Diffusion Coupled with Damage

The incorporation of damage variables within the context of continuum mechanics implies the presence of a recurrent distribution of microcracks or microvoids within a sufficiently small volume element that represents the entire material. 

In any such continuum formulation the magnitude of the representative volume element should be much smaller than any structural dimension of the material volume at hand. For this reason, the following formulation applies to the early stage of fluid-induced damage prior to the coalescence of the individual arc-shaped microcracks to form the long hairline cracks at the fiber matrix interfaces shown in Fig. 4.30a-c. 

The basic formulation follows (5.15)-(5.21), with the presence of damage introduced by incorporating it within the Helmholz free energy j/, namely (Weitsman 1987b) 

Considering transverse isotropy, as applies to unidirectionally fiber reinforced composites, it is necessary to express (5.43) in terms of the individual and joint invariants of s,y and d[, for that particular symmetry. Such an expansion yields an expression for that involves 17 material parameters, all of which are functimis of m, T, d j2 and +dpj. Obviously, such a general formulation is practically intractable. 

A simplification occurs for the realistic isothermal case of diffusion across the thickness direction Xi of a plate, where all fibers run parallel to X2- In this case xj/ contains eight parameters. 

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Because no coupling between the neighboring cell walls is assumed, only the main diagonal of the material matrix has nonzero values and, due to the nonlinear elastic-plastic struchual behavior with damage of the cell walls, their components are functions of the corresponding strain components. As mentioned above, the overall stress-strain relation of the foam is obtained by adding both contributions of the skeleton and the diffuse continuum ... 

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