Damage model for plastic materials at finite strains

We introduce a model for elastoplasticity at finite strains coupled with damage. The internal energy of the deformed elastoplastic body depends on the deformation, the plastic strain, and the unidirectional isotropic damage. The main novelty is a dissipation distance allowing the description of coupled dissipative behavior of damage and plastic strain. Moving from time‐discretization, we prove the existence of energetic solutions to the quasistatic evolution problem.

In what follows, we adopt the format of generalized standard materials [28] and assume that the material behavior is governed We rely on the concept of energetic solutions, and consider a quasistatic evolution, namely a trajectory [0, ] ∋  → ( ( ), ( ), ( )) on a time interval [0, ] satisfying at every time a stability condition and an energy balance, see Definition 3.1. Our main result, Theorem 3.2 in Section 3, asserts the existence of an energetic solution for any compatible (stable) initial datum. To prove this result we apply a standard time-discretization scheme introduced by MIELKE and co-workers. [38,41] This scheme has shown to be very successful in order to achieve existence of energetic solution to rate-independent systems. It is versatile, as it has been employed in many different settings, like in problems of nonlinear plasticity (e.g., [34,39]), damage, [40,50,51] cracks growth (e.g. [18,30]), delamination, [46] dislocations evolution, [49] and many others (see [41] and references therein for a more detailed discussion and a more exhaustive bibliography).
In many applications, thanks to the solid theory of MIELKE, the existence of energetic solutions is easily obtained by checking a series of standard hypotheses. In the present paper, due to our particular model which couples plasticity with damage, we borrow ingredients coming from both these fields. The proof of our main result relies on checking that the coupled dissipation defined in Section 2.3 is a lower semicontinuous quasidistance (C1-C2), the loading power is energetically controlled (C3), sublevels of the energy are compact (C4), and the set of stable states is closed (C5). In order to verify these conditions, we need to proof the linear growth of the plastic dissipation (Proposition 2.2) and a formula showing that the (abstract) coupled dissipation distance (21) splits additively into a damage-dissipation and a damage-weighted plastic dissipation (Proposition 2.4). We consider this part the main novelty of the present work.
The paper is organized as follows. Section 2 introduces our model, emphasizing the treatment of the dissipation potential that accounts for damage and plastic processes. The existence proof, based on incremental energy minimization, is presented in Section 3. We note in passing that our analysis rests on the conditions of global stability; alternative solution concepts like viscous approximation, employed in a similar context by CRISMALE and LAZZARONI, [15] or semistability, used by ROUBÍČEK and VALDMAN, [47,48] are excluded from consideration. Finally, in Section 4, we discuss possible extensions and generalizations.

Preliminaries
We first describe the setting of our model and then introduce some basic concepts of linear algebra and geodesic calculus which help to understand the model.
Reference configuration. In the sequel we work on a bounded connected open set Ω ⊂ ℝ , ≥ 2, with Lipschitz boundary representing the reference configuration of an elastoplastic body. We assume that the boundary of Ω is the union of a Dirichlet and Neumann part, namely Ω ∶= Γ ∪ Γ , and suppose Γ has strictly positive ( − 1)-Hausdorff measure. Once we have fixed a Dirichlet boundary condition for the deformation ∶ Ω → ℝ , we can make use of the Poincaré inequality ‖ ‖ 1, ≤ ‖∇ ‖ , which holds true for this domain since  −1 (Γ ) > 0. Throughout the paper we use the letter to denote a generic positive constant that may change from line to line.
Matrices and groups. We denote by ℝ × the vector space of × matrices with real entries. The standard Euclidean inner product is denoted by double dots, namely ∶ = (summation convention). The symbols ℝ × sym and ℝ × anti denote the subspaces of ℝ × consisting of symmetric and anti-symmetric matrices, respectively. The symbol ℝ × dev stands for deviatoric matrices, where deviatoric means tracefree. We employ the following notation for common matrix groups where ∈ ℝ × denotes the identity matrix.
Norms. We consistently use the notation | ⋅ | for norms of tensors and scalars, e.g. | | = ( ∶ ) 1∕2 . This notation is employed in general for -tensors of every order. On the other hand, we make use of the double-bar notation ‖ ⋅ ‖ for norms on function spaces, e.g.
Polar decomposition. For all ∈ ( ) there exists a unique decomposition with ∈ ( ) and ∈ ℝ × sym positive definite. If, moreover, ∈ ( ) then it is easy to see that both and must have determinant equal to 1. Furthermore, as is symmetric, there exists an orthogonal matrix and a diagonal matrix Λ such that The diagonal matrix Λ has the positive eigenvalues of on the diagonal. The matrix = diag(log 1 , … , log ) then satisfies where the last equality follows from the fact that is invertible and −1 = −1 for all ∈ ℝ × . Geodesic exponential map vs. matrix exponential. Let be a (matrix) Lie group, e.g.
( ) or ( ). The (geodesic) exponential map is defined by where is the unique geodesic starting from the identity ∈ with initial velocity lying in the tangent space to at the identity. It is easy to show that the tangent space of ( ) (resp. ( )) at the identity is ℝ × dev (resp. ℝ × anti ), see [11,Example I.9.4., Exercise I. 17(b)]. It is important to remark that in general the geodesic exponential map defined above differs from the algebraic exponential of a matrix used above and denoted by . In fact it was shown in [36, Theorem 6.1] that for the leftinvariant metric induced by the standard Euclidean scalar product ∶ the geodesics on ( ) starting from (0) in direction of ∈ ℝ × dev are given by Notice that for tracefree matrices in general ⊤ ≠ ⊤ , that is why Exp( ) ≠ . For antisymmetric matrices however, the product commutes. This implies that on ( ) the geodesics are exactly given by ( ) = (0) for ∈ ℝ × anti . Rotations. Since the set of rotations ( ) is a compact connected Lie group, the exponential map is surjective [27,Corollary 11.10.]. Therefore, for every ∈ ( ) there exists ∈ ℝ × anti such that = . We can use the spectral theory for real skew-symmetric matrices to bring to a block diagonal form. Namely, there exists an orthogonal matrix such that = Σ ⊤ with where either = 0 ∈ ℝ or = ∈ ℝ 2×2 anti and ∈ ℝ. Since Σ is block diagonal its exponential is easily computed as Using the periodicity of Sinus and Cosinus, the rotation can be written as where Σ is defined as in (4), but with ∈ [0, 2 ).

Plastic dissipation
The (plastic) dissipation potential is a mapping which is measurable in ∈ Ω and convex and positively 1-homogeneous in the rate, i.e., We further assume plastic indifference which corresponds to requiring that This property implies that there exists a measurable, 1-homogeneous functionΔ ∶ Ω × ℝ × → [0, +∞] such that Δ( , ,̇) =Δ( ,̇− 1 ), see [37] or [41, Section 4.2.1.1]. We assume there exist constants 0 , 1 > 0, independent of ∈ Ω, such that With the potential at disposal, we define the induced plastic dissipation distance on ( ) for any pair 1 , 2 ∈ ( ) by Notice that due to plastic indifference we have that p ( , Due to (8), the dissipation distancêp is equivalent to the standard Riemannian distance induced by the Euclidean scalar product on the Lie algebra ℝ × dev . In particular, we have that for every ∈ Ω As it was pointed out in [36], the geodesics with respect toΔ in direction are in general not known and even in the specific Riemannian case geodesics of̂S L connecting the identity to are not given by  → . In particular, it might happen that SL ( ) < | |. However, from standard theory of Riemannian manifolds it is known that is a metric on ( ), see pp. 19-20 of [11]. We conclude this introduction with the following results which are employed in Section 3: Lemma 2.1 ( p is a quasi-distance). For every 1 , 2 , 3 , ∈ ( ) and all ∈ Ω the following properties hold: Proof. The implication (i) follows from the previous remark that SL is a metric on ( ) which by (10) is equivalent to p . Condition (ii) is easily checked, while (iii) follows from (7). □ Notice that p might not be symmetric. We now show that the quasi-distancêp has sublinear growth. To prove this upper bound the most important observation is that, if ∈ ( ) is such that = for some , then we may test the definition of p ( ) with the path  → , ∈ [0, 1] and get Proposition 2.2. There exists a positive constant = ( ) > 0 such that for every 1 , 2 ∈ ( ) and all ∈ Ω Proof. In the following, not to overburden notation, we drop the explicit dependence on ∈ Ω. For the Reader's convenience the proof is split into several steps. In Steps 1-3 we show that for every ∈ ( ) for some constant = ( 1 , ) > 0 ( 1 being the constant in (8)). In Step 4 we deduce the general statement of the proposition.

State space
In order to deal with time-dependent boundary conditions of the form where Dir ∶ [0, ] × ℝ → ℝ represents a Dirichlet datum, we use the so-called multiplicative splitting technique [19,21,30,34] replacing the variable by Dir ( )• , see e.g. [21,Section 5]. More precisely, we set This results in a multiplicative split of the deformation gradient We refer to the next section for the hypotheses on Dir . The space of admissible states, denoted by , is the triple for some coefficients > and p , z > 1. The space  is endowed with the weak topologies of the Sobolev spaces, namely, By Poincaré's inequality, weak convergence in  is equivalent to weak convergence of gradients, i.e., Notice that the space  is not a linear subspace of 1, p (Ω; ℝ × ) because the target space is the manifold ( ). Nevertheless weak limits of sequences ( ) ∈ℕ ⊂  are again in . This follows since weak convergence in  implies strong convergence in p (Ω; ℝ × ). We introduce the short notation = ( , , ) for elements in  and occasionally use the variable in the dissipation distance  although it depends only on the internal variables and is independent of .

Energy
We consider the following total energy for the system: for some material parameters , > 0, where the mapping  → ( ) represents external loading of the mechanical system and is defined as where is a prescribed bulk force and is a prescribed traction on the Neumann boundary Γ . The quantity is the elastic energy of the system and the term represents the energy related to kinematic hardening instead. The terms in (25) involving ∇ and ∇ are higher order energetic terms which have the role of regularizations introducing internal length scales. Notice that the elastic energy density depends on the elastic strain ∇ Dir ( , )∇ −1 whereas the hardening energy depends on the plastic strain . It is convenient to denote the total bulk energy (density) without regularization by The presence of the time-dependent Dirichlet datum is reflected in the power of external forces given by where This motivates the assumptions on the Kirchhoff stress el ( , , ) ⊤ explained below and used in [34].
For our analysis, we ask the following conditions to hold: • Control on the Kirchhoff stress: • Polyconvexity: We assume that h is a normal integrand, meaning h (⋅, , ) is measurable for every ∈ ( ), ∈ [0, 1] and h ( , ⋅, ⋅) is lower semicontinuous for a.e. ∈ Ω. Moreover, we assume that the elastic energy density el is finite just on + ( ) and polyconvex, [6] namely where conv is a normal integrand, conv ( , ⋅, ) is convex for a.e. ∈ Ω and every ∈ [0, 1], and ( ) denotes the vector of all minors of the elastic strain . In dimension = 3, for instance, • Coercivity: Furthermore, we assume the coercivity bounds for some constants 1 , 2 > 0 and exponents satisfying • Monotonicity and continuity: We further assume continuity and monotonicity in . More precisely, we ask and for every ∈ + ( ), ∈ ( ) and a.e. ∈ Ω.
• Regularity of Dirichlet data and loading: Moreover, one needs to assume that Dir and are sufficiently regular. Precisely, one requires Here, the multiplicative stress control (28) Moreover, polyconvexity (30), coercivity (31), and continuity (33) are used to show lower semicontinuity and compactness of the energy, whereas monotonicity (34) is needed for the construction of recovery sequences in Section 3.1. We would like to emphasize that these conditions are compatible with frame-indifference (objectivity) and non-interpenetrability of matter, namely • Objectivity: • Non-interpenetrability: see Example 2.6. As these conditions are essential in modeling continuous media, it is certainly desirable to include them into the model. However, they are not needed for the analysis. (39) it is clear that a finite energy solution satisfies the local noninterpenetration det ∇ > 0 a.e. in Ω. It is possible to guarantee global non-self-interpenetration involving the so-called Ciarlet-Nečas condition, [12] which reads

Remark 2.5 (Ciarlet-Nečas condition). With assumption
where  denotes the Lebesgue measure on ℝ . In order to achieve this we would change the state space  to It can be shown that supposing Ciarlet-Nečas condition for instead of is equivalent under the assumption that Dir ( ) is an orientation-preserving diffeomorphism [41, Lemma 4.1.1]. Moreover, due to the condition > , convergence of ∇ ⇀ ∇ in (Ω) implies convergence of det(∇ ) ⇀ det(∇ ) in 1 (Ω). This shows that  is weakly closed in .

QUASISTATIC EVOLUTION
We follow the concept of energetic solutions, which is solely based on the energy functional , the dissipation distance  and the state space  introduced above. Given initial conditions ( 0 , 0 , 0 ) ∈  we look for an energetic solution ( , , ) ∶ [0, ] → . We first introduce the concept of stable states at a given time ∈ [0, ]: this is defined via the subset ( ) of  defined as An energetic solution is asked to satisfy the following energy balance (E) and global stability condition (S).
We now formulate the main results of the paper. The fineness of a partition is defined as max | − −1 |.

Theorem 3.3 (Existence via incremental minimization).
For every stable initial data 0 ∈ (0) and every sequence of partitions ∈ Π of [0, ] with fineness tending to zero as → ∞, we can find a trajectory ∶ [0, ] →  with (0) = 0 which is piecewise constant on the partition, right-continuous, and satisfies Notice that the statement of Theorem 3.3 is actually stronger than that of Theorem 3.2 because it additionally provides a way to construct energetic solutions using incremental minimization and convergence results.
In order to show existence of energetic solutions, we resort in applying the theory introduced and developed by MIELKE and co-authors in a series of papers and books (see [38] or more recently [41] and references therein). Along the existence proof in Section 3.2 below we use that, under the assumptions stated in Section 2, the following conditions are satisfied: (C1) The dissipation  satisfies the following two properties: (C3) There exists a function ∈ 1 (0, ) such that for all ∈  the following implication holds true: Aiming to prove Theorem 3.3, we start by checking that conditions (C1)-(C4) are indeed satisfied by our model introduced in Section 2. The proof of (C5) is typically the hardest part and we establish it separately in Section 3.1 by arguing as in THOMAS. [50,51] The process in finding mutual recovery sequences used therein is directly applicable to our setting.
In particular, we have checked that which is nothing but sequential precompactness.
It remains to show the lower semicontinuity of , which is equivalent to closedness of sublevels. Take a sequence ⇀ in  where = ( , , ) and assume without loss of generality that sup ( , ) ≤ . We can use estimate (48) and choose a (not relabeled) subsequence such that ( , ) converges to lim inf →∞ ( , ) and ∇ ⇀ ∇ weakly in (Ω), for every ∈ [1,̄p), ∈ [1, ∞). Now in order to use polyconvexity (30) we need to show that This result was established in [39] and applied in [34,Proposition 5.1] (see also [41,Lemma 4.1.3]). The convergence is proven under the assumption that > and which is indeed satisfied here sincēp > > and therefore can be chosen larger than . The lower semicontinuity of ( , , )  → ( , , , ) now follows from classical theory due to the polyconvexity assumptions in Section 2.5. It was pointed out in [34] that the classical assumption of being a Carathéodory function can be relaxed to the one of a normal integrand using a Yosida-Moreau regularization.

Closedness of stable states (C5)
This closedness relies on finding suitable recovery sequences for the damage variable . In [40] this was achieved in the framework of damage in nonlinear elasticity for z > , in which case damage is continuous in space. In the papers, [50,51] it was generalized to 1 < z < . We apply the strategy of [50,51] to our model. In particular, the choice of recovery sequences is the same as in the mentioned works.
We want to prove that, if ( , ) is a sequence such that ∈ ( ), → , and ⇀ in , then ∈ ( ). Thus we need to ensure that for everŷ∈  In order to show this we provide a so-called mutual recovery sequencê(see [40,43]) satisfying Indeed, by stability of , we have for everŷ∈  Then the lim sup bound (49) together with (50) implies ∈ ( ).
By usinĝ≤̂and monotonicity (34) we get the uniform bound Therefore, (56) follows from the Dominated Convergence Theorem. We are left with showing (57). We define Since ⊂ {| − | ≥ } thanks to (53), we can use Markov's inequality to show that As we want this to go to 0 we impose that Now we can write, by the definition of̂, We take the lim sup as → ∞ and use that ∇ ⇀ ∇ weakly in z (Ω) (here , the characteristic function of , converges to 1 strongly in (Ω) for any ∈ [1, +∞), while ∇ tends to ∇ weakly in (Ω) for all < z ; the equiboundedness of ∇ on z implies the claim) to get by weak lower semicontinuity of the norm. □

Proof of Theorem 3.3
We are now in position to prove the main result. We proceed in several steps following the general scheme showed in [38] (see also, e.g. [34] for the treatment of the boundary datum). Since many steps are standard we do not enter into much detail and refer to [38,41]. Nevertheless, for completeness all crucial steps of the proof are mentioned.
Step 1: Approximation via incremental minimization. Let = {0 = 0 < 1 < ⋯ < ( ) = } ∈ Π, ∈ ℕ, be a sequence of partitions such that the fineness tends to zero as tends to ∞. For fixed we iteratively solve for Note that (C2) and (C4) guarantee the existence of minimizers. This selection satisfies = ( , , ) ∈ ( ). This can be seen by using the minimum property in (58) and the triangle inequality (C1 (ii)). Arguing in a standard way (testing the minimum in (58) by −1 ) we arrive at the inequality for every , ∈ , where we have defined the right-continuous piecewise constant approximation We have just established (40) and (41). The next goal is to pass this inequality to the limit.
Step 2: A priori estimates. Using (59) In this step we have exploited the uniform continuity of (⋅, ) guaranteed by conditions (35) and (36).
Notice that we did not construct a limit for the deformation yet because we are only able to use Helly's selection principle on the dissipative variables. We can still use the fact that is controlled by the energy for every fixed time . We define the limit deformation for every ∈ [0, ]. We refer to [21,Proposition 3.3] for further details on the convergence of the first term.

Nonlinear loading
In our discussion, for simplicity and not to overburden notation, we assume that the external loading acts linearly on the system, see (26). In this case, the force densities per unit volume (or area) in the reference configuration are independent of the deformation. Such loads are also called dead loads and are quite standard and commonly used also in nonlinear settings, see e.g. [21,34]. They describe external loads that are for instance determined by experimental devices. Nevertheless, it is possible to extend Theorem 3.2 to the case of nonlinear loading functionals, in the spirit of [18].

BV-regularizations
For consistency, we fixed the condition p , z > 1 throughout this paper. However, without any further assumptions, one can allow p = 1 or z = 1 or both. This extension for the setting of damage in elastic materials has been introduced and dealt with by THOMAS. [52] In plasticity theory, this has already been studied in the linear and nonlinear settings; see e.g. [25] for a quasistatic evolution to the Gurtin-Anand model, at small deformations, where the -control of the plastic strain is compensated by an 2 -control of its curl (the macroscopic Burgers tensor). In this contribution the authors show that, as the regularization term coefficients vanish, the solutions approach a quasistatic evolution to perfect plasticity. [17] The Gurtin-Anand model has been coupled with damage in [13,14], where the regularization term for the variable controls its 1 -norm; this is necessary to ensure lower semicontinuity of the plastic potential. In spirit of [25], the quasistatic perfectly plastic limit is achieved with the aid of a higher order regularization for the damage, which was later improved in [16].
In the nonlinear setting, it appears to be much harder to proof existence of energetic solutions without regularization terms, even for solely elastoplastic models without damage. We refer to [34] for the use of regularization terms controlling the 1,norm, if > 1, or the -norm, if = 1. The main idea to deal with the case = 1 is based on the contribution, [52] which can indeed also be used to cover the case of -regularization in our model, see discussion below. It remains an open problem to show existence of quasistatic evolutions to the model introduced in [39], which uses the term ( curl ) ⊤ as a regularization.
We can consider the general cases p ≥ 1, z ≥ 1; however, for simplicity of discussion we restrict to detail a bit the special case p = z = 1 (the cases when only one exponent is 1 is treated similarly). We define where | (Ω)| and | (Ω)| denote the total variation of and , respectively. The proof only changes slightly compared to the one we presented in Section 3; instead of Sobolev embedding theorems, we use that (Ω) compactly embeds into 1 (Ω). To proof the existence of mutual recovery sequences (C5) as in Lemma 3.5, we notice that Step 1 of the proof can exactly be copied, whereas for Steps 2-3 we argue as in [52].