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Representative volume element model of triply periodic minimal surfaces (TPMS)-based electrostrictive composites for numerical evaluation of effective properties

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Abstract

A model for numerical homogenization of triply periodic minimal surfaces (TPMS) based on electrostrictive composites is presented. This electrostrictive composite consists of TPMS (a three-dimensional continuous structure) implanted in a soft non-electrostrictive matrix. A representative volume element (RVE)-based approach is used to homogenize the electrostrictive composites and determine all the effective electrostrictive, mechanical, and electrical coefficients. Finite element formulation is employed to solve the nonlinear electrostrictive constitutive equations. Special attention is paid to designing the boundary conditions that permit the fast calculation based on simulations of overall deformation-induced due to mechanical and electrical loads. Interestingly, the value of the effective electrostrictive coefficient of the composite surpasses that of the inclusion Pb (mg1/3Nb2/3)O3-PbTO3-BaTiO3 (PMN-PT-BT), even though the matrix is non-electrostrictive due to the additional flexibility imparted by the matrix. This electrostrictive response of TPMS-based composite is independent of the type of TPMS structures used. It is prudent to say that these composites will find their place in practical application owing to their salient features of more flexibility and high electrostrictive coefficient.

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Authors and Affiliations

  1. School of Engineering, Indian Institute of Technology Mandi, Himachal Pradesh, Mandi, 175075, India

    Diwakar Singh, Saurav Sharma, Rajeev Kumar & Rahul Vaish

  2. Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

    Saurav Sharma

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Appendix A

Appendix A

Shape function \({N}_{i}\) for linear tetrahedral element is

$${N}_{i}=\frac{\left({\alpha }_{i}+{\beta }_{i}{x}_{1}+{\gamma }_{i}{x}_{2}+{\delta }_{i}{x}_{3}\right)}{6V},$$
(A.1)

where i varies from 1 to 4, and

$${\alpha }_{1}=\left|\begin{array}{c}{x}_{1}^{2}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{2}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{2}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\beta }_{1}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{2}^{2}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{2}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\gamma }_{1}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{2}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{3}^{2}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\delta }_{1}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{2}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{2}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array}\right|,$$
(A.2)
$${\alpha }_{2}=-\left|\begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\beta }_{2}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\gamma }_{2}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|, {\delta }_{2}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array}\right|,$$
(A.3)
$${\alpha }_{3}=\left|\begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{4}\end{array}\right|, {\beta }_{3}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{4}\end{array}\right|, {\gamma }_{3}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{4}\end{array}\right|, {\delta }_{3}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{4}\end{array}\right|,$$
(A.4)
$${\alpha }_{4}=-\left|\begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{3}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{3}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{3}\end{array}\right|, {\beta }_{3}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{3}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{3}\end{array}\right|, {\gamma }_{3}=-\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{3}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{3}\end{array}\right|, {\delta }_{3}=\left|\begin{array}{c}1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{3}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{3}\end{array}\right|,$$
(A.5)
$$6V=\left|\begin{array}{c}1\\ 1\\ 1\\ 1\end{array} \begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\\ {x}_{1}^{3}\\ {x}_{1}^{4}\end{array} \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\\ {x}_{2}^{3}\\ {x}_{2}^{4}\end{array} \begin{array}{c}{x}_{3}^{1}\\ {x}_{3}^{2}\\ {x}_{3}^{3}\\ {x}_{3}^{4}\end{array}\right|,$$
(A.6)

where the superscript (i.e., 1, 2, 3, and 4) represents the node number and the subscript (i.e., 1, 2, and 3) represents the direction.

The elemental strain displacement matrix is

$${\left[B\right]}_{e}=\frac{1}{6V}\left[\begin{array}{c}{\beta }_{1}\\ 0\\ 0\\ 0\\ {\delta }_{1}\\ {\gamma }_{1}\end{array} \begin{array}{c}0\\ {\gamma }_{1}\\ 0\\ {\delta }_{1}\\ 0\\ {\beta }_{1}\end{array} \begin{array}{c}0\\ 0\\ {\delta }_{1}\\ {\gamma }_{1}\\ {\beta }_{1}\\ 0\end{array} \begin{array}{c}{\beta }_{2}\\ 0\\ 0\\ 0\\ {\delta }_{2}\\ {\gamma }_{2}\end{array} \begin{array}{c}0\\ {\gamma }_{2}\\ 0\\ {\delta }_{2}\\ 0\\ {\beta }_{2}\end{array} \begin{array}{c}0\\ 0\\ {\delta }_{2}\\ {\gamma }_{2}\\ {\beta }_{2}\\ 0\end{array} \begin{array}{c}{\beta }_{3}\\ 0\\ 0\\ 0\\ {\delta }_{3}\\ {\gamma }_{3}\end{array} \begin{array}{c}0\\ {\gamma }_{3}\\ 0\\ {\delta }_{3}\\ 0\\ {\beta }_{3}\end{array} \begin{array}{c}0\\ 0\\ {\delta }_{3}\\ {\gamma }_{3}\\ {\beta }_{3}\\ 0\end{array} \begin{array}{c}{\beta }_{4}\\ 0\\ 0\\ 0\\ {\delta }_{4}\\ {\gamma }_{4}\end{array} \begin{array}{c}0\\ {\gamma }_{4}\\ 0\\ {\delta }_{4}\\ 0\\ {\beta }_{4}\end{array} \begin{array}{c}0\\ 0\\ {\delta }_{4}\\ {\gamma }_{4}\\ {\beta }_{4}\\ 0\end{array}\right].$$
(A.7)

The elemental electric field and electric potential matrix is

$${\left[{B}_{\phi }\right]}_{e}=\left[\begin{array}{c}\frac{\partial {N}_{1}}{\partial {x}_{1}}\\ \frac{\partial {N}_{1}}{\partial {x}_{2}}\\ \frac{\partial {N}_{1}}{\partial {x}_{3}}\end{array} \begin{array}{c}\frac{\partial {N}_{2}}{\partial {x}_{1}}\\ \frac{\partial {N}_{2}}{\partial {x}_{2}}\\ \frac{\partial {N}_{2}}{\partial {x}_{3}}\end{array} \begin{array}{c}\frac{\partial {N}_{3}}{\partial {x}_{1}}\\ \frac{\partial {N}_{3}}{\partial {x}_{2}}\\ \frac{\partial {N}_{3}}{\partial {x}_{3}}\end{array} \begin{array}{c}\frac{\partial {N}_{4}}{\partial {x}_{1}}\\ \frac{\partial {N}_{4}}{\partial {x}_{2}}\\ \frac{\partial {N}_{4}}{\partial {x}_{3}}\end{array}\right].$$
(A.8)

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Singh, D., Sharma, S., Kumar, R. et al. Representative volume element model of triply periodic minimal surfaces (TPMS)-based electrostrictive composites for numerical evaluation of effective properties. Acta Mech 234, 355–375 (2023). https://doi.org/10.1007/s00707-022-03404-2

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