Tensor for expansion and tension

Question

I have a simple (and maybe silly...) question, but: How can I determine for an isotropic, linear elastic material the tensors for the expansion and the tension for $$\boldsymbol u(\boldsymbol x) = \gamma x_2 \boldsymbol e_1$$ where $\gamma \in \mathbb R$ ?

Answer 1

We usually don't speak of tension in the context of linear elastic bodies, but rather when speaking of wires, cables, chains, etc.

The present displacement field is a pure shear displacement field: only $u_{1,2} = \gamma$ is nonzero. Therefore, if we consider that $\lbrace\boldsymbol e_1, \boldsymbol e_2, \boldsymbol e_3\rbrace$ is an orthonormal basis of a Cartesian coordinate system, the infinitesimal strain tensor $\boldsymbol\varepsilon$ with coordinates $\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i})$ has only two nonzero components out of the diagonal. Applying Hooke's law of linear elasticity $\sigma_{ij} = \lambda \delta_{ij}\varepsilon_{kk} + 2\mu\varepsilon_{ij}$ where $(\lambda,\mu)$ are the Lamé parameters, one gets the following components of the Cauchy stress tensor $\boldsymbol\sigma$: $$ [σ_{ij}] = μγ \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}. $$ The traction vector $\boldsymbol t$ along the surface with normal $\boldsymbol n$ is given by $\boldsymbol t = \boldsymbol\sigma\cdot \boldsymbol n$.