On the Incompressible Behavior in Weakly Nonlinear Elasticity

Abstract

This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses.

Appendices

Appendix A: Isotropic Tensors of Elastic Stiffness and Compliance

For isotropic materials, the elastic modulus tensors are linear combinations of products of Kronecker delta functions,

$$\begin{array}{rll}& c_{\mathit{ijkl}}=c_{12}{\delta }_{ij}{\delta }_{kl}+2c_{44}{I}_{\mathit{ijkl}},& \\ & c_{\mathit{ijklmn}}=c_{123}{\delta }_{ij}{\delta }_{kl}{\delta }_{mn}+2c_{144}({\delta }_{ij}{I}_{\mathit{klmn}}+{\delta }_{kl}{I}_{\mathit{ijmn}}+{\delta }_{mn}{I}_{\mathit{ijkl}})\\ & \phantom{c_{ijklmn}}+2c_{456}({\delta }_{in}{I}_{\mathit{jmkl}}+{\delta }_{jn}{I}_{\mathit{imkl}}+{\delta }_{im}{I}_{\mathit{jnkl}}+{\delta }_{jm}{I}_{\mathit{inkl}}),\text{ and}\\ & c_{\mathit{ijklmnpq}}=(c_{1123}-2c_{1255}+4c_{1456}){\delta }_{ij}{\delta }_{kl}{\delta }_{mn}{\delta }_{pq}\\ & \phantom{C_{ijklmnpq}}+2(c_{1255}-2c_{1456})(\phantom{{\delta }_{ij}}{\delta }_{ij}{\delta }_{kl}{I}_{\mathit{mnpq}}+{\delta }_{ij}{\delta }_{mn}{I}_{\mathit{klpq}}+{\delta }_{ij}{\delta }_{pq}{I}_{\mathit{klmn}}\\ & \phantom{C_{ijklmnpq}}+{\delta }_{kl}{\delta }_{mn}{I}_{\mathit{ijpq}}+{\delta }_{kl}{\delta }_{pq}{I}_{\mathit{ijmn}}+{\delta }_{mn}{\delta }_{pq}{I}_{\mathit{ijkl}}\phantom{{\delta }_{ij}})\\ & \phantom{C_{ijklmnpq}}+2c_{1456}[\phantom{{\delta }_{ij}}{\delta }_{ij}(\phantom{{\delta }_{ij}}{\delta }_{kq}{I}_{\mathit{lpmn}}+{\delta }_{kp}{I}_{\mathit{lqmn}}+{\delta }_{lq}{I}_{\mathit{kpmn}}+{\delta }_{lp}{I}_{\mathit{kqmn}}\phantom{{\delta }_{ij}})\\ & \phantom{C_{ijklmnpq}}+{\delta }_{kl}(\phantom{{\delta }_{iq}}{\delta }_{iq}{I}_{\mathit{jpmn}}+{\delta }_{ip}{I}_{\mathit{jqmn}}+{\delta }_{jq}{I}_{\mathit{ipmn}}+{\delta }_{jp}{I}_{\mathit{iqmn}}\phantom{{\delta }_{ij}})\\ & \phantom{C_{ijklmnpq}}+{\delta }_{mn}(\phantom{{\delta }_{ij}}{\delta }_{iq}{I}_{\mathit{jpkl}}+{\delta }_{ip}{I}_{\mathit{jqkl}}+{\delta }_{jq}{I}_{\mathit{ipkl}}+{\delta }_{jp}{I}_{\mathit{iqkl}}\phantom{{\delta }_{ij}})\\ & \phantom{C_{ijklmnpq}}+{\delta }_{pq}(\phantom{{\delta }_{ij}}{\delta }_{in}{I}_{\mathit{jmkl}}+{\delta }_{im}{I}_{\mathit{jnkl}}+{\delta }_{jn}{I}_{\mathit{imkl}}+{\delta }_{jm}{I}_{\mathit{inkl}}\phantom{{\delta }_{ij}})\phantom{{\delta }_{ij}}]\\ & \phantom{C_{ijklmnpq}}+4c_{4455}(I_{\mathit{ijkl}}{I}_{\mathit{mnpq}}+I_{\mathit{ijmn}}{I}_{\mathit{klpq}}+I_{\mathit{ijpq}}{I}_{\mathit{klmn}})\end{array}$$

(36)

where ${I_{\mathit{ijkl}}} = \left ( {{\delta _{ik}}{\delta _{jl}} + {\delta _{il}}{ \delta _{jk}}} \right )/2$ is the fourth-rank identity tensor. The form of $c_{\mathit{ijkl}}$ is well-known whereas the forms for $c_{\mathit{ijklmn}}$ and $c_{\mathit{ijklmnpq}}$ are less common. Using Eqs. (36), it is easy to show that [16] $c_{11}=c_{12}+2c_{44}$, $c_{111}=c_{123}+6c_{144}+8c_{456}$, $c_{112}=c_{123}+2c_{144}$, $c_{155}=c_{144}+2c_{456}$, $c_{1111}=c_{1123}+10c_{1255}+12c_{1456}+12c_{4455}$, $c_{1112}=c_{1123}+4c_{1255}$, $c_{1155}=c_{1255}+2c_{1456}+2c_{4455}$. Note that the analogous isotropic tensors for compliance constants take the same form as Eqs. (36), thus, only a notation change from $c\rightarrow s$ is needed.

Conversions to Landau-Lifshitz notation follow [12, 15, 16] $c_{11}=\lambda +2\mu$, $c_{12}=\lambda$, $c_{44}=\mu$, $c_{111}=2\mathcal{A}+6\mathcal{B}+2\mathcal{C}$, $c_{112}=2\mathcal{B}+2\mathcal{C}$, $c_{123}=2\mathcal{C}$, $c_{144}=\mathcal{B}$, $c_{155}=\mathcal{A}/2+\mathcal{B}$, $c_{456}=\mathcal{A}/4$, $c_{1111}=24(\mathcal{E}+\mathcal{F}+\mathcal{G}+\mathcal{H})$, $c_{1112}=6\mathcal{E}+12\mathcal{F}+24\mathcal{H}$, $c_{1122}=8\mathcal{F}+8\mathcal{G}+24\mathcal{H}$, $c_{1123}=4\mathcal{F}+24\mathcal{H}$, $c_{1255}=3\mathcal{E}/2+2\mathcal{F}$, $c_{1456}=3\mathcal{E}/4$, $c_{1144}=2\mathcal{F}+4\mathcal{G}$, $c_{1155}=3\mathcal{E}+2\mathcal{F}+4\mathcal{G}$, $c_{1244}=3\mathcal{E}/2+2\mathcal{F}$, $c_{1266}=3\mathcal{E}+2\mathcal{F}$, $c_{4455}=2\mathcal{G}$, $c_{4444}=6\mathcal{G}$. Thus, in terms of the [15] convention,

(37)

Appendix B: Stiffness/Compliance Relationships

The derivative of $\boldsymbol{S}$ defined in Eq. (1) with respect to itself is

$$\begin{aligned} \frac{\partial S_{ij}}{\partial S_{kl}}&=I_{\mathit{ijkl}}=c_{\mathit{ijmn}} \frac{\partial E_{mn}}{\partial S_{kl}}+c_{\mathit{ijmnpq}} \frac{\partial E_{mn}}{\partial S_{kl}} E_{pq}+\frac{1}{2}c_{\mathit{ijmnpqrs}} \frac{\partial E_{mn}}{\partial S_{kl}}E_{pq}E_{rs}, \end{aligned}$$

(38)

where $I_{\mathit{ijkl}}=\left (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk} \right )/2$ is the fourth-rank identity. The term $\partial E_{mn}/\partial S_{kl}$ is found by taking the derivative of $\boldsymbol{E}$ in Eq. (8) with respect to $\boldsymbol{S}$,

$$\begin{aligned} \frac{\partial E_{mn}}{\partial S_{kl}}&=s_{\mathit{mnkl}}+s_{\mathit{mnklpq}}S_{pq}+ \frac{1}{2}s_{\mathit{mnklpqrs}}S_{pq}S_{rs}, \end{aligned}$$

(39)

which was obtained by using $\partial S_{ij}/\partial S_{kl}=I_{\mathit{ijkl}}$. Substituting Eqs. (8) and (39) into Eq. (38) and keeping terms up to second-order in $\boldsymbol{S}$ gives

$$\begin{aligned} I_{\mathit{ijkl}}&=c_{\mathit{ijmn}}s_{\mathit{mnkl}}+c_{\mathit{ijmn}}s_{\mathit{mnklpq}}S_{pq}+c_{\mathit{ijmnrs}}s_{\mathit{mnkl}}s_{\mathit{rspq}}S_{pq} \\ &\quad +\frac{1}{2}c_{\mathit{ijmn}}s_{\mathit{mnklpqrs}}S_{pq}S_{rs}+\frac{1}{2}c_{\mathit{ijmntu}}s_{\mathit{mnkl}}s_{\mathit{tupqrs}}S_{pq}S_{rs} \\ &\quad +c_{\mathit{ijmntu}}s_{\mathit{mnklpq}}s_{\mathit{turs}}S_{pq}S_{rs}+\frac{1}{2}c_{\mathit{ijmntuxy}}s_{\mathit{mnkl}}s_{\mathit{tupq}}s_{\mathit{xyrs}}S_{pq}S_{rs}. \end{aligned}$$

(40)

The relationships between the elastic stiffness and compliance tensors are found by equating constant, linear, and quadratic terms of $\boldsymbol{S}$,

$$\begin{aligned} s_{\mathit{ijmn}}c_{\mathit{mnkl}}&=I_{\mathit{ijkl}},~s_{\mathit{ijklmn}}=-s_{\mathit{ijpq}}c_{\mathit{pqrstu}}s_{\mathit{rskl}}s_{\mathit{tumn}}, \text{ and} \\ s_{\mathit{ijklmnpq}}&=-s_{\mathit{ijrs}}c_{\mathit{rstuwxyz}}s_{\mathit{tukl}}s_{\mathit{wxmn}}s_{\mathit{yzpq}}-s_{\mathit{ijrs}}c_{\mathit{rstuwx}}s_{\mathit{tukl}}s_{\mathit{wxmnpq}} \\ & \hphantom{s_{ijklmnpq}=} -2s_{\mathit{ijrs}}c_{\mathit{rstuwx}}s_{\mathit{tuklmn}}s_{\mathit{wxpq}}. \end{aligned}$$

(41)

The stiffness/compliance relations for isotropic materials follow from Eqs. (41),

$$\begin{aligned} &s_{12}=-\frac{c_{12}}{2c_{44}\left (3c_{12}+2c_{44}\right )},~s_{44}= \frac{1}{4c_{44}}, \\ &s_{123}=-a_{1}^{3}\left (c_{123}+2c_{144}+\frac{8}{9}c_{456}\right )+ \frac{8}{3}a_{1}s_{44}^{2}\left (3c_{144}+4c_{456}\right )- \frac{128}{9}s_{44}^{3}c_{456}, \\ &s_{144}=-\frac{4}{3}a_{1}s_{44}^{2}\left (3c_{144}+4c_{456}\right )+ \frac{32}{3}s_{44}^{3}c_{456},~s_{456}=-8c_{456}s_{44}^{3}, \\ &s_{1123}=-a_{1}^{4}\left (c_{1123}+2c_{1255}+\frac{4}{3}c_{4455}- \frac{4}{9}c_{1456}\right ) \\ & \hphantom{s_{1123}=} \quad +\frac{1}{3}a_{1}^{2}\left [8s_{44}^{2}\left (3c_{1255}+2c_{4455}+2c_{1456} \right )-\left (3s_{123}+4s_{144}\right )\left (9c_{123}+18c_{144}+8c_{456} \right )\right ] \\ & \hphantom{s_{1123}=} \quad +\frac{16}{3}s_{44}a_{1}\left [s_{144}\left (3c_{144}+4c_{456} \right )-\frac{8}{3}s_{44}^{2}c_{1456}\right ], \\ &s_{1255}=-\frac{4}{3}a_{1}^{2}\left [s_{44}^{2}\left (3c_{1255}+2c_{1456}+2c_{4455} \right )-\frac{1}{12}\left (3s_{144}+4s_{456}\right )\left (9c_{123}+18c_{144}+8c_{456} \right )\right ] \\ & \hphantom{s_{1123}=} \quad +\frac{16}{3}a_{1}s_{44}\left [s_{44}^{2}c_{1456}-\frac{1}{6} \left (3s_{144}-s_{456}\right )\left (3c_{144}+4c_{456}\right ) \right ] \\ & \hphantom{s_{1123}=} \quad +\frac{32}{3}s_{44}^{4}c_{4455}+\frac{8}{9}s_{44}^{2}\left (9c_{144}s_{144}+12c_{144}s_{456}+8c_{456}s_{456} \right ), \\ &s_{1456}=-4a_{1}s_{44}\left [2s_{44}^{2}c_{1456}+s_{456}\left (3c_{144}+4c_{456} \right )\right ],\text{ and} \\ &s_{4455}=-16s_{44}^{4}c_{4455}-4s_{44}^{2}\left (3c_{144}s_{144}+4c_{144}s_{456}+4c_{456}s_{144}+8c_{456}s_{456} \right ), \end{aligned}$$

(42)

where $a_{1}=3s_{12}+2s_{44}$.