Rediscovering G.F. Becker's early axiomatic deduction of a multiaxial nonlinear stress-strain relation based on logarithmic strain

Abstract

We discuss a completely forgotten work of the geologist G.F. Becker on the ideal isotropic nonlinear stress-strain function. In doing this we provide the original paper from 1893 newly typeset in LaTeX and with corrections of typographical errors as well as an updated notation. Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker's work, combining his approach with the current framework of the theory of nonlinear elasticity. Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker's deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation. In the special case where Poisson's number is zero, the formulation is hyperelastic and the corresponding strain energy has the form of the maximum entropy function.