On the Application of the Mean-Field Homogenization for Non-isotropic Matrix

Abstract

Mean field homogenization methods like the Mori–Tanaka formulation are used to determine the effective response of heterogeneous materials. Typically, these methods are deployed for inclusions within the isotropic matrix. This is due to the availability of closed-form solutions for Eshelby’s tensor when the surrounding medium is isotropic. However, in real life, the matrix can often be transversely isotropic or even orthotropic. This paper proposes a model for the implementation of the Mori–Tanaka formulation for all types of matrices. The proposed method is implemented and validated against full finite element models for three length scales: effective properties (RVE level), phase average stresses (constituent level) and interface stresses. The proposed models are seen to deliver reasonable results at the different length scales for transversely isotropic as well as orthotropic matrices.

Appendices

Appendix A

The calculation of Eshelby’s tensor for prolate spheroid inclusions ($a_{1} > a_{2} = a_{3}$) in an isotropic medium has been elaborated here. For the calculation of the Eshelby terms, the integrals are calculated initially that become elementary functions for special shapes of inclusions.

$$\begin{aligned} I_{2} & = I_{3} = \frac{{2\pi a_{1} a_{3}^{2} }}{{(a_{1}^{2} - a_{3}^{2} )^{3/2} }}\left( {\frac{{a_{1} }}{{a_{3} }}\left( {\frac{{a_{1}^{2} }}{{a_{3}^{2} }} - 1} \right)^{1/2} - cosh^{ - 1} \frac{{a_{1} }}{{a_{3} }}} \right) \\ I_{1} & = 4\pi - 2I_{2} \\ I_{12} & = \left( {I_{2} - I_{1} } \right)/\left( {a_{1}^{2} - a_{2}^{2} } \right) \\ 3I_{11} & = 4\pi /a_{1}^{2} - 2I_{12} \\ I_{22} & = I_{33} = I_{23} \\ 3I_{22} & = 4\pi /a_{2}^{2} - I_{23} - \left( {I_{2} - I_{1} } \right)/\left( {a_{1}^{2} - a_{2}^{2} } \right) \\ I_{23} & = \pi /a_{2}^{2} - \left( {I_{2} - I_{1} } \right)/4\left( {a_{1}^{2} - a_{2}^{2} } \right) \\ \end{aligned}$$

(A1)

After the integrals are calculated, the Eshelby terms are expressed in terms of these integrals as shown below.

$$\begin{aligned} S_{1111} & = \frac{3}{{8\pi \left( {1 - \nu } \right)}}a_{1}^{2} I_{11} + \frac{1 - 2\nu }{{8\pi \left( {1 - \nu } \right)}}I_{1} \\ S_{1122} & = \frac{1}{{8\pi \left( {1 - \nu } \right)}}a_{2}^{2} I_{12} + \frac{1 - 2\nu }{{8\pi \left( {1 - \nu } \right)}}I_{1} \\ S_{1133} & = \frac{1}{{8\pi \left( {1 - \nu } \right)}}a_{3}^{2} I_{13} + \frac{1 - 2\nu }{{8\pi \left( {1 - \nu } \right)}}I_{1} \\ S_{1133} & = \frac{1}{{8\pi \left( {1 - \nu } \right)}}a_{3}^{2} I_{13} + \frac{1 - 2\nu }{{8\pi \left( {1 - \nu } \right)}}I_{1} \\ \end{aligned}$$

(A2)

All other non-zero components are obtained by the cyclic permutation of the 1,2 and 3 terms. The components which cannot be obtained by the cyclic permutation such as $S_{1112} , \,S_{1223} \,and\, S_{1232}$ are zero. For more details, the reader is referred to [28].

Appendix B

The expressions of the Eshelby’s tensor for inclusion in a transversely isotropic medium is reproduced here. In each case, the inclusion has been assumed to be rotationally symmetric (i.e., $a_{1} = a_{2} = a; a_{3} = c$) about the axis normal to the plane of isotropy. The notations here denote their usual meaning as identical to those used by Withers et al. [20]. For a detailed derivation of the tensor, the reader is referred to [20].

$$S_{3333} = 2\mathop \sum \limits_{i = 1}^{2} v_{i}^{3} k_{i} A_{i}^{\prime} \left( {C_{13} - C_{33} k_{i} v_{i}^{2} } \right)I_{2} \left( i \right)$$

(B1)

$$S_{1111} = 2\mathop \sum \limits_{i = 1}^{2} C_{44} \left( {1 + k_{i} } \right)A_{i}^{\prime} v_{i}^{3} I_{1} \left( i \right) - C_{66} \mathop \sum \limits_{i = 1}^{2} A_{i}^{\prime} v_{i}^{3} I_{1} \left( i \right) + \frac{1}{2}DC_{66} I_{1} \left( 3 \right)$$

(B2)

By symmetry,

$$\begin{aligned} S_{2222} & = S_{1111} \\ S_{1122} & = 2\mathop \sum \limits_{i = 1}^{2} C_{44} \left( {1 + k_{i} } \right)A_{i}^{\prime} v_{i}^{3} I_{1} \left( i \right) - 3C_{66} \mathop \sum \limits_{i = 1}^{2} A_{i}^{\prime} v_{i} I_{1} \left( i \right) - \frac{1}{2}DC_{66} I_{1} \left( 3 \right) \\ \end{aligned}$$

(B3)

By symmetry,

$$\begin{aligned} S_{2211} & = S_{1122} \\ S_{1133} & = 2\mathop \sum \limits_{i = 1}^{2} A_{i}^{\prime} v_{i} \left( {C_{13} - C_{33} k_{i} v_{i}^{2} } \right)I_{1} \left( i \right) \\ \end{aligned}$$

(B4)

By symmetry,

$$\begin{aligned} S_{2233} & = S_{1133} \\ S_{3311} & = 2\mathop \sum \limits_{i = 1}^{2} C_{44} v_{i}^{5} k_{i} A_{i}^{\prime} \left( {1 + k_{i} } \right)I_{2} \left( i \right) - 2\mathop \sum \limits_{i = 1}^{2} C_{66} v_{i}^{3} k_{i} A_{i}{\prime} I_{2} \left( i \right) \\ \end{aligned}$$

(B5)

By symmetry,

$$\begin{aligned} S_{3322} & = S_{3311} \\ S_{1212} & = C_{66} \mathop \sum \limits_{i = 1}^{2} A_{i}^{\prime} v_{i} I_{1} \left( i \right) + \frac{1}{2}C_{66} DI_{1} \left( 3 \right) \\ \end{aligned}$$

(B5)

By symmetry,

$$\begin{gathered} S_{1212} = S_{1221} = S_{2112} = S_{2121} \hfill \\ S_{1313} = \frac{1}{2}C_{44} \mathop \sum \limits_{i = 1}^{2} A_{i}^{\prime} v_{i}^{3} \left( {1 + k_{i} } \right)\left[ {I_{2} \left( i \right) - 2k_{i} I_{1} \left( i \right)} \right] + \frac{1}{4}DC_{44} I_{2} \left( 3 \right)v_{3}^{2} \hfill \\ \end{gathered}$$

(B6)

By symmetry,

$$S_{1313} = S_{1331} = S_{3113} = S_{3131} = S_{2323} = S_{2332} = S_{3223} = S_{3232}$$

(B7)

The I integrals are given by

$$\begin{array}{*{20}l} {} \hfill & {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}c} {I_{1} \left( i \right) = \left( {{\raise0.7ex\hbox{${2\pi c}$} \!\mathord{\left/ {\vphantom {{2\pi c} {G^{3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G^{3} }$}}} \right)\left( {v_{i} cG - a^{2} F} \right)} \\ {I_{2} \left( i \right) = \left( {{\raise0.7ex\hbox{${4\pi a^{2} c}$} \!\mathord{\left/ {\vphantom {{4\pi a^{2} c} {G^{3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G^{3} }$}}} \right)(F - {\raise0.7ex\hbox{$G$} \!\mathord{\left/ {\vphantom {G {v_{i} c}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${v_{i} c}$}}} \\ \end{array} } \right\}} \hfill & {if\, v_{i} c > a} \hfill \\ \end{array} } \hfill \\ {and} \hfill & {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}c} {I_{1} \left( i \right) = - \left( {{\raise0.7ex\hbox{${2\pi c}$} \!\mathord{\left/ {\vphantom {{2\pi c} {G^{3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G^{3} }$}}} \right)\left( {v_{i} cG - a^{2} F} \right)} \\ {I_{2} \left( i \right) = - \left( {{\raise0.7ex\hbox{${4\pi a^{2} c}$} \!\mathord{\left/ {\vphantom {{4\pi a^{2} c} {G^{3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G^{3} }$}}} \right)(F - {\raise0.7ex\hbox{$G$} \!\mathord{\left/ {\vphantom {G {v_{i} c}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${v_{i} c}$}}} \\ \end{array} } \right\}} \hfill & {if\, v_{i} c < a} \hfill \\ \end{array} } \hfill \\ \end{array}$$

(B8)

Where

$$G = \sqrt {v_{i}^{2} c^{2} - a^{2} } \quad F = \frac{1}{2}{\text{ln}}\left( {\frac{{v_{i} c + G}}{{v_{i} c - G}}} \right)$$

(B9)

$$C_{13}^{*} = \sqrt {C_{11} C_{33} }$$

(B10)

$$k_{i} = \frac{{{\raise0.7ex\hbox{${C_{11} }$} \!\mathord{\left/ {\vphantom {{C_{11} } {v_{i}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${v_{i}^{2} }$}} - C_{44} }}{{C_{13} + C_{44} }}$$

(B11)

$$\rho^{2} = x_{1}^{2} + x_{2}^{2}$$

(B12)

$$\begin{aligned} v_{1} & = \sqrt {\frac{{\left( {C_{13}^{*} - C_{13} } \right)\left( {C_{13}^{*} + C_{13} + 2C_{44} } \right)}}{{4C_{33} C_{44} }}} + \sqrt {\frac{{\left( {C_{13}^{*} + C_{13} } \right)\left( {C_{13}^{*} - C_{13} - 2C_{44} } \right)}}{{4C_{33} C_{44} }}} \\ v_{2} & = \sqrt {\frac{{\left( {C_{13}^{*} - C_{13} } \right)\left( {C_{13}^{*} + C_{13} + 2C_{44} } \right)}}{{4C_{33} C_{44} }}} - \sqrt {\frac{{\left( {C_{13}^{*} + C_{13} } \right)\left( {C_{13}^{*} - C_{13} - 2C_{44} } \right)}}{{4C_{33} C_{44} }}} \\ \end{aligned}$$

(B13)

$$v_{3} = \sqrt {\frac{{C_{66} }}{{C_{44} }}}$$

(B14)

$$D = \frac{1}{{4\pi C_{44} v_{3} }}$$

(B15)

$$C_{66} = \frac{1}{2}\left( {C_{11} - C_{22} } \right)$$

(B16)

$$\begin{aligned} A_{1}^{\prime} & = - \frac{{\left( {C_{44} - C_{33} v_{1}^{2} } \right)}}{{8\pi C_{33} C_{44} \left( {v_{1}^{2} - v_{2}^{2} } \right)v_{1}^{2} }} \\ A_{2}^{\prime} & = \frac{{\left( {C_{44} - C_{33} v_{2}^{2} } \right)}}{{8\pi C_{33} C_{44} \left( {v_{1}^{2} - v_{2}^{2} } \right)v_{2}^{2} }} \\ \end{aligned}$$

(B17)

$$A_{i} = - 2v_{i} k_{i} A_{i}^{\prime}$$

(B18)