Models for isotropic materials based on the equivalent stress concept are discussed. At first, so-called classical models which are useful in the case of absolutely brittle or ideal ductile materials are presented. Tests for basic stress states are suggested. At second, standard models describing the intermediate range between the absolutely brittle and ideal-ductile behavior are introduced. Any criterion is expressed by various mathematical equations formulated, for example, in terms of invariants. At the same time the criteria can be visualized which simplifies the application. At third, in the main part pressure-insensitive, pressure-sensitive and combined models are separated. Fitting methods based on mathematical, physical and geometrical criteria are necessary. Finally, three examples (gray cast iron, poly(oxymethylene) (POM) and poly(vinyl chloride) (PVC) hard foam) demonstrates the application of different approaches in modeling certain limit behavior. Two appendices are necessary for a better understanding of this chapter: in Chap. 2 applied invariants are briefly introduced and a table of discussed in this chapter criteria with references is given.

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The analysis of well-known strength hypotheses leads to the derivation of a generalized model, which contains a number of known hypotheses as special cases and could be used for the description of the 3D-failure of hard foams. This model in the case of the strength hypothesis for hard foams is characterized by a closed surface in the principal stress space. In order to fit the model to the experimental data certain objective functions are formulated. The optimization results are shown in the Pareto-diagram (optimal solutions for several targets). The results of the fitting are plotted in the Burzyński-plane. It can be seen that reliable modeling requires the knowledge of the material behavior under hydrostatic tension and compression.

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