On anisotropic elasticity and questions concerning its Finite Element implementation

Abstract

We give conditions on the strain–energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric–deviatoric separation of the strain–energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric–deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere.

Appendix A: monoclinic elasticity

Here we extend the results of Sects. 2 and 3 to the case of monoclinic symmetry, for which, in the linear theory, there are 13 independent elastic constants. For this purpose it suffices to consider an isotropic matrix material reinforced with two families of fibres, with fibre directions defined by $\varvec{M}$ and $\varvec{M}^{\prime }$ in the reference configuration, the fibres being in general neither at right angles nor mechanically equivalent. First we establish the general equations of compatibility between nonlinear and linear anisotropic elasticity.

With two fibre directions, the strain–energy function $W$ is a function of three isotropic strain invariants ($I_1,\,I_2,\,I_3$) and five anisotropic invariants ($I_4,\,I_5,\,I_6,\,I_7,\,I_8).$ Thus,

$$\begin{aligned} W = W\left( I_1,\,I_2,\,I_3,\,I_4,\,I_5,\,I_6,\,I_7,\,I_8\right) , \end{aligned}$$

(55)

where

$$\begin{aligned} I_1&= \text{ tr }\,\mathbf C ,\quad I_2=\frac{1}{2}\left[ (\text{ tr }\,\varvec{C})^2 -\text{ tr }\left( \mathbf C ^2\right) \right] ,\nonumber \\ I_3&= \det \mathbf C ,\quad I_4=\varvec{M} \cdot \mathbf C \varvec{M},\quad I_5=\varvec{M}\cdot \mathbf C ^2 \varvec{M},\nonumber \\ I_6&= \varvec{M}^{\prime } \cdot \mathbf C \varvec{M}^{\prime }, \quad I_7=\varvec{M}^{\prime }\cdot \mathbf C ^2 \varvec{M}^{\prime },\quad I_8=\varvec{M} \cdot \mathbf C \varvec{M}^{\prime }. \end{aligned}$$

(56)

Note that, to simplify the ensuing analysis, we are using the invariant $I_8$ as defined above, rather than one of its strictly invariant forms $I_8\varvec{M}\cdot \varvec{M}^{\prime }$ or $I_8^2,$ which do not depend on the sense of either $\varvec{M}$ or $\varvec{M}^{\prime }.$

For the expression for the Cauchy stress $\varvec{\sigma }$ in the incompressible case we refer to Merodio and Ogden [13]. Here we use its compressible counterpart

$$\begin{aligned} J\varvec{\sigma }&= 2W_1\mathbf B +2W_2\left( I_1\mathbf B -\mathbf B ^2\right) +2I_3W_3\mathbf I \nonumber \\&+\,2W_4\varvec{m}\otimes \varvec{m}+2W_5(\mathbf B \varvec{m}\otimes \varvec{m} +\varvec{m}\otimes \mathbf B \varvec{m})\nonumber \\&+\,2W_6\varvec{m}^{\prime }\otimes \varvec{m}^{\prime } +2W_7(\mathbf B \varvec{m}^{\prime }\otimes \varvec{m}^{\prime } +\varvec{m}^{\prime }\otimes \mathbf B \varvec{m}^{\prime })\nonumber \\&+W_8(\varvec{m}\otimes \varvec{m}^{\prime }+\varvec{m}^{\prime }\otimes \varvec{m}). \end{aligned}$$

(57)

In the reference configuration the invariants take the values

$$\begin{aligned} I_1=I_2=3,~~ I_3=I_4=I_5=I_6=I_7=1,~~ I_8= \varvec{M}\cdot \varvec{M}^{\prime }.\!\!\!\!\!\!\nonumber \\ \end{aligned}$$

(58)

Assuming that the reference configuration is stress free, it follows from (57), when evaluated in the reference configuration, that the conditions

$$\begin{aligned}&W_1+2W_2+W_3=0,\nonumber \\&W_4+2W_5=0, \quad W_6+2W_7=0,\quad W_8=0, \end{aligned}$$

(59)

must hold there.

Now let $\mathbf e$ be the infinitesimal strain tensor and let $e=\text{ tr }\mathbf e .$ Then, to the first order in $\mathbf e ,$ we obtain

$$\begin{aligned} I_1&= 3+2e,\quad I_2=3+4e,\quad I_3=1+2e,\nonumber \\ I_4&= 1+2\varvec{M}\cdot \mathbf e \varvec{M},\quad I_5=1+4\varvec{M}\cdot \mathbf e \varvec{M},\nonumber \\ I_6&= 1+2\varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime },\quad I_7=1+4\varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime },\nonumber \\ I_8&= \varvec{M}\cdot \varvec{M}^{\prime }+2\varvec{M}\cdot \mathbf e \varvec{M}^{\prime }, \end{aligned}$$

(60)

and $\mathbf B =\mathbf I +2\mathbf e .$ Then, using the restrictions (59) and linearizing (57) in $\mathbf e ,$ we obtain, after a lengthy but straightforward process,

$$\begin{aligned}&\varvec{\sigma }= \alpha \mathbf e +[(\beta -\alpha ) e+\gamma (\varvec{M}\cdot \mathbf e \varvec{M})+\gamma ^{\prime } (\varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime })\\ \nonumber&\qquad +\zeta \varvec{M}\cdot \mathbf e \varvec{M}^{\prime }]\mathbf I +4W_5(\varvec{M}\otimes \mathbf e \varvec{M} +\mathbf e \varvec{M}\otimes \varvec{M})\\ \nonumber&\qquad +4W_7(\varvec{M}^{\prime }\otimes \mathbf e \varvec{M}^{\prime } +\mathbf e \varvec{M}^{\prime }\otimes \varvec{M}^{\prime })\\ \nonumber&\qquad +(\gamma e+\delta \varvec{M}\cdot \mathbf e \varvec{M}+\epsilon \varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime }+\eta \varvec{M}\cdot \mathbf e \varvec{M}^{\prime })\varvec{M}\otimes \varvec{M}\\ \nonumber&\qquad + (\gamma ^{\prime } e +\epsilon \varvec{M}\cdot \mathbf e \varvec{M}+\delta ^{\prime } \varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime }+\eta ^{\prime }\varvec{M}\cdot \mathbf e \varvec{M}^{\prime })\varvec{M}^{\prime }{\otimes }\varvec{M}^{\prime }\\ \nonumber&\qquad +\tfrac{1}{2}(\zeta e+\eta \varvec{M}\cdot \mathbf e \varvec{M}+\eta ^{\prime }\varvec{M}^{\prime }\cdot \mathbf e \varvec{M}^{\prime }+4W_{88}\varvec{M}\cdot \mathbf e \varvec{M}^{\prime })\\ \nonumber&\qquad \times (\varvec{M}\otimes \varvec{M}^{\prime }+\varvec{M}^{\prime }\otimes \varvec{M}), \end{aligned}$$

where we have introduced the notations

$$\begin{aligned} \alpha&= 4\left( W_1+W_2\right) ,\nonumber \\ \beta&= 4\left( W_{11}+4W_{12}+4W_{22}+2W_{13}+4W_{23}+W_{33}\right) , \nonumber \\ \gamma&= 4\left( W_{14}+2W_{24}+W_{34}+2W_{15}+4W_{25}+2W_{35}\right) , \nonumber \\ \gamma ^{\prime }&= 4\left( W_{16}+2W_{26}+W_{36}+2W_{17}+4W_{27}+2W_{37}\right) , \nonumber \\ \delta \,&= 4\left( W_{44}+4W_{45}+4W_{55}\right) ,\nonumber \\ \nonumber \delta ^{\prime }&= 4\left( W_{66}+4W_{67}+4W_{77}\right) , \nonumber \\ \epsilon&= 4\left( W_{46}+2W_{47}+ 2W_{56}+4W_{57}\right) ,\nonumber \\ \zeta&= 4\left( W_{18}+2W_{28}+W_{38}\right) , \nonumber \\ \eta&= 4\left( W_{48}+2W_{58}\right) ,\quad \eta ^{\prime }=4\left( W_{68}+2W_{78}\right) , \end{aligned}$$

(61)

for the combinations of derivatives of $W$ evaluated in the reference configuration. These constants, together with $W_5,\,W_7$ and $W_{88}$ constitute the 13 independent elastic constants of monoclinic symmetry.

By comparing with the general expression for the Cauchy stress in linear anisotropic elasticity in terms of the Voigt notation we obtain 21 expressions for elastic constants, only 13 of which are independent. These are summarized as

$$\begin{aligned} c_{ii}&= \beta +2\left( \gamma +4W_5\right) M_i^2+2\left( \gamma ^{\prime }+4W_7\right) {M_i^{\prime }}^2+2\zeta M_iM_i^{\prime }\nonumber \\&+\delta M_i^4+ 2\eta M_i^3M_i^{\prime }+\left( 2\epsilon +4W_{88}\right) M_i^2{M_i^{\prime }}^2\nonumber \\&+2\eta ^{\prime }M_i{M_i^{\prime }}^3+\delta ^{\prime } {M_i^{\prime }}^4,\end{aligned}$$

(62)

$$\begin{aligned} c_{ij}&= \beta -\alpha +\gamma \left( M_i^2+M_j^2\right) +\gamma ^{\prime } \left( {M_i^{\prime }}^2+{M_j^{\prime }}^2\right) \nonumber \\&+\zeta \left( M_iM_i^{\prime }+M_jM_j^{\prime }\right) +\delta M_i^2M_j^2+\delta ^{\prime } {M_i^{\prime }}^2{M_j^{\prime }}^2\nonumber \\&+\epsilon \left( M_i^2{M_j^{\prime }}^2+ {M_i^{\prime }}^2M_j^2\right) +4W_{88}M_iM_jM_i^{\prime }M_j^{\prime }\nonumber \\&+\left( \eta M_iM_j+\eta ^{\prime }M_i^{\prime }M_j^{\prime }\right) \left( M_iM_j^{\prime }+M_i^{\prime }M_j\right) , \end{aligned}$$

(63)

for $i,\,j\in \{1,\,2,\,3\},\,i\ne j,$

$$\begin{aligned} c_{14}&= \left( \gamma +\delta M_1^2 +\epsilon {M_1^{\prime }}^2+\eta M_1M_1^{\prime }\right) M_2M_3\nonumber \\&+ \left( \gamma ^{\prime }+\epsilon M_1^2 +\delta ^{\prime }{M_1^{\prime }}^2+\eta ^{\prime } M_1M_1^{\prime }\right) M^{\prime }_2M^{\prime }_3\nonumber \\&+\tfrac{1}{2}\left( \zeta +\eta M_1^2+\eta ^{\prime } {M_1^{\prime }}^2+4W_{88}M_1M_1^{\prime }\right) \nonumber \\&\times \left( M_2M_3^{\prime }+M_2^{\prime }M_3\right) ,\end{aligned}$$

(64)

$$\begin{aligned} c_{15}&= \left( \gamma +4W_5+\delta M_1^2 +\epsilon {M_1^{\prime }}^2+\eta M_1M_1^{\prime }\right) M_1M_3\nonumber \\&+ \left( \gamma ^{\prime }+4W_7+\epsilon M_1^2 +\delta ^{\prime }{M_1^{\prime }}^2+\eta ^{\prime } M_1M_1^{\prime }\right) M^{\prime }_1M^{\prime }_3\nonumber \\&+\tfrac{1}{2}\left( \zeta +\eta M_1^2+\eta ^{\prime } {M_1^{\prime }}^2+4W_{88}M_1M_1^{\prime }\right) \nonumber \\&\times \left( M_1M_3^{\prime }+M_1^{\prime }M_3\right) ,\end{aligned}$$

(65)

$$\begin{aligned} c_{16}&= \left( \gamma +4W_5+\delta M_1^2 +\epsilon {M_1^{\prime }}^2+\eta M_1M_1^{\prime }\right) M_1M_2\nonumber \\&+ \left( \gamma ^{\prime }+4W_7+\epsilon M_1^2 +\delta ^{\prime }{M_1^{\prime }}^2+\eta ^{\prime } M_1M_1^{\prime }\right) M^{\prime }_1M^{\prime }_2\nonumber \\&+\tfrac{1}{2}\left( \zeta +\eta M_1^2+\eta ^{\prime } {M_1^{\prime }}^2+4W_{88}M_1M_1^{\prime }\right) \nonumber \\&\times \left( M_1M_2^{\prime }+M_1^{\prime }M_2\right) . \end{aligned}$$

(66)

Note, in particular, that $W_5$ and $W_7$ do not appear in $c_{14}.$ This is because the index 4 corresponds to the pair of indices 23, which are different from the first index 1 in this case. The constants $c_{24},\,c_{25},\,c_{26}$ and $c_{34},\,c_{35},\,c_{36}$ follow the same pattern, with the index 1 in the bracketed terms replaced by 2 and 3, respectively. Then, $c_{25}$ and $c_{36}$ do not contain $W_5$ and $W_7.$

We also have

$$\begin{aligned} c_{44}&= \tfrac{1}{2}\alpha +2W_5\left( M_2^2+M_3^2\right) +2W_7\left( {M_2^{\prime }}^2+{M_3^{\prime }}^2\right) \nonumber \\&+\delta M_2^2M_3^2+2\epsilon M_2M_3M_2^{\prime }M_3^{\prime }+\delta ^{\prime } {M_2^{\prime }}^2{M_3^{\prime }}^2\nonumber \\&+W_{88}\left( M_2M_3^{\prime }+M_2^{\prime }M_3\right) ^2\nonumber \\&+\left( \eta M_2M_3+\eta ^{\prime }M_2^{\prime }M_3^{\prime }\right) \left( M_2M_3^{\prime }+M_2^{\prime }M_3\right) . \end{aligned}$$

(67)

Then, $c_{55}$ and $c_{66}$ are obtained by replacing the index 2 by 1 and the index 3 by 1, respectively.

Finally, we have

$$\begin{aligned} c_{45}&= \left( 2W_5+\delta M_3^2+\eta M_3M_3^{\prime }\right) M_1M_2\nonumber \\&+\left( 2W_7+\delta {M_3^{\prime }}^2+\eta ^{\prime } M_3M_3^{\prime }\right) M^{\prime }_1M^{\prime }_2\nonumber \\&+\tfrac{1}{2}\left( \eta M_3^2+\eta ^{\prime }{M_3^{\prime }}^2+2\epsilon M_3M_3^{\prime }\right) \left( M_1M_2^{\prime }+M_1^{\prime }M_2\right) \nonumber \\&+W_{88}\left( M_1M_3^{\prime }+M_1^{\prime }M_3\right) \left( M_2M_3^{\prime }+M_2^{\prime }M_3\right) , \end{aligned}$$

(68)

and $c_{46}$ and $c_{56}$ are obtained simply by re-ordering the indices appropriately.

It is now convenient to let the two fibre directions in the reference configuration define the $(x_1,\,x_2)$ coordinate plane, so that $M_3=M_3^{\prime }=0$ and the 21 constants reduce to the 13 appropriate for monoclinic symmetry, with $c_{14}=c_{24}=c_{34}=c_{15}=c_{25}=c_{35}=c_{46}=c_{56}=0.$

We now turn to the formulation of the strain–energy function based on the invariants $I^*_1,\,I^*_2,\,I_3,\,I^*_4,\,I^*_5,\,I^*_6,\,I^*_7$ defined in Sect. 3, supplemented by their counterpart for $I_8,$ namely

$$\begin{aligned} I^*_8=J^{-2/3}I_8. \end{aligned}$$

(69)

Thus, $W^*=W^*(I^*_1,\,I^*_2,\,I_3,\,I^*_4,\,I^*_5,\,I^*_6,\,I^*_7,\,I^*_8)$ and it is easy to show that the conditions (59) holding in the reference configuration become

$$\begin{aligned} W_3^*=0,\quad W^*_4+2W^*_5=0, \quad W^*_6+2W^*_7=0, \quad W^*_8=0.\nonumber \\ \end{aligned}$$

(70)

It also follows that

$$\begin{aligned}&W^*_{34}+2W^*_{35}=\tfrac{1}{12}\left( 3\gamma +8W_5+\delta +\epsilon +\eta \varvec{M}\cdot \varvec{M}^{\prime }\right) ,\nonumber \\&W^*_{36}+2W^*_{37}=\tfrac{1}{12}\left( 3\gamma ^{\prime }+8W_7+\delta ^{\prime }+\epsilon +\eta ^{\prime } \varvec{M}\cdot \varvec{M}^{\prime }\right) ,\nonumber \\&W^*_{38}=\tfrac{1}{12}\left( 3\zeta +\eta +\eta ^{\prime }+4W^*_{88}\varvec{M}\cdot \varvec{M}^{\prime }\right) , \end{aligned}$$

(71)

and

$$\begin{aligned} W^*_{33}&= \tfrac{1}{4}\beta +\tfrac{1}{6}(\gamma +\gamma ^{\prime }) +\tfrac{2}{9}\left( W_5+W_7\right) +\tfrac{1}{36}(\delta +\delta ^{\prime }+2\epsilon )\nonumber \\&+\tfrac{1}{18}(3\zeta +\eta +\eta ^{\prime })\varvec{M}\cdot \varvec{M}^{\prime }\nonumber \\&+\tfrac{1}{9}W^*_{88}(\varvec{M}\cdot \varvec{M}^{\prime })^2, \end{aligned}$$

(72)

while the terms in (61) that do not involve a derivative with respect to $I_3$ are unaffected by the change $W\rightarrow W^*.$ Note that $W^*_{13}$ and $W^*_{23}$ do not appear.

The terms in (71) can now be simply related to the Voigt constants. Specifically, we obtain

$$\begin{aligned} \tfrac{1}{12}(c_{11}+c_{12}-c_{33}-c_{23})&= \left( W^*_{34}+2W^*_{35}\right) M_1^2\nonumber \\&+\left( W^*_{36}+2W^*_{37}\right) {M_1^{\prime }}^2\nonumber \\&+W^*_{38}M_1M_1^{\prime }, \end{aligned}$$

(73)

$$\begin{aligned} \tfrac{1}{12}(c_{22}+c_{12}-c_{33}-c_{13})&= \left( W^*_{34}+2W^*_{35}\right) M_2^2 \nonumber \\&+\left( W^*_{36}+2W^*_{37}\right) {M_2^{\prime }}^2\nonumber \\&+W^*_{38}M_2M_2^{\prime }, \end{aligned}$$

(74)

$$\begin{aligned} \tfrac{1}{12}\left( c_{16}+c_{26}+c_{36}\right)&= \left( W^*_{34}+2W^*_{35}\right) M_1M_2\nonumber \\&+\left( W^*_{36}+2W^*_{37}\right) M_1^{\prime }M_2^{\prime }\nonumber \\&+\tfrac{1}{2}W^*_{38}\left( M_1M_2^{\prime }+M_1^{\prime }M_2\right) . \end{aligned}$$

(75)

For a decoupled model of the form

$$\begin{aligned} W^*=f(J)+\mathcal{W }\left( I^*_1,\,I^*_2,\,I^*_4,\,I^*_5,\,I^*_6,\,I^*_7,\,I^*_8\right) , \end{aligned}$$

(76)

it follows that $W^*_{34}=W^*_{35}=W^*_{36}=W^*_{37}=W^*_{38}=0$ and hence the Voigt constants must be interrelated according to

$$\begin{aligned}&c_{11}+c_{12}-c_{33}-c_{23}=0, \nonumber \\&c_{22}+c_{12}-c_{33}-c_{13}=0, \nonumber \\&c_{16}+c_{26}+c_{36}=0, \end{aligned}$$

(77)

and the 13 constants are reduced to 10. Thus the material is not fully monoclinic in the linearized limit. It follows that materials with two families of non-orthogonal fibres for which the strain–energy function can be decomposed additively as (76) do not behave like monoclinic solids when subject to infinitesimal deformations. It can also be checked that a monoclinic material for which the restrictions (77) hold deforms in pure dilatation under hydrostatic stress.

By switching the indices 1 and 3 in the first two results in (77) the results (25) in Sect. 3 are recovered (here $\varvec{M}$ and $\varvec{M}^{\prime }$ define the $(x_1,\,x_2)$ plane whereas in Sect. 3 they define the $(x_2,\,x_3)$ plane).