Elasticity Namespace Reference

Namespaces
	StVenantKirchhoff

Classes
class	Eigen
class	Matrix
class	PolynomialError
class	PolynomialRHS
class	Residual

Detailed Description

Discretization of elastic deformation. Formulation of stress and strain relationship follows the book of Ciarlet (Volume I).

Here a list of quantities of the deformation \( x \mapsto \phi(x) \) and their meaning:

\( u(x) = \phi(x)-x \)

Displacement

\( F(x) = \nabla \phi(x) = I + \nabla u(x) \)

Deformation gradient

\( C = F^TF = I + \nabla u + (\nabla u)^T + (\nabla u)^T(\nabla u)\)

The right Cauchy-Green strain tensor

\( E = \tfrac12 (C-I) = \tfrac12\bigl(\nabla u + (\nabla u)^T + (\nabla u)^T(\nabla u)\bigr)\)

The Green-St. Venant strain tensor

\( J = \operatorname{det} F \)

The deformed volume element

The quantity describing the forces in an elastically deformed body is the stress. Depending on whether the stress is measured in deformed or undeformed coordinates, we have the Cauchy or the two Piola-Kirchhoff stress tensors. Material laws are typically given as either the Cauchy stress \(\hat T\) or the second Piola-Kirchhoff stress tensor \( \Sigma = J F^{-1} \hat T F^{-T}\). Using the second, we can write the equations of nonlinear elasticity in weak form as

\[ \int_\Omega \bigl((I+F) \Sigma\bigr) \colon \nabla v \,dx = \int_\Omega f \cdot v \,dx + \int_{\Gamma_N} \sigma_n \cdot v \,ds. \]

This equation is complemented by a stress-strain relation \(\Sigma = \Sigma(E)\) depending on the material, for instance Hooke's law (with Lamé-Navier coefficients)

\[ \Sigma = \lambda (\operatorname{tr} E) I + 2 \mu E. \]

The model is called geometrically linear if in this equation \(F\) is replaced by zero, and the quadratic term in the Green-St. Venant stress tensor is neglected. Combining this with Hooke's law and the fact that \(\tfrac12(A+A^T)\) is the projection of a matrix \(A\) to the subspace of symmetric matrices and \( (\operatorname{tr} A) I \) its projection on the subspace spanned by the identity, we obtain the Lamé-Navier equations

\[ 2\mu \int_\Omega \epsilon(u):\epsilon(v) \,dx + \lambda \int_\Omega \nabla\!\cdot\!u \nabla\!\cdot\!v \,dx = \int_\Omega f \cdot v \,dx + \int_{\Gamma_N} \sigma_n \cdot v \,ds. \]

The local integrators for this equation a re part of the deal.II library. The integrators in the namespace StVenantKirchhoff extend this to the geometrically nonlinear weak formulation above, but with the linear stress-strain relation established by Hooke's law.