| It is shown that the Gurson model (Gurson, 1975 and 1977), a widely-used constitutive equation for both ductile and quasi-brittle materials, allows for a change in character of the differential equations fairly early in many loading paths. Since this result has rather broad implications, considerable effort is expended in documenting the analysis, and in illustrating just where ellipticity (or hyperbolicity for dynamic problems) is lost for many traditional stress and strain paths. The analysis is a classical analysis in which the conditions for the existence of a discontinuous bifurcation (DB) in strain are identified.;It is proposed that just prior to the change of equation character, a discrete constitutive equation be introduced to describe the development of a crack. Such a constitutive equation relates traction to displacement discontinuity and is also known as a decohesive model, or a cohesive crack model. Part of the unique contributions of this segment of the work involves relating parameters of the discrete model to the parameters that evolve from the ellipticity analysis of the continuum model. The procedure is illustrated through an implementation in a conventional finite element code and solutions to small model problems. |
