An optimization structure for frictional plasticity

Abstract

A theoretical framework of rigid plasticity is presented that is based on optimization and includes frictional dissipation. It has been used in this paper as a foundation for existing and modified models of granular materials consisting of rigid granules. It has the major advantage that it enables yield criteria to be created numerically, which is particularly useful when analytical expressions cannot be found. This framework is constructed by first postulating: (i) a dissipation function that can depend on the current components of stress, but is always homogeneous of degree one and positive definite, (ii) a volume constraint function that is also homogeneous of degree one, and (iii) a balance of the rate of doing work and the rate at which energy needs to be dissipated. A mathematical process similar to the construction of a dual norm in convex analysis then leads to: a flow rule; a single natural representation of the yield surface; and a useful constitutive inequality involving the components of stress and strain rate. Dissipation functions appropriate for the Drucker–Prager and Matsuoka–Nakai yield surfaces are investigated when a simple dilation rule is the volume constraint. These include a case where an explicit expression for the yield function is not found and, instead, the yield function is found numerically. Such numerical yield functions have been checked graphically against carefully constructed envelopes and found to be consistent with them.

Including friction in the mathematics of classical plasticity

“…As shown in [4,8,14] and [17], the yield surface can be considered as an envelope in stress space. In order to find the envelope, consider the power balance for a rigid plastic material in the form of the rate of doing work minus the rate of energy dissipation.…”

“…The problem of finding the envelope can be formulated as an optimization problem with a constraint [4]. The power balance, at yield, can again be written…”

“…[3]). But at the same time, there has been a continued interest [4,5] in providing a wider mathematical framework for classical plasticity that manages to include granular materials, together with the well-established metal plasticity. This work has been motivated by the expectation that it would help the modeller include physical insights more easily and so minimize the number of adjustable parameters needed in constructing a model.…”

“…4. The graphical method has been reformulated as a constrained optimization problem [4]. This allows the yield function to be generated numerically.…”

Simulating pressure-induced compaction by grain rearrangement

“…This process, described in detail elsewhere (Chandler, 1985;Chandler & Sands, 2007a, 2010, starts with a kinematic constraint and dissipation function and produces appropriate flow rules and yield functions. This article focuses on a mechanism, called 'self-cancelling shear strains', whereby compaction of loose granular assemblies can be produced by hydrostatic pressure alone.…”