Deformation

Our initial objective is to provide a concise mathematical description of the deformation that an elastic body has sustained. An important physical quantity derived directly from $\vec\phi(\vec{X})$, whose utility will become apparent in the next sections, is the deformation gradient tensor $\mathbf{F}\in{\mathbf{R}}^{3\times3}$.

Strain Tensor

Consider the Green strain tensor $$ \mathbf{E}=\frac{1}{2}(\mathbf{F}^{T}\mathbf{F}-I) $$

Strain Energy Density

The Strain Energy Density Function is a mathematical expression that describes the amount of energy per unit volume that a material stores during deformation. This function depends on the amount of strain that the material undergoes and plays a crucial role in determining the material’s properties. Invariants are denoted by $I_1$, $I_2$, $I_3$ and defined as: $$ \DeclareMathOperator{\tr}{tr} I_1(\mathbf{F})=\tr(\mathbf{F}^{T}\mathbf{F}),\quad I_2(\mathbf{F})=\tr[(\mathbf{F}^{T}\mathbf{F})^{2}],\quad I_3(\mathbf{F})=\det(\mathbf{F}^{T}\mathbf{F})=(\det{\mathbf{F}})^{2} $$ In such case, we can use the chain rule to compute the stress $P$ as: $$ \begin{align} \mathbf{P} & = \frac{\partial\Psi(I_1,I_2,I_3)}{\partial\mathbf{F}}=\frac{\partial\Psi}{\partial{I_1}}\frac{\partial{I_1}}{\partial{\mathbf{F}}}+\frac{\partial\Psi}{\partial{I_2}}\frac{\partial{I_2}}{\partial{\mathbf{F}}}+\frac{\partial\Psi}{\partial{I_3}}\frac{\partial{I_3}}{\partial{\mathbf{F}}} \ & = \frac{\partial\Psi}{\partial{I_1}}\cdot2\mathbf{F}+\frac{\partial\Psi}{\partial{I_2}}\cdot4\mathbf{F}\mathbf{F}^{T}\mathbf{F}+\frac{\partial\Psi}{\partial{I_3}}\cdot2I_3\mathbf{F}^{-T} \end{align} $$ Finally, we note the additional invariant: $$ J=\det{\mathbf{F}}=\sqrt{I_3} $$ that is often used in replacement of $I_3$ while defining certain constitutive models. This quantity has an important physical interpretation as it represents the fraction of volume change due to deformation: a value of $J=1$ implies that volume is preserved exactly while, $J=2$ would indicate an expansion to twice the undeformed volume and $J = 0.2$ would be a compression down to 20% of the rest volume.

Stress Tensor

An example of an isotropic constitutive model defined via isotropic invariants is Neohookean elasticity: $$ \Psi(I_{1},J)=\frac{\mu}{2}(I_1-3)-{\mu}\log(J)+\frac{\lambda}{2}\log^{2}(J) $$ From this definition, we can easily compute $$ \mathbf{P}(\mathbf{F})=\mu(\mathbf{F}-{\mathbf{F}^{-T}})+\lambda\log(J)\mathbf{F}^{-T} $$

Linear tetrahedral elements

Tetrahedral meshes are among the most popular discrete volumetric geometry representations. where $$ \mathbf{D}_s= \begin{bmatrix} x_1-x_0 & x_2-x_0 & x_3-x_0\ y_1-y_0 & y_2-y_0 & y_3-y_0\ z_1-z_0 & z_2-z_0 & z_3-z_0\ \end{bmatrix} $$ is the deformed shape matrix and $$ \mathbf{D}_m= \begin{bmatrix} X_1-X_0 & X_2-X_0 & X_3-X_0\ Y_1-Y_0 & Y_2-Y_0 & Y_3-Y_0\ Z_1-Z_0 & Z_2-Z_0 & Z_3-Z_0\ \end{bmatrix} $$ is called the reference shape matrix. $$ \mathbf{D}_s = \mathbf{F}\mathbf{D}_m $$ $$ \mathbf{F}=\mathbf{D}_s\mathbf{D}_m^{-1} $$ $$ \mathbf{H}= \begin{bmatrix}
\vec{f_1} & \vec{f_2} & \vec{f_3} \end{bmatrix} =-W\mathbf{P}(\mathbf{F})\mathbf{D}m^{-T} $$ $$ \vec{f{4}}=-\vec{f_1}-\vec{f_2}-\vec{f_3} $$ $$ \mu=\frac{k}{2(1+\nu)} $$ $$ \lambda=\frac{k\nu}{(1+\nu)(1-2\nu)} $$