A polyconvex extension of the logarithmic Hencky strain energy

Abstract

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function W : GL + (n) → R which is equal to the classical Hencky strain energyin a neighborhood of the identity matrix 1; here, GL + (n) denotes the set of n × n-matrices with positive determinant, F ∈ GL + (n) denotes the deformation gradient, U = √ F T F is the corresponding stretch tensor, log U is the principal matrix logarithm of U , tr is the trace operator, X is the Frobenius matrix norm and devn X is the deviatoric part of X ∈ R n×n . The extension can also be chosen to be coercive, in which case Ball's classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions WVL in the so-called Valanis-Landel formwith w : (0, ∞) → R, where λ1, . . . , λn denote the singular values of F .

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

“…2 It has been first shown by Richter [23,53,57,58] (see also Flory [18] and Sansour [61]) that the accompanying symmetric nonlinear Kirchhoff stress tensor τ = det F • σ (but not the Cauchy stress as repeatedly claimed in [44,45]) admits a similar additive structure in the sense that 12) in which the deviatoric part of the Kirchhoff stress only depends on W iso and the spherical part only depends on W vol . A typical example of the foregoing volumetric-isochoric format is the geometrically nonlinear quadratic Hencky energy [29,36,52,65]…”

“…The loss of ellipticity is generally related to instability phenomena (separation into arbitrary fine phase mixtures [26,67], shear banding) and possible discontinuous equilibrium solutions [59]. Checking rank-one convexity for a given nonlinear elastic material can be quite challenging, see e.g., [8,35,47,56] and Appendix D, although John Ball's polyconvexity [6] as an easy-to-verify sufficient condition can often be helpful [36,62]. A certain simplification occurs in the isotropic case.…”

Sharp rank-one convexity conditions in planar isotropic elasticity for the additive volumetric-isochoric split

“…in which the deviatoric part of the Kirchhoff stress only depends on W iso and the spherical part only depends on W vol . A typical example of the foregoing volumetric-isochoric format is the geometrically nonlinear quadratic Hencky energy [47,29,65,36]…”

“…Checking rank-one convexity for a given nonlinear elastic material can be quite challenging, see e.g. [35,8,56,48] and Appendix D, although John Ball's polyconvexity [6] as an easy-to-verify sufficient condition can often be helpful [63,36]. A certain simplification occurs in the isotropic case.…”

Polyconvex anisotropic hyperelasticity with neural networks

“…The theoretical aspects of polyconvexity are still subject to current research [23,54,55], and the formulation of polyconvex models remains a challenging task [52,53]. For a long time after its initial conception, the polyconvexity requirement was practically restricted to isotropic material response, as no anisotropic formulation was available that ensured at the same time: polyconvexity, objecticity, material symmetry and a stress-free reference configuration.…”