Comment on Hyperbolicity of Wave Problem for Valanis’ Global Damage Theory
In a very interesting and stimulating recent paper, Valanis (199l) proposed a new type of nonlocal damage theory, called the global damage theory. Some conclusions of this paper, however, require further analysis, which is the purpose of this technical note. Offering as motivation some questionable and unfounded critical comments on the nonlocal damage theory originally proposed by Pijaudier-Cabot and Baiant (1987) and Baiant and Pijaudier-Cabot ( 1988), Valanis formulated a different theory and cltrimed it to be superior because, as he stated, it exhibits unconditional hyperbolicity of the wave problem. On closer scrutiny, though, Valanis' claim is found to be invalid, and for further development of nonlocal damage models it is important to understand why.
Nonlocal damage theory
In the usual local finite element analysis, strain softening causes spurious mesh sensitivity and incorrect convergence when the element is refined to vanishing size. In a previous continuum formulation, these incorrect features were overcome by the imbricate nonlocal continuum, which, however, introduced some unnecessary computational complications due to the fact that all response was treated as nonlocal. The key idea of the present nonlocal damage theory is to subject to nonlocal treatment only those variables that control strain softening, and to treat the elastic part of the strain as local. The continuum damage mechanics formulation, convenient for separating the nonlocal treatment of damage from the local treatment of elastic behavior, is adopted in the present work. The only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of the material. Avoidance of spurious mesh ...
Localization limiters in transient problems
Abstract In materials with strain softening, closed form solutions have shown that the energy dissipation associated with the localization and damage processes vanishes. For finite element solutions, the dissipation depends on the size of the element in the localized zone. To remedy this situation, non-local and imbricate formulations have previously been proposed. In this paper, a simple remedy based on introducing additional higher order terms in the governing equations in the strain-softening portions of the domain is proposed. It is shown that the resulting formulation provides solutions which are independent of element size with a finite energy dissipation. Example solutions are given for one-dimensional rod problems and one-dimensional spherically symmetric problems.