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Exact solution for cavitated bifurcation for compressible hyperelastic materials. (English) Zbl 1210.74025

Summary: A new exact analytic solution for spherical cavitated bifurcation is presented for a class of compressible hyperelastic materials. The strain energy density of the materials is assumed to be a linear function of three strain invariants, which may be regarded as a first-order approximation to the general strain energy density near the reference configuration, and also may satisfy certain constitutive inequalities of hyperelastic materials. An explicit formula for the critical stretch for the cavity nucleation and a simple bifurcation solution for the deformed cavity radius which describes the cavity growth are obtained. The potential energy associated with the cavitated deformation is examined. It is always lower than that associated with the homogeneous deformation, thus the state of cavitated deformation is relatively stable. On the basis of the presented analytic solutions for the stretches and stresses, the catastrophic transition of deformation and the jumping of stresses for the cavitation are discussed in detail. The boundary layers of the displacements, the strain energy distribution and stresses near the formed cavity wall are observed. These investigations illustrate that cavitation reflects a local behaviour of materials.

MSC:

74B20 Nonlinear elasticity
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