Numerical Solution of a Non-smooth Variational Problem Arising in Stress Analysis : the Scalar Case

The numerical solution of a non-smooth minimization problem arising in the stress analysis of a plastic body is investigated. The problem of interest is the computation of the maximal stress of an elastic three-dimensional body under external forces [3, 18]. The computation of that threshold allows one to determine if the material can support the forces applied to it or if the traction on the material or its boundary is too large. Consider a homogeneous isotropic elastic body Λ ⊂ R. The load capacity ratio, as defined in [16], is the maximal positive number C such that the body will not collapse under any external traction field bounded by CY0, where Y0 is the elastic limit. It is independent of the distribution of external loading. If we neglect the body forces and consider only forces on the boundary, it implies that no collapse will occur for any field g on ∂Λ as long as ess supy∈∂Λ |g(y)| < CY0. The load capacity ratio is a quantity that depends on the geometry of the domain Λ. The load capacity ratio is a quantity that also appears in the limit analysis problem in Temam (1985) [18], and which represents, from the mechanical standpoint, a criterion to determine if the material can support the imposition of both body and surface external forces. Load capacity ratio and stress concentration factors are used by engineers to compare the maximal stress for a given body with the stress computed analytically for simplified geometries [13, 15]. Such problems also lead to the computation of optimal stresses as in [17, 18].