Violation of the Complementing Condition and Local Bifurcation in Nonlinear Elasticity
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The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic differential operator and the principal part of the corresponding boundary operators. We study the implications of failure of the CC in the context of nonlinear elasticity. In particular we show that for axisymmetric deformations of cylinders and for any homogeneous isotropic material, failure of the CC is equivalent to the existence of sequences of possible bifurcation points accumulating at the point where the CC fails. For non axisymmetric deformations and for Hadamard–Green type materials, we show for axial compressions of the cylinder that the CC fails on a full interval of values of the loading parameter, and for the lateral compression problem it fails at least once.
KeywordsNonlinear elasticity Complementing condition Global bifurcation Wrinkling
Mathematics Subject Classification (2000)74B20 74G60 35J57 35Q74
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