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By Claude Brezinski

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2. Srtes update procedure The stress state in a path dependent material can be readily treated using Lagrangian meshes, because the quadrature points coincide with material points regardless of the deformation of the continuum. 4. Thus, it is necessary to update the stress field with respect to the convective velocity e. In nonlinear problems, the material rate of stress is usually related to the deformation history and current stress state with large variety of constitutive models. 93) For updating the stresses, the spatial derivative of stress must be evaluated.

3. 16) where c and r are the cohesion and internal friction, and ~" and o-n are the shear and normal stresses, respectively. From the conventional Mohr circle diagram, we have _ 1 (o-l, o-3 )cos ~ = c . 18) results in ( o - l - o - 3 ) / 2 - c = 0 , which represents the Tresca yield criterion of maximum shear stress. The Mohr-Coulomb criterion neglects the influence of the intermediate principal stress. In addition, numerical difficulties will be encountered for the Mohr-Coulomb yield surface around the comers, which lead to singularities and cause numerical ill-conditioning.

A number of yield criteria have been developed and tested to describe the behavior of granular material with consideration of yielding and densification characteristics (Desai and Siriwardane 1984, Chen and Baladi 1985 and Lewis and Schrefler 1987). These models generally comprise two surfaces, one to reflect shear failure and the second to capture densification. The solution yields details on the powder displacement from which it is possible to establish the stress state in the powder and the densification can be derived from consideration of the elemental volumetric strain.

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