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Growth and instability in elastic tissues. (English) Zbl 1120.74336
Summary: The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.

MSC:
74B20 Nonlinear elasticity
74G60 Bifurcation and buckling
74K25 Shells
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