 Linear elasticity

Linear elasticity describes small deformations using a stress that depends linearly on the infinitesimal strain tensor. If $\mathbf{u}$ is the displacement field, thenFor an unstressed isotropic material, the constitutive equation iswhere $\lambda ,\mu$ are the Lamé parameters, ${\delta }_{ij}$ is the Kronecker delta and repeated indices are summed. In particular, $\mu$ is the shear modulus.

The small-displacement-gradient approximation permits superposition: for the same linear material and linear boundary conditions, the sum of two displacement solutions responds to the sum of their applied loads. This makes it possible to calculate bending, vibration and elastic waves from common material coefficients. Small strain alone is not sufficient when rotations are large; a slender elastic rod can turn through a large angle while stretching very little, requiring geometrically nonlinear kinematics.

Linear elasticity is a foundational concept in the field of mechanics of materials and structural analysis that describes how solid materials deform under applied loads. It assumes that the relationship between stress (internal forces) and strain (deformation) in a material is linear and reversible within the elastic limit of the material. This means that if the applied load is removed, the material will return to its original shape without permanent deformation.