1.4.2 Nonlinear Elastic Behavior

If an elastic material exhibits nonlinear behavior, the stress-strain relationship is often cast in incremental form relating some increments of strains to stress, or vice versa ds = Dide (1.38)

where ds is the incremental vector of stresses, de is the incremental vector of strains, DI is the incremental material stiffness matrix, and CI is the incremental material compliance matrix. If the experimental stress-strain curves of a material are known, the terms in these matrices can be taken as the values of the tangential or secant slopes of these curves. The analysis of structures made of materials that exhibit nonlinear elastic behavior has to be performed numerically in incremental steps as well.

Alternatively, if the nonlinear relationship between any given components of stress (or strain) can be expressed as a mathematical function of strains (or stresses) and material constants k1, k2, k3, etc., as follows:

aij = fij(en, en, e33, eu, e23, e13, k1, k2, k3,...) (1.40)

ej = gij (S11, S22, S33, S12, S23, S13, k1, k2, k},. . .) (1.41)

such relationships can be incorporated directly into the analysis to obtain closed-form solutions. However, this type of analysis can be performed only if both the structure and the loading conditions are very simple.

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