A multisurface constitutive model for highly cross-linked polymers with yield data obtained from molecular dynamics simulations

Abstract

Constitutive properties for highly cross-linked glassy polymers are currently determined by molecular dynamics (MD) simulations. This avoids the need for ad-hoc experimentation. Constitutive data in functional form, such as yield surfaces, still require identification and correspondence to an existing function. In addition, loss of information occurs with fitting procedures. The present alternative consists in directly defining piecewise-linear yield functions from a set of points obtained by MD simulations. To prevent the algorithmic issues of multisurface plasticity, we propose an alternative to active-set strategies by simultaneously including all yield functions regardless of being active. We smooth the complementarity conditions using the Chen–Mangasarian function. In addition, extrapolation is proposed for slowly-evolving quantities such as the effective plastic strain while fully implicit integration is adopted for rapidly-evolving constitutive quantities. Since polymers exhibit finite-strain behavior, we propose a semi-implicit integration algorithm which allows a small number of steps to be used up to very large strains. Experimentally-observed effects herein considered are: thermal effects on strain (i.e. thermal expansion), Young’s modulus dependence on temperature and the effects of strain rate and temperature on the yield stress. A prototype model is first studied to assess the performance of the integration algorithm, followed by a experimental validation and a fully-featured, thermally-coupled 2D example.

Appendix

Concerning the result (12), and ignoring the body force term for conciseness, we have the integration by parts formula:

$$\begin{aligned} \int _{\Omega _{b}}\nabla _{b}\cdot (\varvec{F}_{ab}\varvec{S}_{ab})^{T}\cdot \dot{\varvec{u}}\mathrm {d}\Omega _{b} & = -\int _{\Omega _{b}}\left( \varvec{F}_{ab}\varvec{S}_{ab}\right) ^{T}\cdot \nabla _{b}\dot{\varvec{u}}\mathrm {d}\Omega _{b}+\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t}\Leftrightarrow \\ & = {-{\int }}_{\Omega _{b}}\left( \varvec{S}_{ab}\varvec{F}_{ab}^{T}\right) \cdot \nabla _{b}\dot{\varvec{u}}\mathrm {d}\Omega _{b}+\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t}\quad (\text {since }\varvec{S}_{ab}=\varvec{S}_{ab}^{T})\Leftrightarrow \\ & = {-{\int }}_{\Omega _{b}}\left( \left[ \varvec{S}_{ab}\right] _{ij}\left[ \varvec{F}_{ab}\right] _{kj}\right) \left[ \nabla _{b}\dot{\varvec{u}}\right] _{ik}\mathrm {d}\Omega _{b}+\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t}\quad (\text {in components}){{\Leftrightarrow }}\\ & = {-{\int }}_{\Omega _{b}}\left[ \varvec{S}_{ab}\right] _{ij}\left[ \nabla _{b}\dot{\varvec{u}}\right] _{ik}\left[ \varvec{F}_{ab}\right] _{kj}\mathrm {d}\Omega _{b}+\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t} \end{aligned}$$

since $\varvec{e}_{ab}=\frac{1}{2}\left( \varvec{F}_{ab}^{T}\varvec{F}_{ab}-\varvec{I}\right)$ and $\dot{\varvec{e}}_{ab}=\frac{1}{2}\left( \dot{\varvec{F}}_{ab}^{T}\varvec{F}_{ab}+\varvec{F}_{ab}^{T}\varvec{F}_{ab}\right)$ but $\dot{\varvec{F}}_{ab}{={\nabla }}_{b}\dot{\varvec{u}}_{a}$ we have

$$\begin{aligned} {\int }_{\Omega _{b}}\left[ \varvec{S}_{ab}\right] _{ij}\left[ \dot{\varvec{F}}_{ab}\right] _{ik}\left[ \varvec{F}_{ab}\right] _{kj}\mathrm {d}\Omega _{b}&=\int _{\Gamma _{i}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t} \end{aligned}$$

(60)

However, symmetry of $\varvec{S}_{ab}$ results in: $\left[ \varvec{S}_{ab}\right] _{ij}\left[ \dot{\varvec{F}}_{ab}\right] _{ik}\left[ \varvec{F}_{ab}\right] _{kj}=\left[ \varvec{S}_{ab}\right] _{ij}\left[ \dot{\varvec{F}}_{ab}\right] _{jk}\left[ \varvec{F}_{ab}\right] _{ki}$ which implies that

$$\begin{aligned} {\int }_{\Omega _{b}}\left[ \varvec{S}_{ab}\right] _{ij}\left[ \dot{\varvec{F}}_{ab}\right] _{ik}\left[ \varvec{F}_{ab}\right] _{kj}\mathrm {d}\Omega _{b}&= {\int }_{\Omega _{b}}\left[ \varvec{S}_{ab}\right] _{ij}\underbrace{\frac{1}{2}\left\{ \left[ \dot{\varvec{F}}_{ab}\right] _{ik}\left[ \varvec{F}_{ab}\right] _{kj}+\left[ \dot{\varvec{F}}_{ab}\right] _{jk}\left[ \varvec{F}_{ab}\right] _{ki}\right\} }_{\left[ \dot{\varvec{e}}_{ab}\right] _{ij}}\mathrm {d}\Omega _{b}\Leftrightarrow \end{aligned}$$

(61)

$$\begin{aligned} {\int }_{\Omega _{b}}\text {tr}\left[ \varvec{S}_{ab}^{T}\dot{\varvec{e}}_{ab}\right] \mathrm {d}\Omega _{b}&=\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t} \end{aligned}$$

(62)

Finally, by using Voigt notation for $\varvec{S}_{ab}$ and $\dot{\varvec{e}}_{ab}$,

$$\begin{aligned} \int _{\Omega _{b}}\mathbf {S}_{ab}^{T}\dot{\mathbf {e}}_{ab}\mathrm {d}\Omega _{b}=\int _{\Gamma _{t}}J_{ab}\overline{\varvec{t}}\cdot \dot{\varvec{u}}\mathrm {d}\Gamma _{t} \end{aligned}$$

(63)