5.3 Nearly-Incompressible Materials

Mooney-Rivlin Hyperelasticity

This material model is a hyperelastic Mooney-Rivlin type with uncoupled deviatoric and volumetric behavior. The uncoupled strain energy W is given by:

\[ \begin{equation} W=c_{1}\left(\tilde{I}_{1}-3\right)+c_{2}\left(\tilde{I}_{2}-3\right)+\frac{1}{2}K\left(\ln J\right)^{2}\,.\label{eq449} \end{equation} \]

Here, \(c_{1}\) and \(c_{2}\) are the Mooney-Rivlin material coefficients, \(\tilde{I}_{1}\) and \(\tilde{I}_{2}\) are the invariants of the deviatoric part of the right Cauchy-Green deformation tensor, \(\mathbf{\tilde{C}}=\mathbf{\tilde{F}}^{T}\cdot\mathbf{\tilde{F}}\), where \(\mathbf{\tilde{F}}=J^{-1/3}\mathbf{F}\), _\(\mathbf{F}\)_ is the deformation gradient and \(J=\det\mathbf{F}\) is the Jacobian of the deformation. When \(c_{2}=0\), this model reduces to an uncoupled version of the incompressible neo-Hookean constitutive model.

The Cauchy stress is given by

\[ \begin{equation} \boldsymbol{\sigma}=p\mathbf{I}+\frac{2}{J}\left[\left(c_{1}+c_{2}\tilde{I}_{1}\right)\mathbf{\tilde{b}}-c_{2}\mathbf{\tilde{b}}^{2}-\frac{1}{3}\left(c_{1}\tilde{I}_{1}+2c_{2}\tilde{I}_{2}\right)\mathbf{I}\right]\,.\label{eq450} \end{equation} \]

The spatial elasticity tensor is given by

\[ \begin{equation} \boldsymbol{\mathcal{C}}=p\left(\mathbf{I}\otimes\mathbf{I}-2\mathbf{I}\odot\mathbf{I}\right)-\frac{2}{3}\left(\dev\boldsymbol{\sigma}\otimes\mathbf{I}+\mathbf{I}\otimes\dev\boldsymbol{\sigma}\right)+\boldsymbol{\mathcal{C}}_{w}\,,\label{eq451} \end{equation} \]

where,

\[ \begin{equation} \begin{aligned}\boldsymbol{\mathcal{C}}_{w} & =\frac{4}{3J}\left(c_{1}\tilde{I}_{1}+2c_{2}\tilde{I}_{2}\right)\left(\mathbf{I}\odot\mathbf{I}-\frac{1}{3}\mathbf{I}\otimes\mathbf{I}\right)+\frac{4c_{2}}{J}\left(\mathbf{\tilde{b}}\otimes\mathbf{\tilde{b}}-\mathbf{\tilde{b}}\odot\mathbf{\tilde{b}}\right)\\ & -\frac{4c_{2}}{3J}\left[\left(\tilde{I}_{1}\mathbf{\tilde{b}}-\mathbf{\tilde{b}}^{2}\right)\otimes\mathbf{I}+\mathbf{I}\otimes\left(\tilde{I}_{1}\mathbf{\tilde{b}}-\mathbf{\tilde{b}}^{2}\right)\right]+\frac{8c_{2}\tilde{I}_{2}}{9J}\mathbf{I}\otimes\mathbf{I}\,. \end{aligned} \label{eq452} \end{equation} \]

This material model uses a three-field element formulation, interpolating displacements as linear field variables and pressure and volume ratio as piecewise constant in each element ¹.

Ogden Hyperelastic

The Ogden material is defined using the following hyperelastic strain energy function:

\[ \begin{equation} W\left(\lambda_{1},\lambda_{2},\lambda_{3},J\right)=\sum\limits_{i=1}^{N}\frac{c_{i}}{m_{i}^{2}}\left(\tilde{\lambda}_{1}^{m_{i}}+\tilde{\lambda}_{2}^{m_{i}}+\tilde{\lambda}_{3}^{m_{i}}-3\right)+U\left(J\right)\,.\label{eq453} \end{equation} \]

Here, \(\tilde{\lambda}_{i}\) are the deviatoric principal stretches and \(c_{i}\) and \(m_{i}\) are material parameters. The term \(U\left(J\right)\) is the volumetric component and \(J\) is the determinant of the deformation gradient.

Note that the neo-Hookean and Mooney-Rivlin models can also be obtained from the general Ogden strain energy function using special choices for \(c_{i}\) and \(m_{i}\).

Veronda-Westmann Hyperelasticity

This model is similar to the Mooney-Rivlin model in that it also uses an uncoupled strain energy. However, in this case the strain energy is given by an exponential form:

\[ \begin{equation} W=C_{1}\left[e^{\left(C_{2}\left(\tilde{I}_{1}-3\right)\right)}-1\right]-\frac{C_{1}C_{2}}{2}\left(\tilde{I}_{2}-3\right)+U\left(J\right)\,.\label{eq454} \end{equation} \]

The dilatational term \(U\) is identical to the Mooney-Rivlin model.

The Cauchy stress \(\boldsymbol{\sigma}\) is found from

\[ \begin{equation} \boldsymbol{\sigma}=p\mathbf{I}+\dev\tilde{\boldsymbol{\sigma}}\,,\label{eq455} \end{equation} \]

where

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=\frac{2}{J}\left[\left(W_{1}+I_{1}W_{2}\right)\tilde{\mathbf{b}}-W_{2}\tilde{\mathbf{b}}^{2}\right]\,.\label{eq456} \end{equation} \]

The strain energy derivatives are given by

\[ \begin{equation} W_{1}=C_{1}C_{2}e^{C_{2}\left(I_{1}-3\right)}\,,\label{eq457} \end{equation} \]

\[ \begin{equation} W_{2}=-\frac{C_{1}C_{2}}{2}\,.\label{eq458} \end{equation} \]

This material model was the result from the research of the elastic response of skin tissue ².

Arruda-Boyce Hyperelasticity

Arruda and Boyce proposed a model for the deformation of rubber materials ³. Their main motivation was to develop a model that accurately captures the behavior of rubbers in different loading scenarios and that can be described with a limited number of physically motivated parameters. Their model is based on the Langevin chain statistics, which models a rubber chain segment between chemical crosslinks as a number \(N\) of rigid links of equal length \(l\). The parameter \(N\) is related to the locking stretch \(\lambda_{L}\), the stretch at which the chains reach their full extended state, \(\lambda_{L}=\sqrt{N}\).

Their proposed strain-energy is a truncated Taylor series of the inverse Langevin function. A formulation that retains the first five terms of this function takes on the following form:

\[ \begin{equation} \tilde{W}=\mu\sum\limits_{i=1}^{5}\frac{\alpha_{i}}{N^{i-1}}\left(\tilde{I}_{1}^{i}-3^{i}\right)+U\left(J\right)\,,\label{eq459} \end{equation} \]

where \(\mu\) is a shear-modulus like parameter and the coefficients \(\alpha_{i}\) are

\[ \begin{equation} \alpha_{1}=\frac{1}{2}\,,\quad\alpha_{2}=\frac{1}{20}\,,\quad\alpha_{3}=\frac{11}{1050}\,,\quad\alpha_{4}=\frac{19}{7000}\,,\quad\alpha_{5}=\frac{519}{673750}\,.\label{eq460} \end{equation} \]

The Cauchy stress is given by

\[ \begin{equation} \boldsymbol{\sigma}=p\mathbf{I}+\frac{2}{J}\dev\left(W_{1}\mathbf{\tilde{b}}\right)=p\mathbf{I}+\frac{2W_{1}}{J}\left(\mathbf{\tilde{b}}-\frac{1}{3}\tilde{I}_{1}\mathbf{I}\right)\,,\label{eq461} \end{equation} \]

where,

\[ \begin{equation} W_{1}=\frac{\partial\tilde{W}}{\partial\tilde{I}_{1}}=\mu\sum\limits_{i=1}^{5}\alpha_{i}i\left(\frac{\tilde{I}_{1}}{N}\right)^{i-1}\,.\label{eq462} \end{equation} \]

Transversely Isotropic Hyperelastic

This constitutive model can be used to represent a material that has a single preferred fiber direction and was developed for application to biological soft tissues ⁴⁵⁶. It can be used to model tissues such as tendons, ligaments and muscle. The elastic response of the tissue is assumed to arise from the resistance of the fiber family and an isotropic matrix. It is assumed that the strain energy function can be written as follows:

\[ \begin{equation} W=F_{1}\left(\tilde{I}_{1},\tilde{I}_{2}\right)+F_{2}\left(\tilde{\lambda}\right)+\frac{K}{2}\left[\ln\left(J\right)\right]^{2}\,.\label{eq463} \end{equation} \]

Here, \(\tilde{I}_{1}\) and \(\tilde{I}_{2}\) are the first and second invariants of the deviatoric version of the right Cauchy Green deformation tensor \(\mathbf{\tilde{C}}\) and \(\tilde{\lambda}\) is the deviatoric part of the stretch along the fiber direction (\(\tilde{\lambda}^{2}=\mathbf{A}\cdot\mathbf{\tilde{C}}\cdot\mathbf{A}\), where \(\mathbf{A}\) is the initial fiber direction). The function \(F_{1}\) represents the material response of the isotropic ground substance matrix, while \(F_{2}\)represents the contribution from the fiber family. The strain energy of the fiber family is as follows:

\[ \begin{equation} \tilde{\lambda}\frac{\partial F_{2}}{\partial\tilde{\lambda}}=\begin{cases} 0 & \tilde{\lambda}\leqslant1\\ C_{3}\left(e^{C_{4}\left(\tilde{\lambda}-1\right)}-1\right) & 1<\tilde{\lambda}<\lambda_{m}\\ C_{5}+C_{6}\tilde{\lambda} & \tilde{\lambda}\geqslant\lambda_{m} \end{cases}\,.\label{eq464} \end{equation} \]

Here, \(\lambda_{m}\) is the stretch at which the fibers are straightened, \(C_{3}\) scales the exponential stresses, \(C_{4}\) is the rate of uncrimping of the fibers, and \(C_{5}\) is the modulus of the straightened fibers. \(C_{6}\) is determined from the requirement that the stress is continuous at \(\lambda_{m}\),

\[ \begin{equation} C_{6}=\frac{1}{\lambda_{m}}\left[C_{3}\left(e^{C_{4}\left(\lambda_{m}-1\right)}-1\right)-C_{5}\right]\,.\label{eq464b} \end{equation} \]

It also follows that

\[ \begin{equation} F_{2}\left(\tilde{\lambda}\right)=\begin{cases} 0 & \tilde{\lambda}\leqslant1\\ C_{3}\left(e^{-C_{4}}\left[\Ei\left(C_{4}\tilde{\lambda}\right)-\Ei\left(C_{4}\right)\right]-\ln\tilde{\lambda}\right) & 1<\tilde{\lambda}<\lambda_{m}\\ \left(C_{5}+\frac{C_{6}}{2}\tilde{\lambda}\right)\tilde{\lambda}+C_{7} & \tilde{\lambda}\geqslant\lambda_{m} \end{cases}\,,\label{eq464c} \end{equation} \]

where

\[ \begin{equation} C_{7}=C_{3}\left(e^{-C_{4}}\left[\Ei\left(C_{4}\lambda_{m}\right)-\Ei\left(C_{4}\right)\right]-\ln\lambda_{m}\right)-\left(C_{5}+\frac{C_{6}}{2}\lambda_{m}\right)\lambda_{m}\,.\label{eq464d} \end{equation} \]

This material model uses a three-field element formulation, interpolating displacements as linear field variables and pressure and volume ratio as piecewise constant on each element ¹.

Ellipsoidal Fiber Distribution

This constitutive model describes a material that is composed of an ellipsoidal continuous fiber distribution in an uncoupled formulation. The deviatoric part of the stress is given by ⁷⁸⁹,

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(\tilde{I}_{n}-1\right)\tilde{\boldsymbol{\sigma}}_{n}\left(\mathbf{n}\right)\sin\varphi\,d\varphi\,d\theta\,,\label{eq465} \end{equation} \]

and the corresponding elasticity tensor is

\[ \begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(\tilde{I}_{n}-1\right)\tilde{\boldsymbol{\mathcal{C}}}_{n}\left(\mathbf{n}\right)\sin\phi\,d\phi\,d\theta\,.\label{eq466} \end{equation} \]

\(\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{N}\cdot\mathbf{\tilde{C}}\cdot\mathbf{N}\) is the square of the fiber stretch \(\mathbf{F}\), \(\mathbf{N}\) is the unit vector along the fiber direction (in the reference configuration), which in spherical angles is directed along \(\left(\theta,\varphi\right)\), \(\mathbf{n}=\mathbf{\tilde{F}}\cdot\mathbf{N}/\tilde{\lambda}_{n}\) and \(H\left(\cdot\right)\) is the unit step function that enforces the tension-only contribution. The fiber stress is determined from a fiber strain energy function in the usual manner:

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}_{n}\left(\mathbf{n}\right)=2J^{-1}\tilde{I}_{n}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}\,,\label{eq467} \end{equation} \]

whereas the fiber elasticity tensor is

\[ \begin{equation} \tilde{\boldsymbol{\mathcal{C}}}_{n}\left(\mathbf{n}\right)=4J^{-1}\tilde{I}_{n}^{2}\frac{\partial^{2}\tilde{\Psi}}{\partial\tilde{I}_{n}^{2}}\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\,,\label{eq468} \end{equation} \]

where in this material

\[ \begin{equation} \tilde{\Psi}\left(\mathbf{n},\tilde{I}_{n}\right)=\xi\left(\mathbf{n}\right)\left(\tilde{I}_{n}-1\right)^{\beta\left(\mathbf{n}\right)}\,.\label{eq469} \end{equation} \]

The materials parameters \(\beta\) and \(\xi\) are determined from:

\[ \begin{equation} \begin{aligned}\xi\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\xi_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\xi_{2}^{2}}+\frac{\cos^{2}\varphi}{\xi_{3}^{2}}\right)^{-1/2}\,,\\ \beta\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\beta_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\beta_{2}^{2}}+\frac{\cos^{2}\varphi}{\beta_{3}^{2}}\right)^{-1/2}\,. \end{aligned} \label{eq470} \end{equation} \]

Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a material that acts as the ground matrix. The total stress is then given by the sum of the fiber stress and the ground matrix stress:

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=\tilde{\boldsymbol{\sigma}}_{m}+\tilde{\boldsymbol{\sigma}}_{f}\,.\label{eq471} \end{equation} \]

Fiber with Exponential Power Law Uncoupled

This material model describes a constitutive model for fibers, where a single fiber family follows an exponential power law strain energy function. The deviatoric part of the Cauchy stress is given by:

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=2J^{-1}H\left(\tilde{I}_{n}-1\right)\tilde{I}_{n}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}\,,\label{eq472} \end{equation} \]

and the corresponding spatial elasticity tensor is

\[ \begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=4J^{-1}H\left(\tilde{I}_{n}-1\right)\tilde{I}_{n}^{2}\frac{\partial^{2}\tilde{\Psi}}{\partial\tilde{I}_{n}^{2}}\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\,,\label{eq473} \end{equation} \]

where \(\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{N}\cdot\mathbf{\tilde{C}}\cdot\mathbf{N}\) is the square of the fiber stretch, \(\mathbf{N}\) is the fiber orientation in the reference configuration,

\[ \begin{equation} \mathbf{N}=\sin\varphi\cos\theta\,\mathbf{e}_{1}+\sin\varphi\sin\theta\,\mathbf{e}_{2}+\cos\varphi\,\mathbf{e}_{3}\,,\label{eq474} \end{equation} \]

and \(\mathbf{n}=\mathbf{\tilde{F}}\cdot\mathbf{N}/\tilde{\lambda}_{n}\) and \(H\left(\cdot\right)\) is the unit step function that enforces the tension-only contribution. The fiber strain energy density is given by

\[ \begin{equation} \tilde{\Psi}=\frac{\xi}{\alpha\beta}\left(\exp\left[\alpha\left(\tilde{I}_{n}-1\right)^{\beta}\right]-1\right)\,,\label{eq475} \end{equation} \]

where \(\xi>0\), \(\alpha\geqslant0\)and \(\beta\geqslant2\).

Note: In the limit when \(\alpha\to0\), this expression produces a power law,

\[ \begin{equation} \lim\limits_{\alpha\to0}\tilde{\Psi}=\frac{\xi}{\beta}\left(\tilde{I}_{n}-1\right)^{\beta}\,.\label{eq476} \end{equation} \]

Note: When \(\beta>2\), the fiber modulus is zero at the strain origin (\(\tilde{I}_{n}=1)\). Therefore, use \(\beta>2\)when a smooth transition in the stress is desired from compression to tension.

Fung Orthotropic

The hyperelastic strain energy function for a Fung Orthotropic material is given by ¹⁰¹¹

\[ \begin{equation} \tilde{\Psi}=\frac{1}{2}c\left(e^{\tilde{Q}}-1\right)+U\left(J\right)\,,\label{eq477} \end{equation} \]

where

\[ \begin{equation} \tilde{Q}=c^{-1}\sum\limits_{a=1}^{3}\left[2\mu_{a}\mathbf{M}_{a}:\mathbf{\tilde{E}}^{2}+\sum\limits_{b=1}^{3}\lambda_{ab}\left(\mathbf{M}_{a}:\mathbf{\tilde{E}}\right)\left(\mathbf{M}_{b}:\mathbf{\tilde{E}}\right)\right]\,.\label{eq478} \end{equation} \]

Here, \(\mathbf{\tilde{E}}=\frac{1}{2}\left(\mathbf{\tilde{C}}-\mathbf{I}\right)\) and \(\mathbf{M}_{a}=\mathbf{A}_{a}\otimes\mathbf{A}_{a}\), where \(\mathbf{A}_{a}\) are orthonormal vectors that define the initial direction of material axes. The orthotropic Lamé coefficients should be chosen such that the stiffness matrix,

\[ \begin{equation} \left[\begin{array}{cccccc} \lambda_{11}+2\mu_{1} & \lambda_{12} & \lambda_{13} & 0 & 0 & 0\\ \lambda_{12} & \lambda_{22}+2\mu_{2} & \lambda_{23} & 0 & 0 & 0\\ \lambda_{13} & \lambda_{23} & \lambda_{33}+2\mu_{3} & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{2}\left(\mu_{1}+\mu_{2}\right) & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{2}\left(\mu_{2}+\mu_{3}\right) & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2}\left(\mu_{1}+\mu_{3}\right) \end{array}\right]\label{eq479} \end{equation} \]

is positive definite.

Tension-Compression Nonlinear Orthotropic

This material model is based on the following uncoupled hyperelastic strain energy function ¹²:

\[ \begin{equation} \Psi\left(\mathbf{C},\lambda_{1},\lambda_{2},\lambda_{3}\right)=\tilde{\Psi}_{\text{iso}}\left(\mathbf{\tilde{C}}\right)+\sum\limits_{i=1}^{3}\tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)+U\left(J\right)\,.\label{eq480} \end{equation} \]

The isotropic strain energy \(\tilde{\Psi}_{\text{iso}}\) and the dilatational energy \(U\) are the same as for the Mooney-Rivlin material (Section Mooney-Rivlin Hyperelasticity). The tension-compression term is defined as follows:

\[ \begin{equation} \tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)=\begin{cases} \xi_{i}\left(\tilde{\lambda}_{i}-1\right)^{\beta_{i}} & \tilde{\lambda}_{i}>1\\ 0 & \tilde{\lambda}_{i}\leqslant1 \end{cases}\,,\xi_{i}\geqslant0\quad\mbox{(no sum over }i\text{).}\label{eq481} \end{equation} \]

The \(\tilde{\lambda}_{i}\) parameters are the deviatoric fiber stretches of the local material fibers,

\[ \begin{equation} \tilde{\lambda}_{i}=\left(\mathbf{A}_{i}\cdot\mathbf{\tilde{C}}\cdot\mathbf{A}_{i}\right)^{1/2}\,.\label{eq482} \end{equation} \]

The local material fibers are defined (in the reference frame) as an orthonormal set of vectors \(\mathbf{A}_{i}\). The corresponding deviatoric part of the Cauchy stress is

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=J^{-1}\sum\limits_{i=1}^{3}\frac{1}{\tilde{\lambda}_{i}}\frac{\partial\tilde{\Psi}}{\partial\tilde{\lambda}_{i}}\mathbf{a}_{i}\otimes\mathbf{a}_{i}\,,\label{eq483} \end{equation} \]

and the spatial elasticity tensor is

\[ \begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=J^{-1}\sum\limits_{i=1}^{3}\frac{1}{\tilde{\lambda}_{i}}\frac{\partial}{\partial\tilde{\lambda}_{i}}\left(\frac{1}{\tilde{\lambda}_{i}}\frac{\partial\tilde{\Psi}}{\partial\tilde{\lambda}_{i}}\right)\mathbf{a}_{i}\otimes\mathbf{a}_{i}\otimes\mathbf{a}_{i}\otimes\mathbf{a}_{i}\,,\label{eq484} \end{equation} \]

where \(\mathbf{a}_{i}=\mathbf{\tilde{F}}\cdot\mathbf{A}_{i}\).

Holmes-Mow Uncoupled

The uncoupled hyperelastic strain-energy function for this material is given by ¹³,

\[ \begin{equation} \Psi\left(\mathbf{C}\right)=\tilde{\Psi}\left(\tilde{\mathbf{C}}\right)+U\left(J\right)\,,\label{eq:HMU-SED} \end{equation} \]

where

\[ \begin{equation} \tilde{\Psi}\left(\tilde{\mathbf{C}}\right)=\frac{1}{2}\frac{\mu}{\beta}\left(e^{\tilde{Q}}-1\right)\,,\label{eq:HMU-DSED} \end{equation} \]

and

\[ \begin{equation} \tilde{Q}=\beta\left(\tilde{I}_{1}-3\right)\,.\label{eq:HMU-Q} \end{equation} \]

Here, \(\mu\) is the shear modulus and \(\beta\) is the exponential nonlinearity coefficient. The corresponding spatial stress and elasticity tensors are

\[ \begin{equation} \tilde{\boldsymbol{\sigma}}=\frac{\mu}{J}e^{\tilde{Q}}\tilde{\mathbf{b}}\,,\label{eq:HMU-stress} \end{equation} \]

and

\[ \begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=2\beta\frac{\mu}{J}e^{\tilde{Q}}\tilde{\mathbf{b}}\otimes\tilde{\mathbf{b}}\label{eq:HMU-elasticity} \end{equation} \]

respectively. Note that \(\tilde{\boldsymbol{\sigma}}\) does not reduce to zero when \(\tilde{\mathbf{b}}=\mathbf{I}\), but \(\dev\tilde{\boldsymbol{\sigma}}\) does. These expressions can be substituted into (2.6-61) and (2.6-68) to evaluate the final expressions for the Cauchy stress and spatial elasticity tensors, respectively.

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