A class of transversely isotropic non-linear elastic bodies that is not Green elastic

Abstract

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered.

Appendix: A generalization of the problem of biaxial extension/compression and shear of a slab

For the problem of the biaxial extension/compression and shear of a slab described in Sect. 5.1, the following deformation that is more general than (34) is also possible for the stress given by (32):

$$\begin{aligned} x_1=\lambda _1 X_1+\kappa _1 X_2,\quad x_2=\lambda _2 X_2+\kappa _2 X_1,\quad x_3=\lambda _3X_3, \end{aligned}$$

(131)

where $\lambda _i$, $i=1,2,3$, and $\kappa _1$, $\kappa _2$ are constants. The deformation gradient is

$$\begin{aligned} \mathbf {F}=\sum _{i=1}^3\lambda _i\mathbf {e}_i\otimes \mathbf {E}_i+\kappa _1\mathbf {e}_1\otimes \mathbf {E}_2+\kappa _2\mathbf {e}_2\otimes \mathbf {E}_1, \end{aligned}$$

(132)

from which we can obtain $J=(\lambda _1\lambda _2-\kappa _1\kappa _2)\lambda _3$ and

$$\begin{aligned} \mathbf {E}= & {} \frac{1}{2}\left[ \big (\lambda _1^2+\kappa _2^2-1\big )\mathbf {E}_1\otimes \mathbf {E}_1+\big (\lambda _2^2+\kappa _1^2-1\big )\mathbf {E}_2\otimes \mathbf {E}_2+\big (\lambda _3^2-1\big )\mathbf {E}_3\otimes \mathbf {E}_3\right. \nonumber \\&\left. +\left( \kappa _1\lambda _2+\kappa _2\lambda _2\right) \left( \mathbf {E}_1\otimes \mathbf {E}_2+\mathbf {E}_2\otimes \mathbf {E}_1\right) \right] . \end{aligned}$$

(133)

From (3) and (32) and using the above expression for J we obtain for the non-zero components of the second Piola–Kirchhoff stress tensor

$$\begin{aligned} S_{11}=\frac{\lambda _3\left( \lambda _2^2\sigma _1+\kappa _1^2\sigma _2\right) }{\left( \lambda _1\lambda _2-\kappa _1\kappa _2\right) },\quad S_{12}=\frac{\lambda _3\left( \kappa _2\lambda _2\sigma _1+\kappa _1\lambda _1\sigma _2\right) }{\left( \kappa _1\kappa _2-\lambda _1\lambda _2\right) },\quad S_{22}=-\frac{\lambda _3\left( \kappa _2^2\sigma _1+\lambda _1^2\sigma _2\right) }{\left( \kappa _1\kappa _2-\lambda _1\lambda _2\right) }. \end{aligned}$$

(134)

The above expressions for $\mathbf {S}$, $\mathbf {E}$ and $\mathbf {a}_0$ (that is given in (33)) can be replaced in (6) obtaining implicit relations for unconstrained solids that are more general than (38)–(41).

In the case of an inextensible slab using (133) and (33) in (9) we obtain (compare with (54))

$$\begin{aligned} 2cs\left( \kappa _1\lambda _1+\kappa _2\lambda _2\right) +c^2\big (\lambda _1^2-1+\kappa _2^2\big )+s^2\big (\lambda _2^2-1+\kappa _1^2\big )=0, \end{aligned}$$

(135)

and the non-zero components of $\mathbf {S}_\mathrm {a}$ from (134) considering (15) are

$$\begin{aligned} S_{\mathrm {a}_{11}}= & {} -\frac{c^2}{2}\left[ c^2\big (\kappa _2^2+\lambda _1^2-1\big )+2cs\left( \kappa _1\lambda _1+\kappa _2\lambda _2\right) +s^2\big (\kappa _1^2+\lambda _2^2-1\big )\right] +\frac{\lambda _3\left( \lambda _2^2\sigma _1+\kappa _1^2\sigma _2\right) }{\left( \lambda _1\lambda _2-\kappa _1\kappa _2\right) }, \end{aligned}$$

(136)

$$\begin{aligned} S_{\mathrm {a}_{22}}= & {} -\frac{s^2}{2}\left[ c^2\big (\kappa _2^2+\lambda _1^2-1\big )+2cs\left( \kappa _1\lambda _1+\kappa _2\lambda _2\right) +s^2\big (\kappa _1^2+\lambda _2^2-1\big )\right] +\frac{\lambda _3\left( \lambda _1^2\sigma _2+\kappa _2^2\sigma _1\right) }{\left( \lambda _1\lambda _2-\kappa _1\kappa _2\right) }, \end{aligned}$$

(137)

$$\begin{aligned} S_{\mathrm {a}_{12}}= & {} -\frac{cs}{2}\left[ c^2\big (\kappa _2^2+\lambda _1^2-1\big )+2cs\left( \kappa _1\lambda _1+\kappa _2\lambda _2\right) +s^2\big (\kappa _1^2+\lambda _2^2-1\big )\right] +\frac{\lambda _3\left( \kappa _2\lambda _2\sigma _1+\kappa _1\lambda _1\sigma _2\right) }{\left( \kappa _1\kappa _2-\lambda _1\lambda _2\right) }. \end{aligned}$$

(138)

These expressions for the components of $\mathbf {S}_\mathrm {a}$ and (133) must be used in (26) to obtain a generalization of (50)–(53).