Minimal kinematic boundary conditions for simulations of disordered microstructures

Abstract

In simulations of representative volume elements (RVEs) of materials with disordered microstructures, commonly used rigid and periodic boundary conditions (BCs) introduce additional constraints, causing: (i) boundary effects, (ii) unrealistic stiff response, (iii) artificial wavelengths in the solution fields, and (iv) suppression of solutions with localized deformation that otherwise may occur in the simulation. In this paper we define the minimal kinematic boundary conditions such that only the desired overall strain is imposed on the RVE, with no other undesirable constraints. We prove that such BCs result in a unique solution for the linear elastic case, and that the uniqueness for nonlinear problems is dependent on the pointwise positive definiteness of the incremental stiffness tensor. Upon incorporating the minimal BCs into the finite element framework, we consider, as an example, two-dimensional, linear elastic, disordered polycrystals and perform a systematic study of the effects of boundary conditions while varying the RVE size and controlling the sampling error. The results demonstrate that the minimal BCs, applicable to a RVE of any shape, are superior to other BCs, in that they give more realistic overall behaviour, reduce the required size of the RVE, and eliminate the superficial wavelengths in the solution field, ubiquitous in simulations with other boundary conditions.

Notes

Note that we postulate only the existence of the overall stress tensor for the RVE, not its relation to the microscopic stress field. A more detailed discussion on the overall stress is given by Bishop and Hill (1951) and Hill (1952).

The functions u _i (x) and w _i (x) are further required to be in the Sobolev functional space H ¹(V), i.e. the functions and their first partial derivatives must be square-integrable on V.

Alternatively, one can formulate the variational problem by minimizing the functional

, where λ _ij  = λ _ji are Lagrange multipliers. The uniqueness then follows from the principle of minimum potential energy.

Hill (1958, 1962) defined the rate potentials and extremal principles analogous to the minimum potential energy principle in elasticity. As in the linear case, an alternative proof of uniqueness can be constructed using those principles.