Crack analysis of circular bars reinforced by a piezoelectric layer under torsional transient loading

Abstract

In this article, the transient response of a cylinder, with a piezoelectric coating, weakened by multiple radial cracks is investigated. The problem is under torsional transient loading. First, the solution of the problem, weakened by a Volterra-type screw dislocation, is achieved by using Laplace and the finite Fourier sine transform. The solution is obtained for displacement and stress fields in the bar with a piezoelectric layer. At the next step, the dislocation solution is used to derive a set of Cauchy singular integral equations for analysis of bars with a circular cross section containing some radial cracks. The solution of the singular integral equations is used to determine the torsional rigidity of the cross section and also the stress intensity factors of the crack tips. In addition, several examples are presented to show the effect of the piezoelectric coating and torsional transient loading on the stress intensity factors and torsional rigidity of the system.

Appendices

Appendix A

Kernels of the integral equations are:

$$\begin{aligned}&k_{ij} ({p,q,s})=\sqrt{({r_j^{\prime }})^{2} + ({r_j \theta _j^{\prime }})^{2}} \left\{ \frac{G^{2}}{2J_0} ({R_1^2 -r_j^2})r_i +\frac{G}{2\pi r_i} \mathop \sum \limits _{n=1}^\infty {\Omega }_n\right. \\&\quad \left( {\left( {\frac{r_j}{R_1}} \right) ^{N_1} -1} \right) \left( -\left( {\frac{r_i}{R_2}} \right) ^{N_1} -\left( {\frac{r_i}{r_j}} \right) ^{N_1} +C_\mathrm{{eq}} \left( {\left( {\frac{r_i}{R_1}} \right) ^{N_1} +\left( {\frac{r_i R_1}{r_j R_2}} \right) ^{N_1}} \right) \right) \, \mathrm{{cos}} ({n \theta }) \\&\qquad -\frac{G}{2\pi r_i} \mathop \sum \limits _{n=1}^\infty \frac{1}{1-\left( {\frac{R_1}{R_2}} \right) ^{2n}C_\mathrm{{eq}}} \\&\quad \left( {\left( {\frac{r_j}{R_1}} \right) ^{n}-1} \right) \left( -\left( {\frac{r_i}{R_2}} \right) ^{n}-\left( {\frac{r_i}{r_j}} \right) ^{n}+C_\mathrm{{eq}} \left( {\left( {\frac{r_i}{R_1}} \right) ^{n}+\left( {\frac{r_i R_1}{r_j R_2}} \right) ^{n}} \right) \right) \mathrm{{cos}} ({n \theta }) \\&\qquad +\frac{G}{4\pi r_i ({C_\mathrm{{eq}} +1})}\mathop \sum \limits _{m=0}^\infty \eta ^{m} \left[ {2\left( {F_m \left( {\frac{r_i}{R_1} ,\theta } \right) -F_m \left( {\frac{R_1 r_i}{R_2^2} ,\theta } \right) } \right) } \right. \\&\qquad +\,({C_\mathrm{{eq}} -1})\left( {F_m \left( {\frac{r_j r_i}{R_1^2} ,\theta } \right) -F_m \left( {\frac{R_1^2 r_i}{R_2^2 r_j} ,\theta } \right) } \right) \\&\qquad \left. \left. +\,({1+C_\mathrm{{eq}}})\left( {F_m \left( {\frac{r_j r_i}{R_2^2} ,\theta } \right) -F_m \left( {\frac{r_i}{r_j} ,\theta } \right) } \right) \right] \right\} , \quad 0\le r_i \le r_j \\&k_{ij} ({p,q,s})=\sqrt{\left( {r_j^{\prime }} \right) ^{2}+\left( {r_j \theta _j^{\prime }} \right) ^{2}} \\&\quad \left\{ {\frac{G^{2}}{2J_0} ({R_1^2 -r_j^2})r_i } +\frac{G}{2\pi r_i} \mathop \sum \limits _{n=1}^\infty \right. \\&\quad \left( {\Omega }_n \left( {\left( \left( {\frac{r_i r_j}{R_2^2}} \right) ^{N_1} +C_\mathrm{{eq}} \left( {\frac{r_i r_j}{R_1^2}} \right) ^{N_1} +2\left( {\left( {\frac{R_1}{R_2}} \right) ^{2N_1} -1} \right) \left( {\frac{r_i}{R_1}} \right) ^{N_1} {\overline{C_\mathrm{{eq}}}} \right) } \right) \right. \\&\qquad \left. -\frac{n^{2}}{N_1^2} \left( {\frac{r_j}{r_i}} \right) ^{N_1} \right) \, \mathrm{{cos}} ({n \theta }) \\&\qquad -\frac{G}{2\pi r_i} \mathop \sum \limits _{n=1}^\infty \left( \frac{1}{1-\left( {\frac{R_1}{R_2}} \right) ^{2n}C_\mathrm{{eq}}} \left( \left( {\frac{r_i r_j}{R_2^2}} \right) ^{n}+C_\mathrm{{eq}} \left( {\frac{r_i r_j}{R_1^2}} \right) ^{n} \right. \right. \\&\qquad \left. \left. +\,2\left( {\left( {\frac{R_1}{R_2}} \right) ^{2n}-1} \right) \left( {\frac{r_i}{R_1}} \right) ^{n}{\overline{C_\mathrm{{eq}}}} \right) -\left( {\frac{r_j}{r_i}} \right) ^{n} \right) \mathrm{{cos}} ({n \theta }) +\frac{G}{4\pi r_i ({C_\mathrm{{eq}} +1})} \\&\quad \left[ {({C_\mathrm{{eq}} +1})F_0 \left( {\frac{r_j}{r_i} ,\theta } \right) } \left. -\mathop \sum \limits _{m=0}^\infty \eta ^{m}\left( 2F_m \left( {\frac{R_1 r_i}{R_2^2} ,\theta } \right) -2F_m \left( {\frac{r_i}{R_1} ,\theta } \right) \right. \right. \right. \\&\qquad \left. \left. \left. -\,({C_\mathrm{{eq}} -1})F_m \left( {\frac{r_j r_i}{R_1^2} ,\theta } \right) -({C_\mathrm{{eq}} +1})F_m \left( {\frac{r_j r_i}{R_2^2} ,\theta } \right) \right) \right] \right\} , \quad r_j \le r_i \le R_1 \end{aligned}$$

The left side of the integral equation (33) is

$$\begin{aligned} Q_i ({r_i, \theta _i, s})=-\frac{4e_{15} R_2}{\epsilon _{11} \pi rs}D_0 \mathop \sum \limits _{n=1}^\infty {\int }_{-1}^1 {\varLambda }_n \left( {\frac{r}{R_2}} \right) ^{N_1} \mathrm{{cos}} ({n \theta })\mathrm{{d}}t-\frac{GM_0}{J_0 s}r_i \, \mathrm{{cos}}\varphi _i \end{aligned}$$

where all the parameters with index i are functions of p and those with index j are functions of q.

Appendix B

1.1 Proof of Eq. (26)

We use a formula given in the appendix of Ref. [26]

$$\begin{aligned} \mathop \sum \limits _{n=1}^\infty k^{n}\mathrm{{cos}}({n \theta })=\frac{-k^{2}+k\mathrm{{cos}}(\theta )}{1+k^{2}-2 k\mathrm{{cos}}(\theta )} \end{aligned}$$

(B1)

Equation (B1) can be rewritten as

$$\begin{aligned} \mathop \sum \limits _{n=1}^\infty k^{n}\mathrm{{cos}}({n \theta }) = \frac{1}{2}\frac{-k+ \mathrm{{cos}}(\theta )}{\left( {\frac{1}{k}+k} \right) /2-\mathrm{{cos}}(\theta )} \end{aligned}$$

(B2)

With the aid of formulas between hyperbolic functions, we can achieve Eq. (26)

$$\begin{aligned} \mathop \sum \limits _{n=1}^\infty k^{n}\mathrm{{cos}}({n \theta })= & {} \frac{1}{2}\frac{-\mathrm{{sin}}h\left( {\ln \left( k \right) } \right) -\mathrm{{cos}}h\left( {\mathrm{{ln}}\left( k \right) } \right) + \mathrm{{cos}}(\theta )}{\mathrm{{cos}}h\left( {\mathrm{{ln}}\left( k \right) } \right) -\mathrm{{cos}}(\theta )} \nonumber \\= & {} -\frac{1}{2}\left( {\frac{\mathrm{{sin}}h\left( {\ln \left( k \right) } \right) }{\mathrm{{cos}}h\left( {\ln \left( k \right) } \right) -\mathrm{{cos}}(\theta )}+1} \right) \end{aligned}$$

(B3)

1.2 Proof of Eq. (27)

When $n\rightarrow \infty$, $N_1$ can be approximated with n. Then, we can use expansion of ${\Omega }_n$ as

$$\begin{aligned} \mathop {\lim } \limits _{N_1 \rightarrow n} {\Omega }_n= & {} \mathop {\hbox {lim}}\limits _{N_1 \rightarrow n} \frac{n^{2}}{N_1^2 \left( {1-\left( {\frac{R_1}{R_2}} \right) ^{2N_1} C_\mathrm{{eq}}} \right) }=\frac{1}{1-\left( {\frac{R_1}{R_2}} \right) ^{2n}C_\mathrm{{eq}}} =\mathop \sum \limits _{m=0}^\infty \left( {\left( {\frac{R_1}{R_2}} \right) ^{2n}C_\mathrm{{eq}}} \right) ^{m} \nonumber \\= & {} \mathop \sum \limits _{m=0}^\infty \left( {\left( {\frac{R_1}{R_2}} \right) ^{2n}\frac{\frac{C_{44}}{G}+\frac{e_{15}^2}{G \epsilon _{11}} -1}{\frac{C_{44}}{G}+\frac{e_{15}^2}{G \epsilon _{11}} +1}} \right) ^{m}=\mathop \sum \limits _{m=0}^\infty \left( {\left( {\frac{R_1}{R_2}} \right) ^{2n}\frac{\frac{C_{44}}{G}+\frac{e_{15}^2}{G \epsilon _{11}} -1}{\frac{C_{44}}{G}+\frac{e_{15}^2}{G \epsilon _{11}} +1}} \right) ^{m} \end{aligned}$$

(B4)

By defining $\eta =\frac{e_{15}^2 + \epsilon _{11} C_{44} -G \epsilon _{11}}{e_{15}^2 + \epsilon _{11} C_{44} +G \epsilon _{11}}$ we have

$$\begin{aligned} \mathop {\lim } \limits _{N_1 \rightarrow n} {\Omega }_n =\mathop \sum \limits _{m=0}^\infty \eta ^{m}\left( {\left( {\frac{R_1}{R_2}} \right) ^{2n}} \right) ^{m} \end{aligned}$$

(B5)

With the aid of Eq. (B5), we follow the proof of (27) as

$$\begin{aligned} \mathop {\hbox {lim}}\limits _{N_1 \rightarrow n} \mathop \sum \limits _{n=1}^\infty {\Omega }_n x^{n} \mathrm{{cos}} ({n \theta })=\mathop \sum \limits _{m=0}^\infty \mathop \sum \limits _{n=1}^\infty \eta ^{m}\left( {\left( {\frac{R_1}{R_2}} \right) ^{2m}x} \right) ^{n} \mathrm{{cos}} ({n \theta })=\mathop \sum \limits _{m=0}^\infty \mathop \sum \limits _{n=1}^\infty \eta ^{m} ({\kappa _m^2 x})^{n} \mathrm{{cos}} ({n \theta }) \end{aligned}$$

(B6)

By using Eq. (B3), Eq. (B6) is rewritten as

$$\begin{aligned} \mathop {\hbox {lim}}\limits _{N_1 \rightarrow n} \mathop \sum \limits _{n=1}^\infty {\Omega }_n x^{n} \mathrm{{cos}} ({n \theta })=-\frac{1}{2}\mathop \sum \limits _{m=0}^\infty \left( {\frac{\mathrm{{sin}}h ({\ln ({({R_1 /R_2})^{2m}x})})}{\mathrm{{cos}}h ({\ln ({({R_1 /R_2})^{2m}x})})- \mathrm{{cos}}(\theta )}+1} \right) \end{aligned}$$

(B7)

In the following discussion, Eq. (B7) can be expressed in terms of $F_m ({x, \theta })$ as

$$\begin{aligned} \mathop {\hbox {lim}}\limits _{N_1 \rightarrow n} \mathop \sum \limits _{n=1}^\infty {\Omega }_n x^{n} \mathrm{{cos}} ({n \theta })=-\frac{1}{2}\mathop \sum \limits _{m=0}^\infty ({F_m ({x, \theta })+1}) \end{aligned}$$

(B8)

1.3 Proof of the Eq. (34):

The domain of integration (33) is divided into three subregions $r<a$, $R_1 >r\ge a$ and $R_2 \ge r\ge R_1$. Substituting Eqs. (3) and (14) into Eq. (33) results in

$$\begin{aligned} M= & {} J\alpha =\alpha G\int _0^{2\pi } \int _0^{R_1} r^{2}\left( {\frac{1}{r}\frac{\partial \omega ({r, \theta })}{\partial \theta } +r} \right) \mathrm{{d}}r\mathrm{{d}}\theta \nonumber \\&+\int _0^{2\pi } \int _{R_1} ^{R_2} r^{2}\left( {\alpha \left( {C_{44} \left( {\frac{1}{r}\frac{\partial \omega }{\partial \theta } +r} \right) +e_{15} \frac{1}{r}\frac{\partial \phi }{\partial \theta }} \right) } \right) \mathrm{{d}}r\mathrm{{d}}\theta \nonumber \\= & {} \alpha \left[ G\int _0^{2\pi } \frac{\partial \omega ({r, \theta })}{\partial \theta } \int _0^a r\mathrm{{d}}r\mathrm{{d}}\theta +G\int _0^{2\pi } \frac{\partial \omega ({r, \theta })}{\partial \theta } \int _a^{R_1} r\mathrm{{d}}r\mathrm{{d}}\theta +\frac{1}{2}\pi GR_1^4 \right. \nonumber \\&\quad \left. +\frac{1}{2}\pi C_{44} ({R_2^4 -R_1^4})+C_{44} \int _0^{2\pi } \frac{\partial \omega _3}{\partial \theta } \int _{R_1} ^{R_2} r\mathrm{{d}}r\mathrm{{d}}\theta +e_{15} \int _0^{2\pi } \frac{\partial \phi }{\partial \theta } \int _{R_1} ^{R_2} r\mathrm{{d}}r\mathrm{{d}}\theta \right] \end{aligned}$$

(B9)

We use some relations according to the definition of the dislocation cut as

$$\begin{aligned} \omega ({r, 2\pi })-\omega ({r,0})= & {} 0, \quad 0<r\le a \nonumber \\ \omega ({r, 2\pi })-\omega ({r,0})= & {} b_{z}/\alpha , \quad \alpha< r \le R_{1} \nonumber \\ \omega ({r, 2\pi })-\omega ({r,0})= & {} 0, \quad R_1<r\le R_2 \nonumber \\ \phi ({r, 2\pi })-\phi ({r,0})= & {} 0, \quad 0<r\le R_2 \end{aligned}$$

(B10)

Using the above equations, the proof is completed as

$$\begin{aligned} J= & {} J_0 -\frac{1}{2}({C_{44} ({R_2^2 -R_1^2})+G({R_1^2 -a^{2}})})\frac{b_z}{\alpha } \nonumber \\ J_0= & {} \frac{1}{2}\pi ({GR_1^4 +C_{44} ({R_2^4 -R_1^4})}) \end{aligned}$$

(B11)