Remarks on rank-one convexity and quasiconvexity for planar functions with an additive volumetric-isochoric split

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We study convexity properties of isotropic energy functions in planar nonlinear elasticity in the context of Morrey’s conjecture, which states that rank-one convexity does not imply quasiconvexity in the two-dimensional case. Recently, it has been shown that for the special case of isochoric energy functions on GL+(2) = {F ∈ R2×2 | det F > 0}, i.e. for any isotropic function W : GL+(2) → R with W (aF ) = W (F ) for all a > 0, these two notions of generalized convexity are, in fact, equivalent. Here, we consider the more general case of functions on GL+(2) with an additive volumetric-isochoric split of the form

W (F ) = W_iso(F ) + W_vol(det F ) = W_iso( F/√det F) + W_vol(det F )

with an isochoric function W_iso on GL^+(2) and a function W_vol on (0, ∞). In particular, we investigate the importance of the function

W^+_magic : GL+(2) → R , W^+_magic(F ) = λ_max / λ_min − log λ_max / λ_min + log det F = λ_max / λ_min + 2 log λ_min

and its convexity properties; here, λ_max ≥ λ_min > 0 are the ordered singular values of the deformation gradient F ∈ GL^+(2). This function arises naturally as an “extremal” case in the class of volumetric-isochorically split energies with respect to rank-one convexity.


FachzeitschriftProceedings in applied mathematics and mechanics : PAMM
PublikationsstatusAngenommen/Im Druck - 22 Okt. 2022