Abstract
We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type W(F)=frac{mu }{2} hspace{0.07em} frac{lVert F rVert ^{2}}{det F}+f(det F); such an energy is rank-one convex if and only if the function f is convex.
Highlights
One of the most important constitutive requirements in nonlinear elasticity is the rank-one convexity of the elastic energy W : GL+(n) → R
Our investigation clearly demonstrates that for planar isotropic hyperelasticity, assuming a volumetric-isochoric split of the elastic energy endows the theory with a lot of additional mathematical structure which can be exploited
We have shown how the classical Knowles-Sternberg planar ellipticity criterion - which is represented by a family of two-dimensional differential inequalities - can be reduced to a family of only onedimensional coupled differential inequalities for split energies, which allows for a much more accessible rank-one convexity criterion
Summary
One of the most important constitutive requirements in nonlinear elasticity is the rank-one convexity of the elastic energy W : GL+(n) → R. Knowles and Sternberg have provided a criterion for rank-one convexity in terms of g, conclusively reducing the problem to the singular values representation [14, 31,32,33, 69], see [11, 15, 55, 71, 73, 75]. Rank-one convexity in isotropic planar incompressible elasticity W : SL(2) → R can be checked [22]: for an energy of the form. These results allow for an explicit calculation of the quasiconvex relaxation for conformally invariant and incompressible isotropic planar hyperelasticity [37, 38]
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