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C. Adam, F. Ziegler:
"Moderately large forced oblique vibrations of elastic-viscoplastic deteriorating slightly curved beams";
Archive of Applied Mechanics, 67 (1997), 375 - 392.

An integral equation formulation for the dynamic biaxial response of slightly curved elastic-viscoplastic beams is presented in the context of a multiple field analysis, which takes into account the geometrically nonlinear influence of moderately large deflections. Materials are considered in the regime of rate-dependent plasticity and are subjected to accumulated ductile damage. The latter is modeled by the growth of voids in the plastic zones of an initially porous elastic material. Inelastic defects of the material are considered in the linear elastic background beam by a second imposed strain field (eigenstrains). Geometric nonlinear effects of large deflections under conditions of immovable supports are approximately taken into account and, by inspection, they render another "strain field" to be imposed on the linear background beam. Superposition applies in the linear elastic background in an incremental formulation and linear methods as those based on Green's functions and Duhamel's integral are used to account for the given loads as well as for the resultants of the imposed strain fields. The intensity and the distribution of the imposed strain fields are calculated incrementally in a time stepping procedure and they are determined by the constitutive law and by application of the nonlinear geometric relations. The numerical procedure resulting from the multiple fields in the elastic background is illustrated for two cases: A preloaded viscoplastic beam of rectangular cross-section is subjected to oblique flexural vibrations when forced by a sinusoidal load and an I-beam with a prescribed initial curvature is severely impacted and thus driven into the plastic regime.

An integral equation formulation for the dynamic biaxial response of slightly curved elastic-viscoplastic beams is presented in the context of a multiple field analysis, which takes into account the geometrically nonlinear influence of moderately large deflections. Materials are considered in the regime of rate-dependent plasticity and are subjected to accumulated ductile damage. The latter is modeled by the growth of voids in the plastic zones of an initially porous elastic material. Inelastic defects of the material are considered in the linear elastic background beam by a second imposed strain field (eigenstrains). Geometric nonlinear effects of large deflections under conditions of immovable supports are approximately taken into account and, by inspection, they render another "strain field" to be imposed on the linear background beam. Superposition applies in the linear elastic background in an incremental formulation and linear methods as those based on Green's functions and Duhamel's integral are used to account for the given loads as well as for the resultants of the imposed strain fields. The intensity and the distribution of the imposed strain fields are calculated incrementally in a time stepping procedure and they are determined by the constitutive law and by application of the nonlinear geometric relations. The numerical procedure resulting from the multiple fields in the elastic background is illustrated for two cases: A preloaded viscoplastic beam of rectangular cross-section is subjected to oblique flexural vibrations when forced by a sinusoidal load and an I-beam with a prescribed initial curvature is severely impacted and thus driven into the plastic regime.