Micromechanical modeling of the biaxial behavior of strain-induced crystallizable polyethylene terephthalate-clay nanocomposites

Abstract

The present work addresses the question of the quantitative prediction of the biaxial response of polymer–clay nanocomposites experiencing strain-induced crystallization. Polyethylene terephthalate is taken as material model to represent the continuous amorphous phase of nanocomposites. A continuum-based micromechanical model is developed to predict the combined effect of strain-induced phase transformation and nanocomposite structural characteristics on the overall elastic-viscoplastic response. Comparisons with available experimental data are presented to illustrate the capabilities of the model in relation to various loading parameters in terms of loading path, loading rate and loading temperature. The model is used to provide a better understanding of the relationship between nanocomposite structural characteristics, phase transformation, intrinsic properties and loading parameters.

Appendices

Appendix A

The intercalated cluster of clay is replaced by an equivalent homogeneous nanoparticle having transversely isotropic properties. Each intercalated clay stack is seen as a laminated composite sub-structure, consisting of several clay platelets separated by polymer, for which the elastic tensor is estimated using the laminate theory [30]. The elastic stiffness tensor C_clay is given from the six elastic constants written as:

$$\begin{aligned} E_{11} & = E_{33} = \phi {}_{S/P}E_{S} + \left( {1 - \phi_{S/P} } \right)E_{{\text{G}}} \\ E_{22} & = \frac{{E_{{\text{S}}} E_{{\text{G}}} }}{{\phi_{S/P} E_{{\text{G}}} + \left( {1 - \phi_{S/P} } \right)E_{{\text{S}}} - \phi_{S/P} \left( {1 - \phi_{S/P} } \right)\eta_{1} E_{{\text{G}}} E_{{\text{S}}} }} \\ \nu_{12} & = \nu_{23} = \phi_{S/P} \nu_{{\text{S}}} + \left( {1 - \phi_{S/P} } \right)\nu_{{\text{G}}} \\ \nu_{13} & = \frac{{\nu_{{\text{S}}} \phi_{S/P} E_{{\text{S}}} \left( {1 - \nu_{{\text{G}}}^{{2}} } \right) + \nu_{{\text{G}}} \left( {1 - \phi_{S/P} } \right)E_{{\text{G}}} \left( {1 - \nu_{{\text{S}}}^{{2}} } \right)}}{{\phi_{S/P} E_{{\text{S}}} \left( {1 - \nu_{{\text{G}}}^{{2}} } \right) + \nu_{{\text{G}}} \left( {1 - \phi_{S/P} } \right)E_{{\text{G}}} \left( {1 - \nu_{{\text{S}}}^{{2}} } \right)}} \\ G_{12} & = G_{23} = \frac{{G_{{\text{S}}} G_{{\text{G}}} }}{{\phi_{S/P} G_{{\text{G}}} + \left( {1 - \phi_{S/P} } \right)G_{{\text{S}}} - \phi_{S/P} \left( {1 - \phi_{S/P} } \right)\eta_{2} G_{{\text{G}}} G_{{\text{S}}} }} \\ G_{13} & = \frac{{E_{11} }}{{2\left( {1 + \nu_{13} } \right)}} \\ \end{aligned}$$

(A1)

The two parameters $\eta_{1}$ and $\eta_{2}$ are given by:

$$\begin{aligned} \eta_{1} & = \frac{{\nu_{{\text{S}}}^{2} {{E_{{\text{G}}} } \mathord{\left/ {\vphantom {{E_{{\text{G}}} } {E_{{\text{S}}} }}} \right. \kern-\nulldelimiterspace} {E_{{\text{S}}} }} + \nu_{{\text{G}}}^{2} {{E_{{\text{S}}} } \mathord{\left/ {\vphantom {{E_{{\text{S}}} } {E_{{\text{G}}} }}} \right. \kern-\nulldelimiterspace} {E_{{\text{G}}} }} - 2\nu_{{\text{S}}} \nu_{{\text{G}}} }}{{\phi_{S/P} E_{{\text{S}}} + \left( {1 - \phi_{S/P} } \right)E_{{\text{G}}} }} \\ \eta_{2} & = \frac{{\nu_{{\text{S}}}^{2} {{G_{{\text{G}}} } \mathord{\left/ {\vphantom {{G_{{\text{G}}} } {G_{{\text{S}}} }}} \right. \kern-\nulldelimiterspace} {G_{{\text{S}}} }} + \nu_{{\text{G}}}^{2} {{G_{{\text{S}}} } \mathord{\left/ {\vphantom {{G_{{\text{S}}} } {G_{{\text{G}}} }}} \right. \kern-\nulldelimiterspace} {G_{{\text{G}}} }} - 2\nu_{{\text{S}}} \nu_{{\text{G}}} }}{{\phi_{S/P} G_{{\text{S}}} + \left( {1 - \phi_{S/P} } \right)G_{{\text{G}}} }} \\ \end{aligned}$$

(A2)

where E is the Young’s modulus, G is the shear modulus and $\nu$ is the Poisson’s ratio. The subscripts S and G refer to the silicate and the gallery (confined polymer matrix in the intersilicate layers whose elastic constants are taken equal to those of the amorphous PET), respectively. The term $\phi_{S/P}$ is the volume fraction of silicate in the intercalated clay stack:

$$\phi_{S/P} = \frac{{Nd_{S} }}{t}$$

(A3)

where t is the thickness of the intercalated clay stack:

$$t = \left( {N - 1} \right)d_{001} + d_{S}$$

(A4)

The quantities N, d₀₀₁ and d_S are the clay structural parameters schematically defined in Fig. 

Fig. 10

Intercalated clay stack

10. They denote, respectively, the average number of silicate layers per clay stack, the average silicate interlayer spacing and the thickness of the silicate layer, respectively. When the intercalated morphology is invoked in the main body of the paper, the employed average structural parameters are N = 8, L_S = 200 nm, d_S = 1 nm and d₀₀₁ = 2 nm.

The volume fraction of intercalated clay stacks can be expressed as:

$$\phi_{{\rm clay}} = \frac{{\rho_{m} }}{{\rho_{S} \phi_{S/P} }}W_{S}$$

(A5)

where W_S is the silicate weight fraction, $\rho_{S}$ is the silicate density and $\rho_{m}$ is the density of the polymer matrix:

$$\rho_{m} = \phi_{{\rm am}} \rho_{{\rm am}} + \phi_{{\rm cry}} \rho_{{\rm cry}}$$

(A6)

The PET, crystal and clay densities were taken equal to 1.335 g/cm³, 1.445 g/cm³ and 2.3 g/cm³, respectively.

The aspect ratio of the intercalated clay stack $\alpha_{{\rm clay}}$ is given by:

$$\alpha_{{\rm clay}} = \frac{{\left( {N - 1} \right)d_{001} + d_{S} }}{{L_{S} }}$$

(A7)

where L_S is the clay layer length.

The crystal volume fraction $\phi_{{\rm cry}}$ is calculated from the crystal weight fraction W_cry as follows:

$$\phi_{{\rm cry}} = \frac{{W_{{\rm cry}} }}{{W_{{\rm cry}} + \left( {\frac{{\rho_{{\rm cry}} }}{{\phi_{{\rm am}} \rho_{{\rm am}} + \phi_{{\rm clay}} \rho_{{\rm clay}} }}} \right)\left( {1 - W_{{\rm cry}} } \right)}}$$

(A8)

Appendix B

The parameters $B_{IK}^{\left( 1 \right)}$ and $B_{IJ}^{\left( 2 \right)}$ are given by:

$$B_{IK}^{\left( 1 \right)} = - \frac{1}{3} + \frac{2}{{4725\left( {1 - \nu_{{\rm am}} } \right)^{2} }}\left\{ {\begin{array}{*{20}c} {\left[ \begin{gathered} 3\left( {35\nu_{{\rm am}}^{2} - 70\nu_{{\rm am}} + 36} \right)\Delta_{IK} \hfill \\ + 7\left( {50\nu_{{\rm am}}^{2} - 59\nu_{{\rm am}} + 8} \right)\left( {\Delta_{I} + \Delta_{K} } \right) \hfill \\ - 2\left( {175\nu_{{\rm am}}^{2} - 343\nu_{{\rm am}} + 103} \right) \hfill \\ \end{gathered} \right]\left( {\frac{{\phi_{{\rm clay}} }}{{D_{II} D_{KK} }} + \frac{{\phi_{{\rm cry}} }}{{E_{II} E_{KK} }}} \right)} \\ { + 21\left( {25\nu_{{\rm am}} - 2} \right)\left( {1 - 2\nu_{{\rm am}} } \right)\left( \begin{gathered} \phi_{{\rm clay}} \frac{{\left( {G_{II} + G_{KK} } \right)}}{{D_{II} D_{KK} }} \hfill \\ + \phi_{{\rm cry}} \frac{{\left( {H_{II} + H_{KK} } \right)}}{{E_{II} E_{KK} }} \hfill \\ \end{gathered} \right)} \\ { + 21\left( {25\nu_{{\rm am}} - 23} \right)\left( {1 - 2\nu_{{\rm am}} } \right)\left( \begin{gathered} \phi_{{\rm clay}} \frac{{\left( {G_{II} \Delta_{K} + G_{KK} \Delta_{I} } \right)}}{{D_{II} D_{KK} }} \hfill \\ + \phi_{{\rm cry}} \frac{{\left( {H_{II} \Delta_{K} + H_{KK} \Delta_{I} } \right)}}{{E_{II} E_{KK} }} \hfill \\ \end{gathered} \right)} \\ { + 1575\left( {1 - 2\nu_{{\rm am}} } \right)^{2} \left( {\phi_{{\rm clay}} \frac{{G_{II} G_{KK} }}{{D_{II} D_{KK} }} + \phi_{{\rm cry}} \frac{{H_{II} H_{KK} }}{{E_{II} E_{KK} }}} \right)} \\ \end{array} } \right\}$$

(B1)

$$B_{IJ}^{\left( 2 \right)} = \frac{1}{2} + \frac{1}{{1575\left( {1 - \nu_{{\rm am}} } \right)^{2} }}\left( {\frac{{\phi_{{\rm clay}} }}{{D_{IJ} D_{IJ} }} + \frac{{\phi_{{\rm cry}} }}{{E_{IJ} E_{IJ} }}} \right)\left[ \begin{gathered} \left( {70\nu_{{\rm am}}^{2} - 140\nu_{{\rm am}} + 72} \right)\Delta_{IJ} \hfill \\ - \left( {175\nu_{{\rm am}}^{2} - 266\nu_{{\rm am}} + 75} \right)\frac{{\left( {\Delta_{I} + \Delta_{J} } \right)}}{2} \hfill \\ + \left( {350\nu_{{\rm am}}^{2} - 476\nu_{{\rm am}} + 164} \right) \hfill \\ \end{gathered} \right]$$

where $\Delta_{I}$, $\Delta_{IJ}$, $D_{IJ}$, $E_{IJ}$, $G_{IJ}$ and $H_{IJ}$ are defined by:

$$\begin{aligned} \Delta_{1} & = \frac{{3\left[ {1 - \alpha^{4} g\left( {\alpha^{2} } \right)} \right]}}{{1 - \alpha^{4} }}, \\ \Delta_{2} & = \Delta_{3} = \frac{1}{2}\left( {3 - \Delta_{1} } \right), \\ \Delta_{11} & = \frac{{5\left[ {2 + \alpha^{4} - 3\alpha^{4} g\left( {\alpha^{2} } \right)} \right]}}{{2\left( {1 - \alpha^{4} } \right)^{2} }}, \\ \Delta_{12} & = \Delta_{21} = \Delta_{13} = \Delta_{31} = \frac{{15\alpha^{4} \left[ { - 3 + \left( {1 + 2\alpha^{4} } \right)g\left( {\alpha^{2} } \right)} \right]}}{{4\left( {1 - \alpha^{4} } \right)^{2} }}, \\ \Delta_{22} & = \Delta_{23} = \Delta_{32} = \Delta_{33} = \frac{1}{8}\left( {15 - 3\Delta_{11} - 4\Delta_{12} } \right) \\ \end{aligned}$$

(B2)

with

$$g\left( \alpha \right) = \left\{ \begin{gathered} \frac{{\cosh^{ - 1} \alpha }}{{\alpha \sqrt {\alpha^{2} - 1} }}\quad {\text{if}}\;\alpha > 1 \hfill \\ \frac{{\cos^{ - 1} \alpha }}{{\alpha \sqrt {1 - \alpha^{2} } }}\quad {\text{if}}\;\alpha < 1 \hfill \\ \end{gathered} \right.$$

(B3)

$$\begin{aligned} D_{IJ} & = 2\left( {V_{IJ} + NP_{IJ} } \right) \\ E_{IJ} & = 2\left( {V_{IJ} + NI_{IJ} } \right) \\ \end{aligned}$$

(B4)

and

$$\begin{gathered} \left\{ {\left. {\begin{array}{*{20}c} {G_{I1}^{{}} } \\ {G_{I2}^{{}} } \\ {G_{I3}^{{}} } \\ \end{array} } \right\}} \right. = \left[ {\begin{array}{*{20}c} {U_{11} + 2V_{11} + WP_{11}^{{}} } & {U_{21} + MP_{21}^{{}} } & {U_{31} + MP_{31}^{{}} } \\ {U_{12} + MP_{12}^{{}} } & {U_{22} + 2V_{22} + WP_{22}^{{}} } & {U_{32} + MP_{32}^{{}} } \\ {U_{13} + MP_{13}^{{}} } & {U_{23} + MP_{23}^{{}} } & {U_{33} + 2V_{33} + WP_{33}^{{}} } \\ \end{array} } \right]^{ - 1} \left\{ {\left. {\begin{array}{*{20}c} {U_{I1} + MP_{I1}^{{}} } \\ {U_{I2} + MP_{I2}^{{}} } \\ {U_{I3} + MP_{I3}^{{}} } \\ \end{array} } \right\}} \right. \hfill \\ \left\{ {\left. {\begin{array}{*{20}c} {H_{I1}^{{}} } \\ {H_{I2}^{{}} } \\ {H_{I3}^{{}} } \\ \end{array} } \right\}} \right. = \left[ {\begin{array}{*{20}c} {U_{11} + 2V_{11} + WI_{11}^{{}} } & {U_{21} + MI_{21}^{{}} } & {U_{31} + MI_{31}^{{}} } \\ {U_{12} + MI_{12}^{{}} } & {U_{22} + 2V_{22} + WI_{22}^{{}} } & {U_{32} + MI_{32}^{{}} } \\ {U_{13} + MI_{13}^{{}} } & {U_{23} + MI_{23}^{{}} } & {U_{33} + 2V_{33} + WI_{33}^{{}} } \\ \end{array} } \right]^{ - 1} \left\{ {\left. {\begin{array}{*{20}c} {U_{I1} + MI_{I1}^{{}} } \\ {U_{I2} + MI_{I2}^{{}} } \\ {U_{I3} + MI_{I3}^{{}} } \\ \end{array} } \right\}} \right. \hfill \\ \end{gathered}$$

(B5)

with

$$\begin{aligned} U_{11} & = \left( {4\nu_{{\rm am}} + \frac{2}{{\alpha^{2} - 1}}} \right)f\left( \alpha \right) + 4\nu_{{\rm am}} + \frac{4}{{3\left( {\alpha^{2} - 1} \right)}}, \\ U_{12} & = U_{13} = \left( {4\nu_{{\rm am}} - \frac{{2\alpha^{2} + 1}}{{\alpha^{2} - 1}}} \right)f\left( \alpha \right) + 4\nu_{{\rm am}} - \frac{{2\alpha^{2} }}{{\alpha^{2} - 1}}, \\ U_{21} & = U_{31} = \left( { - 2\nu_{{\rm am}} - \frac{{2\alpha^{2} + 1}}{{\alpha^{2} - 1}}} \right)f\left( \alpha \right) - \frac{{2\alpha^{2} }}{{\alpha^{2} - 1}}, \\ U_{22} & = U_{23} = U_{32} = U_{33} = \left( { - 2\nu_{{\rm am}} + \frac{{4\alpha^{2} - 1}}{{4\left( {\alpha^{2} - 1} \right)}}} \right)f\left( \alpha \right) + \frac{{\alpha^{2} }}{{2\left( {\alpha^{2} - 1} \right)}}, \\ \end{aligned}$$

(B6)

$$\begin{aligned} V_{11} & = \left( { - 4\nu_{{\rm am}} + \frac{{4\alpha^{2} - 2}}{{\alpha^{2} - 1}}} \right)f\left( \alpha \right) - 4\nu_{{\rm am}} + \frac{{12\alpha^{2} - 8}}{{3\left( {\alpha^{2} - 1} \right)}}, \\ V_{12} & = V_{21} = V_{13} = V_{31} = \left( { - \nu_{{\rm am}} - \frac{{\alpha^{2} + 2}}{{\alpha^{2} - 1}}} \right)f\left( \alpha \right) - 2\nu_{{\rm am}} - \frac{2}{{\alpha^{2} - 1}}, \\ V_{22} & = V_{23} = V_{32} = V_{33} = \left( {2\nu_{{\rm am}} - \frac{{4\alpha^{2} - 7}}{{4\left( {\alpha^{2} - 1} \right)}}} \right)f\left( \alpha \right) + \frac{{\alpha^{2} }}{{2\left( {\alpha^{2} - 1} \right)}}, \\ \end{aligned}$$

(B7)

$$\begin{aligned} MP_{IJ} & = \frac{{\lambda_{{\rm am}} \mu_{{\rm clay}} - \lambda_{{\rm clay}} \mu_{{\rm am}} }}{{\left( {\mu_{{\rm clay}} - \mu_{{\rm am}} } \right)\left[ {2\left( {\mu_{{\rm clay}} - \mu_{{\rm am}} } \right) + 3\left( {\lambda_{{\rm clay}} - \lambda_{{\rm am}} } \right)} \right]}}, \\ NP_{IJ} & = \frac{{\mu_{{\rm am}} }}{{2\left( {\mu_{{\rm clay}} - \mu_{{\rm am}} } \right)}}, \\ MI_{IJ} & = \frac{{\lambda_{{\rm am}} \left( {1 - \Omega_{IK}^{\left( i \right)} \delta_{kk} } \right) - 2\mu_{{\rm am}} \Omega_{IJ}^{\left( i \right)} }}{{2\left( {\mu_{{\rm cry}} - \mu_{{\rm am}} } \right)}}, \\ NI_{IJ} & = \frac{{\mu_{{\rm am}} }}{{2\left( {\mu_{{\rm cry}} - \mu_{{\rm am}} } \right)}}, \\ WP_{II} & = MP_{II}^{{}} + 2NP_{II} , \\ WI_{II} & = MI_{II}^{{}} + 2NI_{II}. \\ \end{aligned}$$

(B8)

in which $\lambda$ and $\mu$ are the Lame’s constants and

$$\left\{ {\left. {\begin{array}{*{20}c} {\Omega_{I1}^{{}} } \\ {\Omega_{I2}^{{}} } \\ {\Omega_{I3}^{{}} } \\ \end{array} } \right\}} \right. = \left[ {\begin{array}{*{20}c} \gamma & {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } & {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } \\ {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } & \gamma & {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } \\ {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } & {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } & \gamma \\ \end{array} } \right]^{ - 1} \left\{ {\left. {\begin{array}{*{20}c} {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } \\ {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } \\ {\lambda_{{\rm cry}} - \lambda_{{\rm am}} } \\ \end{array} } \right\}} \right.$$

(B9)

with $\gamma = \lambda_{{\rm cry}} - \lambda_{{\rm am}} + 2\left( {\mu_{{\rm cry}} - \mu_{{\rm am}} } \right)$.

Appendix C

The Duvaut-Lions approach was employed to transform plasticity to viscoplasticity [47, 48]:

$${\dot{\varvec{\varepsilon }}}^{vp} = \frac{1}{\eta }{\mathbf{C}}_{{\rm am}}^{ - 1} :\left( {{\overline{\varvec{\sigma }}} - {\overline{\overline{\varvec{\sigma }}}}} \right)$$

(C1)

$${\dot{\mathbf{e}}}^{vp} = \frac{1}{\eta }\left( {{\overline{\mathbf{e}}}^{vp} - {\overline{\overline{\varvec{e}}}}^{p} } \right)$$

(C2)

where $\eta$ is a viscosity parameter, ${\overline{\varvec{\sigma }}}$ and ${\overline{\overline{\varvec{\sigma }}}}$ are the total average viscoplastic stress tensor and the overall inviscid plastic stress tensor, respectively, and, ${\overline{\mathbf{e}}}^{vp}$ and ${\overline{\overline{\varvec{e}}}}^{p}$ are the viscoplastic strain tensor and the inviscid plastic strain tensor, respectively. The inviscid solution, in terms of the actual stress tensor ${\overline{\overline{\varvec{\sigma }}}}_{n + 1}$ and the internal variable ${\overline{\overline{\varvec{e}}}}_{n + 1}^{p}$, is updated at each increment allowing the calculation of the new stress ${\overline{\varvec{\sigma }}}_{n + 1}$ and the viscoplastic strain ${\overline{\mathbf{e}}}^{vp}_{n + 1}$ by integrating the two previous equations using a backward Euler algorithm:

$${\overline{\varvec{\sigma }}}_{n + 1} = \frac{{\left( {{\overline{\varvec{\sigma }}}_{n} + {\mathbf{C}}_{{\rm am}} :\Delta {\overline{\varvec{\varepsilon }}}_{n + 1} } \right) + \frac{{\Delta t_{n + 1} }}{\eta }{\overline{\overline{\varvec{\sigma }}}}_{n + 1} }}{{1 + \frac{{\Delta t_{n + 1} }}{\eta }}}$$

(C3)

$${\overline{\mathbf{e}}}^{vp}_{n + 1} = \frac{{{\overline{\mathbf{e}}}^{vp}_{n} + \frac{{\Delta t_{n + 1} }}{\eta }{\overline{\overline{\varvec{e}}}}^{p}_{n + 1} }}{{1 + \frac{{\Delta t_{n + 1} }}{\eta }}}$$

(C4)

where $\Delta t_{n + 1}$ is the time step. When ${{\Delta t_{n + 1} } \mathord{\left/ {\vphantom {{\Delta t_{n + 1} } \eta }} \right. \kern-\nulldelimiterspace} \eta } \to \infty$ the inviscid solution is recovered and when ${{\Delta t_{n + 1} } \mathord{\left/ {\vphantom {{\Delta t_{n + 1} } \eta }} \right. \kern-\nulldelimiterspace} \eta } \to 0$ the elastic solution is achieved.