Derivation of heterogeneous material laws via data-driven principal component expansions

Abstract

A new data-driven method that generalizes experimentally measured and/or computational generated data sets under different loading paths to build three dimensional nonlinear elastic material law with objectivity under arbitrary loadings using neural networks is proposed. The proposed approach is first demonstrated by exploiting the concept of representative volume element (RVE) in the principal strain and stress spaces to numerically generate the data. A computational data-training algorithm on the generalization of these principal space data to three dimensional objective isotropic material laws subjected to arbitrary deformation is given. To validate these data-driven derived material laws, large deformation and buckling analysis of nonlinear elastic solids with reference material models and engineering structure with microstructure are performed. Numerical experiments show that only seven sets of data under different stress loading paths on RVEs are required to reach reasonable accuracy. The requirements for constitutive law such as objectivity are preserved approximately. The consistent tangent modulus is also derived. The proposed approach also shows a great potential to obtain the material law between different scales in the multiscale analysis by pure data.

Appendices

Appendix A: Homogenization in principal space

At the material point inside the RVE, the deformation gradient tensor $\varvec{F}$$\left( =\partial \varvec{x}/\partial \varvec{X}\right)$ and the first Piola-Kirchhoff stress tensor $\varvec{P}$ constitute one basic work-conjugate pair. Homogenization of the micro-variables $\varvec{F}$ and $\varvec{P}$ at finite strains yields the macro-variables $\varvec{\bar{F}}$ and $\varvec{\bar{P}}$. They are governed by surface data of their microscopic fields [39].

$$\begin{aligned} \varvec{\bar{F}}= & {} \frac{1}{V_{0}}\int _{\partial \Omega _{0}}\varvec{x}\otimes \varvec{ N}\,dS, \end{aligned}$$

(14)

$$\begin{aligned} \varvec{\bar{P}}= & {} \frac{1}{V_{0}}\int _{\partial \Omega _{0}}\varvec{t}\otimes \varvec{ X}\,dS \end{aligned}$$

(15)

where $\varvec{x}$ is the current position, $V_{0}$ is the volume of the body $\Omega _{0}$ at the original configuration, $\varvec{N}$ the outward normal vector to surface $\partial \Omega _{0}$ and $\varvec{t}=\varvec{P} \cdot \varvec{N}$ the traction vector. The homogenized Cauchy stress can also be expressed:

$$\begin{aligned} \varvec{\bar{\Sigma }}= & {} \frac{1}{V}\int _{\partial \Omega _{0}}\varvec{t}\otimes \varvec{ x}\,dS \end{aligned}$$

(16)

where V is the volume of the body $\Omega _{0}$ at the current configuration. The homogenized secondary PK stress $\bar{\mathbf { S}}$ is defined by

$$\begin{aligned} \bar{\varvec{S}}=\varvec{\bar{F}}^{-1}\varvec{\cdot \bar{P}} \end{aligned}$$

(17)

Fig. 16

Two issues on artificial neural network to train material law for the materials. a The order of the inputs is switched, the output stress can be completely different. b Although the two inputs (stretch) are exactly the same, the output (stress) can be different

We only consider the principal components on RVE. The boundary conditions on the RVE are given by

$$\begin{aligned}&\begin{array}{ll} u_{1}=0,\quad t_{2}=0\qquad t_{3}=0&\quad \text {at }x=0, \end{array} \end{aligned}$$

(18)

$$\begin{aligned}&\begin{array}{ll} u_{2}=0,\quad t_{1}=0\qquad t_{3}=0&\quad \text {at }y=0, \end{array} \end{aligned}$$

(19)

$$\begin{aligned}&\begin{array}{ll} u_{3}=0,\quad t_{1}=0\qquad t_{2}=0&\quad \text {at }z=0, \end{array} \end{aligned}$$

(20)

where $t_{1},t_{2}$ and $t_{3}$ are the components of the traction vector, $\varvec{t}=t_{1}\mathbf {e}_{1}+t_{2}\mathbf {e}_{2}+t_{3}\mathbf {e}_{3}$. At the boundary of the RVE, the homogenous deformation is applied

$$\begin{aligned} \varvec{x}=\varvec{\bar{F}}\cdot \varvec{X} \end{aligned}$$

(21)

where

$$\begin{aligned} \varvec{\bar{F}}=\lambda _{1}\mathbf {e}_{1}\otimes \mathbf {e} _{1}+\lambda _{2}\mathbf {e}_{2}\otimes \mathbf {e}_{2}+\lambda _{3}\mathbf {e} _{3}\otimes \mathbf {e}_{3} \end{aligned}$$

(22)

in which $\lambda _{1}$, $\lambda _{2},$$\lambda _{3}$ are the principal stretches in the x, y and z directions respectively. With the displacement on the outer boundary of RVE (Eq. 1), substituting (Eq. 22) into (Eq. 17) and invoking (Eq. 15) yields the stress expressed by the traction force on the outer boundary.

Appendix B: Data training by ANN in principal space

For general ANN, a functional transformation is defined

$$\begin{aligned} \Phi \left( c_{i}^{n},W_{ij}^{n+1},b_{j}^{n+1}\right) =\tanh \left( c_{i}^{n}W_{ij}^{n+1}+b_{j}^{n+1}\right) \end{aligned}$$

on $n\hbox {th}$ layer (superscript n represents the layer number), which can map the data unit on $i\hbox {th}$ neuron ($c_{i}^{n}$) from the $n\hbox {th}$ layer to $n+1\hbox {th}$. $W_{ij}^{n+1}$ are the weights for the link between $i\hbox {th}$ neuron on layer $c^{n}$ and $j\hbox {th}$ neuron on layer $c^{n+1}$. $b_{j}^{n+1}$ are the biases on $j\hbox {th}$ neuron on layer $c^{n+1}$ ($\mathbf{W}^{\mathbf{n+1}},\mathbf{b}^{\mathbf{n+1}}$ in vector form).

In terms of stress-strain relationship in principal space, the training of data tries to minimize Eq. (8), which starts at the input layer with the initial value:

$$\begin{aligned}{}[c_{1}^{1},c_{2}^{1},c_{3}^{1}]=[a_{1},a_{2},a_{3}] \end{aligned}$$

The mapping from $n\hbox {th}$ layer to $n+1\hbox {th}$ ($n=1\cdots N$) layer takes:

$$\begin{aligned} c_{j}^{n+1}=\Phi \left( c_{i}^{n},W_{ij}^{n+1},b_{j}^{n+1}\right) \end{aligned}$$

The output at the final layer is the principal stress:

$$\begin{aligned} S_{j}=c_{i}^{N}W_{ij}^{N+1}+b_{j}^{N+1} \end{aligned}$$

(23)

It can be seen from Eq. (11) that $\frac{\partial S_{i}}{ \partial a_{j}}$ needs to be computed for tangent modulus. It can be obtained through a recursion formula

$$\begin{aligned} \frac{\partial S_{i}}{\partial a_{j}}=W_{ik}^{N+1}\frac{\partial c_{k}^{N} }{\partial a_{j}} \end{aligned}$$

(24)

where

$$\begin{aligned} \frac{\partial c_{k}^{n}}{\partial a_{j}}=B_{ki}^{n-1}W_{im}^{n}\frac{ \partial c_{m}^{n-1}}{\partial a_{j}} \end{aligned}$$

can be obtained from above layer. Here

$$\begin{aligned} B_{ki}^{n}=\left( 1-\tanh ^{2}\left( c_{m}^{n}W_{mk}^{n+1}+b_{k}^{n+1}\right) \right) \delta _{ki} \end{aligned}$$

(25)

no summation on k and $\delta _{ki}$ is Kronecker delta. The process will return and end at the first input layer:

$$\begin{aligned} \frac{\partial c_{i}^{1}}{\partial a_{j}}=\delta _{ij}. \end{aligned}$$

Even with the data in the principal space, data training using ANN still faces some difficulties (demonstrated in Fig. 16). In Fig. 16a, with the input stretch ($a_{1},a_{2},a_{3}$), we can obtain the output stress ($S_{1}^{\diamond },S_{2}^{\diamond },S_{3}^{\diamond }$). When we switch $a_{2}$ and $a_{3}$ with input ($a_{1},a_{3},a_{2}$), we cannot get the symmetric result ($S_{1}^{\diamond },S_{3}^{\diamond },S_{2}^{\diamond }$). ($S_{1}^{^{*}}\ne S_{1}^{\diamond },S_{2}^{*}\ne S_{2}^{\diamond },S_{3}^{*}\ne S_{3}^{\diamond }$). In Fig. 16b, with the input ($a_{1},a_{2},a_{3}$) with $a_{2}=a_{3}$, the output data does not have $S_{2}\ne S_{3}$. This is the reason why the permutation is carried out in the data training and take the average to get ($S_{1},S_{2},S_{3}$). Then the coefficient 1 / 6 is introduced in Eq. (13). It also implies that the data training with the full component of stress and stretch is more difficult, demonstrating the advantage of the present method in principal space.