Small strain crystal plasticity based on the primal‐dual interior point method

The paper presents a novel approach for rate‐independent single crystal plasticity based on the Infeasible Primal‐Dual Interior Point Method in a small strain framework. Therein, the principle of maximum dissipation together with the yield functions on the slip systems in the crystal are considered as the constrained optimization problem. The constraint conditions adapted using slack variables and a Lagrangian is formulated using barrier functions. The equations are linearized and solved using Newton's method. Numerical examples are presented for slip on the individual slip systems in a single crystal under global rotation.


Introduction
Single crystal plasticity, which plays a major role in the analysis of material anisotropy and texture evolution, treats each crystalline grain, having a distinct orientation, individually. The polycrystalline material response is obtained upon considering a structure consisting of various individual grains, often also considering interface effects at the grain boundaries. On the individual grain level, single crystal plasticity can be treated in the mathematical framework of multisurface plasticity, leading to a constrained optimization problem wherein multiple constraints are defined as yield criteria on the different slip systems. Different approaches have been established in this field, see, e.g., [2], [3]. In rate-independent models, the set of active slip systems in the grain is possibly nonunique and is identified in, e.g., an active set search. Rate dependent approaches are based on power-type creep laws which do not differentiate into active or inactive slip systems. However, the constitutive equations of these formulations are often very stiff and require a small time increment.
Here, a new algorithm for the solution of the constrained optimization problem based on the primal dual interior point method (PDIPM), [1], involving slack variables is presented for the framework of small strain single crystal plasticity. The use of slack variables therein stabilizes the conventional method and allows for a temporary violation of the constraint during the optimization. The optimization is solved using a Lagrange functional, wherein the nonlinear system of equations resulting from the derivation of the Lagrange functional is linearized using taylor expansion and solved by a Newton Raphson scheme. All slip systems are considered simultaneously, omitting an iterative active set search. PDIPM has been found to lead to very efficient algorithms and better convergence rates than barrier or penalty methods. The stability of the algorithm would be especially beneficial in complex material models, such as a multiscale description of polycrystalline materials. Numerical examples for a face-centered cubic single crystal are presented.

Theoretical framework
A classical constitutive framework for single crystal plasticity considering small strains the basis of the formulation. With the classical assumption of the linear strain tensor ε = sym[∇u] as the symmetric part of the displacement gradient and further its additive decomposition ε = ε e + ε p into an elastic part and a plastic part, we assume an associative framework with a decoupled free energy ψ(ε e , γ α ) = ψ e (ε e ) + ψ p (γ α ) ∀ α and an associated structure of crystal plasticity for the modelling of hardening phenomena, g α = ∂ψ p /∂γ α . The reduced dissipation inequality is then given by with the classical Kuhn-Tucker conditionsγ α ≥ 0, Φ α ≤ 0, Φ αγα = 0 where Φ α (σ, g α ) = τ α − g α for α = 1, 2, ..., m.
The nonlinear constrained optimization problem is then defined by max D red subject to Φ α ≤ 0,γ α ≥ 0 ∀ α = 1, 2, ..., m and treated in an Infeasible Primal-Dual Interior Point Method, leading to the Lagrangian 2 of 2 Section 6: Material modelling in solid mechanics with the logarithmic barrier functions, Lagrange multipliers λ α and the slack variables s α . The stationary point is found from the derivation with respect to the plastic slip variables γ α , the lagrange multipliers λ α and the slack variables s α . The Taylor series expansion of the equations is linearized and solved using Newton's method. Classically, a line-search algorithm is applied in the numerical algorithm.

Single crystal under simple shear
Based on the presented algorithm, a face-centered cubic (fcc) single crystal structure is simulated under global rotation using Euler angles. Fig.1a shows the octahedral arrangement of the slip systems in the fcc crystal and the table in Fig.1b  The results are compared to a single crystal analysis in the literature, cf. [4]. The accumulated slip A shown in Fig. 2a agrees well. Fig.2b shows that the slip is not activated at the same time on all slip systems and does not evolve uniformly.