Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow

This work rigorously implements a recent model of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear, that is, the total deformation gradient splits multiplicatively into elastic and plastic parts, and the elastic energy density is polyconvex. There are two internal variables: The system of all dislocations is modeled via $1$-dimensional boundaryless integral currents, whereas the history of plastic flow is encoded in a plastic distortion matrix-field. As our main result we construct an energetic solution in the case of a rate-independent flow rule. Besides the classical stability and energy balance conditions, our notion of solution also accounts for the movement of dislocations and the resulting plastic flow. Because of the path-dependence of plastic flow, a central role is played by so-called ``slip trajectories'', that is, the surfaces traced out by moving dislocations, which we represent as integral $2$-currents in space-time. The proof of our main existence result further crucially rests on careful a-priori estimates via a nonlinear Gronwall-type lemma and a rescaling of time. In particular, we have to account for the fact that the plastic flow may cause the coercivity of the elastic energy functional to decay along the evolution, and hence the solution may blow up in finite time.


INTRODUCTION
Dislocation flow is the principal mechanism behind macroscopic plastic deformation in crystalline materials such as metals [1,6,35]. The mathematical theories of large-strain elasto-plasticity and of crystal dislocations have seen much progress recently. Notably, a number of works have investigated phenomenological models of large-strain elasto-plasticity [28,44,45,47,48,51,64] by utilizing so-called "internal variables". This area has a long tradition and we refer to [1,32,33,42,43,65] for recent expositions and many historical references. However, the internal variables are usually conceived in a somewhat ad hoc manner (e.g., total plastic strain) and do not reflect the microscopic physics, at least not directly.
In parallel, the theory of dislocations has developed rapidly over the last years, but usually macroscopic plastic effects are neglected in this area. On the static (non-evolutionary) side we mention [7, 17-19, 26, 29, 30, 38, 39] for some recent contributions. On the evolutionary side, the field of discrete dislocation dynamics (DDD) considers discrete systems of dislocations moving in a crystal; see [14] for a recent monograph on the computational side and [63] for a mathematical approach. In the case of fields of dislocation we also refer to the works [2][3][4]9].
The recent article [34] introduced a model of large-strain elasto-plastic evolution in crystalline materials with the pivotal feature that the plastic flow is driven directly by the movement of dislocations. In the case of a rate-independent flow rule, the present work places this model on a rigorous mathematical foundation, defines a precise notion of (energetic) solution, and establishes an existence theorem (Theorem 4.9) for such evolutions under physically meaningful assumptions. Such a theorem may in particular be considered a validation of the model's mathematical structure.
In the following we briefly outline the model from [34], our approach to making the notions in it precise, and some aspects of the strategy to prove the existence of solutions.
Kinematics. The reference (initial) configuration of a material specimen is denoted by Ω ⊂ R 3 , which is assumed to be a bounded Lipschitz domain (open, connected, and with Lipschitz boundary). It is modelled as a macroscopic continuum with total deformation y : [0, T ] × Ω → R 3 , for which we require the orientation-preserving condition det ∇y(t) > 0 pointwise in Ω (almost everywhere) for any time t ∈ [0, T ]. We work in the large-strain, geometrically nonlinear regime, where the deformation gradient splits according to the Kröner decomposition [15,26,31,32,37,38,40,41,57,58] ∇y = EP into an elastic distortion E : [0, T ] × Ω → R 3×3 and a plastic distortion P : [0, T ] × Ω → R 3×3 (with det E, det P > 0 pointwise a.e. in Ω). We refer in particular to the justification of this relation in [34], which is based on a description of the crystal lattice via the "scaffold" Q = P −1 . However, neither E nor P can be assumed to be a gradient itself and P is treated as an internal variable, that is, P is carried along the plastic flow.
In line with much of the literature, we impose the condition of plastic incompressibility det P(t) = 1 a.e. in Ω, that is, the plastic distortion P(t) = P(t, ) is volume-preserving, which is realistic for many practically relevant materials [1,32].
Dislocations and slips. As mentioned before, in crystalline materials the dominant source of plasticity is the movement of dislocations, that is, 1-dimensional topological defects in the crystal lattice [1,6,35]. Every dislocation has associated with it a (constant) Burgers vector from a finite set B = {±b 1 , . . . , ±b m } ⊂ R 3 \ {0}, which is determined by the crystal structure of the material. We collect all dislocation lines with Burgers vector b ∈ B that are contained in our specimen at time t, in a 1-dimensional integral current T b (t) on Ω (see [17,18,63] for similar ideas and [27,36] as well as Section 2.3 for the theory of integral currents). This current is boundaryless, i.e., ∂ T b (t) = 0 since dislocation lines are always closed loops inside the specimen Ω; for technical reasons we assume that all dislocation lines are in fact closed globally (one may need to add "virtual" lines on the surface ∂ Ω to close the dislocations). When considering the evolution of t → T b (t), several issues need to be addressed: First, in order to rigorously define the dissipation, that is, the energetic cost to move the dislocations from T b (s) to T b (t) (s < t), we need a notion of "traversed area" between T b (s) and T b (t). Indeed, in a rate-independent model, where only the trajectory, but not the speed of movement, matters, this area, weighed in a manner depending on the state of the crystal lattice, corresponds to the dissipated energy. Second, only evolutions t → T b (t) that can be understood as "deformations" of the involved dislocations should be admissible. In particular, jumps are not permitted (at least not without giving an explicit jump path). Finally, on the technical side, we need a theory of evolutions of integral currents t → T b (t) based on the trajectory's variation in time. For instance, we require a form of the Helly selection principle to pick subsequences of sequences (t → T b n (t)) n for which T b n (t) converges for every t ∈ [0, T ]. It is a pivotal idea of the present work that all of the above requirements can be fulfilled by considering as fundamental objects not the dislocations T b (t) themselves, but the associated slip trajectories, which contain the whole evolution of the dislocations in time. We represent a slip trajectory as a 2-dimensional integral current S b (for the Burgers vector b ∈ B) in the space-time cylinder [0, T ] × R 3 with the property that i.e., the pushforward under the spatial projection p(t, x) := x of the slice S b | t of S b at time t (that is, with respect to the temporal projection t(t, x) := t). The theory of integral currents entails that T b (t) is a 1-dimensional integral current and ∂ T b (t) = 0 for almost every t ∈ (0, T ).
The total traversed slip surface from T b (s) to T b (t) can be seen to be the integral 2-current in R 3 given by , that is, the pushforward under the spatial projection of the restriction of S b to the time interval [s,t]. Note, however, that S b | t s does not contain a "time index", which is needed to describe the plastic flow (see below), and also that multiply traversed areas may lead to cancellations in S b | t s . This will require us to define the dissipation as a function of S b ([s,t] × R 3 ) directly, and not of S b | t s .
Plastic flow. With a family (S b ) b of slip trajectories at hand, we can proceed to specify the resulting plastic effect. To give the discrete dislocations a non-infinitesimal size we convolve S b with the dislocation line profile η ∈ C ∞ c (R 3 ; [0, ∞)), which is assumed to satisfy´η dx = 1, to obtain the thickened slip trajectory S b η := η * S b (with " * " the convolution in space). For kinematic reasons detailed in [34], the plastic distortion P follows the plastic flow equation Here, the spatial 2-vector γ b (t, x) ∈ 2 R 3 is the density of the measure p(S b η ) := p( S b η ) S b η at (t, x), which takes the role of the geometric slip rate, and "⋆" denotes the Hodge star operation, so that ⋆γ b (t, x) is the normal to the (thickened) slip surface at (t, x). Moreover, κ := 1 2 ∑ b∈B δ b is the Burgers measure (so the above integral with respect to κ is half the sum over all Burgers vectors b ∈ B).
Note that the projection in the definition of D has the effect of disregarding dislocation climb, so that P represent the history of dislocation glide only (see Section 6.2 in [34] for more on this). It turns out that for technical reasons we cannot enforce that ⋆γ b is orthogonal to P −1 b for admissible slip trajectories (which would obviate the need for the projection in (1.1)); see Remark 6.5 for an explanation.
Energy functionals. For the elastic energy we use W e (y, P) :=ˆΩ W e (∇yP −1 ) dx and make the hyperelasticity assumption that y(t) is a minimizer of W e ( , P(t)) for all t ∈ [0, T ]. This is justified on physical grounds by the fact that elastic movements are usually much faster than plastic movements [8,13,24]. For the elastic energy density W e we require polyconvexity [11,12] as well as (mild) growth and continuity conditions. In particular, our assumptions will be satisfied for the prototypical elastic energy densities of the form W e (E) := W (E) + Γ(det E), where W : R 3×3 → [0, ∞) is convex, has r-growth, and is r-concave with a sufficienly large r > 3 (depending on the other exponents in the full setup), and Γ : R → [0, +∞] continuous, convex, and Γ(s) = +∞ if and only if s ≤ 0; see Example 4.1 for details.
Further, we introduce the core energy as where ζ > 0. Here, M(T b ) = T b (R 3 ) is the mass of the current T b , i.e., the total length of all lines contained in T b . This core energy represents an atomistic potential energy "trapped" in the dislocations [1,6,35] (also see Section 6.4 in [34]). The present work could be extended to also incorporate more complicated (e.g., anisotropic) core energies, but we refrain from doing so for expository reasons. Given further an external loading f : [0, T ] × Ω → R 3 (in the easiest case), the total energy is then E (t, y, P, It is interesting to note that we do not need to employ a hardening term in the energy that gives coercivity in P or ∇P, like in all previous works on (phenomenological) elasto-plastic evolution in the large-strain regime, see e.g. [28,44,45,47,48,51,64].

Dissipation.
A key role in the formulation of the dynamics is played by the dissipation, i.e., the energetic cost associated with a trajectory between T (s) and T (t) (s < t). In the model introduced in [34], the dissipation along a slip trajectory S b from s to t may be expressed as (utilizing the formulation involving multi-vectors)ˆ[ Here, the function R b : 2 R 3 → [0, ∞) is the (potentially anisotropic and b-dependent) convex and 1-homogeneous dissipation potential (expressing the dissipational cost of a unit slip surface), which we require to satisfy ) (which is simple and has unit length) and the total variation measure The premultiplication with P here actually means the pushforward under P, i.e., P(v ∧ w) = (Pv) ∧ (Pw) for simple 2-vectors, and for non-simple 2-vectors extended by linearity. We refer to Section 2 for details on these notions.
The precise form of the dissipation we employ in this work, to be found in Section 4.1, is in fact a bit more involved due to the mathematical necessity of introducing a form of hardening (on the level of dislocations, not on the plastic distortion P). Otherwise, the specimen could rip immediately, preventing the existence of solutions for any non-trivial time interval.
The dissipation defined above controls, in the presence of suitable hardening/coercivity assumptions, a type of variation of S b in the interval [s,t], which is defined as This naturally leads to a theory of integral currents with bounded variation (in time), which was developed in [60]. The required aspects of this theory are recalled in Section 2.4 as the basis upon which our rigorous modeling of dislocations and slip trajectories in Section 3 is built.
Energetic solutions. In [34], the relation linking plastic distortion rates (velocities) and the corresponding stresses is given by the flow rule (in the multi-vector version) Euler-Lagrange equation are two of the most vexing open problems in the mathematical theory of elasticity [12]. Consequently, we need to formulate our whole system in a completely derivativefree setup, where X b and M do not appear. For this we employ an energetic framework based on the Mielke-Theil theory of rate-independent systems introduced in [53][54][55]; see [52] for a comprehensive monograph, which also contains many more references. The basic idea is to replace the flow rule by a (global) stability relation and an energy balance, stated precisely in Section 4.2, which employ only the total energy and dissipation functionals.
However, our framework differs from the classical energetic theory, as presented in [52], in a number of significant ways. Most notably, the central idea of the energetic theory to use a dissipation distance between any two states of the system [28,45,51,63] is modified here. This is a consequence of the fact that in order to define the change in plastic distortion associated with the movement of a dislocation we do not merely need the endpoints, but the whole trajectory. We will associate two "forward operators" to a slip trajectory, which determine the endpoint of the evolution for the dislocations and for the plastic distortion, respectively. The definition of the dislocation forward operator is straightforward (see Section 3.3), but for the plastic forward operator some effort needs to be invested (see Section 3.4). Further, we need to avoid the formation of jumps in the evolution since, for the reasons discussed above, we cannot define the plastic distortion associated with these jumps. As rate-independent evolutions can develop jumps naturally, we need to introduce a rescaling of time to keep the jump paths resolved.
The precise definition of our notion of solution is given in Definition 4.5, after all the above objects have been rigorously defined and all the assumptions have been stated. Our main existence result is Theorem 4.9. We will construct solutions as limits of a time-stepping scheme, where we minimize over "elementary" slip trajectories at every step. While we employ a number of ideas of the classical energetic theory, we will give a complete and essentially self-contained proof.
Decay of coercivity. An important argument in the limit passage, as the step size tends to zero in the time-stepping scheme, is to establish sufficient a-priori estimates on the total energy. This is, however, complicated by the fact that the integrand of W e depends on ∇yP −1 and hence the coercivity of W e in ∇y may decay as P evolves. Indeed, while we add some "hardening" to the dissipation to make it comparable to the geometric variation of the slip trajectories (see Section 4.1), we do not add a regularization in P or ∇P (like, for instance, in [45,51]). Thus, we can only obtain a differential estimate of the form where α N is the energy plus dissipation of the N'th approximate solution. The above differential inequality (or, more precisely, the associated difference inequality) does not fall into the situation covered by the classical Gronwall lemma and finite-time blowup to +∞ is possible as N → ∞. Indeed, the ODEu = e u , u(0) = u 0 has the solution u(t) = − log(e −u 0 − Ct), which blows up for t → e −u 0 /C. However, using nonlinear Gronwall-type lemma (see Lemma 5.4), we can indeed show an N-independent interval of boundedness for all the α N . Physically, if the time interval of existence is bounded, then the material fails (rips) in finite time.
Outline of the paper. We begin by recalling notation, basic facts, and the theory of space-time integral currents of bounded variation in Section 2. In Section 3 we define rigorously the basic kinematic objects of our theory, namely dislocation systems, slip trajectories, and the forward operators. The following Section 4 details our assumptions on the energy and dissipation functionals, defines our notion of solutions, and states the main existence result, Theorem 4.9. The time-incremental approximation scheme to construct a solution is introduced in Section 5. Finally, Section 6 is devoted to the limit passage and the proof of the existence theorem.
agreement No 757254 (SINGULARITY). The author would like to thank Thomas Hudson for discussions related to this work.

NOTATION AND PRELIMINARIES
This section recalls some notation and results, in particular from geometric measure theory.
2.1. Linear and multilinear algebra. The space of (m × n)-matrices R m×n is equipped with the Frobenius inner product A : The k-vectors in an n-dimensional real Hilbert space V are contained in k V and the k-covectors in k V , k = 0, 1, 2, . . .. For a simple k-vector ξ = v 1 ∧ · · · ∧ v k and a simple k-covector α = w 1 ∧ · · · ∧ w k the duality pairing is given as ξ , α = det (v i · w j ) i j ; this is then extended to nonsimple k-vectors and k-covectors by linearity. The inner product and restriction of η ∈ k V and α ∈ l V are η α ∈ l−k V and η α ∈ k−l V , respectively, which are defined as We will exclusively use the mass and comass norms of η ∈ k V and α ∈ k V , given via where we call a simple k-vector η = v 1 ∧ · · · ∧ v k a unit if the v i can be chosen to form an orthonormal system in V . For a k-vector η ∈ k V in an n-dimensional Hilbert space V with inner product ( , ) and fixed ambient orthonormal basis {e 1 , . . . , e n }, we define the Hodge dual ⋆η ∈ n−k V as the unique vector satisfying ξ ∧ ⋆η = (ξ , η) e 1 ∧ · · · ∧ e n , ξ ∈ k V.
In the special case n = 3 we have the following geometric interpretation of the Hodge star: ⋆η is the normal vector to any two-dimensional hyperplane with orientation η. In fact, for a, b ∈ 1 R 3 the identities hold, where "×" denotes the classical vector product. Indeed, for any v ∈ R 3 , the triple product v · (a × b) is equal to the determinant det(v, a, b) of the matrix with columns v, a, b, and so Hence, the first identity follows. The second identity follows by applying ⋆ on both sides and using ⋆ −1 = ⋆ (since n = 3). A linear map S : V → W , where V,W are real vector spaces, extends (uniquely) to a linear map and extending by (multi-)linearity to k V .

2.2.
Spaces of Banach-space valued functions. Let w : [0, T ] → X (T > 0) be a process (i.e., a function of "time") that is measurable in the sense of Bochner, where X is a reflexive and separable Banach space; see, e.g., [52,Appendix B.5] for this and the following notions. We define the corresponding X -variation for [σ , τ] ⊂ [0, T ] as Its elements are called (X -valued) functions of bounded variation. We further denote the space of Lipschitz continuous functions with values in a Banach space X by Lip([0, T ]; X ). Note that we do not identify X -valued processes that are equal almost everywhere (with respect to "time"). By a repeated application of the triangle inequality we obtain the Poincaré-type inequality exist (only the left limit at 0 and only the right limit at T ). For all but at most countably many jump points t ∈ (0, T ), it also holds that w(t+) = w(t−) =: w(t).

2.3.
Integral currents. We refer to [36] and [27] for the theory of currents and in the following only recall some basic facts that are needed in the sequel. We denote by H k R the k-dimensional Hausdorff measure restricted to a (countably) krectifiable set R; L d is the d-dimensional Lebesgue measure. The Lebesgue spaces L p (Ω; R N ) and the Sobolev spaces W k,p (Ω; R N ) for p ∈ [1, ∞] and k = 1, 2, . . . are used with their usual meanings. Let ) be the space of (smooth) differential k-forms with compact support in an open set U ⊂ R d . The exterior differential of a 0-form (i.e., a function) element of the dual canonical basis); for a simple k-form ω = f dx j 1 ∧ · · · ∧ dx j k the exterior differential is given as dω : for all other forms this definition is extended by linearity.
The dual objects to differential k-forms, i.e., elements of the dual space D k (U ) := D k (U ) * (k ∈ N∪{0}) are the k-currents. There is a natural notion of boundary for a k-current For a 0-current T , we formally set ∂ T := 0. where: x ∈ R the k-vector T (x) is simple, has unit length (| T (x)| = 1), and lies in the (k-times wedge product of) approximate tangent space T x R to R at x; (iii) m ∈ L 1 loc (H k R; N); The map T is called the orientation map of T and m is the multiplicity.
Let T = T T be the Radon-Nikodým decomposition of T with the total variation measure Let Ω ⊂ R d be a bounded Lipschitz domain, i.e., open, connected and with a (strong) Lipschitz boundary. We define the following sets of integral k-currents (k ∈ N ∪ {0}): The boundary rectifiability theorem, see [27, 4.2.16] or [36,Theorem 7.9.3], entails that for T ∈ For we define the product current of T 1 , T 2 as For its boundary we have the formula Let θ : Ω → R m be smooth and let T = m T H k R ∈ I k (Ω). The (geometric) pushforward θ * T (often also denoted by "θ # T " in the literature) is where θ * ω is the pullback of the k-form ω. We say that a sequence ( For T ∈ I k (R d ), the (global) Whitney flat norm is given by and one can also consider the flat convergence F(T − T j ) → 0 as j → ∞. Under the mass bound  [36,Section 7.6] or [27,Section 4.3]) entails that a given integral current S = m S H k+1 R ∈ I k+1 (R n ) can be sliced with respect to a Lipschitz map f : R n → R as follows: Set R| t := f −1 ({t}) ∩ R. Then, R| t is (countably) H k -rectifiable for almost every t ∈ R. Moreover, for H k -almost every z ∈ R| t , the approximate tangent spaces T z R and T z R| t , as well as the approximate gradient ∇ R f (z), i.e., the projection of ∇ f (z) onto T z R, exist and Also, ξ (z) is simple and has unit length. Set where D R f (z) is the restriction of the differential D f (z) to T z R, and Then, the slice is an integral k-current, S| t ∈ I k (R n ). We recall several important properties of slices: First, the coarea formula for slices,ˆR Third, the cylinder formula and, fourth, the boundary formula 2.4. BV-theory of integral currents and deformations. In this section we briefly review some aspects of the theory on space-time currents of bounded variation, which was developed in [60]. In the space-time vector space R 1+d ∼ = R × R d we denote the canonical unit vectors as e 0 , e 1 , . . . , e d with e 0 the "time" unit vector. The orthogonal projections onto the "time" component and "space" component are respectively given by by t : The variation and boundary variation of a (1 + k)-integral current S ∈ I 1+k ([σ , τ] × Ω) in the interval I ⊂ [σ , τ] are defined as If For L 1 -almost every t ∈ [σ , τ], is the slice of S with respect to time (i.e., with respect to t).
is not defined and we say that S has a jump at t. In this case, the vertical piece S ({t} × R d ) takes the role of a "jump transient". This is further elucidated by the following lemma, which contains an estimate for the mass of an integral (1 + k)-current in terms of the masses of the slices and the variation.
More relevant to the present work is the following: where (0, 1) is the canonical current associated with the interval (0, 1) (with orientation +1 and multiplicity 1). Then, according to the above definition, Thus, the S H so defined can be understood as deforming T via H into H(1, ) * T . We refer to Lemma 4.3 in [60] for estimates relating to the variation of such homotopical deformations.
Next, we turn to topological aspects. For this, we say that 3) The following compactness theorem for this convergence in the spirit of Helly's selection principle is established as Theorem 3.7 in [60].
We can use the variation to define the (Lipschitz) deformation distance between T 0 , The key result for us in this context is the following "equivalence theorem"; see Theorem 5.1 in [60] for the proof.
Moreover, in this case, for all j from a subsequence of the j's, there are S j ∈ Lip([0, 1]; I k (Ω)) with Here, the constant C > 0 depends only on the dimensions and on Ω.

DISLOCATIONS AND SLIPS
This section introduces the key notions that we need in order to formulate the model from [34] rigorously, most notably dislocation systems and slip trajectories. Dislocation systems are collections of dislocation lines, indexed by their (structural) Burgers vector, which is constant along a dislocation line. Slip trajectories describe the evolution of a dislocation system. Crucially, they also provide a way to obtain the evolution of the plastic distortion. To this aim we will introduce suitable "forward operators", one for dislocation systems and one for plastic distortions.
3.1. Burgers measure. Assume that we are given a set of Burgers vectors Then, the Burgers measure κ ∈ M + (S 2 ) is the purely atomic measure In the following we will often use the convenient notation for any expression F defined on B. We also write "κ-almost every b" instead of "for every b ∈ B".

Dislocation systems.
The set of (discrete) dislocation systems is defined to be where I 1 (Ω) is the set of all integral 1-currents supported in Ω (see Section 2.3 for notation). We interpret this definition as follows: T b contains all dislocation lines with Burgers vector b ∈ B. The symmetry condition T −b = −T b for κ-a.e. b means that the sign of a Burgers vector can be flipped when accompanied by a change of line orientation (this also explains the factor 1 2 in the definition of κ). The dislocation lines are assumed to be closed (globally). While usually one only assumes closedness inside the specimen Ω, in all of the following we require global closedness, essentially for technical reasons. This can always be achieved by adding "virtual" dislocation lines on ∂ Ω.

Slips and dislocation forward operator.
To describe evolutions (in time) of dislocation systems, we define the set of Lipschitz slip trajectories as , with respect to t = t). We then have We let the L ∞ -(mass-)norm and the (joint) variation of Σ ∈ Lip([0, T ]; DS(Ω)) be defined for any interval I ⊂ [0, T ] as, respectively, In the following, we will also make frequent use of the space of elementary slip trajectories 1];DS(Ω)) and Var(Σ) := Var(Σ; [0, 1]). The idea here is that an elementary slip trajectory Σ ∈ Sl(Φ) gives us a way to transform a dislocation system Φ into a new dislocation system in a progressive-in-time manner. The additional condition in the definition of Sl(Φ) entails that S b starts at T b , for which we could equivalently require S b (0) = T b for κ-almost every b.
We may then define the dislocation forward operator for , be a family of Lipschitz-homotopies satisfying 3.4. Plastic evolution. We now consider how slip trajectories give rise to an evolution of the plastic distortion. For this, consider a dislocation system Φ = (T b ) b ∈ DS(Ω) and a slip trajectory ) be a dislocation line profile satisfying´η dx = 1 (i.e., η is a standard mollifier), which is globally fixed. We define the thickened slip trajectory with the convolution " * " acting in space only and ω being understood as extended by zero outside Ω.
is absolutely continuous with respect to Lebesgue measure. Its density, called the geometric slip rate, is equal to and for all k = 0, 1, 2, . . . there is a constant C k > 0, which only depends on η, such that Proof. Fix b ∈ B. We first observe by linearity that since also |η * ω| ≤ 1 by the properties of the mollification. Thus, for where L > 0 is a universal Lipschitz constants of the scalar functions t → Var( From the coarea formula for slices, see (2.1), we then get Thus, the density of p(S b η ) has been identified as the expression for γ b given in (3.1). Via Young's convolution inequality and (2.1) again, it satisfies for for almost every x, and (3.2) holds for k = 0. The higher differentiability follows by pushing the derivatives onto the mollifier and estimating analogously.
We also define the normal slip rate for which we have by the preceding Lemma 3.
where the (total) plastic drift D(t, x, R) for t ∈ [0, T ] and R ∈ R 3×3 with det R > 0 is given as with g b corresponding to γ b for S b as above. By proj R −1 b ⊥ we here denote the orthogonal projection onto the orthogonal complement to the line R −1 b. We will show in Lemma 3.3 below that this ODE indeed has a solution for almost every x ∈ Ω. We then define the plastic distortion path P Σ starting at P induced by the slip trajectory We first consider the question of well-definedness:

) has a unique solution for almost every x ∈ Ω and P
in Ω as well as where is uniformly bounded and Lipschitz in U M , and the projection  To show (3.7), we estimate for all t ∈ [0, T ], Taking the L s -norm in x, this gives .
The same arguments hold also when starting the evolution at t 0 ∈ [0,t). Thus, the Lipschitz continuity of t → Var(Σ; [0,t]) in conjunction with the additivity of the variation yield the Lipschitz continuity of t → P Σ (t), considered with values in L s (Ω; R 3×3 ).
The claimed incompressibility property det P Σ (t, x) = 1 for all t ∈ [0, T ] and almost every x follows directly from (3.8).
The next lemma shows the transportation of regularity along the plastic evolution. Lemma 3.4. Assume that additionally P ∈ W 1,q (Ω; R 3×3 ) for a q ∈ (3, ∞]. Then,

9)
and P Σ is a Lipschitz function with values in W 1,q (Ω; R 3×3 ), where C > 0 and the Lipschitz constant depend (monotonically) on P W 1,q and Var(Σ; Note that while the growth of C in P W 1,q and Var(Σ; [0, T ]) may be very fast, we will always apply this result in the presence of a uniform bound for those quantities; then the estimates are of the same type as the ones in Lemma 3.3 and in particular additive in the variation.
Proof. By Lemma 3.3 (for s = ∞) and the embedding of We have by the chain rule that where ∇ denotes the (weak) x-gradient. Then, since time derivative and weak gradient commute, we get that ∇P Σ satisfies the ODE Integrating in time from 0 to t, taking the L q -norm in x, and applying (3.10), we get The integral form of Gronwall's lemma now yields Combining this with (3.2) in Lemma 3.2, where we have absorbed some terms into the constant C > 0. Together with (3.7) this yields (3.9). Further, varying the starting point and employing the Lipschitz continuity of t → Var(Σ; [0,t]) in conjunction with the additivity of the variation gives for all s < t that where L depends on Var(Σ; [0, T ]) and P W 1,q (which bounds P(s) W 1,q by (3.9)). This gives the Lipschitz continuity of t → P Σ (t) with values in W 1,q (Ω; R 3×3 ).
Next, we show that we may dispense with the pointwise definition of a solution to (3.4).
and for such t it holds that Proof. We have seen above that where the limit is in W 1,q and the last equality follows via the Lipschitz continuity of P Σ in time with respect to values in W 1,q and the fact that this implies x-uniform pointwise Lipschitz continuity by the embedding W 1,q (Ω; R 3×3 ) c ֒→ C(Ω; R 3×3 ). Thus, (3.11) has been established.

3.5.
Operations on slip systems. We now introduce useful operations on (elementary) slip trajectories, namely rescalings and concatenations, and we also define the so-called "neutral" slip trajectory.
Define (using the notation of Lemma 2.5) and also define the plastic drift D ′ analogously to D, but with a * Σ in place of Σ. Then, for the solution P a * Σ of (3.11) the rate-independence property Proof. The fact that a * Σ = (a * S b ) b ∈ Sl(Φ; [0, T ′ ]) follows from Lemma 2.5. Turning to (3.12), we denote by a * γ b the geometric slip rate defined in Lemma 3.2 with respect to a * S b . Note that for ω ∈ D 2 (R 3 ) and all 0 ≤ s < t ≤ T we obtain in the same way as in the proof of Lemma 2.5 (which can be found in Lemma 3.4 of [60]) using the area formula thatˆt where we changed variables in the last line. Thus, ) .
By the uniqueness of the solution to (3.11) we thus obtain P ′ = P a * Σ , which implies (3.12). The additional statements are then clear (using also Lemma 2.5).

13) and
where the rescaling r α and the translation t τ (α = 0, τ ∈ R) are given by From Lemma 2.5 we see that Σ 2 • Σ 1 ∈ Sl(Φ) and that (3.14), (3.15) hold. The validity of the first statement in (3.13) follows in a straightforward manner since, if Σ 1 The second statement in (3.13) is a direct consequence of (3.12) in Lemma 3.6.
There exists a slip trajectory Id Φ ∈ Sl(Φ), called the neutral slip trajectory, such that Then, we say that Σ j converges weakly* to Σ, in symbols "Σ j * ⇀ Σ". As the main compactness result we have the following: Moreover, Then, Fatou's lemma implies The lower semicontinuity of the L ∞ -norm follow directly from the corresponding statement in Proposition 2.6.
For later use we also state the compactness for elementary slips explicitly: Proposition 3.10. Let Φ ∈ DS(Ω) and assume that the sequence with L j the maximum (in b) of the Lipschitz constants of the scalar maps t → Var(S b j ; [0,t]). Then, there exists Σ ∈ Sl(Φ) and a (not relabelled) subsequence such that Moreover, Proof. By Proposition 3.9 we obtain the convergence in Lip([0, 1]; DS(Ω)) and the lower semicontinuity assertions. From Proposition 2.6 we further obtain that also the condition Finally, we have the following continuity properties.
. This directly implies the assertion.
To see the uniform convergence in [0, T ] × Ω, observe first that from Lemma 3.4 we know that the (P j ) Σ j are uniformly Lipschitz continuous in time when considered with values in W 1,q (note that the norms P j W 1,q and the variations Var(Σ j ; [0, T ]) are uniformly bounded by the Uniform Boundedness Principle). Hence, by the (generalized) Arzelà-Ascoli theorem we may select a subsequence of j's (not specifically labeled) such that for some Here we also used the compact embedding W 1,q (Ω; R 3×3 ) c ֒→ C(Ω; R 3×3 ). On the other hand, let γ b j , γ b and g b j , g b be defined as in Section 3.3 for the slip trajectories Σ j and Σ, respectively. Since Σ j * Rewriting the ODE (3.4) as an integral equation and multiplying by a test function ϕ ∈ C ∞ c (Ω), we see that (P j ) Σ j solves (3.4) if and only if where H b is as in the proof of Lemma 3.4. As j → ∞, the above convergences in conjunction with the Lipschitz continuity of H b and the (strong × weak*)-continuity of the integral, givê Hence, P * solves (3.4). By Lemma 3.3, the solution of (3.4) for Σ is unique, whereby P * = P Σ .

ENERGETIC EVOLUTIONS
In this section we list our precise assumptions, translate the model from [34] into the energetic formulation, and then state our main result, Theorem 4.9, which establishes the existence of an energetic solution.
The core energy of the dislocation system Φ ∈ DS(Ω) is defined as where ζ > 0 was specified in Assumption (A1). More complicated expressions (for instance, with anisotropy or dependence on the type of dislocation) are possible, but we will only use the above to keep the exposition as simple as possible. We can then define for y ∈ W 1,p g (Ω; R 3 ), P ∈ W 1,q (Ω; R 3×3 ) with det P = 1 a.e. in Ω, and Φ = (T b ) b ∈ DS(Ω) the total energy where f is the external loading specified in Assumption (A4) and , is the duality product between W 1,p (Ω; R 3 ) * and W 1,p (Ω; R 3 ). We next turn to the dissipation. For this, we first introduce a convenient notation for a path in the full internal variable space induced by a slip trajectory. Let z = (P, Φ) ∈ W 1,q (Ω; R 3×3 ) × DS(Ω) with det P = 1 a.e. in Ω.
Here, P Σ is understood as a continuous map from [0, T ] × Ω to R 3×3 and, as before, we let the Burgers measure κ be given by If Σ ∈ Sl(z), i.e., Note that Diss(Σ; I) depends on P (from z = (P, Φ)) through P Σ . However, we think of Σ as "attached" at the starting point z and from the context it will always be clear where it is attached, usually through the notation "Σ ∈ Sl(z; [0, T ])". While this constitutes a slightly imprecise use of notation, it improves readability and hence we will adopt it in the following.
The next example presents a concrete dissipational cost similar to the one in [34].
be convex, positively 1-homogeneous, Lipschitz, and satisfy the bounds C −1 |ξ | ≤ R b (ξ ) ≤ C|ξ | for all ξ ∈ 2 R 3 and a b-uniform constant C > 0. We remark that the (global) Lipschitz continuity is in fact automatic in this situation, see, e.g., [61, Lemma 5.6]. Assume furthermore that for all b ∈ B we are given a "hardening factor" h b : [1, ∞) → (0, ∞) that is locally Lipschitz continuous, increasing, and satisfies , where p(ξ ) denotes the pushforward of the 2-vector ξ under the spatial projection p(t, x) := x, which is then further pushed forward under P. Note that |P| ≥ 1 since det P = 1 (e.g., by Weyl's matrix inequality), so the above expression is well-defined. The first three points in Assumption (A3) are easily verified. For the fourth point (coercivity), we observe that P −1 = (cof P) T since det P = 1, and so, by Hadamard's inequality, for some C > 0, where for the second inequality we have also used τ 4 ≤ (C + C 2 / min h b )h b (τ) for all τ ≥ 1, which is an elementary consequence of (4.4). Then, which is the claim.
In the previous example, the hardening factor h b (P) can be interpreted as making it more energetically expensive for dislocations to glide if |P| becomes large. This is physically reasonable since after a large amount of plastic distortion has taken place, the crystal will have many point defects and so dislocation glide is impeded [6,35]. It is also necessary for our mathematical framework: Without a hardening factor the dissipation may no longer control the variation and no solution may exist for positive times (see the proof of Proposition 5.2 and also of Lemma 4.14 below). This corresponds to instantaneous ripping of the specimen. For instance, even if det P = 1, a principal minor of P may blow up, e.g., for P ε := diag(ε, ε, ε −2 ) with ε ↓ 0. Remark 4.3. More generally, in Assumption (A3) one could require R b to be only semielliptic instead of convex in the second argument ξ , see, e.g., [36,Section 8.3] for a definition of this generalized convexity notion. This allows for more general dissipation potentials, but semiellipticity is hard to verify in general.
Remark 4.4. The present theory extends to E incorporating an additional (additive) hardening or softening energy of the form W h (P, Φ) for P ∈ W 1,q (Ω; R 3×3 ) with det P = 1 a.e. in Ω, and Φ ∈ DS(Ω). In order for this to be compatible, the modified E still needs to satisfy the conclusions of Lemma 4.17 below.

Energetic formulation.
In general, jumps in time cannot be excluded for rate-independent systems [52]. Thus, we will work with a rescaled time s in which the process does not have jumps (or, more precisely, the jumps are resolved). By the rate-independence, this rescaling does not change the dynamics besides a reparameterization of the external loading. In the existence theorem to follow, we will construct a Lipschitz rescaling function ψ :  Our notion of solution then is the following: is called an energetic solution to the system of dislocation-driven elasto-plasticity with rescaling function ψ : Here and in the following, we use the notation L ∞ (I; X ) for the set of (Bochner-)measurable and uniformly norm-bounded functions defined on the interval I ⊂ R, but we do not identify maps that are equal almost everywhere in I. In a similar vein, we use the good representative for s → S b (s), so that z(s) = (P(s), Σ(s)) = (P(s), (S b (s)) b ) is well-defined for every s ∈ [0, ∞).
Moreover, Diss( Σ) in (E) is to be interpreted relative to z 0 (recall from Section 4.1 that the starting point is omitted in our notation). In (S), the conditionψ(s) > 0 includes the existence oḟ ψ(s), which is the case for L 1 -almost every s ∈ [0, ∞) by Rademacher's theorem. The differential equation in (P) is to be understood in W 1,q (Ω; R 3×3 ) (see Lemma 3.5).
Let us now motivate how the above formulation (S), (E), (P) corresponds to the model developed in [34], as outlined in the introduction. First, we observe that in general we do not have enough regularity to consider derivatives of the processes or functionals. Instead, we reformulate the model as follows: The condition (P) corresponds directly to (1.1). The stability (S) and energy balance (E) come about as follows: The Free Energy Balance (a consequence of the Second Law of Theormodynamics) in the whole domain Ω reads as (see Section 4 in [34]) d dt W e (y(t), z(t)) + W c (z(t)) − P(t, y(t)) = −∆(t). (4.6) Here, the external power is given as where , is the duality product between W 1,p (Ω; R 3 ) * and W 1,p (Ω; R 3 ), and we neglect the inertial term for the rate-independent formulation (cf. Section 6.1 in [34]). If we integrate (4.6) in time over an interval [0,t] ⊂ [0, T ] and use an integration by parts to observê we arrive at This yields (E) after the rescaling described at the beginning of this section. The stability (S) is a stronger version of the local stability relation which follows from the flow rule (1.3) or, more fundamentally, the Principle of Virtual Power (see Section 4 in [34]). We refer to [52] for more on the equivalence or non-equivalence of (S) & (E) with "differential" models of rate-independent processes.
Remark 4.6. The pieces where ψ is flat correspond to the jump transients, which are therefore explicitly resolved here. Note that there could be several Diss-minimal slip trajectories connecting the end points of a jump, which lead to different evolutions for the plastic distortion. Thus, we cannot dispense with an explicit jump resolution. Moreover, the stability may not hold along such a jump transient and hence we need to requireψ(s) > 0 in (S). We refer to [20,21,46,47,49,50,56,62] for more on this.
Remark 4.7. The stability (S) in particular entails the elastic minimization y(s) ∈ Argmin E ψ (s, y, z(s)) : y ∈ W 1,p g (Ω; R 3 ) as well as the orientation-preserving assertion det ∇y(s) > 0 a.e. in Ω for all s ∈ [0, ∞). This can be seen by testing with Σ := Id Σ(s) ∈ Sl(Σ(s)) (see Lemma 3.8) and also using the properties of W e in Assumption (A2). In this sense, we are in an elastically optimal state. This corresponds to the supposition that elastic movements are much faster than plastic movements, which is true in many materials [8,13,24]. Note that there is no associated Euler-Lagrange equation (elastic force balance) since we cannot differentiate the functional W e ; see [12] for this and related open problems in nonlinear elasticity theory. For (E), the rescaling invariance is an easy consequence of a change of variables and Lemma 4.15 in the following section: For s ′ ∈ [0, S ′ ], we compute For (P) the rate-independence has already been shown in Lemma 3.6.

Existence of solutions.
The main result of this work is the following existence theorem: in Ω is such that the initial stability relation holds for all y ∈ W 1,p g (Ω; R 3 ), Σ ∈ Sl(Φ 0 ).

Then, there exists an energetic solution to the system of dislocation-driven elasto-plasticity in the sense of Definition 4.5 satisfying the initial conditions
Moreover, Var

4.4.
Properties of the energy and dissipation. In preparation for the proof of Theorem 4.9 in the next sections, we collect several properties of the energy and dissipation functionals. We start with the question of coercivity.
Lemma 4.11. For every t ∈ [0, T ], y ∈ W 1,p g (Ω; R 3 ) with det ∇y > 0 a.e. in Ω, P ∈ L s (Ω; R 3×3 ) with det P = 1 a.e. in Ω, and 1 s = 1 p − 1 r , it holds that Proof. For a, b > 0 and all ρ > 1 we have the elementary inequality which follows from Young's inequality for a 1/ρ , b with exponents ρ, ρ/(ρ − 1). Hence, for F, P ∈ R 3×3 with det P = 0 we get with ρ := r/p, whereby ρ − 1 = r/s, that Raising this inequality to the r'th power and using the coercivity in Assumption (A2), we get (combining constants as we go), , where in the last line we further employed the Poincaré-Friedrichs inequality (the boundary values of y are fixed). Moreover, . On the other hand, we have for any ε > 0, by Young's inequality again, Combining the above estimates, and choosing ε > 0 sufficiently small to absorb the last term in (4.7) into the corresponding term originating from W e , the claim of the lemma follows.
The next lemma extends the classical results on the weak continuity of minors [11,59] and in a similar form seems to have been proved first in [45, in Ω and for all j ∈ N. Then, Proof. We have, by Cramer's rule, ∇y j P −1 j = ∇y j · (cof P j ) T , and using, for instance, Pratt's convergence theorem, Then, which is equivalent to our assumption s > 2p p−1 . Next, we recall that cof(∇y j P −1 j ) = cof(∇y j ) · cof(P −1 j ) = cof(∇y j ) · P T j . By the weak continuity of minors (see, e.g., [61, Lemma 5.10]) we know that cof(∇y j ) ⇀ cof(∇y) in L p/2 . Thus, cof(∇y j P −1 j ) ⇀ cof(∇yP −1 ) in L σ ′′ (D; R) if 1 σ ′′ := 2 p + 1 s < 1, which is equivalent to s > p p−2 . Since our assumptions imply s > 2p p−1 > p p−2 , we also obtain convergence in this case.
We can then state a result on the lower semicontinuity of the elastic energy: Proposition 4.13. The functional ( y, P) → W e ( y, P) is weakly (sequentially) lower semicontinuous with respect to sequences (y j ) ⊂ W 1,p g (Ω; R 3 ) satisfying det ∇y j > 0 a.e. in Ω, and (P j ) ⊂ W 1,q (Ω; R 3×3 ) with det P j = 1 a.e. in Ω.
Proof. Let (y j , P j ) be as in the statement of the proposition with y j ⇀ y in W 1,p and P j ⇀ P in W 1,q . Let s > 2p p−1 . By the Rellich-Kondrachov theorem, W 1,q (Ω; R 3×3 ) c ֒→ L s (Ω; R 3×3 ) (since q > 3 this holds for all s ∈ [1, ∞]) and hence P j → P strongly in L s . Then, by Lemma 4.12 all minors of the compound sequence ∇y j P −1 j converge weakly in L σ for some σ > 1. Thus, the lower semicontinuity follows in the usual manner for the polyconvex integrand W e (via strong lower semicontinuity and Mazur's lemma), see, e.g., [61, Theorem 6.5] for this classical argument.
Next, we establish some basic properties of the dissipation.
with a constant C > 0. Moreover, for Σ 1 ∈ Sl(z), and Σ 2 ∈ Sl(Σ 1 ≫ z), it holds that Proof. The first claim follows directly from the properties assumed on R b in Assumption (A3). The second claim (4.9) follows in the same way as (3.15) in Lemma 3.7 (also using (3.13)).
Furthermore, by Lemma 3.12, P Σ j → P Σ uniformly in [0, 1] × Ω. Thus, also using the local Lipschitz continuity of R b (see Assumption (A3)), the fact that both |P Σ j | and | S b j | are uniformly bounded, and Fatou's lemma, we obtain This is the assertion.
For convenient later use, in the following lemma we collect several convergence assertions.
Proof. Ad (i). The first term W e (y, P) in the definition of E , see (4.3), is lower semicontinuous by Proposition 4.13; the second term − f (t), y is in fact continuous since f (t) is continuous in t with values in the dual space to W 1,p (Ω; R 3 ) by (A4); the third term W c (Φ) is weakly* lower semicontinuous by the weak* lower semicontinuity of the mass and Fatou's lemma (as in Lemma 4.16).
Ad (ii). We first prove the continuity property for W e . The compact embedding of W 1,q (Ω; R 3×3 ) into C(Ω; R 3×3 ) (since q > 3) entails that the P j are uniformly bounded and converge uniformly to P. We further observe via (4.2) in (A2) (clearly, PP −1 j ∈ X M for some M ≥ 1) that W e (∇yP −1 j ) ≤ C M (1 +W e (∇yP −1 )) a.e. in Ω Since taking inverses is a continuous operation on matrices from X M , we get P −1 j → P −1 a.e. in Ω. Then, W e (∇yP −1 j ) → W e (∇yP −1 ) a.e. in Ω by the continuity of W e (see (A2)). Thus, as C M (1 + W e (∇yP −1 )) is integrable by assumption, it follows from the dominated convergence theorem that W e (y, P j ) → W e (y, P).
For the power term we argue as in (i).

Ad (iii). This follows again from the properties of the external force, see (A4).
Ad (iv). This was proved in Lemma 4.16.
We also record the following fact, which occupies a pivotal position in this work: It allows us to translate the weak* convergence of dislocation systems into a slip trajectory (of vanishing dissipation) connecting these dislocation systems to their limit. This will be crucially employed later to show stability of the limit process (see Proposition 6.4). Then, In this case there are where the constant C > 0 only depends on the dimensions and on Ω, and Diss(Σ j ) is understood relative to any starting point P ∈ W 1,q (Ω; R 3×3 ) for a q ∈ (3, ∞] with det P = 1 a.e. in Ω. Proof. Since we assume that κ is purely atomic, and also using the growth properties of R b in Assumption (A3), the first claim follows immediately from Proposition 2.7. For the existence of the Σ j as claimed we further obtain S b j ∈ Lip([0, 1]; I 1 (Ω)) with from this result. Then, for Σ j : since P Σ j remains uniformly bounded (in j) by Lemma 3.4 (and the embedding W 1,q (Ω; R 3×3 ) ֒→ C(Ω; R 3×3 )), whereby Assumption (A3) (iii) becomes applicable.
Remark 4.19. Note that we do not claim that any two dislocation systems Φ 1 , Φ 2 ∈ DS(Ω) can be connected by a slip trajectory. Indeed, if Ω is not simply connected and has a hole (with respect to countably 1-rectifiable loops), then there are dislocation systems that cannot be deformed into each other.

TIME-INCREMENTAL APPROXIMATION SCHEME
We start our construction of the energetic solution with a time-discretized problem and corresponding discrete solution. For brevity of notation it will be convenient to define the deformation space Y := W 1,p g (Ω; R 3 ) : det ∇y > 0 a.e. in Ω and the internal variable space Z := (P, Φ) ∈ W 1,q (Ω; R 3×3 ) × DS(Ω) : det P = 1 a.e. in Ω . Set For k = 1, . . . , N, we will in the following construct Here, γ ≥ M(Φ 0 ) (5.1) is a parameter.
Remark 5.1. The assumption Σ L ∞ ≤ γ in the minimization is necessary because we cannot control Σ L ∞ by the variation of Σ alone, see Example 3.6 in [60]. The assumption (5.1) is required for the well-posedness of the time-incremental problem since it makes the neutral slip trajectory admissible (see Lemma 3.8) and hence the candidate set for the minimization in (IP) is not empty. Later, when we have a time-continuous process, we can infer a uniform mass bound from the energy balance (E) and the coercivity of E (Lemma 4.11) and then let γ → ∞.
The existence of discrete solutions is established in the following result. Here and in the following, all constants implicitly depend on the data in Assumptions (A1)-(A5).
Proof. Assume that for k ∈ {1, . . . , N} a solution (y N j , z N j , Σ N j ) j=1,...,k−1 to the time-incremental minimization problem (IP) has been constructed up to step k − 1. This is trivially true for k = 1 by Assumption (A5). In the following, we will show that then also a solution (y N k , z N k , Σ N k ) to (IP) at time step k exists and (5.2) holds.
Step 1: Any solution (y N k , z N k , Σ N k ) to (IP) at time step k, if it exists, satisfies (5.2). To show the claim we assume that (y N k , z N k , Σ N k ) is a solution to (IP) at time step k. Testing with y := y N k−1 and the neutral slip trajectory Σ := Id Φ N k−1 ∈ Sl(z N k−1 ) (see Lemma 3.8), we get To bound the integral, we first estimate for any (t, y, P, Φ) ∈ (0, T ) × Y × Z using Lemma 4.11 (with the constant C potentially changing from line to line) where in the last line we used a 1/p ≤ a for a ≥ 1 and C also absorbs the expressions depending on ḟ L ∞ ([0,T ];[W 1,p ] * ) . Gronwall's lemma then gives that for all τ ≥ t it holds that E (τ, y, P, Φ) + P s L s + 1 ≤ (E (t, y, P, Φ) + P s L s + 1)e C(τ−t) . We may also estimate, using the same arguments as above,

Plugging this into (5.3),
. Next, observe via an iterated application of Lemma 3.3 and (4.8) in Lemma 4.14 that Combining the above estimates, where we used that a s ≤ Ce a for a ≥ 1, and e C∆T N − 1 ≤ 2C∆T N for ∆T N small enough. We remark that we used the exponential function (as opposed to a polynomial expression) here mainly for reasons of convenience. We thus arrive at the claim (5.2) at k.
Step 2: In (IP) at time step k, the minimization may equivalently be taken over y ∈ Y , Σ ∈ Sl(z N k−1 ) satisfying the bounds for a constant C(α N k−1 ) > 0, which only depends on the data from the assumptions besides α N k−1 . Recalling (IP), we immediately have (5.7). To see the claims (5.5), (5.6), observe first that from Step 1 we may restrict the minimization in (IP) at time step k to y, Σ such that for , if it exists, must satisfy (5.2) and hence this bound. From (4.8) in Lemma 4.14 we then immediately get that . Hence, the requirement (5.6) is established after redefining C(α N k−1 ). Next, for all y ∈ Y , Σ ∈ Sl(z N k−1 ) with (5.6), we get by virtue of Lemma 4.11, [W 1,p ] * + 1 for a constant C > 0. We estimate similarly to (5.4), where we also used (5.6). Then, using further Assumption (A4), we see that Hence, we may assume that y satisfies (5.5) after redefining C(α N k−1 ) once more.
From the previous step we know that we may restrict the minimization to all y ∈ Y , Σ ∈ Sl(z N k−1 ) satisfying the bounds (5.5)-(5.7). Clearly, taking y := y N k−1 and Σ := Id Φ N k−1 ∈ Sl(z N k−1 ), the set of candidate minimizers is not empty (also recall (5.1)). We now claim that we may then take a minimizing sequence ( y n , Σ n ) ⊂ Y × Sl(z N k−1 ) for (IP) such that The first of these convergences follows by selecting a subsequence (not relabelled) using (5.5) and the weak compactness of norm-bounded sets in W 1,p g (Ω; R 3 ). For the second convergence, we observe via (5.6), (5.7) that for Σ n it holds that Moreover, a rescaling via Lemma 2.5 shows that we may additionally assume the steadiness property for constants L n ≥ 0 that are bounded by (an n-independent) constant L > 0. Crucially, this rescaling does not change the expression Then we get from Proposition 3.10 that there exists Σ * ∈ Sl(z N k−1 ) and a subsequence (not relabelled) such that Σ n * ⇀ Σ * in Sl(z N k−1 ). Next, we observe that the joint functional is lower semicontinuous with respect to the convergences in (5.8). To see this, we first note that by Lemmas 3.11, 3.12, in Ω. The first and second term in (5.10) are then lower semicontinuous by Lemma 4.17 (i) and (iv), respectively. We note that y * ∈ Y since it must have finite energy by the weak lower semicontinuity of E , whereby also ∇y * > 0 a.e. in Ω by (4.1) in Assumption (A2). Thus, we conclude that (y N k , Σ N k ) := (y * , Σ * ) is the sought minimizer of the time-incremental minimization problem (IP) at time step k. By Step 1, this (y N k , z N k , Σ N k ) satisfies (5.2).

5.2.
Discrete energy estimate and stability. The next task is to establish that our construction indeed yields a "discrete energetic solution".
Similarly, we may test (IP) at time step k with y ∈ Y and Σ where we have used Lemma 3.7 and Diss( Σ • Σ N k ) = Diss( Σ) + Diss(Σ N k ) by the additivity of the dissipation, see (4.9) in Lemma 4.14. Canceling Diss(Σ N k ) on both sides, we arrive at (5.12)

5.3.
A-priori estimates. In this section we establish a bound on α N k that is uniform in N. This is complicated by the fact that in the coercivity estimate of E at time step j, the term P N j s L s occurs with a negative sign (see Lemma 4.11). The exponent s > 1 makes P N j s L s grow superlinearly in ∑ j j=1 Var(Σ N j ), potentially causing blow-up in finite time. In order to deal with this, we first establish a nonlinear Gronwall-type lemma: Then, for all j ∈ {0, . . . , N} with j∆T < T ∞ it holds that a j ≤ A * ( j∆T ).
We remark that the maximal solution to (5.14) is a solution A * : [0, T ∞ ) → R of (5.14) with the property that for any other solution A of (5.14) it holds that A ≤ A * on the intersection of both intervals of definition. It can be shown, see, e.g., [66,Section 8.IX,p.67], that A * exists and can be maximally defined; we assume that our interval [0, T ∞ ) is already such a maximal domain of definition. Obviously, if a unique solution A to (5.14) exists on a maximal time interval [0, T ∞ ), then A * = A.
Proof. First, we remark that we may assume without loss of generality that a j−1 ≤ a j for j = 1, . . . , N. Indeed, we may set which is clearly increasing, satisfies a j ≤ b j , and where we used (5.13) and the monotonicity of h. We then use b j in place of a j . Let a be the piecewise-affine interpolant of a j , namely, Thus, by (5.13) and the fact that a(t) is increasing and h is monotone, By a classical comparison principle for ODEs, see [66,Theorem 8.X and following remarks, p.68], with A * (t) given as the maximal solution to (5.14). This directly implies the conclusion of the lemma.
For the reader's convenience we give a short direct proof of (5.15). First, we claim: with the following two properties: To see this claim, let t 0 ∈ [0, T ′ ] be the first point such that u(t 0 ) = v(t 0 ). By (i), t 0 > 0. For t < t 0 it holds that u(t) < v(t) and then Taking the lower limit as t ↑ t 0 , we obtain D − u(t 0 ) ≥ D − v(t 0 ), which contradicts (ii). This shows the claim.
For 0 < ε ≤ 1 let A ε be a maximally extended solution to We have A ε ′ < A ε for all 0 < ε ′ < ε ≤ 1 by our claim. In particular, A ε (t) is monotonically decreasing as ε ↓ 0 and thus A ε ↓ A * locally uniformly (by equi-continuity) with A * the maximal solution to (5.14). In fact, a similar argument can be used to construct A * .
Fix T ′ < T ∞ . For any ε > 0 we observe a(0) < A ε (0) and Hence, we may apply the claim again (note that in (ii) we only need to check Since the right-hand side converges to A * (t) as ε ↓ 0, we obtain a(t) ≤ A * (t) for t ∈ [0, T ′ ] and then also for t ∈ [0, T ∞ ). This is (5.15).
We can now state a uniform energy bound up to any time before the blow-up point T * (> 0), or up to T if there is no blowup.
the a-priori estimates hold. Moreover, By the same argument as the one at the beginning of the proof of Lemma 5.4 The parts of (5.17) relating to y N k W 1,p , ∑ k j=1 Var(Σ N j ) and Σ N k L ∞ follow from the coercivity of E and Diss in the same way as we proved (5.5)-(5.7) (in particular, using the coercivity estimates of Lemmas 4.11,4.14). From Lemma 3.4 we further get where the constant C from (3.9) and then also the (redefined) constant C(β N k ) depend on P 0 W 1,q and ∑ k j=1 Lemma 4.14). For the bound on M(Φ N k ), we can use again Lemma 4.11, but this time using the coercivity originating from the core energy.
Our next task is to show that for T * defined in (5.18) it holds that T * > 0, for which we apply the preceding Lemma 5.4 with h(s) := Ce s , which is continuous and increasing, and initial value β N 0 = α N 0 to (5.19). The maximal solution is easily seen to be A * (t) = − log(e −α N 0 −Ct), which is defined on the maximal interval [0, T ∞ ) with T ∞ = e −α N 0 /C > 0. Thus, as A * is increasing, for all 0 < τ < T * it holds that , . . . , N}; N ∈ N). Consequently, T * ≥ T ∞ > 0.

PROOF OF THE EXISTENCE THEOREM
At this stage we have an N-uniform bound on ∑ k j=1 Var(Σ N j ) for any t N k ≤ τ < T * . However, when letting N → ∞, this BV-type bound is too weak to prevent the formation of jumps in the dislocation trajectory. Jumps are undesirable because we need the "time index" provided by a Lipschitz trajectory to define the path of the plastic distortion as in Section 3.4. Hence, we now rescale the time to make the discrete evolution uniformly Lipschitz continuous (and move the blow-up time to +∞). Then we will be able to pass to the limit and complete the proof of Theorem 4.9.
6.1. Rescaling of time. Let N ∈ N and set, for k = 0, . . . , N, , as in Proposition 5.2. Then define the increasing sequence The Clearly, ψ N is strictly increasing on [0, σ N ] and We also recall from (4.5) that where (y, P, Φ) = (y, z) ∈ Y × Z . In the new time the time-incremental minimization problem (IP) reads as follows: For k = 0, . . . , N we have in Proposition 5.2 constructed solutions Moreover, we may also assume that Σ N k is steady in the sense that This can be achieved via Lemma 2.5 (see Step 3 in the proof of Proposition 5.2 why this rescaling is allowed).
We will now define a suitable interpolant for the discrete solution.
Proof. The assertions in (6.5) (6. We now show (6.7). Let σ > 0. By (6.1), if s N k ≤ σ (k ∈ {0, . . . , N}, N ∈ N), or equivalently, t N k = ψ N (s N k ) ≤ ψ N (σ ), then the quantity β N k remains bounded by σ + E (0, y 0 , z 0 ) + 1. Hence, by the definition of T * in (5.18), we have T * ≥ lim sup N→∞ ψ N (σ ), and then also On the other hand, if T ′ < T * , then there is λ < ∞ with β N k ≤ λ for all k ∈ {0, . . . , N} such that t N k ≤ T ′ and N ∈ N sufficiently large. From (5.17) and Lemma 4.14 we get that where we have considered the "constant" C as an increasing function. Thus, for the times s N k = [ψ N ] −1 (t N k ) corresponding to the t N k it holds via (6.1) that and we see that s N k remains bounded by σ ′ for those k.
Together with (6.8), this completes the proof of (6.7). We can easily make S → C(S) increasing and then pass to the upper semicontinuous envelope.
Proof. This is a direct translation of Proposition 5.3, noting that we use a change of variables for the external power integral in (i).
6.2. Passage to the limit. We first establish that a limit process exists as N → ∞. Then we will show that this limit process has the required properties. To estimate the variation of P N , take any partition 0 = σ 0 < σ 1 < · · · < σ K = S of the interval [0, S] and apply Lemma 3.4 to the definition (6.4) to see where the constant C depends on P 0 W  To show ψ(∞) = lim s→∞ ψ(s) = T * , let ε > 0. From (6.7) we may find σ > 0 such that for Then we get ψ(s) ≥ T * − ε and ψ(s) ≤ T * + ε. Letting ε → 0, we conclude that ψ(∞) = T * .
Next, we observe that (s, P) → M(s, P) is continuous in the following sense: If s j → s in [0, S] and P j ⇀ P in W 1,q , then for any sequence y j ∈ M(s j , P j ) with y j ⇀ y in W 1,p it holds that y ∈ M(s, P). To see this, it suffices to combine (i), (ii), and (iii) of Lemma 4.17, which together imply that limits of minimizers are minimizers themselves. One can either argue directly or realize that these two statements together imply the Γ-convergence [23] of W e ( , P j ) − f ψ (s j ), , from which the claimed continuity property follows. Note that here we also use the monotonicity and upper semicontinuity of the constant C(s) > 0 from Proposition 6.1 with respect to s. Similarly, we also obtain that M(s, P) is weakly closed, hence weakly compact.
where the error term o(1) vanishes as N → ∞ since ψ N (s) → ψ(s) and the integrand is uniformly bounded by Assumption (A4) and the definition of Y S . We can now apply Fatou's lemma and, in turn, (6.14), (6.15), (6.13) to estimate lim inf This establishes the lower limit inequality in (6.13).
We now prove the stability and energy balance for the limit solution.
Proof. In Proposition 6.2 (ii) we established the time-incremental stability at time step k, namely for all y ∈ Y and Σ ∈ Sl(z N k ) with Σ L ∞ ≤ γ. Using that Σ N (s) * ⇀ Σ(s) in DS(Ω), P(s) ⇀ P(s) in W 1,q , and also the uniform Lipschitz continuity of Σ (with respect to a metric for the weak* convergence, e.g., the flat norm) and P (with respect to W 1,q ), we obtain For y, Σ as in the statement of the proposition we define the following "recovery sequence" for Σ: Σ N s := Σ • Σ N s ∈ Sl(z N k(N) ). We have Σ N s L ∞ = max Σ L ∞ , Σ N s L ∞ ≤ γ for N > N(s) sufficiently large (depending on s, but this will not matter in the following). We also observe from (4.9) in Lemma 4.14 that Diss( Σ N s ) = Diss( Σ N s ) + Diss( Σ) and from Lemma 3.7 that The slip trajectory Σ N s is thus admissible in (6.17) at k = k(N) for N sufficiently large, giving , Σ ≫ Σ(s)) + Diss( Σ N s ) + Diss( Σ). Passing to a (further) subsequence in N (for fixed s, not relabelled) to obtain y N k(N) ⇀ y in W 1,p , we may use the assertions (i), (ii) of Lemma 4.17 as well as the locally uniform convergence ψ N → ψ, to pass to the lower limit N → ∞ in (6.18) at k = k(N), obtaining E ψ (s, y, z(s)) ≤ E ψ (s, y, Σ ≫ z(s)) + Diss( Σ).
Finally observing that E ψ (s, y(s), z(s)) ≤ E ψ (s, y, z(s)) by (6.12), the conclusion (6.16) follows. Remark 6.5. As remarked in the Introduction and explained further in Section 6.2 of [34], the projection in the definition of the total plastic drift in (3.5) has the effect of disregarding climb. The reason why we cannot simply enforce that ⋆γ b is orthogonal to P −1 b for admissible slip trajectories is that this makes it impossible to deform some dislocations into each other via Proposition 4.18. Indeed, such a deformation may require a slip trajectory violating the orthogonality constraint, if only on a trajectory with vanishing variation. In this case the recovery construction in the preceding proposition would fail. Proof. From Proposition 6.2 (i) we have for all k ∈ {1, . . . , N} the discrete lower energy estimate Fix a point s ∈ [0, ∞) and define for N ∈ N the index k(N) to be the largest k ∈ {0, . . . , N} such that s N k ≤ s. Then, by Lemma 6.3 and Lemma 4.17 (i) as well as (6.12), we obtain (by arguments as in the preceding proof of Proposition 6.4) E ψ (s, y(s), z(s)) ≤ lim inf N→∞ E ψ N (s N k(N) , y N k(N) , z N k(N) ).
Here, we note that while the conditionψ(σ ℓ ) > 0 may force gaps in the partition, on these gaps the integrand vanishes and so the above statement is not affected.
Next, we show the plastic flow equation. Proof. The ODE holds for P N , see (6.4). Using the convergence assertions from Lemma 6.3, we can then pass to the limit using (the same technique as in the proof of) Lemma 3.12.
Finally, we record the following regularity estimate: The other estimates follow directly from Proposition 6.1 in conjunction with the assertions of Lemma 6.3. 6.3. Proof of Theorem 4.9. Finally, we dispense with the restriction that Σ L ∞ ≤ γ for the test trajectory Σ in the stability condition (S). From now on we make the dependence on γ explicit and write y γ , P γ , Σ γ , ψ γ for y, P, Σ, ψ.
In particular, we have T * > 0 since the arguments before give a γ-independent lower bound on T * (see Lemma 5.5).
The stability (S), the energy balance (E), and the plastic flow equation (P) follow from the construction and Propositions 6.4, 6.6, 6.7 using the same techniques as in the previous section. We omit the repetitive details. Let us however observe that every Σ ∈ Sl(Σ(s)) (which includes the assumption Σ L ∞ < ∞) becomes admissible for γ sufficiently large. In this way, all parts of Definition 4.5 follow. The initial conditions are satisfied by construction. The proof of Theorem 4.9 is thus complete.