A finite strain fibre-reinforced viscoelasto-viscoplastic model of plant cell wall growth

Abstract

A finite strain fibre-reinforced viscoelasto-viscoplastic model implemented in a finite element (FE) analysis is presented to study the expansive growth of plant cell walls. Three components of the deformation of growing cell wall, i.e. elasticity, viscoelasticity and viscoplasticity-like growth, are modelled within a consistent framework aiming to present an integrative growth model. The two aspects of growth—turgor-driven creep and new material deposition—and the interplay between them are considered by presenting a yield function, flow rule and hardening law. A fibre-reinforcement formulation is used to account for the role of cellulose microfibrils in the anisotropic growth. Mechanisms in in vivo growth are taken into account to represent the corresponding biology-controlled behaviour of a cell wall. A viscoelastic formulation is proposed to capture the viscoelastic response in the cell wall. The proposed constitutive model provides a unique framework for modelling both the in vivo growth of cell wall dominated by viscoplasticity-like behaviour and in vitro deformation dominated by elastic or viscoelastic responses. A numerical scheme is devised, and FE case studies are reported and compared with experimental data.

Appendices

Appendix 1: The update algorithm of ${\overline{\varvec{S}}}$ and $\overline{\mathbb {C}}^\mathrm{v}$

The update criterion (11) for the configuration $\tilde{{\mathcal {B}}}_t$ is implemented as follows:

(102)

where the increments $\Delta {\tilde{\varvec{F}}}_n$ and are defined as follows:

(103)

in which the velocity gradient ${\varvec{L}}_n^\mathrm{ve}$ is computed from $\Delta {\varvec{F}}_n^\mathrm{ve}$ by using the relation

$$\begin{aligned} \Delta {\varvec{F}}_n^\mathrm{ve} = ( {{\varvec{1}}+{\varvec{L}}_n^\mathrm{ve} \Delta t_n }). \end{aligned}$$

(104)

The update of the algorithmic internal variables ${\tilde{\varvec{H}}}_{n+1}$ and is obtained by using the middle rule [52] as follows:

$$\begin{aligned} {\tilde{\varvec{H}}}_{n+1} ={\tilde{\tilde{\varvec{H}}}}_n +{\exp } \bigg ( -\frac{{\Delta t_n }}{{2\tilde{\tau }}} \bigg ){\tilde{\varvec{S}}{}_{n+1}^\circ } , \end{aligned}$$

(105)

(106)

where ${\tilde{\tilde{\varvec{H}}}}_n$ and are defined as

$$\begin{aligned} {\tilde{\tilde{\varvec{H}}}}_n ={\exp }\bigg (- \frac{{\Delta t_n}}{{\tilde{\tau }}} \bigg )\,{\tilde{\varvec{H}}}_n -{\exp }\bigg (- \frac{{\Delta t_n }}{{2\tilde{\tau }}} \bigg ){\tilde{\varvec{S}}_n^\circ } , \end{aligned}$$

(107)

(108)

In order to obtain expressions of ${\overline{\varvec{S}}}$ and $\overline{\mathbb {C}}^\mathrm{v}$, specified functions $W^{\circ }$ and $\tilde{W}^{\circ }$ in Eqs. (38) and (39) are set in the similar form as the deviatoric part of the free energy of wall matrix in Eq. (31) as follows:

$$\begin{aligned} W^{\circ }\big ( {\hat{\tilde{\varvec{C}}}} \big )=\frac{1}{2}\mu \big ( {\hbox {tr}\,({\hat{\tilde{\varvec{C}}}})-3} \big ), \end{aligned}$$

(109)

(110)

Thus, the stress tensors ${\tilde{\varvec{S}}{}_{n+1}^\circ }$ and in Eqs. (105) and (106) is obtained by using Eqs. (38) and (39) as

(111)

(112)

where the deviatoric operators are defined as

$$\begin{aligned} \hbox {DEV}_{n+1}^+ [ { \cdot } ]= & {} \cdot -\frac{1}{3}( {{\tilde{\varvec{C}}}_{n+1} :( \cdot )} ){\tilde{\varvec{C}}{}_{n+1}^{-1}} ,\end{aligned}$$

(113)

$$\begin{aligned} \hbox {DEV}_{n+1} [ { \cdot } ]= & {} \cdot -\frac{1}{3}( {{\overline{\varvec{C}}}_{n+1} :( \cdot )} ){\overline{\varvec{C}}}_{n+1}^{-1} . \end{aligned}$$

(114)

Substituting Eqs. (105), (106), (111), (112) into Eq. (45) yields

(115)

where the two functions $\tilde{g}{}_n^*$ and are defined, respectively, as

(116)

By using Eqs. (115), (24) indicates that the viscoelastic components of tangent modulus is

(117)

where two operators, $\widetilde{\hbox {DEV}}[ { \cdot } ]$ and ${\partial \{ {{\overline{J}{}^{-2/3}}\hbox {DEV}} \}}/{\partial {\overline{\varvec{C}}}}[ { \cdot }]$, are defined, respectively, as

$$\begin{aligned} \widetilde{\hbox {DEV}}[ { \cdot } ]= & {} \cdot -\frac{1}{3}{\overline{\varvec{C}}{}^{-1}}\otimes ( {{\overline{\varvec{C}}}:( \cdot )} ),\end{aligned}$$

(118)

$$\begin{aligned} \frac{\partial \{ {{\overline{J}{}^{-2/3}}\hbox {DEV}} \}}{\partial {\overline{\varvec{C}}}}[ {\cdot } ]= & {} {\overline{J}{}^{-2/3}}\left\{ {-\frac{2}{3}\hbox {DEV}[ { \cdot } ]\otimes {\overline{\varvec{C}}{}^{-1}}-\frac{2}{3}{\overline{\varvec{C}}{}^{-1}}\otimes \hbox {DEV}[ { \cdot } ]} \right. \nonumber \\&\left. {+\frac{2}{3}( {( \cdot ):{\overline{\varvec{C}}}})\bigg [ {{\mathbb {I}}_{{\overline{\varvec{C}}{}^{-1}}} -\frac{1}{3}{\overline{\varvec{C}}{}^{-1}}\otimes {\overline{\varvec{C}}{}^{-1}}} \bigg ]} \right\} . \end{aligned}$$

(119)

The remaining work is to compute two tensors and ${\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }/{\partial {\overline{\varvec{C}}}}$ in Eq. (117).

By using Eqs. (112) and (119), is obtained straightforward as

(120)

On the other hand, by using the criterion (103) (if $\hbox {tr}\,{\varvec{L}}^\mathrm{ve}\ge 0), {\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }/{\partial {\overline{\varvec{C}}}}$ is computed by the chain rule as follows:

$$\begin{aligned} \frac{\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }{\partial {\overline{\varvec{C}}}}=\frac{\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }{\partial {\tilde{\varvec{C}}}}\bigg ( {\frac{\partial {\tilde{\varvec{C}}}}{\partial {\overline{\varvec{C}}}}} \bigg )_n , \end{aligned}$$

(121)

where tensor ${\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }/{\partial {\tilde{\varvec{C}}}}$ is expressed as

(122)

in which by using Eq. (111) we have

$$\begin{aligned} \frac{\partial {\tilde{\tilde{\varvec{S}}}}{}_{n+1}^\circ }{\partial {\tilde{\varvec{C}}}}= & {} \frac{2}{3}\mu \tilde{J}_{n+1}^{-2/3} \left\{ {\bigg ( {\hbox {tr}\,({\tilde{\varvec{C}}}_{n+1} )} \bigg )\bigg [ {{\mathbb {I}}_{{\tilde{\varvec{C}}{}_{n+1}^{-1}} } -\frac{1}{3}{\tilde{\varvec{C}}{}_{n+1}^{-1}} \otimes {\tilde{\varvec{C}}{}_{n+1}^{-1}} } \bigg ]} \right. \nonumber \\&\left. -\bigg [ {{\tilde{\varvec{C}}{}_{n+1}^{-1}} \otimes \hbox {DEV}_{n+1}^+ [ {\varvec{1}} ]+\hbox {DEV}_{n+1}^+ [ {\varvec{1}} ]\otimes {\tilde{\varvec{C}}{}_{n+1}^{-1}} } \bigg ] \right\} . \end{aligned}$$

(123)

The other unknown tensor in Eq. (121), ${\partial {\tilde{\varvec{C}}}}/{\partial {\overline{\varvec{C}}}}$, is deduced as follows. It can be shown that (if $\hbox {tr}\,({\varvec{L}}^\mathrm{ve})\ge 0)$

(124)

By using Eq. (124), two one-parameter families of right Cauchy–Green tensors are constructed in the forms of

$$\begin{aligned} {\overline{\varvec{C}}}_\varepsilon ={\overline{\varvec{C}}}+\varepsilon {\varvec{H}}_n , \end{aligned}$$

(125)

(126)

where $\varepsilon$ is a scalar parameter, tensor ${\varvec{H}}_n$ is defined as

$$\begin{aligned} {\varvec{H}}_n ={\overline{\varvec{C}}}_{n+1} -{\overline{\varvec{C}}}_n . \end{aligned}$$

(127)

The derivative of ${\tilde{\varvec{C}}}_\varepsilon$ with respect to $\varepsilon$ is computed from Eq. (126) as

(128)

The definitions (125) and (126) indicate that

$$\begin{aligned} \left. {\frac{\partial {\tilde{\varvec{C}}}_\varepsilon }{\partial \varepsilon }} \right| _{\varepsilon =0} =\frac{\partial {\tilde{\varvec{C}}}}{\partial {\overline{\varvec{C}}}}\left. {:\frac{\partial {\overline{\varvec{C}}}_\varepsilon }{\partial \varepsilon }} \right| _{\varepsilon =0} =\frac{\partial {\tilde{\varvec{C}}}}{\partial {\overline{\varvec{C}}}}:{\varvec{H}}_n . \end{aligned}$$

(129)

Therefore, Eqs. (128) and (129) yield the expression of ${\partial {\tilde{\varvec{C}}}}/{\partial {\overline{\varvec{C}}}}$ as follows:

(130)

Once tensors ${\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }/{\partial {\tilde{\varvec{C}}}}$ and ${\partial {\tilde{\varvec{C}}}}/{\partial {\overline{\varvec{C}}}}$ are obtained by Eqs. (122) and (130), respectively, Eq. (121) yields the expression of ${\partial {\tilde{\varvec{S}}{}_{n+1}^\circ } }/{\partial {\overline{\varvec{C}}}}$. Then the expression of $\overline{\mathbb {C}}^\mathrm{v}$ is obtained by substituting Eqs. (118–121) into Eq. (117).

Appendix 2: The derivative of function $\phi$

Let $( \cdot )$ be an entity standing for tensor, vector or scalar. By taking the derivative of Eq. (67) with respect to $( \cdot )$ we obtain

$$\begin{aligned} \frac{\partial \theta }{\partial ( \cdot )}=\vartheta _1 \frac{\partial \overline{J}_1 }{\partial ( \cdot )}+\vartheta _2 \frac{\partial \overline{J}_2 }{\partial ( \cdot )}+\vartheta _h \frac{\partial h}{\partial ( \cdot )}, \end{aligned}$$

(131)

where

$$\begin{aligned} \vartheta _1 ={Y_1^2 \frac{\big ({\theta -Y_2^2 }\big )^{2}}{{\vartheta _D }}} , \quad \vartheta _2 = \frac{{Y_2^2 \big ( {\theta -Y_1^2 } \big )^{2}}}{{\vartheta _D }}, \quad \vartheta _h = \frac{{-\big ( {\theta -Y_1^2 } \big )^{2}\big ( {\theta -Y_2^2 } \big )^{2}}}{{\vartheta _D }} \end{aligned}$$

(132)

in which

$$\begin{aligned} \vartheta _D =2\big \{ {\big ( {1+h} \big )\big ( {\theta -Y_1^2 } \big )\big ( {\theta -Y_2^2 } \big )\big ( {2\theta -Y_1^2 -Y_2^2 } \big )-\overline{J}_1 Y_1^2 \big ( {\theta -Y_2^2 } \big )-\overline{J}_2 Y_2^2 \big ( {\theta -Y_1^2 } \big )} \big \}. \end{aligned}$$

(133)

Using the chain rule and substituting Eq. (131) into the derivative of the distance function $\phi$ with respect to $\left( \cdot \right)$ yields

$$\begin{aligned} \frac{\partial \phi }{\partial ( \cdot )}=\bigg ( {\frac{\partial \phi }{\partial \overline{J}_1 }+\vartheta _1 \frac{\partial \phi }{\partial \theta }} \bigg )\frac{\partial \overline{J}_1 }{\partial ( \cdot )}+\bigg ( {\frac{\partial \phi }{\partial \overline{J}_2 }+\vartheta _2 \frac{\partial \phi }{\partial \theta }} \bigg )\frac{\partial \overline{J}_2 }{\partial ( \cdot )}+\vartheta _h \frac{\partial \phi }{\partial \theta }\frac{\partial h}{\partial ( \cdot )} \end{aligned}$$

(134)

in which

$$\begin{aligned} \frac{\partial \phi }{\partial \overline{J}_1 }= & {} \frac{\theta ^{2}}{2\phi \big ( {Y_1^2 -\theta } \big )^{2}}, \quad \frac{\partial \phi }{\partial \overline{J}_2 }=\frac{\theta ^{2}}{2\phi \big ( {Y_2^2 -\theta } \big )^{2}},\nonumber \\ \frac{\partial \phi }{\partial \theta }= & {} \frac{\phi }{\theta }+\frac{\theta ^{2}}{\phi }\left( {\frac{\overline{J}_1 }{\big ( {Y_1^2 -\theta } \big )^{3}}+\frac{\overline{J}_2 }{\big ( {Y_2^2 -\theta } \big )^{3}}} \right) . \end{aligned}$$

(135)