Dugdale plastic zone model of a penny-shaped crack in a magnetoelectroelastic cylinder under magnetoelectroelastic loads

Abstract

This paper extends the Dugdale plastic zone to the penny-shaped crack problem for a magnetoelectroelastic (MEE) cylinder, where the crack surfaces are assumed to be magnetoelectrically permeable. Using potential function theory and Hankel transform method, the present boundary value problem is translated into solving a sectionalized Fredholm integral equation of the second kind. The non-singular field solutions are analyzed, and the effects of the applied MEE loads and geometric dimension on the width of plastic zone and crack opening displacements are evaluated. Numerical results show that for the considered model, each of the far-field MEE combination load $\lambda$, the far-field magnetoelectric combination load $\lambda _{\mathrm{EH}}$ and the purely mechanical load $\sigma _0$ applied in the far field, respectively, plays an important role in the present fracture analysis. For a given mechanical load and a fixed crack configuration, the negative magnetoelectric loads are prone to promote the crack growth and propagation than the positive ones. These should be helpful for the design and manufacture of MEE materials and devices.

Appendices

Appendix A

$D_0$ and $B_0$ appearing in Eq. (11) are, respectively,

$$\begin{aligned} D_0= & {} \frac{2c_{13} e_{31} -e_{33} (c_{11} +c_{12} )}{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}\sigma _0 +\left[ {\frac{4c_{13} e_{31} e_{33} -(c_{11} +c_{12} )e_{33}^2 -2c_{33} e_{31}^2 }{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}+\varepsilon _{33} } \right] E_0 \nonumber \\&+\left\{ {\frac{2e_{31} (c_{13} f_{33} -c_{33} f_{31} )+e_{33} \left[ {2c_{13} f_{31} -f_{33} \left( {c_{11} +c_{12} } \right) } \right] }{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}+g_{33} } \right\} H_0 \end{aligned}$$

(A-1)

$$\begin{aligned} B_0= & {} \frac{2h_{13} e_{31} -h_{33} (c_{11} +c_{12} )}{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}\sigma _0 +\left\{ {\frac{2f_{31} \left( {c_{13} e_{33} -c_{33} e_{31} } \right) +f_{33} \left[ {2c_{13} e_{31} -(c_{11} +c_{12} )e_{33} } \right] }{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}+g_{33} } \right\} E_0 \nonumber \\&+\left[ {\frac{4c_{13} f_{31} f_{33} -2c_{33} f_{31}^2 -(c_{11} +c_{12} )f_{33}^2 }{2c_{13}^2 -c_{33} (c_{11} +c_{12} )}+\mu _{33} } \right] H_0 \end{aligned}$$

(A-2)

Appendix B

$\varOmega _i \left( {\xi ,\alpha } \right) \;\left( {i=1,2,3,4} \right)$ in Eq. (34) are derived as

$$\begin{aligned} \left[ {\varOmega _1 \quad \varOmega _2 \quad \varOmega _3 \quad \varOmega _4 } \right] ^{\mathrm{T}}=\left[ {\Xi _{ij} } \right] ^{\mathrm {-1}}\left[ {\mathrm{X}_1 \quad \mathrm{X}_2 \quad \mathrm{X}_3 \quad \mathrm{X}_4 } \right] ^{\mathrm{T}} \end{aligned}$$

(B-1)

where

$$\begin{aligned} \Xi _{1j} \left( \xi \right)= & {} \frac{1}{s_j^2 }F_{4j} \xi I_0 \left( {\frac{\xi c}{s_j }} \right) -\frac{c_{12} -c_{11} }{c}I_1 \left( {\frac{\xi c}{s_j }} \right) \quad j=1,2,3,4 \end{aligned}$$

(B-2)

$$\begin{aligned} \Xi _{ij} \left( \xi \right)= & {} \frac{1}{s_j^2 }G_{\left( {i-1} \right) j} \xi I_1 \left( {\frac{\xi c}{s_j }} \right) \quad \quad i=2,3,4,\quad j=1,2,3,4 \end{aligned}$$

(B-3)

$$\begin{aligned} \mathrm{X}_1 \left( {\xi ,\alpha } \right)= & {} \sum _{k=1}^4 {\left[ {\frac{c_{11} -c_{12} }{s_k c}K_1 \left( {\frac{c}{s_k }\xi } \right) -\frac{F_{4k} }{s_k^2 }\xi K_0 \left( {\frac{c}{s_k }\xi } \right) } \right] \sinh \left( {\frac{\alpha }{s_k }\xi } \right) } \end{aligned}$$

(B-4)

$$\begin{aligned} \mathrm{X}_j \left( {\xi ,\alpha } \right)= & {} \sum _{k=1}^4 {\frac{1}{s_k^2 }G_{\left( {j-1} \right) k} \sinh \left( {\frac{\alpha }{s_k }\xi } \right) K_1 \left( {\frac{c}{s_k }\xi } \right) } \quad j=2,3,4 \end{aligned}$$

(B-5)

with $K_0 (\cdot )$ and $K_1 (\cdot )$ being, respectively, the modified Bessel functions of the second kind of order zero and order one, and $I_1 (\cdot )$ being the modified Bessel function of the first kind of order one.

Appendix C

$\Theta _i \left( {\xi _1 } \right) \;\left( {i=1,2,3,4} \right)$ in Eq. (43a) are derived as

$$\begin{aligned} \left[ {\Theta _1 \quad \Theta _2 \quad \Theta _3 \quad \Theta _4 } \right] ^{\mathrm{T}}=\left[ {\Sigma _{ij} } \right] ^{\mathrm {-1}}\left[ {\Lambda _1 \quad \Lambda _2 \quad \Lambda _3 \quad \Lambda _4 } \right] ^{\mathrm{T}} \end{aligned}$$

(C-1)

where

$$\begin{aligned} \Sigma _{1j} \left( {\xi _1 } \right)= & {} \frac{1}{s_j^2 }F_{4j} \frac{\xi _1 }{c}I_0 \left( {\frac{\xi _1 }{s_j }} \right) -\frac{c_{12} -c_{11} }{c}I_1 \left( {\frac{\xi _1 }{s_j }} \right) \quad j=1,2,3,4 \end{aligned}$$

(C-2)

$$\begin{aligned} \Sigma _{ij} \left( {\xi _1 } \right)= & {} \frac{1}{s_j^2 }G_{\left( {i-1} \right) j} \frac{\xi _1 }{c}I_1 \left( {\frac{\xi _1 }{s_j }} \right) \quad \quad i=2,3,4,\quad j=1,2,3,4 \end{aligned}$$

(C-3)

$$\begin{aligned} \Lambda _1 \left( {\xi _1 } \right)= & {} \sum _{k=1}^4 {\left[ {\frac{c_{11} -c_{12} }{s_k c}K_1 \left( {\frac{\xi _1 }{s_k }} \right) -\frac{F_{4k} }{s_k^2 }\frac{\xi _1 }{c}K_0 \left( {\frac{\xi _1 }{s_k }} \right) } \right] } \end{aligned}$$

(C-4)

$$\begin{aligned} \Lambda _j \left( {\xi _1 } \right)= & {} \sum _{k=1}^4 {\frac{1}{s_k^2 }G_{\left( {j-1} \right) k} K_1 \left( {\frac{\xi _1 }{s_k }} \right) } \quad j=2,3,4 \end{aligned}$$

(C-5)