Two-dimensional analysis of progressive delamination in thin film electrodes

Abstract

By employing the two-dimensional analysis, i.e., plane strain and plane stress, a semi-analytical method is developed to investigate the interfacial delamination in electrodes. The key parameters are obtained from the governing equations, and their effects on the evolution of the delamination are evaluated. The impact of constraint perpendicular to the plane is also investigated by comparing the plane strain and plane stress. It is found that the delamination in the plane strain condition occurs easier, indicating that the constraint is harmful to maintain the structure stability. According to the obtained governing equations, a formula of the dimensionless critical size for delamination is provided, which is a function of the maximum volumetric strain and the Poisson’s ratio of the active layer.

Appendices

Appendix A: Coefficient functions in the governing equations

The explicit expressions for the kernel functions $\bar{{F}}_p$, $\bar{{F}}_q$, $\bar{{G}}_p$, $\bar{{G}}_q$, $\bar{{f}}_{c},$ and $\bar{{g}}_c$ can be described as

$$\begin{aligned} \bar{{F}}_p \left( {\bar{{x}},\bar{{\eta }}} \right)= & {} -\frac{3\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{2}}{\bar{{h}}^{2}}\bar{{\eta }}^{2}\nonumber \\&+\,\left\{ {{\begin{array}{ll} {-\left( {1-\upsilon _\mathrm{s} } \right) }, &{}\quad {0\leqslant \eta \leqslant x}, \\ {\frac{3\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{2}}{\bar{{h}}^{2}}(\bar{{x}}-\bar{{\eta }})^{2}},&{}\quad {x<\eta \leqslant L}, \\ \end{array} }} \right. \end{aligned}$$

(A1)

$$\begin{aligned} \bar{{F}}_q \left( {\bar{{x}},\bar{{\eta }}} \right)= & {} -\frac{{1}}{{\uppi }}\left[ {\ln \frac{\left( {\bar{{x}}-\bar{{\eta }}} \right) ^{2}}{\left( {\bar{{x}}+\bar{{\eta }}} \right) ^{2}}} \right] \nonumber \\&+\,\left\{ {{\begin{array}{ll} {\frac{4\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}}{\bar{{h}}}\bar{{\eta }}},&{}\quad {0\leqslant \eta \leqslant x}, \\ {\frac{4\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}}{\bar{{h}}}\bar{{x}}},&{}\quad {x<\eta \leqslant L}, \\ \end{array} }} \right. \end{aligned}$$

(A2)

$$\begin{aligned} \bar{{G}}_p \left( {\bar{{x}},\bar{{\eta }}} \right)= & {} -\frac{{1}}{{\uppi }}\ln \left[ {\frac{\left( {\bar{{x}}^{2}-\bar{{\eta }}^{2}} \right) ^{2}}{\bar{{\eta }}^{4}}} \right] \nonumber \\&+\,\left\{ {{\begin{array}{ll} {\frac{2\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{3}}{\bar{{h}}^{3}}\left( {3\bar{{x}}\bar{{\eta }}^{2}-\bar{{\eta }}^{3}} \right) },&{}\quad {0\leqslant \eta \leqslant x}, \\ {\frac{2\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{3}}{\bar{{h}}^{3}}\left( {-\bar{{x}}^{3}+3\bar{{x}}^{2}\bar{{\eta }}} \right) },&{}\quad {x<\eta \leqslant L}, \\ \end{array} }} \right. \nonumber \\ \end{aligned}$$

(A3)

$$\begin{aligned} \bar{{G}}_q \left( {\bar{{x}},\bar{{\eta }}} \right)= & {} \frac{3\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{2}}{\bar{{h}}^{2}}(\bar{{\eta }}^{2}-2\bar{{x}}\bar{{\eta }})^{2}\nonumber \\&+\,\left\{ {{\begin{array}{ll} {\left( {1-\upsilon _\mathrm{s} } \right) },&{}\quad {0\leqslant \eta \leqslant x}, \\ {-\frac{3\left( {1-\upsilon _\mathrm{p} ^{2}} \right) }{\bar{{E}}}\frac{\bar{{L}}^{2}}{\bar{{h}}^{2}}(\bar{{x}}-\bar{{\eta }})^{2}},&{}\quad {x<\eta \leqslant L}, \\ \end{array} }} \right. \nonumber \\ \end{aligned}$$

(A4)

$$\begin{aligned}&\bar{{f}}_c \Big (\bar{{x}},\bar{{t}} \Big )\nonumber \\&\quad =-\left( 1+\upsilon _\mathrm{p} \right) \left[ -\frac{\bar{{x}}}{12}+\frac{{4}}{{\uppi }^{{4}}}\mathop \sum \limits _{n=1}^{\infty } \frac{\left( -1 \right) ^{n}-1}{n^{4}}\exp \left( -n^{2}{\uppi }^{{2}}{\bar{t}} \right) \bar{{x}} \right] -\,\frac{1+\upsilon _\mathrm{p} }{3}\nonumber \\&\qquad \times \left\{ \begin{array}{l} \bar{t}\bar{x}+\frac{\bar{h}}{\bar{L}}\bar{t}\bar{x}+\frac{\bar{L}}{\bar{h}}\left( -\frac{\bar{x}}{6}+\frac{\bar{x}^{3}}{6}\right) \\ -\frac{\bar{L}}{\bar{h}}\frac{{8}}{{\uppi }^{3}}\mathop \sum \limits _{m=1}^{\infty } \frac{1+\left( -1 \right) ^{m}}{m^{3}}\exp \left( -\frac{m^{2}{\uppi }^{{2}}h^{2}}{4L^{2}}{\bar{t}} \right) \left\{ \sin \left[ \frac{m{\uppi }}{2}\left( \bar{{\eta }}+1 \right) \right] -\sin \left( \frac{m{\uppi }}{2} \right) \right\} \end{array}\right\} \!, \nonumber \\ \end{aligned}$$

(A5)

$$\begin{aligned}&\bar{{g}}_c \left( {\bar{{x}},\bar{{t}}} \right) \nonumber \\&\quad =\left( {{1+}\upsilon _\mathrm{p} } \right) \bar{{i}}\frac{\bar{{L}}^{2}}{\bar{{h}}^{2}}\left[ {-\frac{\bar{{x}}^{2}}{{12}}+\frac{{4}}{{\uppi }^{{4}}}\mathop \sum \limits _{n=1}^{\infty } \frac{\left( {-1} \right) ^{n}-{1}}{n^{4}}\exp \left( {-n^{2}{\uppi }^{{2}}{\bar{t}}} \right) \bar{{x}}^{2}} \right] .\nonumber \\ \end{aligned}$$

(A6)

Appendix B: Numerical method for solving the derived integral equations

In order to numerically solve the integral Eqs. (15a)–(17d) and Eq. (20), the length of the active layer L is divided into $\psi$ segments. $\bar{{x}}_i =\bar{{\eta }}_i =x_i /L=(2i-1)/2\psi$ denotes the mid-point of the ith line segment. Hence, the interface can be partitioned into three zones: damage-free zone ($1\leqslant i\leqslant \varsigma )$, cohesive zone ($\varsigma +1\leqslant i\leqslant \xi ),$ and debonding zone ($\xi +1\leqslant i\leqslant \psi )$. Numerical integration of the integrals in Eqs. (15a)–(17d) and Eq. (20) can be made by the first integral mean value theorem. Take $\int _0^1 {\bar{{G}}_p } \bar{{p}}\hbox {d}\bar{{\eta }}$ as an example

$$\begin{aligned} \int _0^1 {\bar{{G}}_p } \bar{{p}}\hbox {d}\bar{{\eta }}= & {} \sum _{j=1}^\psi {\int _{\alpha _j }^{\alpha _{j+1} } {\bar{{G}}_p \bar{{p}}} \hbox {d}\bar{{\eta }}}\nonumber \\\approx & {} \sum _{j=1}^\psi \bar{{p}}\left( {\bar{{\eta }}_j } \right) \int _{\alpha _j }^{\alpha _{j+1} } {\bar{{G}}_p } \hbox {d}\bar{{\eta }} \nonumber \\= & {} \sum _{j=1}^\psi {{\bar{\bar{{G}}}}_p \bar{{p}}\left( {\bar{{\eta }}_j } \right) } , \end{aligned}$$

(B1)

where $\alpha _j \!=\!\left( {j-1} \right) /\psi$ and $\bar{{\bar{{G}}}}_p \left( {\bar{{x}},\bar{{\eta }}_j } \right) \!=\!\int _{\alpha _j }^{\alpha _{j+1} } {\bar{{G}}_p \left( {\bar{{x}},\bar{{\eta }}} \right) } \hbox {d}\bar{{\eta }}$. By using the same numerical integration for all integrals involved, the dimensionless governing Eqs. (15a)–(17d) and Eq. (20) can be discretized into a system of linear equations of $\bar{{p}}$ and $\bar{{q}}$.

Consequently, there are $2\psi +1$ unknowns including $\bar{{p}}\left( {\bar{{\eta }}_j } \right)$, $\bar{{q}}\left( {\bar{{\eta }}_j } \right) \!,$ and $\bar{{L}}_0$ in $2\psi +1$ linear algebraic equations. For a given time, the stresses and the lengths of the cohesive zone and debonding zone can be obtained by solving the system of linear equations repeatedly. In the first iteration, initial trial values of segment numbers $\varsigma$ and $\xi$ are assumed and the corresponding interfacial stresses can be obtained by solving the equations. Accordingly, $\varsigma$ and $\xi$ can be adjusted by comparing the interfacial stresses with cohesive strengths, as well as comparing the open/sliding displacements with critical values. Then the second iteration begins. The iteration will not stop until $\varsigma$ and $\xi$ no longer change.