Energetic Approach to Large Strain Gradient Crystal Plasticity

We formulate a problem of the evolution of elasto-plastic materials subjected to external loads in the framework of large deformations and multiplicative plasticity. We focus on a spontaneous inhomogenization interpreted as a structuralization process. Our model includes gradients of the plastic strain and of hardening variables which provide a relevant length scale of the model. A simple computational experiment interpreted as a hint of a deformation substructure formation is included.


Introduction
The elastic-plastic behavior of crystalline materials poses a challenge for mathematical analysis on the microscopic, the mesoscopic, and the macroscopic scales.
Here, we study a rate-independent model arising in the crystal plasticity.A common and successful approach to the analysis of crystalline materials is by means of energy minimization; see e.g.Ortiz & Repetto [30].This is manifested for various elastic crystals, even for those with the potential of undergoing phase transitions.The applicability of variational methods has been broadened to include rate-independent evolution.Typically, these models are characterized by energy minimization of a functional including macroscopic quantities such as the macroscopic deformation gradient as well as a dissipation functional.In order to introduce a physically relevant scale to our problem we assume, following earlier works of Bažant & Jirásek [3], Dillon & Kratochvíl [9], Gurtin and Gurtin & Anand [15,16], Mainik & Mielke [22] and others, that our energy functional depends also on the gradient of the plastic tensor.The gradient term models nonlocal effects caused by short-range interactions among dislocations.It is not clear, however, which function of the gradient should be used.For further details we refer to Kratochvíl & Sedláček [18], and to Bakó & Groma [1] for attempts to derive it from statistical physics revealing thus complexity of the problem, for more details.A related approach to non-local models in damage and plasticity was undertaken in Bažant & Jirásek [3], see also [8,10,12,23].
In this paper, we formulate the so-called energetic solution due to Mielke et al. [28] to our problem.This concept of a solution is based on two requirements.First, as a consequence of the conservation law for linear momentum, all work put into the system by external forces or boundary conditions is spent on increasing the stored energy or it is dissipated.Secondly, the formulation must satisfy the second law of thermodynamics, which in the present mechanical framework has the form of a dissipation inequality.The last requirement enters the framework as the assumption of the existence of a nonnegative convex potential of dissipative forces.As a consequence, the imposed deformation evolves in such a way that the sum of the stored and dissipated energies is always minimized.The main advantage of this approach is that it allows us to exploit the theory of the modern calculus of variations and suggests a numerical approach to this problem.
To expose the essence of the mathematical structure of the energetic approach, we first analyze a proto-model called here a material with internal variables.It freely follows the exposition of Francfort & Mielke [11] and we recall it here to motivate the notion of the energetic solution.In the second step, the framework is applied to elasto-plastic materials by specifications of some internal variables.One of the main results is that the described energetic approach can be identified with crystal plasticity with strain gradients in the version formulated by Gurtin [15].Gurtin's model is formulated in the mathematical language of differential equations.From the point of view of numerical solution of a boundary value problem of crystal plasticity the energetic formulation is more convenient.
Our results are closely related to [22], where the authors proved the existence of energetic solutions to strain gradient plasticity with polyconvex energy density allowing even for +∞; see [2] and to Giacomini & Lussardi [13] where the linear-elastoplasticity framework is considered.Here we allow for finite quasiconvex stored energy density and large deformations.This is motivated by relaxation theory in the calculus of variation, where the effective macroscopic energy density is quasiconvexification of the microscopic energy density.Thus, our results may be applied to plasticity of materials with developing microstructures, as in shape memory alloys, [4], or [26] for instance.We refer an interested reader to [19] for a model describing cyclic plasticity in these materials.Another related paper is Carstensen et al. [6], where the authors use the energetic approach to plasticity without strain gradients.
In what follows, Ω ⊂ R n , is an open bounded domain and L β (Ω; R n ), 1 ≤ β < +∞ denotes the usual Lebesgue space of mappings Ω → R n whose modulus is integrable with the power β, and L ∞ (Ω; R n ) is the space of measurable and essentially bounded mappings Ω → R n .Further, W 1,β (Ω; R n ) standardly represents the space of mappings which live in L β (Ω; R n ) and their gradients belong to The subdifferential of f will be denoted ∂ sub f and its elements will be called subgradients of f at x 0 .

Materials with internal variables
Consider a material whose elastic properties depend on internal variables z ∈ Z ⊂ R m .The stored energy density is then W = W(F e , z), where F e ∈ R n×n is the elastic strain.We are interested in the rateindependent evolution of the material.To this end, we assume the existence of a nonnegative convex potential δ = δ( ż) of dissipative forces, where ż denotes the time derivative of z.In order to ensure rateindependence, δ must be positively one-homogeneous, i.e., δ(α ż) = αδ( ż) for all α > 0. Finally, we define for z ∈ Z a thermodynamic force The evolution rule is introduced in the form where Maximal monotonicity of the subdifferential implies that for all θ ∈ ∂ sub δ(ξ) we have Remark 2.1.In particular, taking ξ = 0 and realizing that the one-homogeneity of δ yields δ( ż) = ω, ż for all ω ∈ ∂ sub δ( ż) we get for all θ ∈ ∂ sub δ(0).Inequality (4) expresses the socalled maximum dissipation principle (see e.g.Hill [17] or Simo [31]), which says that thermodynamic forces "available" in the so-called elastic domain ∂ sub δ(0) are not strong enough to overcome frictional forces.
In what follows, Ω ⊂ R n , is a bounded Lipschitz domain representing the so-called reference configuration, ν is the outer unit normal to ∂Ω, and ∂Ω ⊃ Γ 0 , Γ 1 which are disjoint.The deformation will be denoted y : Ω → R n .The evolution of the system will be controlled by external forces.Let f (t) : Ω → R n be the (volume) density of the external body forces and g(t) : Γ 1 ⊂ ∂Ω → R n be the (surface) density of the surface forces.The equilibrium equations governing the mechanical behavior of the system are: in Ω, ( 5) The full system characterizing the proto-model consists of ( 5)- (7) supplemented by (2): where z 0 ∈ Z is an initial condition for the internal variable.
In order to regularize our problem we may add the gradient of the internal variable, i.e., for ω ≥ 1 and ε > 0 put The evolution rule changes to so we have the thermodynamic force The potential energy of our system can be written ( := ε/ω) (11) where the work done by external forces is Hence, the integration with respect to time gives We can also consider a more general form of δ which can also depend on (x, z), i.e. δ := δ(x, z, ż).
Typically, however, we do not have enough smoothness in the internal variable to compute the time derivative on the right-hand side of (13).
Following Mielke [24], we define the dissipation distance between two values of internal variables z 0 , z 1 ∈ Z as where z ∈ C 1 [0, 1]; Z , and set where z 1 , z 2 ∈ Z := {z : Ω → R M ; z(x) ∈ Z a.e. in Ω}.We assume that Z is equipped with strong and weak topologies which define the notions of convergence used below.Following [11,22], we impose the following assumptions on D: (i) Weak lower semicontinuity: whenever z k z and zk z.
(ii) Positivity: If {z k } ⊂ Z is bounded and at the same time min{D(z k , z), D(z, z k )} → 0 then

Energetic solution
Suppose that we look for the time evolution of y(t) ∈ Y ⊂ {y : Ω → R n } and z(t) ∈ Z during the time interval [0, T ].The following two properties are the key ingredients of the so-called energetic solution due to Mielke and Theil [27,28].
(ii) Energy balance: ∀t, where Definition 2.2.The mapping t → (y(t), z(t)) ∈ Y × Z is an energetic solution to the problem (I, δ, L) if the stability inequality and the energy balance are satisfied.
Remark 2.3.Note that the stability inequality (i) can be written in the form ∀t ∈ which means that y(t), z(t) always minimizes (ỹ, z) → I(t, ỹ, z)+D(z(t), z).This means that the equilibrium configurations are characterized by energetic minima.Contrary to elasticity theory, the minimized energy is not only the overall elastic energy described by I but the dissipated energy is added.
It is convenient to put Q := Y × Z and to set q := (y, z).Moreover, we define the set of stable states at time t as and Moreover, a sequence {(t k , q k )} k∈N is called stable if q k ∈ S(t k ).

Applications to elasto-plasticity
Now we apply the energetic approach to an elastoplastic problem.

Problem statement
In what follows y : Ω → R n will be a deformation of a body Ω ⊂ R n (in a fixed reference configuration) with the deformation gradient F = ∇y.In particular, y covers both elastic and plastic deformation.We define the multiplicative split, F = F e F p , into an elastic part F e and an irreversible plastic part F p , which belongs to SL(n) := {A ∈ R n×n ; det A = 1}.The socalled plastic strain F p and the vector p ∈ R m of the hardening variables are internal variables influencing elasticity.In other words, z(x) = (F p (x), p(x)) ∈ SL(n) × R m for almost all x ∈ Ω.The energy functional I takes the form I t, y(t), z(t) (22) with L given by (12).In order to ease the notation, we omit the dependence of W on x.However, all the theory developed in this paper may include nonhomogeneous W.
In what follows, we suppose that where Γ 0 ⊂ ∂Ω with a positive surface measure.Moreover, we suppose that Γ 0 ∩ Γ 1 = ∅.Further As q = (y, z) it will be advantageous and will cause no confusion to write D as dependent on q, i.e., D(q 1 , q 2 ) := D(z 1 , z 2 ) if q 1 = (y 1 , z 1 ) and q 2 = (y 2 , z 2 ).Similarly, we may write I in terms of q = (y, z) as where, obviously, L t, q(t) := L t, y(t) .In this situation, Q = (Q 1 , Q 2 ) are conjugate plastic stress and conjugate hardening forces, respectively.
The elastic domain is defined as Remark 3.1.The principle of maximal dissipation asserts that is maximal if Ḟp and ṗ are kept fixed and (Q 1 , Q 2 ) ∈ Q(x, z).This means that for all (A, B) ∈ Q(x, z).
Finally, we conclude with an example covered by our approach.
Example 3.2 (Simple shear carried by a single slip).Consider a single slip system defined by two orthonormal vectors a, b ∈ R 3 such that a is the glide direction and b is the slip-plane normal.Further suppose that we have a particular case of a so-called separable material where: where F p (t) = I + γ(t)a ⊗ b where γ is the plastic slip.The slip system is generally not fixed in the reference configuration.The slip-plane normal b in the reference configuration has the form b = (F p ) b.However, in this special case we have that b = b, so that the slip-plane normal is kept constant during the process [15].Due to the special case of F p we may identify z := (γ, p) because F p depends only on γ.