A non-ellipticity result, or the impossible taming of the logarithmic strain measure. (June 2018) - Martin, Robert J.; Ghiba, Ionel-Dumitrel; Neff, Patrizio
Abstract: Constitutive laws in terms of the logarithmic strain tensor log U, i.e. the principal matrix logarithm of the stretch tensor U = F T F corresponding to the deformation gradient F, have been a subject of interest in nonlinear elasticity theory for a long time. In particular, there have been multiple attempts to derive a viable constitutive law of nonlinear elasticity from an elastic energy potential which depends solely on the logarithmic strain measure ‖ log U ‖ 2, i.e. an energy function W : GL + ( n ) → R of the form (1) W ( F ) = Ψ ( ‖ log U ‖ 2 ) with a suitable function Ψ : [ 0, ∞ ) → R, where ‖ . ‖ denotes the Frobenius matrix norm and GL + ( n ) is the group of invertible matrices with positive determinant. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form(1) is Legendre–Hadamard elliptic. Similarly, we consider the related isochoric strain measure ‖ dev n log U ‖ 2, where dev n log U is the deviatoric part of log U . Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n = 2, we show that for n ≥ 3, no strictly monotone function Ψ : [ 0, ∞ ) → R exists such that F ↦ Ψ ( ‖ dev n log U ‖ 2 ) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F ↦ Ψ ( ‖ dev n log U ‖ 2 ) + W vol ( det F ) cannot be rank-one convex for any function W vol : ( 0, ∞ ) → R if Ψ is strictly monotone. Highlights: For n > 1, there exists no strictly monotone function Ψ : [ 0, ∞ ) → R such that the isotropic energy function F ↦ Ψ ( ‖ log F T F ‖ 2 ) is rank-one convex on the set GL + ( n ) of matrices with positive determinant. Similarly, for n > 2, there is no strictly monotone Ψ such that F ↦ Ψ ( ‖ dev n log F T F ‖ 2 ) is rank-one convex. Thus there is no viable rank-one convex elastic energy potential in terms of either ‖ log F T F ‖ 2 or ‖ dev n log F T F ‖ 2, since for applications in nonlinear hyperelasticity, the strict monotonicity of the energy in terms of these logarithmic strain measures follows from simple physical reasoning. Under sufficient regularity assumptions, these non-existence results apply to volumetric-isochorically decoupled energy functions of the form F ↦ Ψ ( ‖ dev n log U ‖ 2 ) + W vol ( det F ) as well.