Micromechanics methods for evaluating the effective moduli of soft neo-Hookean composites

Abstract

Most biological soft tissues are multiphase composite materials, and the determination of their effective constitutive relations is a major concern in medical engineering. In this paper, we consider a class of soft two-phase composites, in which both phases are isotropic and hyperelastic neo-Hookean materials. For such an isotropic composite consisting of inclusions uniformly distributed but randomly oriented in a matrix, two constitutive parameters are required to characterize its hyperelastic constitutive relation. Micromechanics methods, including dilute concentration method, Mori–Tanaka method, self-consistent method, and differential method are extended to predict the effective properties of this kind of composites. Analytical solutions are given for the hyperelastic neo-Hookean composites reinforced by spherical particles, long fibers, and penny-shaped platelets, respectively. Finite element simulations are performed to evaluate the accuracy of these theoretical methods.

Appendices

Appendix A: Nonlinear elasticity

Refer to $\mathbf{X}$ and $\mathbf{x}$ as the position of a material point in the initial and current configurations, respectively. The displacement of the material point is $\mathbf{u}=\mathbf{x}-\mathbf{X}$. Thus the deformation gradient tensor $\mathbf{F}$ reads

$$\begin{aligned} \mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}=\mathbf{I}+\frac{\partial \mathbf{u}}{\partial \mathbf{X}}=\left( {\delta _{ij} +u_{i,j}} \right) \mathbf{e}_i \mathbf{e}_j, \end{aligned}$$

(55)

where $\delta _{ij}$ is the Kronecker delta, $u_{i,j}={\partial u_i }/{\partial X_j}$ are the displacement gradients, and $\mathbf{e}_i$ are the base vectors.

The strain energy density function of a neo-Hookean hyperelastic material can be expressed as Eq. (1) in the main text. According to the theory of nonlinear elasticity [43], the Cauchy stress tensor can be derived as

$$\begin{aligned} {\varvec{\sigma }}=I_3^{-1/2} \mathbf{F}\cdot \frac{\partial W}{\partial \mathbf{F}}=\mu I_3^{-5/6} \mathbf{F}\cdot \mathbf{F}^{\mathrm{T}}+\left( {-\frac{\mu }{3}I_1 I_3^{-5/6}+\kappa I_3^{1/2}-\kappa } \right) \mathbf{I}, \end{aligned}$$

(56)

where $\mathbf{I}$ is the second-order identical tensor. Substituting Eq. (55) into (56), we obtain the Cauchy stresses

$$\begin{aligned} \sigma _{ij}=\mu I_3^{-5/6} \left( {u_{i,j} +u_{j,i} +u_{i,k} u_{j,k}} \right) +\left( {\mu I_3^{-5/6}-\frac{1}{3}\mu I_1 I_3^{-5/6}+\kappa I_3^{1/2}-\kappa } \right) \delta _{ij}. \end{aligned}$$

(57)

In the principal stretch coordinate system, the deformation gradient tensor becomes $\mathbf{F}=\text {diag}\left\{ {\lambda _1, \lambda _2 ,\lambda _3} \right\}$, with $\lambda _i \left( {i=1,2,3} \right)$ being the principal stretches. Correspondingly, $I_1$ and $I_3$ can be re-expressed as

$$\begin{aligned} I_1= & {} 3+2\left( {\varepsilon _1 +\varepsilon _2 +\varepsilon _3} \right) +\left( {\varepsilon _1^2 +\varepsilon _2^2 +\varepsilon _3^2} \right) , \nonumber \\ I_3= & {} 1+2\left( {\varepsilon _1 +\varepsilon _2 +\varepsilon _3} \right) +\cdots , \end{aligned}$$

(58)

where $\varepsilon _i=\lambda _i-1$ are the principal Cauchy strains.

In the case of infinitesimal deformation, $\left| {u_{i,j}} \right| <<1$, substituting Eq. (58) into (57) produces

$$\begin{aligned} \sigma _{ij}=2\mu \varepsilon _{ij} +\left( {\kappa -\frac{2}{3}\mu } \right) \varepsilon _{kk} \delta _{ij}, \end{aligned}$$

(59)

where $\varepsilon _{ij}$ denote the Cauchy strains.

Therefore, in the cases of infinitesimal deformation, the neo-Hookean constitutive model in Eq. (1) in the main text degenerates to the linear elastic Hooke’s law.

Appendix B: Simplified calculation of transversely isotropic tensors

An arbitrary transversely isotropic tensor $\mathbf{H}$ possesses six independent parameters. When it has rotational symmetry about the $x_1$ axis, $\mathbf{H}$ can be written in a matrix form [39] as

$$\begin{aligned} \mathbf{H}=\left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} d &{} g &{} g &{} &{} &{} \\ h &{} \frac{b+e}{2} &{} \frac{b-e}{2} &{} &{} &{} \\ h &{} \frac{b-e}{2} &{} \frac{b+e}{2}&{} &{} &{} \\ &{} &{} &{} e &{} &{} \\ &{} &{} &{} &{} f &{} \\ &{} &{} &{} &{} &{} f \\ \end{array}} \right] , \end{aligned}$$

(60)

where the six italic alphabets a – f are defined by

$$\begin{aligned} \begin{array}{l} b=2H_{2222} -2H_{2323} \;,\;g=H_{1122} \;,\;h=H_{2211}, \\ d=H_{1111} \;,\;e=2H_{2323} \;,\;f=2H_{1212}. \\ \end{array} \end{aligned}$$

(61)

Using the Walpole’s method, a transversely isotropic tensor $\mathbf{H}$ can be simply expressed as $\mathbf{H}=\left( {b,g,h,d,e,f} \right)$. The algorithms of transversely isotropic tensors can readily be obtained in terms of matrices. For two transversely isotropic tensors $\mathbf{H}_\chi =\left( {b_\chi , g_\chi , h_\chi ,d_\chi , e_\chi , f_\chi } \right)$, $(\chi =1,2)$, for example, their algorithms can be expressed as

$$\begin{aligned} \mathbf{H}_1 \pm \mathbf{H}_2= & {} \left( {b_1 \pm b_2, g_1 \pm g_2, h_1 \pm h_2, d_1 \pm d_2, e_1 \pm e_2, f_1 \pm f_2} \right) , \nonumber \\ \mathbf{H}_1:\mathbf{H}_2= & {} \left( {b_1 b_2 +2h_1 g_2, g_1 b_2 +d_1 g_2, b_1 h_2 +h_1 d_2, d_1 d_2 +2g_1 h_2, e_1 e_2, f_1 f_2} \right) , \nonumber \\ \mathbf{H}^{-1}= & {} \left( {\frac{d}{\Delta },-\frac{g}{\Delta },-\frac{h}{\Delta },\frac{b}{\Delta },\frac{1}{e},\frac{1}{f}} \right) \;\;\;,\;\;\;\Delta =bd-2gh. \end{aligned}$$

(62)

Moreover, if $\mathbf{H}$ is an isotropic tensor, it can be further simplified as $\mathbf{H}=\left[ {\zeta , \eta } \right]$, where $\zeta =3H_{1111}-4H_{1212}$and$\eta =2H_{1212}$. For two isotropic tensors $\mathbf{H}_\chi =\left[ {\zeta _\chi , \eta _\chi } \right] \;\;,\;\;\left( {\chi =1,2} \right)$, one has the relations

$$\begin{aligned} \mathbf{H}_1 \pm \mathbf{H}_2=\left[ {\zeta _1 \pm \zeta _2, \eta _1 \pm \eta _2} \right] \;,\;\mathbf{H}_1:\mathbf{H}_2=\left[ {\zeta _1 \zeta _2, \eta _1 \eta _2} \right] \;,\;\mathbf{H}_1^{-1}=\left[ {\frac{1}{\zeta _1},\frac{1}{\eta _1}} \right] . \end{aligned}$$

(63)

Given the above notations, the algorithms of transversely isotropic tensors will become more intuitive. For example, the effective elastic stiffness tensor for the composites considered in this paper is isotropic and thus can be expressed as

$$\begin{aligned} \mathbf{C}= & {} \left[ {3\kappa , 2\mu } \right] \nonumber \\= & {} \left( {2\kappa +\frac{2}{3}\mu , \kappa -\frac{2}{3}\mu , \kappa -\frac{2}{3}\mu , \kappa +\frac{4}{3}\mu , 2\mu , 2\mu } \right) . \end{aligned}$$

(64)

For the inclusions, one can express their Eshelby tensors in similar forms, which can be used to calculate the LST tensor ${\varvec{\Pi }}$ and the OST tensor ${\varvec{\Omega }}$ easily.