Rediscovering GF 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 GF Becker on the ideal isotropic nonlinear stress–strain function (Am J Sci 1893; 46: 337–356). 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, which in today’s notation can be written as

$$\begin{array}{ll}{T}^{\mathrm{Biot}}\hfill & =2G\cdot {\mathrm{dev}}_3\log (U)+K\cdot \mathrm{tr}[\log (U)]\cdot 11\hfill \\ \hfill & =2G\cdot \log (U)+\Lambda \cdot \mathrm{tr}[\log (U)]\cdot 11,\hfill \end{array}$$

where T^Biot is the Biot stress tensor, log(U) is the principal matrix logarithm of the right Biot stretch tensor $U=\sqrt{F^{\mathrm{T}}F}$ , $X={\displaystyle {\sum }_{i=1}^3{X}_{i,i}}$ denotes the trace and dev₃ X = X − (1/3) tr (X) · 11 denotes the deviatoric part of a matrix $X\in {ℝ}^{3\times 3}$ .

Here, G is the shear modulus and K is the bulk modulus. For Poisson’s number ν = 0 the formulation is hyperelastic and the corresponding strain energy

$$W_{\mathrm{Becker}}^{\nu =0}(U)=2G\left[<U,\log (U)-11>+3\right]$$

has the form of the maximum entropy function.