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 energy
in 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=√FTF is the corresponding stretch tensor,
logU is the principal matrix logarithm of
U,
tr is the trace operator,
∥X∥ is the Frobenius matrix norm and
devnX is the deviatoric part of
X∈Rn×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 form
with
w:(0,∞)→R, where
λ1,…,λn denote the singular values of
F.