Solution of a bonded bimaterial problem of two interfaces subjected to different temperatures

Abstract

A closed-form solution is derived for the bonded bimaterial planes at two interfaces subjected to different temperatures. The bonded planes with two interfaces are symmetric with respect to the interface, which is straight. A rational mapping function and complex stress functions are used for the analysis. The problem is reduced to a Riemann–Hilbert problem. The solution includes an integral term. This integral cannot be carried out. However, the first derivative of complex stress functions which does not include integral terms with regard to the variables of the mapping plane is achieved. Therefore, there is no need for numerical integration to calculate stress components and to determine unknown coefficients in a complex stress function. This is very beneficial. It is more difficult to derive the solution to the two-interface problem compared to the general solution. As a demonstration of geometry, semi-strips bonded in two places at the ends of strips are considered. Unbonded parts are a model of debonding. The solution for different geometrical shapes can be obtained by changing the mapping function. Some stress distributions are shown for different lengths of the interface. The stress intensity of debonding (SID) (corresponding to the root of strain energy release rate) is investigated for the debonding extension. SID is the same as the root of strain energy release rate for the evaluation of the strength at the debonding tip.

Appendices

Appendix 1: Derivation of $\overline{F_{11} (1/\bar{{t}}_1)}$

Taking the conjugate of function $F_{11} (t_1 )$ for $j=1$ in Eq. (38c), the following equation is obtained:

$$\begin{aligned} \overline{F_{11} (1/\bar{{t}}_1 )} =\overline{\chi _1 (1/\bar{{t}}_1 )} \int \nolimits _{\bar{{\beta }}_1}^{\bar{{\alpha }}_2} {\frac{1}{\overline{\chi _1 (\sigma )} \overline{(\sigma -1/\bar{{t}}_1 )}}} \hbox {d}\bar{{\sigma }} \end{aligned}$$

(65)

Using the relations $\bar{{\sigma }}=1/\sigma , \bar{{\alpha }}_2 =1/\alpha _2$, and $\bar{{\beta }}_1 =1/\beta _1$ on the unit circle, the equation above is

$$\begin{aligned} \overline{F_{11} (1/\bar{{t}}_1 )}= & {} \overline{\chi _1 (1/\bar{{t}}_1 )} \int \nolimits _{\beta _1}^{\alpha _2} {\frac{t_1 }{\overline{\chi _1 (\sigma )} \sigma (\sigma -t_1 )}} \hbox {d}\sigma \nonumber \\= & {} \chi _2 (t_1 )\int \nolimits _{\beta _1}^{\alpha _2} {\frac{\sigma }{\chi _2 (\sigma )t_1 (\sigma -t_1 )}} \hbox {d}\sigma \nonumber \\= & {} \chi _2 (t_1 )\int \nolimits _{\beta _1}^{\alpha _2} {\frac{1}{\chi _2 (\sigma )(\sigma -t_1 )}} \hbox {d}\sigma +\frac{\chi _2 (t_1 )}{t_1 }\int \nolimits _{\beta _1}^{\alpha _2} {\frac{1}{\chi _2 (\sigma )}} \hbox {d}\sigma \nonumber \\= & {} F_{21} (t_1 )-\frac{\chi _2 (t_1 )}{t_1}\frac{2i}{\chi _2 (1)}I_{21} (0) \end{aligned}$$

(66)

where the following relation was used:

$$\begin{aligned} \frac{\overline{\chi _1 (1/t_1 )}}{\overline{\chi _1 (\sigma )} }=\frac{\sigma ^{2}\chi _2 (t_1 )}{t_1^2 \chi _2 (\sigma )}. \end{aligned}$$

(67)

$I_{21} (0)$ is derived as follows: When the variable $\sigma$ is transformed on the real variable t by

$$\begin{aligned} \sigma= & {} \frac{(t+i)}{(t-i)}, \end{aligned}$$

(68a)

$$\begin{aligned} t= & {} -i\frac{(1+\sigma )}{(1-\sigma )} \end{aligned}$$

(68b)

the following equation is obtained:

$$\begin{aligned} \int \nolimits _{\beta _1}^{\alpha _2} {\frac{1}{\chi _2 (\sigma )}} \hbox {d}\sigma =\frac{-2i}{\chi _2 (1)}\int \nolimits _{t_{\beta _{_1}}}^{t_{\alpha _{_2} }} {\frac{1}{\chi _{2t} (t)}} \hbox {d}t\equiv \frac{-2i}{\chi _2(1)}I_{21} (0) \end{aligned}$$

(69)

where

$$\begin{aligned}&\displaystyle \hbox {d}\sigma =\frac{-2i}{(t-i)^{2}}\hbox {d}t \end{aligned}$$

(70a)

$$\begin{aligned}&\displaystyle \chi _2 (\sigma )=\frac{\chi _2 (1)}{(t-i)^{2}}\chi _{2t} (t) \end{aligned}$$

(70b)

$$\begin{aligned}&\displaystyle \chi _{2t} (t)=(t-t_{\alpha _1} )^{m_2}(t-t_{\beta _1} )^{1-m_2 }(t-t_{\alpha _2} )^{m_2}(t-t_{\beta _2} )^{1-m_2} \end{aligned}$$

(70c)

$$\begin{aligned}&\displaystyle t_{\alpha _1} =-i\frac{1+\alpha _1}{1-\alpha _1},\quad t_{\beta _1} =-i\frac{1+\beta _1}{1-\beta _1},\quad t_{\alpha _2} =-i\frac{1+\alpha _2}{1-\alpha _2},\quad t_{\beta _2} =-i\frac{1+\beta _2}{1-\beta _2} \end{aligned}$$

(70d)

Values of the elliptical integral type $I_{21} (0)$ must be calculated by a numerical integration regarding real variable t (see Appendix 4).

Appendix 2: Derivation of function $P_1 (t_1 )$

Substituting Eqs. (42) and (44) into Eq. (7), the following equation is derived:

$$\begin{aligned} \psi _1 (t_1 )= & {} -\overline{\phi _1 (1/\overline{t_1} )} -\frac{\overline{\omega (1/\overline{t_1} )}}{{\omega }^{'}(t_1 )}{\phi }_1^{'} (t_1 ) \nonumber \\= & {} -\overline{\chi _1 (1/\overline{t_1} )} \overline{P_1 (1/\overline{t_1} )} -\frac{\chi _2 (t_1 )}{t_1}\left[ {\frac{\overline{C_1}}{\pi \chi _2 (1)}I_{21} (0)+\frac{\gamma _1 }{1+\lambda _1}\sum \limits _l {\left\{ {\frac{\overline{a_{1l}} {\xi }_{1l}^{'2}}{\chi _2 ({\xi }_{1l}^{'} )}+\frac{\overline{b_{1l}} {\eta }_{1l}^{'2}}{\chi _2 ({\eta }_{1l}^{'} )}} \right\} }} \right] \nonumber \\&+\sum \limits _{k=1}^N {\frac{A_{1k} \bar{{B}}_k {\zeta }_k^{'2} }{{\zeta }_k^{'} -t_1}} +(\hbox {regular terms in}\,S_1^+ ) \end{aligned}$$

(71)

The equation above has poles at $t_1 ={\zeta }_k^{'}(k=1,2,3,\ldots ,N)$. When $\overline{\chi _1 (1/\overline{t_1} )} \overline{P_1 (1/\overline{t_1} )}$ is expanded to Laurent series at $t_1 ={\zeta }_k^{'}, \overline{\chi _1 (1/\overline{t_1 } )} \overline{P_1 (1/\overline{t_1} )}$ is expressed as:

$$\begin{aligned} \overline{\chi _1 (1/\overline{t_1} )} \overline{P_1 (1/\overline{t_1} )} =\sum \limits _{k=1}^N {\frac{\overline{\chi _1 (1/\overline{{\zeta }_k^{'}} )} \bar{{q}}_k}{{\zeta }_k^{'} -t_1}} +(\hbox {regular terms at}\,t_1 ={\zeta }_k^{'}) \end{aligned}$$

(72)

The unknown constant $\bar{{q}}_k$ is determined so that $\psi _1 (t_1 )$ is regular at $t_1 ={\zeta }_k^{'}$ as follows:

$$\begin{aligned} \bar{{q}}_k =\frac{A_{1k} \bar{{B}}_k {\zeta }_k^{'2}}{\overline{\chi _1 (1/\overline{{\zeta }_k{'}} )}} \end{aligned}$$

(73)

$\therefore$

$$\begin{aligned} P_1 (t_1 )=-\sum \limits _{k=1}^N {\frac{\overline{A_{1k}} B_k }{\chi _1 (\zeta _k )(\zeta _k -t_1 )}} \end{aligned}$$

(74)

Appendix 3: Determination of $a_{1l} ,\hbox {b}_{1l},\xi _{1l} ,\eta _{1l}$

Using Eqs. (30) and (45), function $\Theta _1 (t_1 )$ expressed by Eq. (16) is expressed as follows:

$$\begin{aligned} \Theta _1 (t_1 )= & {} \phi _1 (t_1 )+\overline{\phi _2 (1/\overline{t_1} )} \nonumber \\= & {} H_1 (t_1 )+\overline{H_2 (1/\bar{{t}}_1 )} +\frac{\gamma _2 }{1+\lambda _2}\sum \limits _l {\left\{ {\overline{a_{2l}} {\xi }_{2l}^{'} +\overline{b_{2l}} {\eta }_{2l}^{'}} \right\} } \nonumber \\&+\frac{\chi _1 (t_1 )}{t_1}\left\{ {\sum \limits _{k=1}^N {\frac{A_{2k} \bar{{B}}_k {\zeta }_k^{'2}}{\chi _1 ({\zeta }_k^{'} )}+\frac{\gamma _2}{1+\lambda _2}\sum \limits _l {\frac{\overline{a_{2l}} {\xi }_{2l}^{'2}}{\chi _1 ({\xi }_{2l}^{'} )}+\frac{\gamma _2 }{1+\lambda _2}\sum \limits _l {\frac{\overline{b_{2l}} {\eta }_{2l}^{'2}}{\chi _1 ({\eta }_{2l}^{'} )}}}}} \right\} \nonumber \\&-\chi _1 (t_1 )\sum \limits _{k=1}^N {\frac{\overline{A_{1k} } B_k}{\chi _1 (\zeta _k )(\zeta _k -t_1 )}} +\chi _1 (t_1 )\sum \limits _{k=1}^N {\frac{A_{2k} \bar{{B}}_k {\zeta }_k^{'2}}{\chi _1 ({\zeta }_k^{'} )({\zeta }_k^{'} -t_1 )}} \nonumber \\&+\frac{\gamma _1}{1+\lambda _1}\sum \limits _l {\left\{ {1-\frac{\chi _1 (t_1 )}{\chi _1 (\xi _{1l} )}} \right\} \frac{a_{1l} }{\xi _{1l} -t_1}} -\frac{\gamma _2}{1+\lambda _2}\sum \limits _l {\left\{ {1-\frac{\chi _1 (t_1 )}{\chi _1 ({\eta }_{2l}^{'} )}} \right\} \frac{\overline{b_{2l}} {\eta }_{2l}^{'2}}{{\eta }_{2l}^{'} -t_1}} \nonumber \\&+\frac{\gamma _1}{1+\lambda _1}\sum \limits _l {\left\{ {1-\frac{\chi _1 (t_1 )}{\chi _1 (\eta _{1l} )}} \right\} \frac{b_{1l}}{\eta _{1l} -t_1}} -\frac{\gamma _2 }{1+\lambda _2}\sum \limits _l {\left\{ {1-\frac{\chi _1 (t_1 )}{\chi _1 ({\xi }_{2l}^{'} )}} \right\} \frac{\overline{a_{2l}} {\xi }_{2l}^{'2}}{{\xi }_{2l}^{'} -t_1}} \end{aligned}$$

(75)

where the following relations were used:

$$\begin{aligned}&\frac{\overline{\chi _2 (1/{\bar{t}}_1 )}}{\overline{\chi _2 (\xi _{2l} )}} =\frac{{\xi }_{2l}^{'2} \chi _1 (t_1 )}{t_1^2 \chi _1 ({\xi }_{2l}^{'} )},\end{aligned}$$

(76a)

$$\begin{aligned}&\frac{\overline{\chi _2 (1/\bar{{t}}_1 )}}{\overline{\chi _2 (\eta _{2l} )}}=\frac{{\eta }_{2l}^{'2} \chi _1 (t_1 )}{t_1^2 \chi _1 ({\eta }_{2l}^{'} )}, \end{aligned}$$

(76b)

$$\begin{aligned}&\frac{\overline{\chi _2 (1/{\bar{t}}_1 )}}{\overline{\chi _2 (\zeta _k )}}=\frac{{\zeta }_k^{'2} \chi _1 (t_1 )}{t_1^2 \chi _1 ({\zeta }_k^{'} )} \end{aligned}$$

(76c)

$$\begin{aligned}&{\xi }_{2l}^{'} \equiv 1/\overline{\xi _{2l}} , \end{aligned}$$

(76d)

$$\begin{aligned}&{\eta }_{2l}^{'} \equiv 1/\overline{\eta _{2l}} \end{aligned}$$

(76e)

It was derived that the following term in Eq. (75) was zero, using Eq. (48):

$$\begin{aligned} H_1 (t_1 )+\overline{H_2 (1/\bar{{t}}_1 )} +\frac{\chi _1 (t_1 )}{t_1}\left\{ {\sum \limits _{k=1}^N {\frac{A_{2k} \bar{{B}}_k {\zeta }_k^{'2}}{\chi _1 ({\zeta }_k^{'} )}+\frac{\gamma _2}{1+\lambda _2}\sum \limits _l {\frac{\overline{a_{2l}} {\xi }_{2l}^{'2}}{\chi _1 ({\xi }_{2l}^{'} )}+\frac{\gamma _2}{1+\lambda _2}\sum \limits _l {\frac{\overline{b_{2l}} {\eta }_{2l}^{'2}}{\chi _1 ({\eta }^{'}_{2l} )}}}}} \right\} =0 \end{aligned}$$

(77)

where $H_1 (t_1)$ and $\overline{H_2 (1/\bar{{t}}_1 )}$ are obtained by Eq. (40). Also, the constant term

$$\begin{aligned} \frac{\gamma _2}{1+\lambda _2}\sum \limits _l {\left\{ {\overline{a_{2l}} {\xi }_{2l}^{'} +\overline{b_{2l}} {\eta }_{2l}^{'}} \right\} } \end{aligned}$$

(78)

is included in the arbitrariness of the stress function. The function $\Theta _1 (t_1 )$ expressed by Eq. (75) is the same as the function $\Theta _1 (t_1 )$ expressed by Eq. (28) and Eq. (18), i.e.,

$$\begin{aligned} \Theta _1 (t_1 )=\sum \limits _l {\frac{a_{1l}}{\xi _{1l} -t_1}} +\sum \limits _l {\frac{b_{1l}}{\eta _{1l} -t_1}} \end{aligned}$$

(79)

Because Eqs. (75) and (79) are the same equation, these equations have the same poles inside and outside regions, $S_1^+$ and $S_1^-$, respectively. From this condition, the following relations are obtained:

$$\begin{aligned} \xi _{1l}= & {} {\eta }_{2l}^{'} =\zeta _k \end{aligned}$$

(80a)

$$\begin{aligned} \eta _{1l}= & {} {\xi }_{2l}^{'} ={\zeta }_k^{'} \end{aligned}$$

(80b)

And coefficients of terms including $\chi _1 (t_1 )$ and not including $\chi _1 (t_1 )$ must equal at the same poles inside and outside regions, $S_1^+$ and $S_1^-$, respectively. Comparing the coefficients, the following relations are obtained:

$$\begin{aligned} a_{1l}= & {} -\bar{{A}}_{1k} B_k \end{aligned}$$

(81a)

$$\begin{aligned} b_{1l}= & {} A_{2k} \bar{{B}}_k {\zeta }_k^{'2} \end{aligned}$$

(81b)

For material II, the same procedure is done, and the coefficients are expressed by exchanging suffixes $j=1$ and 2. Then, Eqs. (49) and (50) are obtained.

Appendix 4: First derivative of functions $F_{j1} (t_j )$

The first derivative of function $F_{j1} (t_j )$ which is defined in Eq. (38c) is obtained as follows [19]:

$$\begin{aligned} {F}_{j1}^{'} (t_j )=\frac{i}{2}\frac{y_j (t_j )}{y_j (1)}\left[ {\begin{array}{l} I_{j1} (2)-2iI_{j1} (1)\left\{ {1-\frac{m_j}{1-\alpha _1 }-\frac{1-m_j}{1-\beta _1}-\frac{m_j}{1-\alpha _2}-\frac{1-m_j }{1-\beta _2}} \right\} \\ \quad -I_{j1} (0)\left\{ {\frac{m_j (1+\alpha _1 )(1+\alpha _2 )}{(1-\alpha _1 )(1-\alpha _2 )}+\frac{(1-m_j )(1+\beta _1 )(1+\beta _2 )}{(1-\beta _1 )(1-\beta _2 )}} \right\} \\ \quad +\frac{4I_{j1} (0)\left\{ {(1-m_j )(t_j -\alpha _1 )(t_j -\alpha _2 )+m_j (t_j -\beta _1 )(t_j -\beta _2 )} \right\} }{(1-\alpha _1 )(1-\beta _1 )(1-\alpha _2 )(1-\beta _2 )} \\ \end{array}} \right] \end{aligned}$$

(82)

where

$$\begin{aligned} y_j (t_j )=\frac{\chi _j (t_j )}{\mathop \Pi \limits _{k=1}^2 (t_j -\alpha _k )(t_j -\beta _k )} \end{aligned}$$

(83)

$\chi _j (t_j )$ is given by Eq. (22).

Variables $t_{\alpha _1} , t_{\beta _1} , t_{\alpha _2}$, and $t_{\beta _2}$ are given in Eq. (70d). $I_{j1} (n)(n=0,1,2)$ is defined by the following equations:

$$\begin{aligned} I_{j1} (n)= & {} \int \nolimits _{t_{\beta _{1}}}^{t_{\alpha _{1}}} {\frac{t^{n}}{\chi _{jt} (t)}} \hbox {d}t\quad (n=0,1,2) \end{aligned}$$

(84)

$$\begin{aligned} \chi _{jt} (t)= & {} (t-t_{\alpha _1} )^{m_j}(t-t_{\beta _1} )^{1-m_j }(t-t_{\alpha _2} )^{m_j}(t-t_{\beta _2} )^{1-m_j} \end{aligned}$$

(85)

The above integrals of elliptical type $I_{j1} (n)(n=0,1,2)$ must be obtained by a numerical integration with respect to the real variable t (see Eq. 68b).