Namespaces
Elasticity Namespace Reference

Items related to solving elasticity problems. More...

Namespaces

 ConstitutiveRelations
 Constitutive (stress-strain) relations that characterize the elastic properties of a material.
 

Detailed Description

Items related to solving elasticity problems.

Details

In elasticity problems we solve for the displacement vector field \(\boldsymbol{u}\) in an elastic material that responds to external forces, stresses or deformations. In this static approximation the equations of motion reduce to the elliptic equations

\[ \nabla_i T^{ij} = f_\mathrm{ext}^j \]

that describes a state of equilibrium between the stresses \(T^{ij}\) within the material and the external body forces \(\boldsymbol{f}_\mathrm{ext}\) (Eqns. 11.13 and 11.14 in [31]). For small deformations (see e.g. [31], Section 11.3.2 for a discussion) the stress is related to the strain \(S_{ij}=\nabla_{(i}u_{j)}\) by a linear constitutive relation \(T^{ij}=-Y^{ijkl}S_{kl}\) (Eq. 11.17 in [31]) that describes the elastic properties of the material (see Elasticity::ConstitutiveRelations::ConstitutiveRelation).