Transfer learning of deep material network for seamless structure–property predictions


Modern materials design requires reliable and consistent structure–property relationships. The paper addresses the need through transfer learning of deep material network (DMN). In the proposed learning strategy, we store the knowledge of a pre-trained network and reuse it to generate the initial structure for a new material via a naive approach. Significant improvements in the training accuracy and learning convergence are attained. Since all the databases share the same base network structure, their fitting parameters can be interpolated to seamlessly create intermediate databases. The new transferred models are shown to outperform the analytical micromechanics methods in predicting the volume fraction effects. We then apply the unified DMN databases to the design of failure properties, where the failure criteria are defined upon the distribution of microscale plastic strains. The Pareto frontier of toughness and ultimate tensile strength is extracted from a large-scale design space enabled by the efficiency of DMN extrapolation.

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The authors give warmly thanks to Dr. John O. Hallquist of LSTC for his support to this research. The support from the Yokohama Rubber Co., LTD under the Yosemite project is also gratefully acknowledged.

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Correspondence to Zeliang Liu.

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Appendix A: Analytical solutions of 2D building block

The 2D DMN framework is originally proposed in our previous work [23]. Analytical solutions are available for the two-layer structure shown in the dashed box within Fig. 1, which are derived based on the equilibrium condition

$$\begin{aligned} \sigma _2^1 = \sigma _2^2, \quad \sigma _3^1 = \sigma _3^2, \end{aligned}$$

and kinematic constraint

$$\begin{aligned} \varepsilon _1^1 = \varepsilon _1^2, \end{aligned}$$

with direction 1 tangential to the interface between the two materials and direction 2 orthogonal to direction 1. Expressions of the components in the compliance matrix \(\bar{\mathbf{D }}^r\) after the homogenization operations are

$$\begin{aligned} \bar{D}_{11}^r= & {} \dfrac{1}{\varGamma }(D_{11}^1D_{11}^2),\nonumber \\ \bar{D}_{12}^r= & {} \dfrac{1}{\varGamma }(f_1D_{12}^1D_{11}^2+f_2D_{12}^2D_{11}^1), \nonumber \\ \bar{D}_{13}^r= & {} \dfrac{1}{\varGamma }(f_1D_{13}^1D_{11}^2+f_2D_{13}^2D_{11}^1), \nonumber \\ \bar{D}_{22}^r= & {} f_1D_{22}^1+f_2D_{22}^2-\dfrac{1}{\varGamma }f_1f_2(D_{12}^1 -D_{12}^2)^2,\nonumber \\ \bar{D}_{23}^r= & {} f_1D_{23}^1+f_2D_{23}^2-\dfrac{1}{\varGamma }f_1f_2(D_{13}^1 -D_{13}^2)(D_{12}^1-D_{12}^2),\nonumber \\ \bar{D}_{33}^r= & {} f_1D_{33}^1+f_2D_{33}^2-\dfrac{1}{\varGamma }f_1f_2(D_{13}^1 -D_{13}^2)^2, \end{aligned}$$


$$\begin{aligned} \varGamma =f_1 D_{11}^2+f_2 D_{11}^1 \quad \text {and}\quad f_2=1-f_1. \end{aligned}$$

After the homogenization operation, the two-layer structure is rotated. The matrix \(\mathbf R \) defines the rotation of a second-order tensor through the angle \(\theta \) under Mandel notation,

$$\begin{aligned} \mathbf R (\theta )= \begin{Bmatrix} \cos ^2\theta&\sin ^2 \theta&\sqrt{2}\sin \theta \cos \theta \\ \sin ^2 \theta&\cos ^2\theta&-\sqrt{2}\sin \theta \cos \theta \\ -\sqrt{2}\sin \theta \cos \theta&\sqrt{2}\sin \theta \cos \theta&\cos ^2\theta -\sin ^2\theta \\ \end{Bmatrix}. \end{aligned}$$

After the rotation operation, the new compliance matrix \(\bar{\mathbf{D }}\) is obtained as

$$\begin{aligned} \bar{\mathbf{D }}=\mathbf g (\bar{\mathbf{D }}^r,\theta )=\mathbf R (- \theta )\bar{\mathbf{D }}^r\mathbf R (\theta ). \end{aligned}$$

In the global network structure, it will become the input of another building block in the upper level.

Similarly, the analytical forms of the residual strain \(\delta \bar{\varvec{\varepsilon }}^r\) after the homogenization operation are

$$\begin{aligned} \delta \bar{\varepsilon }_{11}^r= & {} \dfrac{1}{\varGamma }(f_1D_{11}^2 \delta \varepsilon ^1_{11}+f_2D_{11}^1\delta \varepsilon ^2_{11}), \nonumber \\ \delta \bar{\varepsilon }_{22}^r= & {} f_1\delta \varepsilon ^1_{22}+f_2 \delta \varepsilon ^2_{22}\nonumber \\&-\,\dfrac{1}{\varGamma }f_1f_2(D_{12}^1-D_{12}^2) (\delta \varepsilon ^1_{11}-\delta \varepsilon ^2_{11}), \nonumber \\ \delta \bar{\varepsilon }_{12}^r= & {} f_1\delta \varepsilon ^1_{12}+f_2\delta \varepsilon ^2_{12}\nonumber \\&-\,\dfrac{1}{\varGamma }f_1f_2(D_{13}^1-D_{13}^2)(\delta \varepsilon ^1_{11}-\delta \varepsilon ^2_{11}). \end{aligned}$$

The overall residual strain \(\delta \bar{\varvec{\varepsilon }}\) after the rotation operation is given by

$$\begin{aligned} \delta \bar{\varvec{\varepsilon }}=\mathbf R (-\theta )\delta \bar{\varvec{\varepsilon }}^r. \end{aligned}$$

Appendix B: Design of experiments for DMN training

For the two-phase RVE, the elastic compliance matrices of the two materials are denoted by \(\mathbf D ^{p1}\) and \( \mathbf D ^{p2}\). Both materials are assumed to be orthotropic linear elastic during the sampling. Therefore, each material has four independent design variables: \(E_{11}\), \(E_{22}\), \(\nu _{12}\) and \(G_{12}\). The compliance matrices in Mandel notation can be expressed as

$$\begin{aligned} \mathbf D ^{p1}=\left\{ \begin{array}{ccc} 1/E_{11}^{p1}&{}-\nu _{12}^{p1}/E_{22}^{p1}&{}\\ &{}1/E_{22}^{p1}&{}\\ &{}&{}1/(2G_{12}^{p1})\\ \end{array}\right\} \end{aligned}$$


$$\begin{aligned} \mathbf D ^{p2}=\left\{ \begin{array}{ccc} 1/E_{11}^{p2}&{}-\nu _{12}^{p2}/E_{22}^{p2}&{}\\ &{}1/E_{22}^{p2}&{}\\ &{}&{}1/(2G_{12}^{p2})\\ \end{array}\right\} . \end{aligned}$$

To remove the redundancy due to the scaling effect, we have

$$\begin{aligned} E_{11}^{p1}E_{22}^{p1}=1, \quad \log _{10}(E_{11}^{p2}E_{22}^{p2})\in U[-6, 6]. \end{aligned}$$

The other variables are selected randomly as

$$\begin{aligned}&\log _{10}(E_{22}^{p1}/E_{11}^{p1})\in U[-1, 1],\quad \log _{10}(E_{22}^{p2}/E_{11}^{p2})\in U[-1, 1], \\&\quad \dfrac{G_{12}^{p1}}{\sqrt{E_{22}^{p1}E_{11}^{p1}}} \in U[0.25, 0.5], \quad \dfrac{G_{12}^{p2}}{\sqrt{E_{22}^{p2}E_{11}^{p2}}} \in U[0.25, 0.5], \end{aligned}$$

where U represents the uniform distribution. The Poisson’s ratios are selected to guarantee that the compliance matrices are always positive definite,

$$\begin{aligned} \dfrac{\nu _{12}^{p1}}{\sqrt{E_{22}^{p1}/E_{11}^{p1}}}\in U[0.3,0.7], \dfrac{\nu _{12}^{p2}}{\sqrt{E_{22}^{p2}/E_{11}^{p2}}}\in U[0.3,0.7]. \end{aligned}$$

Design of experiments are performed based on the Monte Carlo sampling.

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Liu, Z., Wu, C.T. & Koishi, M. Transfer learning of deep material network for seamless structure–property predictions. Comput Mech 64, 451–465 (2019).

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  • Multiscale modeling
  • Machine learning
  • Micromechanics
  • Nonlinear plasticity
  • Failure analysis
  • Materials design