Some Remarks on the Weakly Nonlinear Theory of Isotropic Elasticity

Abstract

Third- and fourth-order weakly nonlinear theories of elasticity are widely used by applied mathematicians, physicists and acousticians. Although their introduction is traced back to a paper by Signorini dated 1930, some aspects related to the rigorous mathematical derivation of the strain energy for quasi-incompressible isotropic elastic solids are not satisfactorily clear. In this paper we address these aspects and use our results to discuss the incompressible limit of some fourth-order nonlinear constitutive parameters.

Appendices

Appendix A: Connections Between the Fully and Weakly Nonlinear Models for the Strain Energy Function

1.1 A.1 Compressible Materials

In view of the relations (16) between the invariants of the Cauchy-Green deformation tensors and those of the Green-Lagrange strain tensor, up to terms of the fourth order in the strain, the fully nonlinear model for the strain energy (7) approximates to (15) providing that

$$ \overline{W}_{1}+\overline{W}_{2}=\frac{\mu }{2}, $$

(101a)

$$ \overline{W}_{2}+\overline{W}_{3}+\overline{W}_{11}+4\overline{W}_{12}+4 \overline{W}_{22}+2\overline{W}_{13}+4\overline{W}_{23}+ \overline{W}_{33}=\frac{\lambda }{4}, $$

(101b)

$$ \overline{W}_{3}=\frac{A}{8}, $$

(101c)

$$ \overline{W}_{3}+\overline{W}_{12}+\overline{W}_{13}+2 \overline{W}_{22}+3\overline{W}_{23}+\overline{W}_{33}=-\frac{B}{4}, $$

(101d)

$$\begin{aligned} &\overline{W}_{3}+3\overline{W}_{12}+3\overline{W}_{13}+6 \overline{W}_{22}+9\overline{W}_{23}+3\overline{W}_{33} \\ &{}+\overline{W}_{111}+6\overline{W}_{112}+3\overline{W}_{113}+12 \overline{W}_{122}+12\overline{W}_{123} \\ &{}+3\overline{W}_{133}+8\overline{W}_{222}+12\overline{W}_{223}+6 \overline{W}_{233}+\overline{W}_{333}=\frac{C}{4}, \end{aligned}$$

(101e)

$$ \overline{W}_{13}+2\overline{W}_{23}+\overline{W}_{33}= \frac{3}{16}E, $$

(101f)

$$\begin{aligned} &2\overline{W}_{13}+\overline{W}_{22}+6\overline{W}_{23}+3 \overline{W}_{33}+\overline{W}_{112}+\overline{W}_{113}+4 \overline{W}_{122} \\ &{}+6\overline{W}_{123} +2\overline{W}_{133} +4 \overline{W}_{222}+8 \overline{W}_{223} +5\overline{W}_{233} +\overline{W}_{333}=- \frac{F}{4}, \end{aligned}$$

(101g)

$$ \overline{W}_{22}+2\overline{W}_{23}+\overline{W}_{33}=\frac{G}{2}, $$

(101h)

$$\begin{aligned} &4\overline{W}_{13}+3\overline{W}_{22}+14\overline{W}_{23}+7 \overline{W}_{33}+6\overline{W}_{112}+6\overline{W}_{113}+24 \overline{W}_{122} \\ &{}+36\overline{W}_{123}+12\overline{W}_{133}+24\overline{W}_{222}+48 \overline{W}_{223}+30\overline{W}_{233}+6\overline{W}_{333} \\ &{}+\overline{W}_{1111}+8\overline{W}_{1112}+4\overline{W}_{1113}+24 \overline{W}_{1122}+24\overline{W}_{1123}+6\overline{W}_{1133} \\ &{}+32\overline{W}_{1222}+48\overline{W}_{1223}+24\overline{W}_{1233}+4 \overline{W}_{1333}+16\overline{W}_{2222}+32\overline{W}_{2223} \\ &{}+24\overline{W}_{2233}+8\overline{W}_{2333}+\overline{W}_{3333}= \frac{3}{2} H, \end{aligned}$$

(101i)

where the overbars denote quantities evaluated at the reference configuration $(I_{1},I_{2},I_{3})=(3,3,1)$.

1.2 A.2 Incompressible Materials

On using $\text{(16)}_{1,2}$ we deduce that the fully nonlinear model for an incompressible isotropic hyperelastic solid is consistent with the weakly nonlinear model (29) providing that (101a) and

$$ \left . \textstyle\begin{array}{c} \overline{W}_{1}+2\overline{W}_{2}=-\displaystyle \frac{A}{8}, \\ \overline{W}_{1}+3\overline{W}_{2}+\overline{W}_{11}+2 \overline{W}_{12}+\overline{W}_{22}=\displaystyle \frac{D}{2}, \end{array}\displaystyle \right . $$

(102)

are satisfied.

Appendix B: Inversion of Equations (45a)–(45f)

Inverting the system (45a)–(45f) yields the volumetric strain $\upsilon$, its powers $\upsilon ^{2}$ and $\upsilon ^{3}$, the deviatoric invariants $\mathcal{J}^{\star }_{2}$ and $\mathcal{J}^{\star }_{3}$, and $\upsilon \mathcal{J}^{\star }_{2}$ in terms of the invariants of the second Piola-Kirchhoff stress tensor. Specifically, $\upsilon$ is as in (46), and

$$ \kappa ^{2}\upsilon ^{2}=\frac{S_{1}^{2}}{9}+\left (\frac{2}{9\mu }- \frac{\mu _{1}}{6\mu ^{2}}\right )S_{1}S_{2}+\left ( \frac{\mu _{1}}{18\mu ^{2}}-\frac{2}{81\kappa }-\frac{2}{27\mu }- \frac{\mathcal{E}_{3}}{27\kappa ^{2}}\right )S_{1}^{3}, $$

(103a)

$$\begin{aligned} \mathcal{J}^{\star }_{2}&=-\frac{S_{1}^{2}}{12\mu ^{2}}+ \frac{S_{2}}{4\mu ^{2}}-\frac{A}{8\mu ^{4}}S_{3}+\left ( \frac{A}{8\mu ^{4}}+\frac{1}{6\mu ^{3}}+\frac{1}{9\mu ^{2}\kappa }- \frac{\mu _{1}}{6\mu ^{3}\kappa }\right )S_{1}S_{2} \\ &{}+\left (\frac{\mu _{1}}{18\mu ^{3}\kappa }-\frac{1}{27\mu ^{2}\kappa }- \frac{1}{18\mu ^{3}}-\frac{A}{36\mu ^{4}}\right )S_{1}^{3}, \end{aligned}$$

(103b)

$$ \mathcal{J}^{\star }_{3}=\frac{S_{3}}{8\mu ^{3}}- \frac{S_{1}S_{2}}{8\mu ^{3}}+\frac{S_{1}^{3}}{36\mu ^{3}}, $$

(103c)

$$ \kappa \upsilon \mathcal{J}^{\star }_{2}=\frac{S_{1}S_{2}}{12\mu ^{2}}- \frac{S_{1}^{3}}{36\mu ^{2}}, $$

(103d)

$$ \upsilon ^{2}=\frac{S_{1}^{3}}{27\kappa ^{3}}. $$

(103e)

Appendix C: Inversion of Equations (51a)–(51f)

In terms of the invariants of the second Piola-Kirchhoff stress tensor, the deviatoric invariants $\mathcal{J}^{\star }_{2}$ and $\mathcal{J}^{\star }_{3}$, the Lagrange multiplier $p$ is as in (52), and its powers $p^{2}$ and $p^{3}$, and $p\mathcal{J}^{\star }_{2}$ read

$$ p^{2}=\frac{S_{1}^{2}}{9}-\frac{A+2\mu }{18\mu ^{2}}S_{1}S_{2}+ \frac{A+2\mu }{54\mu ^{2}}S_{1}^{3}, $$

(104a)

$$ \mathcal{J}^{\star }_{2}=-\frac{S_{1}^{2}}{12\mu ^{2}}+ \frac{S_{2}}{4\mu ^{2}}-\frac{A}{8\mu ^{4}}S_{3}+ \frac{3A+4\mu }{24\mu ^{4}}S_{1}S_{2}-\frac{A+2\mu }{36\mu ^{4}}S_{1}^{3}, $$

(104b)

$$ p\mathcal{J}^{\star }_{2}=-\frac{S_{1}S_{2}}{12\mu ^{2}}+ \frac{S_{1}^{3}}{36\mu ^{2}}, $$

(104c)

$$ p^{3}=-\frac{S_{1}^{3}}{27}, $$

(104d)

and $\mathcal{J}^{\star }_{3}$ as in (103c).

Appendix D: Coefficients in (54)

The coefficients $a_{j}^{(i)}$ ($i=0,1,2$ and $j=1,\ldots,6$) in (54) are given by

$$ a_{1}^{(0)}=\frac{1}{3},\quad a_{2}^{(0)}=\frac{A+2\mu }{36\mu ^{2}}, $$

(105a)

$$ a_{3}^{(0)}=-\frac{A+2m}{12\mu ^{2}},\quad a_{4}^{(0)}= \frac{8\mu ^{2}+6\mu A+3A^{2}}{72\mu ^{4}}, $$

(105b)

$$ a_{5}^{(0)}=- \frac{8\mu ^{3}+\mu ^{2}(60\kappa +4A-12\mu _{1}+18A_{1})+6\mu A(5\kappa -2\mu _{1})+9\kappa A^{2}}{216 \kappa \mu ^{4}}, $$

(105c)

$$ a_{6}^{(0)}= \frac{4\mu ^{3}+\mu ^{2}(2A+26\kappa +9A_{1}-6\mu _{1})+6\mu A(2\kappa -\mu _{1})+3\kappa A^{2}}{324\kappa \mu ^{4}}, $$

(105d)

$$ a_{1}^{(1)}=-\frac{6\kappa +4\mu -6\mu _{1}}{9\kappa }, $$

(105e)

$$ a_{2}^{(1)}=-\frac{2\mu +A+6D}{18\mu ^{2}}+ \frac{8\mu ^{2}-18\mu \mu _{1}+9\mu _{1}^{2}}{54\kappa \mu ^{2}} + \frac{14\mu -18\mu _{1}+9\mu _{2}}{81\kappa ^{2}}+ \frac{\mathcal{E}_{3}(2\mu -3\mu _{1})}{27\kappa ^{3}}, $$

(105f)

$$ a_{3}^{(1)}=\frac{2\mu +A+6D}{6\mu ^{2}}- \frac{8\mu ^{2}-18\mu \mu _{1}+9\mu _{1}^{2}}{18\kappa \mu ^{2}}, $$

(105g)

$$ a_{4}^{(1)}=-\dfrac{A^{2}+A(4\mu +6D)-3\mu A_{1}}{12\mu ^{4}}+ \frac{(2\mu -3\mu _{1})(2\mu A-3A\mu _{1}+3\mu A_{1})}{36\kappa \mu ^{4}}, $$

(105h)

$$\begin{aligned} a_{5}^{(1)}&= \frac{12\mu ^{2}+\mu (16A-9A_{1}+24D)+3A(A+6D)}{36\mu ^{4}} \\ &{}+\displaystyle \frac{4\mu -2A+24D-9A_{1}+9\mu _{1}+9\mu _{2}}{54\kappa \mu ^{2}} \\ &{}+ \frac{\mu \mu _{1}(8A-24D+9A_{1}-12\mu _{1})-9A\mu _{1}^{2}}{36\kappa \mu ^{4}} \\ &{}+\displaystyle \frac{40\mu ^{3}-36\mu ^{2}\mu _{1}+54\mu ^{2}\mu _{2}-63\mu \mu _{1}^{2}-54\mu \mu _{1}\mu _{2}+54\mu _{1}^{3}}{162\kappa ^{2}\mu ^{3}} \\ &{}+ \frac{\mathcal{E}_{3}(4\mu -3\mu _{1})(2\mu -3\mu _{1})}{54\kappa ^{3}\mu ^{2}}, \end{aligned}$$

(105i)

$$\begin{aligned} a_{6}^{(1)}&=- \frac{6\mu ^{2}+\mu (6A-3A_{1}+12D)+A(A+6D)}{54\mu ^{4}} \\ &{}-\displaystyle \frac{4\mu ^{3}+3\mu ^{2}(8D-2A_{1}+3\mu _{1}+3\mu _{2})+3\mu \mu _{1}(2A+3A_{1}-6\mu _{1}-12D)-9A\mu _{1}^{2}}{162\kappa \mu ^{4}} \\ &{} -\displaystyle \frac{44\mu ^{3}-36\mu ^{2}\mu _{1}+54\mu ^{2}\mu _{2}-63\mu \mu _{1}^{2}-54\mu \mu _{1}\mu _{2}+54\mu _{1}^{3}}{486\kappa ^{2}\mu ^{3}} \\ &{} -\displaystyle \frac{64\mu ^{3}+\mu ^{2}(54\mathcal{E}_{3}-18\mathcal{E}_{4}-84\mu _{1}+36\mu _{2})-162\mu \mu _{1}\mathcal{E}_{3}+81\mu _{1}^{2}\mathcal{E}_{3}}{1458\mu ^{2}\kappa ^{3}}, \\ &{}+\displaystyle \frac{\mathcal{E}_{4}(2\mu -3\mu _{1})-\mathcal{E}_{3}(16\mu -21\mu _{1}+9\mu _{2})}{243\kappa ^{4}}- \frac{\mathcal{E}_{3}^{2}(2\mu -3\mu _{1})}{81\kappa ^{5}}, \end{aligned}$$

(105j)

$$ a_{j}^{(2)}=\left (4\kappa +3A_{1}-\frac{2}{3}A\right )k_{j} \quad (j=1,\ldots,6), $$

(105k)

with $k_{j}$ ($j=1,\ldots,6$) as in (47).