strainenergy

Strain energy, or elastic strain energy, is the energy stored in a deformable body as a result of deformation under applied forces. It represents the work required to elastically deform the body from a reference configuration to its current state. For a continuous solid, the strain energy density w is defined as w = 1/2 σ_ij ε_ij, with summation over i and j; the total strain energy U is obtained by integrating over the volume, U = ∫_V w dV.

In linear elastic, isotropic materials, the energy density can be expressed in several forms. For uniaxial tension with modulus E and strain ε, w = 1/2 E ε^2. More generally, w = 1/2 λ (tr ε)^2 + μ ε_ij ε_ij, where λ and μ are Lamé constants, and the stress-strain relation is σ_ij = λ δ_ij ε_kk + 2 μ ε_ij. The strain energy is recoverable when the loads are removed, provided the material behaves elastically.

In plastic, viscoelastic, or damage-prone materials, part of the work is dissipated as heat or irrecoverable changes, and only the elastic portion remains recoverable.

Applications of strain energy include energy methods in structural analysis. Castigliano’s theorem relates displacements to derivatives of U with respect to applied loads; energy methods are used to assess buckling, vibration, and stability. In fracture mechanics, the energy release rate G describes how U changes with crack area.

For nonlinear elasticity (hyperelastic materials), the strain energy per unit volume is described by a strain energy density function W(F) depending on the deformation gradient F, governing the material’s nonlinear elastic response.