In continuum mechanics, motion is described by a mapping from material coordinates to spatial coordinates. Small-strain theory uses the strain tensor $$\varepsilon = \frac{1}{2}(\nabla u + \nabla u^T),$$ where $u$ is the displacement field. For large deformation, one uses finite-strain measures such as the deformation gradient $F$ and Green–Lagrange strain.

The local balance of linear momentum is $$\rho \ddot{u} = \nabla \cdot \sigma + b,$$ where $\sigma$ is the Cauchy stress tensor and $b$ is body force per unit volume. In static equilibrium, this reduces to $$\nabla \cdot \sigma + b = 0.$$ The symmetry of $\sigma$ follows from angular momentum balance under standard assumptions.

Field equations are not enough; a constitutive law closes the system. For isotropic linear elasticity, $$\sigma = \lambda \, \text{tr}(\varepsilon) I + 2\mu \varepsilon,$$ where $\lambda$ and $\mu$ are Lamé parameters. For Newtonian fluids, $$\tau = 2\mu D,$$ where $D$ is the symmetric velocity gradient. More advanced models include viscoelasticity, plasticity, hyperelasticity, and anisotropy.