5.1 Linear Elasticity

This chapter describes the theoretical background behind the constitutive models that are available in FEBio. Most materials are derived from a hyperelastic strain-energy function. Please consult Section Hyperelasticity for more background information on this class of materials.

In the theory of linear elasticity the Cauchy stress tensor is a linear function of the small strain tensor \(\boldsymbol{\varepsilon}\):

\[ \begin{equation} \boldsymbol{\sigma}=\boldsymbol{\mathcal{C}}:\boldsymbol{\varepsilon}\,.\label{eq402} \end{equation} \]

Here, \(\boldsymbol{\mathcal{C}}\) is the fourth-order elasticity tensor that contains the material properties. In the most general case this tensor has 21 independent parameters. However, in the presence of material symmetry the number of independent parameters is greatly reduced. For example, in the case of isotropic linear elasticity only two independent parameters remain. In this case, the elasticity tensor is given by \(\boldsymbol{\mathcal{C}}=\lambda\mathbf{I}\otimes\mathbf{I}+2\mu\mathbf{I}\,\overline{\underline{\otimes}}\,\mathbf{I}\), or equivalently,

\[ \begin{equation} \mathcal{C}_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)\,.\label{eq403} \end{equation} \]

The material coefficients \(\lambda\) and \(\mu\) are known as the Lamé parameters. Using this equation, the stress-strain relationship can be written as

\[ \begin{equation} \sigma_{ij}=\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}\,.\label{eq404} \end{equation} \]

If the stress and strain are represented inVoigt notation, the constitutive equation can be rewritten in matrix form as

\[ \begin{equation} \left[\begin{array}{c} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{13} \end{array}\right]=\left[\begin{array}{cccccc} \lambda+2\mu & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda+2\mu & \lambda & 0 & 0 & 0\\ \lambda & \lambda & \lambda+2\mu & 0 & 0 & 0\\ 0 & 0 & 0 & \mu & 0 & 0\\ 0 & 0 & 0 & 0 & \mu & 0\\ 0 & 0 & 0 & 0 & 0 & \mu \end{array}\right]\left[\begin{array}{c} \varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{23}\\ \gamma_{13} \end{array}\right]\,.\label{eq405} \end{equation} \]

The shear strain measures \(\gamma_{ij}=2\varepsilon_{ij}\) are called the engineering strains.

The following table relates the Lamé parameters to the more familiar Young's modulus \(E\) and Poisson's ratio \(\nu\) or to the bulk modulus \(K\) and shear modulus \(G\).

	\(E,\nu\)	\(\lambda,\mu\)	\(K,G\)
\(E,\nu\)		\(\begin{array}{l} E=\frac{\mu}{\lambda+\mu}\left(2\mu+3\lambda\right)\\ \nu=\frac{\lambda}{2\left(\lambda+\mu\right)} \end{array}\)	\(\begin{array}{l} E=\frac{9KG}{3K+G}\\ \nu=\frac{3K-2G}{6K+2G} \end{array}\)
\(\lambda,\mu\)	\(\begin{array}{l} \lambda=\frac{\nu E}{\left(1+\nu\right)\left(1-2\nu\right)}\\ \mu=\frac{E}{2\left(1+\nu\right)} \end{array}\)		\(\begin{array}{l} \lambda=K-\frac{2}{3}G\\ \mu=G \end{array}\)
\(K,G\)	\(\begin{array}{l} K=\frac{E}{3\left(1-2v\right)}\\ G=\frac{E}{2\left(1+v\right)} \end{array}\)	\(\begin{array}{l} K=\lambda+\frac{2}{3}\mu\\ G=\mu \end{array}\)

The theoretical range of the Young's modulus and Poisson's ratio for an isotropic material have the ranges

\[ \begin{equation} 0<E<\infty\,,\label{eq406} \end{equation} \]

\[ \begin{equation} -1\leqslant\nu<0.5\,.\label{eq407} \end{equation} \]

Materials with Poisson's ratio (close to) 0.5 are known as (nearly-) incompressible materials. For these materials, the bulk modulus approaches infinity. Most materials have a positive Poisson's ratio. However there do exist some materials with a negative ratio. These materials are known as auxetic materials and they have the remarkable property that they expand under tension.

The linear stress-strain relationship can also be derived from a strain-energy function such as in the case of hyperelastic materials. In this case the linear strain-energy is given by

\[ \begin{equation} W=\frac{1}{2}\boldsymbol{\varepsilon}:\boldsymbol{\mathcal{C}}:\boldsymbol{\varepsilon}\,.\label{eq408} \end{equation} \]

The stress is then similarly derived from \(\boldsymbol{\sigma}=\frac{\partial W}{\partial\boldsymbol{\mathbf{\varepsilon}}}\). In the case of isotropic elasticity, \eqref{eq408} can be simplified:

\[ \begin{equation} W=\frac{1}{2}\lambda\left(\tr\boldsymbol{\mathbf{\varepsilon}}\right)^{2}+\mu\boldsymbol{\mathbf{\varepsilon}}:\boldsymbol{\mathbf{\varepsilon}}\,.\label{eq409} \end{equation} \]

The Cauchy stress is now given in tensor form by

\[ \begin{equation} \boldsymbol{\sigma}=\lambda\left(\tr\boldsymbol{\varepsilon}\right)\mathbf{I}+2\mu\boldsymbol{\varepsilon}\,.\label{eq410} \end{equation} \]