Discontinuity Layout Optimization Research Articles

Application of discontinuity layout optimization to plane plasticity problems

A new and potentially widely applicable numerical analysis procedure for continuum mechanics problems is described. The procedure is used here to determine the critical layout of discontinuities and associated upper-bound limit load for plane plasticity problems. Potential discontinuities, which interlink nodes laid out over the body under consideration, are permitted to crossover one another giving a much wider search space than when such discontinuities are located only at the edges of finite elements of fixed topology. Highly efficient linear programming solvers can be employed when certain popular failure criteria are specified (e.g. Tresca or Mohr–Coulomb in plane strain). Stress/velocity singularities are automatically identified and visual interpretation of the output is straightforward. The procedure, coined ‘discontinuity layout optimization’ (DLO), is related to that used to identify the optimum layout of bars in trusses, with discontinuities (e.g. slip-lines) in a translational failure mechanism corresponding to bars in an optimum truss. Hence, a recently developed adaptive nodal connection strategy developed for truss layout optimization problems can advantageously be applied here. The procedure is used to identify critical translational failure mechanisms for selected metal forming and soil mechanics problems. Close agreement with the exact analytical solutions is obtained.

The determination of the value of the collapse load for statically indeterminate systems undergoing plastic deformation

The discontinuous slip-line form of upper bound plasticity analysis is considered using an equilibrium of forces approach. It is demonstrated that the underlying basis of the approach can be written in terms of stress functions that provide a continuum stress state interpretation of the upper bound solution. An alternative proof of the upper bound theorem, applicable to both associative and non-associative materials, using stress functions is presented. The broader nature of the equilibrium form and the strict conditions under which it is valid are discussed, including examination of the apparent omission of moment equilibrium and associativity in many equilibrium form solutions. Finally, the relationship of the stress function formulation to the output of the computational limit analysis method discontinuity layout optimisation (DLO) and the potential to use the stress function formulation to derive a form of lower bound solution from an upper bound analysis are demonstrated.