User:Didactohedron/Elasticity

Introduction
The following equations are stated without proof; they are taken from the English third edition of Landau and Lifshitz' Theory of Elasticity. The distance between two infinitesimally close points in a body under deformation is given by

$$dl'^2=dl^2+2 u_{ik} dx_i dx_k\,$$

where $$u_{ik}$$, the strain tensor, is given by

$$u_{ik}={1 \over 2} \left ({\partial u_i \over \partial x_k}+{\partial u_k \over \partial x_i}+{\partial u_l \over \partial x_i}{\partial u_l \over \partial x_k} \right )$$

and $$u_i$$ is defined as $$x'_i-x_i$$. The force exerted by internal stresses at a point in a deformed body is given by

$$F_i={\partial \sigma_{ik} \over \partial x_k}$$

Where $$\sigma_{ik}$$ is a symmetric tensor called the stress tensor. The stress and strain tensors are related through the following linear equations:

$$\sigma_{ik}=K u_{ll} \delta_{ik}+2 \mu (u_{ik} - \textstyle {1 \over 3} \delta_{ik} u_{ll})$$ $$u_{ik}={\delta_{ik} \sigma_{ll} \over 9K}+{\sigma_{ik}-\textstyle {1 \over 3} \delta_{ik} \sigma_{ll} \over 2 \mu}$$

Here, K is the bulk modulus or modulus of compression and μ is the modulus of rigidity.