5 Constitutive Equations using conjugate pairs for nonlinear elastic materials with large deformations: Hyper-elasticity

5.1 Introduction

Formulating the constitutive relation for a material seeks a formula that relates the stress measure to the strain measure. Therefore, using a specific stress measure, the correct strain measure must be used.

Therefore the problem at hand is the following: Given a stress tensor, one of the many stress tensors discussed earlier, how to determine the correct strain tensor to use with it?

To make the discussion general, the stress tensor is designated by \(\mathbf {\tilde {B}}\) and its conjugate pair, the strain tensor, by \(\mathbf {\tilde {A}}\).

The stress measure \(\mathbf {\tilde {B}}\) could be any of the stress measures discussed earlier, such as the Cauchy stress tensor \(\mathbf {\tilde {\tau }}\), the second Piola-kirchhoff stress tensor \(\mathbf {\tilde {s}}_{1}\). Now the strain tensor to use is determined. Let \(\left ( \mathbf {\tilde {B},\tilde {A}}\right )\) be the conjugate pair tensors.

Physics is used in finding of \(\mathbf {\tilde {A}}\) for each specific \(\mathbf {\tilde {B}}\)

Let the current amount of energy stored in a unit volume as a result of the body undergoing deformation be \(W\), then the time rate at which this energy changes will be equal to the stress multiplied by the strain rate. Hence

\[ \boxed { \dot {W}=\mathbf {\tilde {B}\colon }\frac {\partial \mathbf {\tilde {A}} }{\partial t} } \]

Where \(\colon \) is the trace matrix operator. This is the rule used to determine \(\mathbf {\tilde {A}}\).

On a stress-strain diagram the following is drawn

The strain measure \(\mathbf {\tilde {A}}\) (the conjugate pair for the stress measure \(\mathbf {\tilde {B}}\)) must satisfy the relation

\begin{align*} \frac {\partial W}{\partial t} & =\frac {\partial W}{\partial \mathbf {\tilde {A}}}:\frac {\partial \mathbf {\tilde {A}}}{\partial t}\\ \dot {W} & =\mathbf {\tilde {B}}:\mathbf {\dot {A}}\end{align*}

For each stress/strain conjugate pair, the terms \(\frac {\partial W}{\partial t},\frac {\partial \mathbf {\tilde {A}}}{\partial t},\mathbf {\tilde {A}}\) are derived.

5.2 Conjugate pair for Cauchy stress tensor

In the deformed state, the stress tensor is the true stress tensor, which is the cauchy stress \(\mathbf {\tilde {\tau }}\), and the strain rate in this state is known to be [2] \[ \frac {1}{2}\left ( \mathbf {\tilde {e}+\tilde {e}}^{T}\right ) \]

Where \(\mathbf {\tilde {e}}\) is the velocity gradient tensor. It is shown in [2] that \[ \fbox {$\mathbf {\tilde {e}}=\mathbf {\dot {F}\cdot \tilde {F}}^{-1}$}\]

Hence in the deformed state

\[ \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \mathbf {\dot {F}\cdot \tilde {F}}^{-1}+\mathbf {\tilde {F}}^{-T}\mathbf {\cdot \dot {F}}^{-1}\right ) \]

In other words, the conjugate strain for the cauchy stress tensor is given by \(\mathbf {\tilde {A}}\) such that \[ \frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{2}\left ( \mathbf {\dot {F}\cdot \tilde {F}}^{-1}+\mathbf {\tilde {F}}^{-T}\mathbf {\cdot \dot {F}}^{-1}\right ) \]

\(\mathbf {\tilde {A}}\) should come out to be the Almansi strain tensor, which is \[ \fbox {$\mathbf {\tilde {A}}=\frac {1}{2}\left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}}^{-1}-\mathbf {\tilde {I}}\right ) $}\] (check)

5.3 Conjugate pair for second Piola-kirchhoff stress tensor \(\tilde {s}_{1}\)

\begin{align*} \dot {W} & =\mathbf {\tilde {B}\colon }\frac {\partial \mathbf {\tilde {A}}}{\partial t}\\ & =\mathbf {\tilde {\tau }\colon \tilde {e}}\end{align*}

Pre dot multiplying \(\mathbf {\tilde {e}}\) by \(\mathbf {\tilde {I}=}\left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}}^{T}\right ) \) and post dot multiplying it with \(\mathbf {\tilde {I}=}\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right )\) which will make no change in the value, results in

\[ \dot {W}=\mathbf {\tilde {\tau }\colon \left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}}^{T}\right ) \cdot \tilde {e}\cdot }\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right ) \]

Using the properties of \(\colon \) the above is written as

\[ \dot {W}=\left ( \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}}^{-T}\right ) \mathbf {\ \colon \ }\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {e}\cdot \tilde {F}}\right ) \]

It was determined earlier that \(\mathbf {\tilde {s}}_{1}=J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\cdot \mathbf {\tilde {F}}^{-T}\) hence \(\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}^{-T}=}\frac {\mathbf {\tilde {s}}_{1}}{J}\) hence the above equation becomes

\[ \dot {W}=\frac {\mathbf {\tilde {s}}_{1}}{J}\mathbf {\ \colon \ }\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot \tilde {e}\cdot \tilde {F}}\right ) \]

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) therefore

\begin{align*} \dot {W} & =\frac {\mathbf {\tilde {s}}_{1}}{J}\mathbf {\ \colon \ }\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \mathbf {\dot {F}\cdot \tilde {F}}^{-1}\cdot \tilde {F}}\right ) \\ & =\mathbf {\tilde {s}}_{1}\mathbf {\ \colon }\frac {1}{J}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \mathbf {\dot {F}\cdot \tilde {F}}^{-1}\cdot \tilde {F}}\right ) \end{align*}

Therefore \[ \frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{J}\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot \mathbf {\dot {F}\cdot \tilde {F}}^{-1}\cdot \tilde {F}}\right ) \]

This shows that \(\mathbf {\tilde {A}=}\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right )\), therefore \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\mathbf {\mathbf {\dot {F}}}^{T}\mathbf {\cdot \tilde {F}+\mathbf {\tilde {F}}}^{T}\mathbf {\mathbf {\cdot \dot {F}}}\)

Or

\[ \fbox {$\mathbf {\tilde {A}=}\frac {1}{2J}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right ) $}\]

The advantage in using the second Piola Kirchhoff stress tensor instead of the Cauchy or the first Piola Kirchhoff stress tensor, is that with the second Piola Kirchhoff stress tensor, calculations are performed the reference configuration (undeformed state) where the state measurements are known instead of using the deformed configuration where state measurements are not known.

5.4 Conjugate pair for first Piola-kirchhoff stress tensor \(\tilde {t}\)

\begin{align*} \dot {W} & =\mathbf {\tilde {B}\colon }\frac {\partial \mathbf {\tilde {A}}}{\partial t}\\ & =\mathbf {\tilde {\tau }\colon \tilde {e}}\end{align*}

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) hence the above becomes

\[ \dot {W}=\mathbf {\tilde {\tau }\colon \dot {F}\cdot \tilde {F}}^{-1}\]

Using the property of \(\mathbf {\colon }\) \(A\colon B\cdot C\) can be written as \(A\cdot C^{T}\colon B\) hence applying this property to the above expression gives

\[ \dot {W}=\mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}}^{-T}\mathbf {\colon \dot {F}}\]

Applying the property that \(A\cdot C^{T}\colon B\rightarrow C\cdot A\colon B^{T}\) to the above results in

\[ \dot {W}=\mathbf {\mathbf {\tilde {F}}}^{-1}\mathbf {\mathbf {\cdot }\tilde {\tau }\colon \dot {F}}^{T}\]

It was found earlier that \(\mathbf {\tilde {t}=}J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\) hence replacing this into the above gives

\[ \dot {W}=\frac {1}{J}\mathbf {\tilde {t}\colon \dot {F}}^{T}\]

This shows that \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{J}\mathbf {\dot {F}}^{T}\) therefore

\[ \fbox {$\mathbf {\tilde {A}=}\frac {1}{J}\mathbf {F}^{T}$}\]

5.5 Conjugate pair for \(\tilde {\sigma }\) Kirchhoff stress tensor

Since \(\mathbf {\tilde {\sigma }}\) is a scaled version of \(\mathbf {\tilde {\tau }}\) where

\[ \mathbf {\tilde {\sigma }=}J\ \mathbf {\tilde {\tau }}\]

It was found earlier that the strain tensor associated with \(\mathbf {\tilde {\tau }}\) is \(\frac {1}{2J}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right ) \) hence the strain tensor associated with \(\mathbf {\tilde {\sigma }}\) is \(\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right )\)

Therefore

\[ \boxed { \mathbf {\tilde {A}=}\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right ) } \]

5.6 Conjugate pair for \(\tilde {r}^{\ast }\) Biot-Lure stress tensor

\begin{align*} \dot {W} & =\mathbf {\tilde {B}\colon }\frac {\partial \mathbf {\tilde {A}}}{\partial t}\\ & =\mathbf {\tilde {\tau }\colon \tilde {e}}\end{align*}

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) hence the above becomes

\begin{align} \dot {W} & =\mathbf {\tilde {\tau }\colon \dot {F}\cdot \tilde {F}}^{-1}\nonumber \\ & =\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \mathbf {\dot {F}+\dot {F}}\right ) \mathbf {\cdot \tilde {F}}^{-1}\nonumber \\ & =\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \mathbf {\dot {F}\cdot \tilde {I}+\tilde {I}\cdot \dot {F}}\right ) \mathbf {\cdot \tilde {F}}^{-1} \tag {1}\end{align}

But \(\left ( \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {F}}\right ) =\mathbf {\tilde {I}}\). Using this the first \(\mathbf {\tilde {I}}\) in (1) above is replaced. Also \(\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right ) =\mathbf {\tilde {I}}\), and using this, the second \(\mathbf {\tilde {I}}\) in equation (1) above is replaced. Therefore (1) becomes

\begin{align*} \dot {W} & =\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \mathbf {\dot {F}\cdot \left ( \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {F}}\right ) +\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right ) \cdot \dot {F}}\right ) \mathbf {\cdot \tilde {F}}^{-1}\\ & =\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \overbrace {\mathbf {\dot {F}\cdot \mathbf {\tilde {F}}^{-1}}}\mathbf {\cdot \mathbf {\tilde {F}}+}\overbrace {\mathbf {\mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}}}\mathbf {\cdot \dot {F}}\right ) \mathbf {\cdot \tilde {F}}^{-1}\end{align*}

Switching the order of terms selected above by transposing them gives

\[ \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \overbrace {\mathbf {\mathbf {\tilde {F}}}^{-T}\mathbf {\cdot \dot {F}}^{T}}\mathbf {\cdot \mathbf {\tilde {F}}+}\overbrace {\mathbf {\tilde {F}}^{-T}\mathbf {\cdot \mathbf {\tilde {F}}}^{T}}\mathbf {\cdot \dot {F}}\right ) \mathbf {\cdot \tilde {F}}^{-1}\]

Taking \(\mathbf {\mathbf {\tilde {F}}^{-T}}\) as common factor gives

\begin{equation} \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left \{ \mathbf {\tilde {F}}^{-T}\cdot \left ( \mathbf {\mathbf {\dot {F}}}^{T}\mathbf {\cdot \mathbf {\tilde {F}}+\mathbf {\tilde {F}}}^{T}\mathbf {\cdot \dot {F}}\right ) \right \} \mathbf {\cdot \tilde {F}}^{-1} \tag {2}\end{equation}

But \[ \mathbf {\mathbf {\dot {F}}}^{T}\mathbf {\cdot \mathbf {\tilde {F}}+\mathbf {\tilde {F}}}^{T}\mathbf {\cdot \dot {F}=}\frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F}\right ) \]

Hence (2) becomes

\begin{equation} \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left \{ \mathbf {\tilde {F}}^{-T}\cdot \frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F}\right ) \right \} \mathbf {\cdot \tilde {F}}^{-1} \tag {3}\end{equation}

But \(\frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F}\right ) =\frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {U}}}^{2}\right ) \) since \(\mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F=\mathbf {\tilde {U}}}^{2}\)

Therefore (3) becomes

\begin{equation} \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left \{ \mathbf {\tilde {F}}^{-T}\cdot \frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {U}}}^{2}\right ) \right \} \mathbf {\cdot \tilde {F}}^{-1} \tag {4}\end{equation}

But \[ \frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {U}}}^{2}\right ) =2\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}}}\right ) \]

Hence (4) becomes

\begin{equation} \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left \{ \mathbf {\tilde {F}}^{-T}\cdot 2\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}}}\right ) \right \} \mathbf {\cdot \tilde {F}}^{-1} \tag {5}\end{equation}

But \[ \mathbf {\mathbf {\tilde {U}\cdot \dot {U}=\dot {U}\cdot U}}\]

From symmetry of \(\mathbf {\mathbf {U}}\) therefore \[ 2\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}}}\right ) =\mathbf {\mathbf {\tilde {U}\cdot \dot {U}+\dot {U}\cdot U}}\]

And (5) becomes

\[ \dot {W}=\mathbf {\tilde {\tau }\colon }\frac {1}{2}\left ( \mathbf {\tilde {F}}^{-T}\cdot \left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}+\dot {U}\cdot U}}\right ) \mathbf {\cdot \tilde {F}}^{-1}\right ) \]

From property of \(\mathbf {\colon }\) the above can be written as

\[ \dot {W}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}}^{-T}\mathbf {\colon }\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}+\dot {U}\cdot U}}\right ) \]

But from above, \(\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}+\dot {U}\cdot U}}\right ) =\mathbf {\mathbf {\tilde {U}\cdot \dot {U}}}\) Hence

\[ \dot {W}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}}^{-T}\mathbf {\colon \mathbf {\tilde {U}\cdot \dot {U}}}\]

Using property of \(\mathbf {\colon }\) the term \(\mathbf {\mathbf {\tilde {U}}}\) is moved to the left of \(\mathbf {\colon }\) to obtain

\[ \dot {W}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}}^{-T}\cdot \mathbf {\mathbf {\tilde {U}}\colon \mathbf {\dot {U}}}\]

But \(\mathbf {\mathbf {\tilde {F}}}^{-T}\cdot \mathbf {\mathbf {\tilde {U}=\tilde {R}}}\) hence the above becomes

\[ \dot {W}=\overbrace {\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {R}}}}\mathbf {\colon \mathbf {\dot {U}}}\]

But it was found earlier that \(\mathbf {\tilde {r}}^{\ast }=J\ \mathbf {\tilde {F}}^{-1}\mathbf {\cdot \tilde {\tau }}\cdot \mathbf {\tilde {R}}\)

Hence \(\dot {W}=\frac {1}{J}\mathbf {\tilde {r}}^{\ast }\mathbf {\colon \mathbf {\dot {U}}}\) Hence

\[ \dot {W}=\mathbf {\tilde {r}}^{\ast }\mathbf {\colon }\frac {1}{J}\mathbf {\mathbf {\dot {U}}}\]

Therefore \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{J}\mathbf {\mathbf {\dot {U}}}\) which results in

\[ \boxed {\mathbf {\tilde {A}=}\frac {1}{J}\mathbf {\mathbf {U}}} \]

5.7 Conjugate pair for \(\tilde {r}\) Jaumann stress tensor

It was found earlier that \(\mathbf {\tilde {r}=}\frac {\left ( \mathbf {\tilde {r}}^{\ast }+\mathbf {\tilde {r}}^{\ast T}\right ) }{2}\) hence the conjugate pair for \(\mathbf {\tilde {r}}\) is \(\frac {\left ( \frac {1}{J}\mathbf {\mathbf {U}}+\frac {1}{J}\mathbf {\mathbf {U}}^{T}\right ) }{2}\)

Since \(\mathbf {\mathbf {\tilde {U}}}\) is symmetrical, therefore conjugate pair for \(\mathbf {\tilde {r}}\) is \(\frac {1}{J}\mathbf {\mathbf {U}}\) Hence \[ \mathbf {\tilde {A}=}\frac {1}{J}\mathbf {\mathbf {U}}\] The same as strain tensor associated with the Biot-Lure stress.