A.3 Pure Shear

To consider shear-based responses and behaviors of a model, uniaxial loadings are often insufficient. One problem, however, that does investigate shear deformations is that of a pure shear problem. In such problems, only a single shear strain and stress component are non-zero. Such a material state results from a deformation gradient of the form,

\[F_{ij}=\frac{1}{2}\left(\lambda+\lambda^{-1}\right)\left(\delta_{i1}\delta_{j1}+\delta_{i2}\delta_{j2}\right)+\frac{1}{2}\left(\lambda-\lambda^{-1}\right)\left(\delta_{i1}\delta_{j2}+\delta_{i2}\delta_{j1}\right)+\delta_{i3}\delta_{j3},\]

where the shear loading is relative to the \(x_1-x_2\) axis. The logarithmic strain tensor is then simply \(\varepsilon_{ij}=\ln\lambda(\delta_{i1}\delta_{j2}+\delta_{i2}\delta_{j1})\). With such a strain tensor, it is trivial to note that \(\sigma_{12}\) is the only non-zero stress.