A global extremum principle for the analysis of solids composed of softening material

An extremum principle is presented covering problems in solid mechanics equilibrium analysis for piecewise linear softening materials. Problems formulated according to this principle are expressed in a mixed "stress and deformation" form. The mechanics interpretation is limited according to linear deformation kinematics. More specialized models, such as an exfremumprinciple in mixed form for linearly elastic materials, an equivalent to the minimum complementary energy principIe, and a statement of a bound theorem of Limit Analysis are identified as special cases within the general formation. Numerical results are presented for two examples of one-dimensional structures made of inhomogeneous, softening material. The evolution of material degradation is demonstrated via a set of solutions obtained for increasing load. Each solution of the set is produced from a single application of a general purpose computer program for constrained nonlinear programming problems, operating on a finite element interpretation of the nonlinear continuum. I. INTRODUCTION The purpose of this paper is to present a global formulation of equilibrium problems in solid mechanics for systems made of softening material. The formulation is expressed in a way to accommodate arbitrarily inhomogeneous, anisotropic materials. Local "state" is represented in terms of a mixed "stress and defo~ation" measure. Also, material properties enter the formulation partly in implicit form, through a combination of local and global constraints. By virtue of these several features, a substantial flexibility in the representation of constitutive properties is available in the model (how this is accomplished explicitly becomes clear with the presentation of the problem statement given below). The formulation itself comprises an extremum principle, where a load factor is to be maximized w.r.t. the set of stress and deformation fields. As a result, the sometimes most challenging issues related to existence and uniqueness are in the present circumstances moot, i.e. established results from nonlinear optimization analysis of convex problems fully support a math- ematical inte~retation of the fo~ulation. Also, making the association (in the usual way) between convexity of the global problem statement and mechanical stability, a category of problems can be identified for which, independent of the specific form of material softening or degradation, the extremum character and thus the stability is preserved indefinitely with increasing load. The model presented here provides a convenience for the treatment of mechanics problems that require for their interpretation the representation of a set of separate fields, where each field is associated with a separate interval of the region of the solid. Of course problems in the analysis of composites, where the separate fields are identified nominally with the physical structure of constituent materials, belong to this category. Problems involving material degradation, for example where an o~ginally homogeneous medium is rendered through the evolution of structural response into a set of regions each with distinctive constitutive character (e.g. elasto-plasticity), have this property as well. Also, availability of the extremum problem formulation described below is advantageous with respect to the treatment of many of the issues arising routinely in analysis and in the construction of models for computational purposes. Considerations in the treatment of convergence and error analysis, the construction of algorithms, modelling of mechanisms of material degradation, and so on, identify examples of the cited "routinely arising issues". In addition, the present formulation makes it possible to obtain computational solutions