Measures of Finite Strain and Stress-Strain Relations

Abstract

Ordinarily, stress-strain relation of Lagrange’s representation is written in terms of Lagrange’s representation of stress σ^ij and the Love measure e_ij^L. From the above relation, we can derive Reiner’s equation in terms of Euler’s representation of stress T^λμ and the Green measure e_λμ^G.
In this paper, it is shown the σ′^ij and e_ij defined by
σ′^ij=Dx,_i^*λx,_j^*μT^λμ,e_ij=\frac12(δ_ij−g^ij)
can play the role of σ^ij and e_ij^L and starting from stress-strain relation of σ′^ij and e_ij we arrive at Reiner’s equation in terms of T^λμ and the Almansi measure e^A_λμ.
Various relations relating to four measures above mentioned and to their first variations are also shown, as well as various relations among different strain invariants.