High Performance Reduced Order Modeling Techniques Based on Optimal Energy Quadrature: Application to Geometrically Non-linear Multiscale Inelastic Material Modeling

Abstract

A High-Performance Reduced-Order Model (HPROM) technique, previously presented by the authors in the context of hierarchical multiscale models for non linear-materials undergoing infinitesimal strains, is generalized to deal with large deformation elasto-plastic problems. The proposed HPROM technique uses a Proper Orthogonal Decomposition procedure to build a reduced basis of the primary kinematical variable of the micro-scale problem, defined in terms of the micro-deformation gradient fluctuations. Then a Galerkin-projection, onto this reduced basis, is utilized to reduce the dimensionality of the micro-force balance equation, the stress homogenization equation and the effective macro-constitutive tangent tensor equation. Finally, a reduced goal-oriented quadrature rule is introduced to compute the non-affine terms of these equations. Main importance in this paper is given to the numerical assessment of the developed HPROM technique. The numerical experiments are performed on a micro-cell simulating a randomly distributed set of elastic inclusions embedded into an elasto-plastic matrix. This micro-structure is representative of a typical ductile metallic alloy. The HPROM technique applied to this type of problem displays high computational speed-ups, increasing with the complexity of the finite element model. From these results, we conclude that the proposed HPROM technique is an effective computational tool for modeling, with very large speed-ups and acceptable accuracy levels with respect to the high-fidelity case, the multiscale behavior of heterogeneous materials subjected to large deformations involving two well-separated scales of length.

Appendix A: Constitutive Model Equations

In this appendix, we summarize the constitutive equations for modeling the micro-cell components of the ferritic ductile iron utilized in Sect. 5. The $J_2$-plasticity model here adopted is similar to that presented in Sections 50–3 of the book [22].

Such as assumed in Sect. 2.3, we take a multiplicative decomposition of the micro-deformation gradient: $\varvec{F}_{\mu }= \varvec{F}^e_{\mu } \varvec{F}^p_{\mu }$, where $\varvec{F}^e_{\mu }$ and $\varvec{F}^p_{\mu }$ are the elastic and plastic deformation gradients, respectively. We also assume an additive decomposition of the free energy, see Eq. (11). In this context, the elastic free energy part is defined by the Henky model given by

$$\begin{aligned} \varphi _{\mu }^e(\varvec{F}_{\mu }, \varvec{F}_{\mu }^p)= & {} \frac{1}{2} \gamma_\mu \left[ (\epsilon _{\mu }^e)_I+(\epsilon _{\mu }^e)_{II}+(\epsilon _{\mu }^e)_{III} \right] ^2 \\&+G_\mu \left[ (\epsilon _{\mu }^e)^2_I+(\epsilon _{\mu }^e)^2_{II}+(\epsilon _{\mu }^e)^2_{III}\right] \; , \end{aligned}$$

(44)

where $\gamma_\mu$ and G_μ are the Lamè parameters ($\gamma_\mu =\frac{E_\mu\nu_\mu }{(1+\nu_\mu )(1-2\nu_\mu ))}$, $G_\mu=\frac{E_\mu}{2(1+\nu_\mu )}$ ), and $(\epsilon _{\mu }^e)_i$ are the logarithmic stretching

$$\begin{aligned} (\epsilon _{\mu }^e)_i=\log (\lambda ^e_i); \quad {\text {for}}:~i=I,\textit{II},\textit{III}, \end{aligned}$$

(45)

being $\lambda ^e_i$ the stretching satisfying

$$\begin{aligned} \lambda ^e_i= \text {eigenvalues}\left( \sqrt{\varvec{F}^e_{\mu }(\varvec{F}^e_{\mu })^T}\right) \, .\end{aligned}$$

(46)

From Eq. (11), the Kirchhoff stress, $\varvec{\tau }$, is given by

$$\begin{aligned} \varvec{\tau }_{\mu }= \sum _{i=I}^{i=\textit{III}}\tau _i \left( \varvec{n}_i \otimes \varvec{n}_i \right) \, ,\end{aligned}$$

(47)

where the principal stresses $\tau _i$ (for $i=I, \textit{II}, \textit{III}$), are

$$\begin{aligned} \tau _i= \gamma_\mu \left[ (\epsilon _{\mu }^e)_I+(\epsilon _{\mu }^e)_{\textit{II}}+(\epsilon _{\mu }^e)_{\textit{III}} \right] + 2G_\mu (\epsilon _{\mu }^e)_i \qquad {\text {for}}:~\, i=I, \textit{II}, \textit{III}, \end{aligned}$$

(48)

and $\varvec{n}_i$ are the eigenvectors of the tensor $\left( \sqrt{\varvec{F}^e_{\mu }(\varvec{F}^e_{\mu })^T}\right)$. The first Piola-Kirchhoff stress can be computed from $\varvec{P}_{\mu }= \varvec{\tau }_{\mu } \varvec{F}^{-T}_{\mu }$.

The free energy plastic part is defined by

$$\begin{aligned} \varphi _{\mu }^p(\alpha _{\mu }) = \frac{1}{2} H_{\mu }\alpha _{\mu }^2+ (\Delta \sigma{_{_Y}}) \left( \alpha _{\mu } + \frac{1}{\delta _{\mu }} {\exp {(-\delta _{\mu } \alpha _{\mu })}}\right) + \sigma _Y^{0} \alpha _{\mu }\, . \end{aligned}$$

(49)

with the internal variable $\alpha _{\mu }$ being the cumulative plastic strain which rate is defined by

$$\begin{aligned} \dot{\alpha }_{\mu } = \sqrt{\frac{2}{3} \varvec{L}_p^T:\varvec{L}_p }; \quad {\text {with}}:~ \varvec{L}_p= \dot{\varvec{F}}_p{\varvec{F}}^{-1}_p \; . \end{aligned}$$

(50)

Considering the Eq. (49), the radius of the von Mises yield function results

$$\begin{aligned} \sigma{_{_Y}}(\alpha _{\mu }) = H_{\mu }\alpha _{\mu }+ \Delta \sigma_{_{Y}} \left(1 - {\exp {(-\delta _{\mu } \alpha _{\mu })}}\right)+ \sigma _Y^{0}\, . \end{aligned}$$

(51)