Continuum in Lagrangian Framework

Total Lagrangian Formulation for Solid Mechanics

Considering continuum mechanics in the total Lagrangian framework, the kinematics and dynamic equations are expressed in terms of the initial, undeformed reference configuration $\Omega^0 \subset \boldsymbol{R}^d$ with $d$ denoting the dimension. A deformation map $\varphi$ between the initial configuration $\Omega^0$ and current deformed configuration $\Omega = \varphi \left( \Omega^0 \right)$ describes the body deformation at time $t$ as

\[\begin{equation} \boldsymbol{r} = \varphi \left( \boldsymbol{r}^0, t \right), \label{deformation-map} \end{equation}\]

where $\boldsymbol{r}^0$ and $\boldsymbol{r}$ are the initial and current positions of a material point, respectively. Subsequently, the deformation gradient tensor $\boldsymbol{F}$ is given by

\[\begin{equation} \boldsymbol{F} = \nabla^0 \boldsymbol{r} = \nabla^0 \boldsymbol{u} + \boldsymbol{I}, \label{deformation-gradient} \end{equation}\]

where $\boldsymbol{u} = \boldsymbol{r} - \boldsymbol{r}^0$ is the displacement, $\nabla^0 \equiv \frac{\partial}{\partial \boldsymbol{r}^0}$ the gradient operator with respect to the initial configuration $\Omega^0$ and $\boldsymbol{I} $ the identity matrix.

The conservation equations for mass and momentum in the total Lagrangian formulation can be expressed as

\[\begin{equation} \begin{cases} \rho = J^{-1}\rho^0 \\ \rho^0 \ddot {\boldsymbol{u}} = \nabla^0 \cdot \boldsymbol{P}^{\operatorname{T}}, \end{cases} \label{conservation-equations} \end{equation}\]

where $\rho^0$ and $\rho$ are the initial and current densities, respectively, $J = \det(\boldsymbol{F})$, $\ddot {\boldsymbol{u}}$ the acceleration, $\boldsymbol{P}$ the first Piola-Kirchhoff stress tensor, and $\operatorname{T}$ the matrix transposition operator. While $\boldsymbol{P}$ can be obtained directly by

\[\begin{equation} \boldsymbol{P} = \boldsymbol{F}\boldsymbol{S}, \label{piola-stress} \end{equation}\]

where $\boldsymbol{S}$ is the second Piola-Kirchhoff stress tensor, $\boldsymbol{P}$ is obtained by the alternative Kirchhoff-stress approach in this work as

\[\begin{equation} \boldsymbol{P} = \boldsymbol{\tau}\boldsymbol{F}^{-\operatorname{T}}. \label{kirchhoff-stress} \end{equation}\]

Here, the Kirchhoff stress $\boldsymbol{\tau}$ is decomposed into volumetric and deviatoric components, and can be derived form the following strain energy function \cite{simo2006computational}

\[\begin{equation} \mathfrak{W}_e = \mathfrak{W}_v \left( J \right) + \mathfrak{W}_s \left(\bar {\boldsymbol{b}} \right). \label{strain-energy} \end{equation}\]

Here, the volume-preserving left Cauchy-Green deformation gradient tensor $\bar {\boldsymbol{b}} = J^ {-\frac{2}{d}} \boldsymbol{b} = \left| \boldsymbol{b} \right|^{ - \frac{1}{d}} \boldsymbol{b}$ with $\boldsymbol{b} = \boldsymbol{F}\boldsymbol{F}^{\operatorname{T}}$. For neo-Hookean materials, the volume-dependent strain energy $\mathfrak{W}_v \left( J \right)$ weighted by the bulk modulus $K$ can be expressed as

\[\begin{equation} \mathfrak{W}_v \left( J \right) = \frac{1}{2}K\left[ \frac{1}{2}\left( J^2 - 1 \right) - \ln J \right], \label{volumetric-energy} \end{equation}\]

whereas the shear-dependent strain energy $\mathfrak{W}_s \left(\bar {\boldsymbol{b}} \right)$ weighted by the shear modulus $G$ \cite{yue2015continuum} is given by

\[\begin{equation} \mathfrak{W}_s \left( \bar{ \boldsymbol{b}} \right) = \frac{1}{2} G \left( \operatorname{tr} \left( \bar {\boldsymbol{b}} \right) - d \right). \label{deviatoric-energy} \end{equation}\]

Then, the Kirchhoff stress tensor $\boldsymbol{\tau}$ can be derived as

\[\begin{equation} \boldsymbol{\tau} = \frac{\partial \mathfrak{W}_e}{\partial \boldsymbol{F} } \boldsymbol{F}^{\operatorname{T}} = \frac{K}{2}\left( J^2 - 1 \right) \boldsymbol{I} + G \operatorname{dev} \left( \bar {\boldsymbol{b}} \right), \label{Kirchhoff_stress} \end{equation}\]

where \begin{equation} \operatorname{dev} \left( \bar{ \boldsymbol{b}} \right) = \bar {\boldsymbol{b}} - \frac{1}{d} \operatorname{tr} \left( \bar{ \boldsymbol{b}} \right) \mathbb{I} = J^ {-\frac{2}{d}} \left[ \boldsymbol{b} - \frac{1}{d} \operatorname{tr} \left( \boldsymbol{b} \right) \mathbb{I} \right]. \label{deviatoric-stress} \end{equation}

The deviatoric operator $\operatorname{dev}\left( \bar{ \boldsymbol{b}} \right)$ returns the trace-free part of $\bar{ \boldsymbol{b}}$, i.e., $\operatorname{tr} \left( \operatorname{dev}\left( \bar{ \boldsymbol{b}} \right) \right)$ is equal to zero. Note that while the volumetric component of the constitutive Eq. $\eqref{Kirchhoff_stress}$ can be modified depending on the material property and all counterparts are appropriate for this study, only the Eq. $\eqref{Kirchhoff_stress}$ is utilized in this study.