A commented translation of Hans Richter's early work "The isotropic law of elasticity"

A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity

“…a mapping ϕ : Ω ⊂ R n → R n given as the composition of finitely many reflections at hyperplanes and/or spheres. 2 Remark 2.1. For n ≥ 3, every non-trivial conformal mapping is a Möbius transformation [1], while in the planar case n = 2, the orientation-preserving Möbius transformations correspond to the complex functions of the form…”

“…In nonlinear elasticity, conformally invariant energy functions are generally not directly suited for modeling the elastic behaviour of a material due to their invariance under purely volumetric scaling. However, they are commonly coupled with a volumetric energy term of the form F → f (det(F )) for some function f : (0, ∞) → R. Energy functions of this type, also known as an additive volumetric-isochoric split [13,2], will be used in Section 4 in order to establish our main results.…”

“…with dev n X = X − 1 n tr(X) • 1 n , hence the quadratic potential for linearized elasticity corresponding to W is given by W lin (∇u) = µ dev 2 sym ∇u 2 (3.10) with µ = ψ ′ (1). Moreover,…”

“…(tr(∇u)) 2 , and W has a unique stress-free state. It is also easy to see that the volumetric term f can be modified such that the resulting energy W is C ∞ -regular (and thus strictly Legendre-Hadamard elliptic as well).…”