k,r_k)\}_{k\in I}\) be a finite covering of \(\partial \Omega \) made of open cubes with centers \(x_k\in \partial \Omega \), side \(2r_k\), with \(r_k>0\), and a face orthogonal to \(\nu _k\in {\mathbb {R}}^2\) such that \(\Omega \cap Q_{\nu _k}(x_k,r_k)\) is a Lipschitz subgraph in the direction \(\nu _k\). Let \(\{\phi _k\}_{k\in I}\) be an associated partition of unity of \(\partial \Omega \). Then
and the last term has a support compactly contained in \(\Omega \). Set
so that \({\widehat{e}}_k\in L^2({\mathbb {R}}^2; {\mathbb {M}^{2{\times } 2}_{\mathrm{sym}}})\) (indeed \({\widehat{u}}\in BD({\mathbb {R}}^2)\subset L^2({\mathbb {R}}^2;{\mathbb {R}}^2)\)) and \({\widehat{p}}_k\in {M_b}({\mathbb {R}}^2;{\mathbb {M}}^{2{\times }2}_D)\) with
For \(h\in {\mathbb {N}}\) so large that the support of the functions \({\widehat{\phi }}_k(x):=\phi _k\left( x+\frac{\nu _k}{h}\right) \) is compactly contained in \(Q_{\nu _k}(x_k,r_k)\) for every \(k\in I\), let us define
we also define \(e_{k,h}\), \(p_{h,k}\) following (6.4). Set
Notice that
with
and that \(u_h\), \(e_h\), \(p_h\) vanish outside a compact subset of \(\Omega \). This last condition and fact that we have only used local translations imply that restricting to \(\overline{\Omega }\)
Moreover, if we regularize \((u_h,e_h,p_h)\) by convolution with a sequence of mollifiers \((\varrho _{\frac{1}{m}})_m\), we get for m sufficiently large that
using again that \(u_h\), \(e_h\), \(p_h\) have compact support in \(\Omega \). Recalling that the regularization by convolution of a measure entails strict convergence on open subsets whose boundaries are not charged by the measure itself, and that \(p_h=0\) on \(\partial \Omega \), we have
and then we conclude by a diagonal argument. \(\square \)
We are now ready to prove Theorem 6.1.
Proof of Theorem 6.1
The proof is divided into two steps.
Step 1: Compactness and global stability By definition of \({\mathcal {H}}_k\) we have that for every \(\beta \in {W^{1, \gamma }(\Omega )}\), \(q\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\), and \(k\in {\mathbb {N}}\)
and then
with \({\mathcal {V}}_1(p_k;0,t)\) the variation of \(p_k\) with respect to \(L^1(\Omega ; {\mathbb {M}}^{n{\times }n}_D)\) in (0, t). Then, by 6.1, the fact that \({\mathcal {Q}}_1\) is quadratic, and Korn’s inequality, we get that there exists a constant C independent of k and t such that
Let \({\widetilde{\Omega }}\) be a smooth open set such that \(\overline{\Omega }\subset {\widetilde{\Omega }}\), and let us define for every k and t the functions \({\widehat{u}}_k(t)\in W^{1,\frac{n}{n-1}}({\widetilde{\Omega }};{{\mathbb {R}}}^n)\), \({\widehat{e}}_k(t)\in {L^2({\widetilde{\Omega }}; {\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})}\), and \({\widehat{p}}_k(t)\in {BV({\widetilde{\Omega }};{\mathbb {M}}^{n{\times }n}_D)}\) as
The \(\alpha _k\) are nonincreasing in time and \(\alpha _k(t,x)\in [0,1]\) with \(\Vert \alpha _k(t)\Vert _{1,\gamma }\le C\) and the functions \(p_k\) from \({[0, T]}\) to \(L^1({\widetilde{\Omega }};{\mathbb {M}}^{n{\times }n}_D)\) have uniformly bounded variations; therefore, taking into account 6.1 we get the existence of two functions \(\alpha :{[0, T]}\rightarrow {W^{1, \gamma }(\Omega ; [0,1])}\) nonincreasing in time and \({\widehat{p}}:{[0, T]}\rightarrow {M_b}({\widetilde{\Omega }};{\mathbb {M}}^{n{\times }n}_D)\) with bounded variation such that up to a subsequence (not relabeled)
for every \(t\in {[0, T]}\). Notice that we have applied [10, Theorem 7.2] considering \({M_b}({\widetilde{\Omega }};{\mathbb {M}}^{n{\times }n}_D)\) as a subspace of \(L^1({\widetilde{\Omega }};{\mathbb {M}}^{n{\times }n}_D)\).
Let us fix \(t\in {[0, T]}\). By the a priori estimate (6.5), we deduce that there exist an increasing sequence \((k_j)_j\) (that could depend on t) and two functions \({\widehat{u}}\in BD({\widetilde{\Omega }})\) and \({\widehat{e}}\in L^2({\widetilde{\Omega }};{\mathbb {M}^{n{\times } n}_{\mathrm{sym}}})\) such that
As in [18, Lemma 9.1] (that holds in our assumptions on \(\Omega \)), we obtain that
and
where p(t) denotes the restriction of \({\widehat{p}}(t)\) to \(\overline{\Omega }\) and we have not relabeled the restrictions of \({\widehat{u}}\), \({\widehat{e}}\) to \(\Omega \). We claim that the quadruple \((\alpha (t),{\widehat{u}},{\widehat{e}},p(t))\) satisfies the stability condition (qs1)\(_\mathrm {pp}\), namely
for every \((\beta , (v, \eta , q))\in {\mathcal {A}}_{\mathrm {pp}}(\alpha (t),w(t))\). Then, since \(({\widehat{u}},{\widehat{e}})\) minimizes the functional \((v,\eta )\mapsto {\mathcal {E}}_{{\mathrm {pp}}}(\beta ,\eta )\) on the convex set \(\{(v,e):(v,e,p(t))\in A_{{\mathrm {pp}}}(w(t))\}\), we have that \(({\widehat{u}},{\widehat{e}})= (u(t), e(t))\) and
for the whole subsequence. We have already shown that
Let us now prove the claim (6.7); since we work with a given t, we can neglect the dependence on j in (6.6). By assumption, for every k we have the stability condition:
for every \((\beta ,v, \eta , q)\in {\mathcal {A}}(\alpha _k(t), w_k(t))\).
Let us fix \((\beta ,v_0, \eta _0, q_0)\in {\mathcal {A}}(\alpha (t),0)\), and test (6.9) by
Arguing as in Theorem 3.6, we deduce that
To get the above inequality, we have also used that
and that for every \(\alpha \in {W^{1, \gamma }(\Omega )}\) and \(p\in {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\)
By 6.1 and the energy balance for \((\alpha _k,u_k,e_k, p_k)\) we get
for C independent of k; by the Hölder inequality and the monotonicity of \(\mu \) it follows that
Thus, letting \(k\rightarrow 0\) in (6.10) we obtain as in Theorem 3.6 the inequality
Let us consider a triple \((v,\eta ,q)\in A_\mathrm {pp}(w(t))\); then \((v-{\widehat{u}},\eta -{\widehat{e}},q-p(t))\in A_\mathrm {pp}(0)\). By Lemmas 6.3 and 6.4 there exist triples \((v_k,\eta _k,q_k)\in A(0)\) such that
In particular Reshetnyak’s Continuity Theorem (cf. [3, Theorem 2.39]) implies that
Therefore, considering \((v_k,\eta _k,q_k)\) in place of \((v_0,\eta _0,q_0)\) in (6.11) and taking the limit of the right-hand side as \(k\rightarrow \infty \) we deduce (6.7).
Step 2: Energy balance From (6.8b) it follows that
Indeed, for every \(\beta _k \rightharpoonup \beta \) in \({W^{1, \gamma }(\Omega )}\) and \((q_k)_k\subset {BV(\Omega ;{\mathbb {M}}^{n{\times }n}_D)}\) with \(q_k \mathrel {\mathop {\rightharpoonup }\limits ^{*}}q\) in \({M_b}(\overline{\Omega };{\mathbb {M}}^{n{\times }n}_D)\), it holds
and then we get (6.12) by the definition of \({\mathcal {V}}_{{\mathcal {H}}_{{\mathrm {pp}}}}\) and \({\mathcal {V}}_{{\mathcal {H}}_k}\). By lower semicontinuity and the fact that \({\mathcal {Q}}_2(\alpha _k(t),{\mathrm {curl}}\,p_k(t))\) is nonnegative it follows that
Collecting 6.1, (6.12), and (6.13), we deduce that
From the stability condition (qs1)\(_\mathrm {pp}\), with arguments similar to those in Lemma 4.3 (cf. [10, Theorem 4.7]), we can prove that the opposite energy imbalance holds and then \((\alpha ,u,e,p)\) is a quasistatic evolution for the coupled perfect plasticity-damage model. By (6.1), (6.12), (6.13), and the energy balance (evaluated in [0, t]), it follows that for every \(t\in {[0, T]}\)
which implies
and then (6.2b) and (6.8a). This concludes the proof. \(\square \)
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Acknowledgments
The author wishes to thank Gianni Dal Maso and Giuliano Lazzaroni for many helpful suggestions and discussions. He also gratefully acknowledges a discussion with Stefano Almi, who brought to his attention A. Giacomini’s paper on Gurtin–Anand model. This work is part of the Project “Calculus of Variations” 2010–11, supported by the Italian Ministry of Education, University, and Research. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Crismale, V. Globally stable quasistatic evolution for strain gradient plasticity coupled with damage. Annali di Matematica 196, 641–685 (2017). https://doi.org/10.1007/s10231-016-0590-7
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DOI: https://doi.org/10.1007/s10231-016-0590-7
Keywords
- Variational models
- Quasistatic evolution
- Energetic solutions
- Strain gradient plasticity
- Damage models
- Incomplete damage
- Softening