Applications of a generalized complementary energy principle for the equilibrium analysis of softening structures

An interpretation for computational solution is given for a new global extremum principle that models a generalized form of elastic/softening structural behavior. The principle and its interpretation are expressed in a form that accommodates arbitrary heterogeneity and anisotropy in the structural material. Also the stress-strain properties that reflect evolution of local softening are represented in the model by a set of parameters defined over the field of the structure. Thus. the model may be used to predict the general behavior of solid structures having non-uniform stress/strain fields that evolve with change in external load. A discretized version of the principle used for computation is based on a consistent. mixed-form finite element interpretation of the principle as stated for the general softening continuum. Example computational solutions are provided covering the evolution of softening for a uniformly loaded homogeneous sheet with a hole, and simulations of a sheet with various configurations of softer or stiffer inclusions in an otherwise uniform structure. The purpose of this paper is to demonstrate the application of a recently developed extremum principle covering equilibrium problems for structures made of softening materials (as described for a one-dimensional stress state, a material for which ‘change in stress per unit change in strain’ diminishes as the magnitude of the load increases is referred to here as a softening material) [ 11. The formulation makes use of a superposition of an arbitrary number of fields, here termed ‘constituent fields,’ to represent total stress, and this feature provides for a constructive approach to the simulation of stress-strain properties associated with any softening material. This representation of the total stress tensor is in contrast to a number of other approaches to the analysis of non-linear material response (e.g. [2-41). With this construction, the non-linear problem is established in a form that in effect affords all of the advantages for analysis generally associated with having the continuum problem interpreted in terms of potentials. For convenience, we first present a brief description of the principle stated for the non-linear continuum. An interpretation of the continuum model into discretized form is presented next. The discrete-form expression for the extremum problem statement is obtained using a self-consistent, mixed-form finite element construction. This leads to a convex constrained nonlinear programming problem. Computational solutions are obtained conveniently using standard software written for such problems. Numerical results are presented showing, for several example problems involving twodimensional softening continua, the evolution of softening and the corresponding stress and/or deformation fields under increasing load. These results are demonstrated for a rectangular sheet of bilinear softening material that has variously located sets of stiffer or less-stiff inclusions, and also for a rectangular sheet with a central hole, solved for each of two choices of softening materials.