The behaviour of pre-stressed, elastic, orthotropic and incompressible materials is analysed in both the static and dynamic regimes. Perturbations caused by dipoles, either static or pulsating, are considered to investigate material instabilities arising near the boundary of ellipticity loss. The perturbation approach is capable of revealing aspects which may remain undetected using methods for material instabilities based on weak discontinuity surfaces. In the static case, for instance, the approach reveals shear band formation for a Mooney-Rivlin material, a circumstance not detected by the conventional approach. In the dynamic case, the perturbative approach provides a basis for the analysis of propagation of disturbances near the boundary of loss of ellipticity. Depending on the level of pre-stress and anisotropy, wave patterns are shown to emerge, with focussing of signals in the direction of shear bands. Varying the direction of the dynamic perturbation excites different wave patterns, which tend to degenerate to families of plane waves parallel to the shear bands, when the elliptic boundary is approached. At the base of the perturbation approach are infinite-body Green’s functions for incremental displacements and in-plane hydrostatic stress obtained by the authors for small isochoric and plane deformation superimposed upon a nonlinear elastic and homogeneous strain. The same functions are employed to develop a boundary element technique for the solution of boundary value incremental problems. In this technique, “static” and “dynamic” contributions are uncoupled in the Green function for incremental tractions: the dynamic contributions are regular whereas the static terms are strongly singular and are solved in closed-form expressions, particularly useful for numerical calculations. The formulation is used to examine the influence of pre-stress on the vibrational response of elastic structures.

Perturbations and boundary integral equations for pre-stressed elastic materials

CAPUANI, Domenico;
2009

Abstract

The behaviour of pre-stressed, elastic, orthotropic and incompressible materials is analysed in both the static and dynamic regimes. Perturbations caused by dipoles, either static or pulsating, are considered to investigate material instabilities arising near the boundary of ellipticity loss. The perturbation approach is capable of revealing aspects which may remain undetected using methods for material instabilities based on weak discontinuity surfaces. In the static case, for instance, the approach reveals shear band formation for a Mooney-Rivlin material, a circumstance not detected by the conventional approach. In the dynamic case, the perturbative approach provides a basis for the analysis of propagation of disturbances near the boundary of loss of ellipticity. Depending on the level of pre-stress and anisotropy, wave patterns are shown to emerge, with focussing of signals in the direction of shear bands. Varying the direction of the dynamic perturbation excites different wave patterns, which tend to degenerate to families of plane waves parallel to the shear bands, when the elliptic boundary is approached. At the base of the perturbation approach are infinite-body Green’s functions for incremental displacements and in-plane hydrostatic stress obtained by the authors for small isochoric and plane deformation superimposed upon a nonlinear elastic and homogeneous strain. The same functions are employed to develop a boundary element technique for the solution of boundary value incremental problems. In this technique, “static” and “dynamic” contributions are uncoupled in the Green function for incremental tractions: the dynamic contributions are regular whereas the static terms are strongly singular and are solved in closed-form expressions, particularly useful for numerical calculations. The formulation is used to examine the influence of pre-stress on the vibrational response of elastic structures.
2009
Material instability; pre-stress; wave propagation; shear band; nonlinear elasticity
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1379436
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