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Data-driven computational method for growth-induced deformation problems of soft materials

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Abstract

This paper presents a data-driven solver (DDS) for the growth-induced deformation problems of soft materials. Contrary to the over-reliance of traditional numerical methods on material constitutive models, the DDS requires only a discrete material dataset of stress–strain pairs to describe the stress–strain relationship. With the multiplicative decomposition of the gradient tensor, the growth effects are introduced. The growth-induced deformation problems are then solved as the pre-strain problems. To overcome the difficulty in determining the growth-induced pre-stress without an explicit constitutive model, a strain-driven searching scheme is thus proposed. Several numerical examples demonstrate the robustness and its good performance of the proposed DDS. In particular, the reason for choosing the strain-driven searching scheme is also illustrated according to the numerical example results.

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Acknowledgments

The supports from the National Key R&D Program of China (No. 2022YFB4201200) and the National Natural Science Foundation of China (Nos. 12072062, 12072061 and 11972108) are gratefully acknowledged.

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  1. State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Zhangcheng Zheng, Yisong Qiu, Hongfei Ye, Hongwu Zhang & Yonggang Zheng

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  1. Zhangcheng Zheng
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  2. Yisong Qiu
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Appendix 1 The strain–displacement matrix in the nonlinear elasticity

Appendix 1 The strain–displacement matrix in the nonlinear elasticity

In the nonlinear elasticity, the strain–displacement matrix concludes linear and nonlinear parts. The strain–displacement matrix for the 2D plane-strain problems have been formulated in the work of Nguyen [38]. In this work, the strain–displacement matrix for the case of 3D continuum body is further rpovided. Discretizing the 3D continuum body with the eight-node brick elements, and the displacement \(\mathbf{u}={\left\{\begin{array}{ccc}{u}_{x}& {u}_{y}& {u}_{z}\end{array}\right\}}^{\mathrm{T}}\) of any point in element \(e\) can be expressed in the interpolation form of node displacement \({\mathbf{u}}^{e}={\left\{\begin{array}{ccc}\begin{array}{ccc}{{u}_{x}^{e}}_{1}& {{u}_{y}^{e}}_{1}& {{u}_{z}^{e}}_{1}\end{array}& \cdots & \begin{array}{ccc}{{u}_{x}^{e}}_{8}& {{u}_{y}^{e}}_{8}& {{u}_{y}^{e}}_{8}\end{array}\end{array}\right\}}^{\mathrm{T}}\) as

$${u}_{x}=\sum_{i=1}^{8}{N}_{i}{{u}_{x}^{e}}_{i},$$
$${u}_{y}=\sum_{i=1}^{8}{N}_{i}{{u}_{y}^{e}}_{i},$$
$${u}_{z}=\sum_{i=1}^{8}{N}_{i}{{u}_{z}^{e}}_{i}.$$
(A1.1)

Writing the Green–Lagrange strain tensor \(\mathbf{E}\) in the Voigt representation (the vector dimension is \(6\times 1\)), and the components of \(\mathbf{E}\) can be expressed as

$${E}_{1}={\mathbf{Q}}_{1}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{1}{\mathbf{u}}^{e},$$
$${E}_{2}={\mathbf{Q}}_{2}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{2}{\mathbf{u}}^{e},$$
$${E}_{3}={\mathbf{Q}}_{3}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{3}{\mathbf{u}}^{e},$$
$$2{E}_{4}={\mathbf{Q}}_{4}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{4}{\mathbf{u}}^{e},$$
$$2{E}_{5}={\mathbf{Q}}_{5}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{5}{\mathbf{u}}^{e},$$
$$2{E}_{6}={\mathbf{Q}}_{6}^{\mathrm{T}}{\mathbf{u}}^{e}+\frac{1}{2}{{\mathbf{u}}^{e}}^{\mathrm{T}}{\mathbf{H}}_{6}{\mathbf{u}}^{e},$$
(A1.2)

in which the \({\mathbf{Q}}_{n}\) and \({\mathbf{H}}_{n}\) \((n=1,\cdots ,6)\) are the combinatorial operations of the directional derivatives of shape functions. \({\mathbf{Q}}_{n}\) can be written as

$$\begin{gathered} {\mathbf{Q}}_{1}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {N_{11}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{12}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{13}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{14}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad \left. {\begin{array}{*{20}c} {N_{15}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{16}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{17}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{18}^{\prime} } & 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{Q}}_{2}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{21}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{22}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{23}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{24}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{25}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{26}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{27}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{28}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{Q}}_{3}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{31}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{32}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{33}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{34}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{35}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{36}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{37}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {N_{38}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{Q}}_{4}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{21}^{\prime} } & {N_{11}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} {N_{22}^{\prime} } & {N_{12}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {N_{23}^{\prime} } & {N_{13}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {N_{24}^{\prime} } & {N_{14}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{25}^{\prime} } & {N_{15}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} {N_{26}^{\prime} } & {N_{16}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {N_{27}^{\prime} } & {N_{17}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {N_{28}^{\prime} } & {N_{18}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{Q}}_{5}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{31}^{\prime} } \\ \end{array} } & {N_{21}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{32}^{\prime} } & {\begin{array}{*{20}c} {N_{22}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{33}^{\prime} } & {\begin{array}{*{20}c} {N_{23}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{34}^{\prime} } & {N_{24}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{35}^{\prime} } \\ \end{array} } & {N_{25}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{36}^{\prime} } & {\begin{array}{*{20}c} {N_{26}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{37}^{\prime} } & {\begin{array}{*{20}c} {N_{27}^{\prime} } & {\begin{array}{*{20}c} 0 & {N_{38}^{\prime} } & {N_{28}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{Q}}_{6}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{31}^{\prime} } & 0 \\ \end{array} } & {N_{11}^{\prime} } & {\begin{array}{*{20}c} {N_{32}^{\prime} } & 0 & {\begin{array}{*{20}c} {N_{12}^{\prime} } & {\begin{array}{*{20}c} {N_{33}^{\prime} } & 0 & {\begin{array}{*{20}c} {N_{13}^{\prime} } & {\begin{array}{*{20}c} {N_{34}^{\prime} } & 0 & {N_{14}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{35}^{\prime} } & 0 \\ \end{array} } & {N_{15}^{\prime} } & {\begin{array}{*{20}c} {N_{36}^{\prime} } & 0 & {\begin{array}{*{20}c} {N_{16}^{\prime} } & {\begin{array}{*{20}c} {N_{37}^{\prime} } & 0 & {\begin{array}{*{20}c} {N_{17}^{\prime} } & {\begin{array}{*{20}c} {N_{38}^{\prime} } & 0 & {N_{18}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
(A1.3)

in which,

$$N_{1i}^{\prime} = \frac{{\partial N_{i} }}{{\partial X_{1} }}, N_{2i}^{\prime} = \frac{{\partial N_{i} }}{{\partial X_{2} }},{ }N_{3i}^{\prime} = \frac{{\partial N_{i} }}{{\partial X_{3} }},{ }\left( {i = 1 \cdots 8} \right).$$
(A1.4)

Then, the linear part of the strain–displacement matrix can be expressed as.

$${\mathbf{B}}_{0} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{Q}}_{1} } & {{\mathbf{Q}}_{2} } & {{\mathbf{Q}}_{3} } \\ \end{array} } & {\begin{array}{*{20}c} {{\mathbf{Q}}_{4} } & {{\mathbf{Q}}_{5} } & {{\mathbf{Q}}_{6} } \\ \end{array} } \\ \end{array} } \right]^{{\text{T}}} .$$
(A1.5)

As for the nonlinear part, \({\mathbf{H}}_{n}\) can be written as

$${\mathbf{H}}_{1} = {\mathbf{r}}_{1}^{{\text{T}}} {\mathbf{r}}_{1} + {\mathbf{r}}_{4}^{{\text{T}}} {\mathbf{r}}_{4} + {\mathbf{r}}_{7}^{{\text{T}}} {\mathbf{r}}_{7} ,$$
$${\mathbf{H}}_{2} = {\mathbf{r}}_{2}^{{\text{T}}} {\mathbf{r}}_{2} + {\mathbf{r}}_{5}^{{\text{T}}} {\mathbf{r}}_{5} + {\mathbf{r}}_{8}^{{\text{T}}} {\mathbf{r}}_{8} ,$$
$${\mathbf{H}}_{3} = {\mathbf{r}}_{3}^{{\text{T}}} {\mathbf{r}}_{3} + {\mathbf{r}}_{6}^{{\text{T}}} {\mathbf{r}}_{6} + {\mathbf{r}}_{9}^{{\text{T}}} {\mathbf{r}}_{9} ,$$
$${\mathbf{H}}_{4} = {\mathbf{r}}_{1}^{{\text{T}}} {\mathbf{r}}_{2} + {\mathbf{r}}_{4}^{{\text{T}}} {\mathbf{r}}_{5} + {\mathbf{r}}_{7}^{{\text{T}}} {\mathbf{r}}_{8} + \left( {{\mathbf{r}}_{1}^{{\text{T}}} {\mathbf{r}}_{2} + {\mathbf{r}}_{4}^{{\text{T}}} {\mathbf{r}}_{5} + {\mathbf{r}}_{7}^{{\text{T}}} {\mathbf{r}}_{8} } \right)^{{\text{T}}} ,$$
$${\mathbf{H}}_{5} = {\mathbf{r}}_{2}^{{\text{T}}} {\mathbf{r}}_{3} + {\mathbf{r}}_{5}^{{\text{T}}} {\mathbf{r}}_{6} + {\mathbf{r}}_{8}^{{\text{T}}} {\mathbf{r}}_{9} + \left( {{\mathbf{r}}_{2}^{{\text{T}}} {\mathbf{r}}_{3} + {\mathbf{r}}_{5}^{{\text{T}}} {\mathbf{r}}_{6} + {\mathbf{r}}_{8}^{{\text{T}}} {\mathbf{r}}_{9} } \right)^{{\text{T}}} ,$$
$${\mathbf{H}}_{6} = {\mathbf{r}}_{3}^{{\text{T}}} {\mathbf{r}}_{1} + {\mathbf{r}}_{6}^{{\text{T}}} {\mathbf{r}}_{4} + {\mathbf{r}}_{9}^{{\text{T}}} {\mathbf{r}}_{7} + \left( {{\mathbf{r}}_{3}^{{\text{T}}} {\mathbf{r}}_{1} + {\mathbf{r}}_{6}^{{\text{T}}} {\mathbf{r}}_{4} + {\mathbf{r}}_{9}^{{\text{T}}} {\mathbf{r}}_{7} } \right)^{{\text{T}}} .$$
(A1.6)

where

$$\begin{gathered} {\mathbf{r}}_{1} = \left[ {\begin{array}{*{20}c} {N_{11}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{12}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{13}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{14}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {N_{15}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{16}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{17}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{18}^{\prime} } & 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{2} = \left[ {\begin{array}{*{20}c} {N_{21}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{22}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{23}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{24}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {N_{25}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{26}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{27}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{28}^{\prime} } & 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{3} = \left[ {\begin{array}{*{20}c} {N_{31}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{32}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{33}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{34}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad \left. {\begin{array}{*{20}c} {N_{35}^{\prime} } & 0 & {\begin{array}{*{20}c} 0 & {N_{36}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{37}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{38}^{\prime} } & 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{4} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{11}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{12}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{13}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{14}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{15}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{16}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{17}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{18}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{5} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{21}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{22}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{23}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{24}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{25}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{26}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{27}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{28}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{6} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{31}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{32}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{33}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{34}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {N_{35}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{36}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{37}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {\begin{array}{*{20}c} {N_{38}^{\prime} } & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{7} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{11}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{12}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{13}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{14}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{15}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{16}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{17}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {N_{18}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{8} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{21}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{22}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{23}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{24}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{25}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{26}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{27}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {N_{28}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} {\mathbf{r}}_{9} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{31}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{32}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{33}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{34}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right. \hfill \\ \qquad\quad\left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {N_{35}^{\prime} } \\ \end{array} } & 0 & {\begin{array}{*{20}c} 0 & {N_{36}^{\prime} } & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & {N_{37}^{\prime} } & {\begin{array}{*{20}c} 0 & 0 & {N_{38}^{\prime} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]. \hfill \\ \end{gathered}$$
(A1.7)

Thus, the nonlinear part of the strain–displacement matrix can be expressed as

$${\mathbf{B}}_{1} \left( {{\mathbf{u}}^{e} } \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{H}}_{1}^{{\text{T}}} {\mathbf{u}}^{e} } & {{\mathbf{H}}_{2}^{{\text{T}}} {\mathbf{u}}^{e} } & {{\mathbf{H}}_{3}^{{\text{T}}} {\mathbf{u}}^{e} } \\ \end{array} } & {\begin{array}{*{20}c} {{\mathbf{H}}_{4}^{{\text{T}}} {\mathbf{u}}^{e} } & {{\mathbf{H}}_{5}^{{\text{T}}} {\mathbf{u}}^{e} } & {{\mathbf{H}}_{6}^{{\text{T}}} {\mathbf{u}}^{e} } \\ \end{array} } \\ \end{array} } \right]^{{\text{T}}} .$$
(A1.8)

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Zheng, Z., Qiu, Y., Ye, H. et al. Data-driven computational method for growth-induced deformation problems of soft materials. Acta Mech 235, 441–466 (2024). https://doi.org/10.1007/s00707-023-03742-9

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