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Surface Zener–Stroh crack model to slip band due to contact


A surface slip band due to contact by a rectangular rigid flat-ended indenter is investigated. An inclined Zener–Stroh crack model is proposed to simulate the slip band. By using the fundamental solution of a single dislocation in a half plane as Green’s function, the Zener–Stroh crack is modeled with continuously distributed dislocations. It leads to a singular integral equation of the first kind, which is solved with the Gauss–Chebyshev numerical quadrature, and then stress intensity factors (SIFs) at the crack tips are evaluated. It is demonstrated that the Zener–Stroh crack model can efficiently capture micro deformation behavior of the surface slip band due to contact. With this model, the corresponding relations of the applied load, the slip band length, the relative sliding displacement of slip band and SIFs are obtained. Compared with the experimental results, it is shown that the surface Zener–Stroh crack model to contact slip band can well address such kind of contact damage problems.

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This work is partially supported by National Natural Science Foundation of China (Grant No. 41630634).

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Correspondence to Lifeng Ma.

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Appendix A: The influence function and transformation relationship in two coordinate systems

The influence function due to a single dislocation in a half plane was given by Hills et al. [46]

$$\begin{aligned} G_{xxx} \left( {x,y;\xi ,\eta } \right)= & {} \left( {y-\eta } \right) \left\{ {-\frac{1}{r_1^2 }-\frac{2x_1 ^{2}}{r_1^4 }+\frac{1}{r_2^2 }+\frac{2x_2^2 }{r_2^4 }-\frac{4\xi x_2 }{r_2^4 }+\frac{4\xi ^{2}}{r_2^4 }+\frac{16\xi x_2^3 }{r_2^6 }-\frac{16\xi ^{2}x_2^2 }{r_2^6 }} \right\} , \nonumber \\ G_{xyy} \left( {x,y;\xi ,\eta } \right)= & {} \left( {y-\eta } \right) \left\{ {-\frac{1}{r_1^2 }+\frac{2x_1 ^{2}}{r_1^4 }+\frac{1}{r_2^2 }-\frac{2x_2^2 }{r_2^4 }+\frac{12\xi x_2 }{r_2^4 }-\frac{4\xi ^{2}}{r_2^4 }-\frac{16\xi x_2^3 }{r_2^6 }+\frac{16\xi ^{2}x_2^2 }{r_2^6 }} \right\} ,\nonumber \\ G_{xxy} \left( {x,y;\xi ,\eta } \right)= & {} -\frac{x_1 }{r_1^2 }+\frac{2x_1 ^{3}}{r_1^4 }+\frac{x_2 }{r_2^2 }-\frac{2\xi }{r_2^2 }-\frac{2x_2^3 }{r_2^4 }+\frac{16\xi x_2^2 }{r_2^4 }-\frac{12\xi ^{2}x_2 }{r_2^4 }-\frac{16\xi x_2^4 }{r_2^6 }+\frac{16\xi ^{2}x_2^3 }{r_2^6 }, \nonumber \\ G_{yxx} \left( {x,y;\xi ,\eta } \right)= & {} -\frac{x_1 }{r_1^2 }+\frac{2x_1 ^{3}}{r_1^4 }+\frac{x_2 }{r_2^2 }-\frac{2\xi }{r_2^2 }-\frac{2x_2^3 }{r_2^4 }-\frac{8\xi x_2^2 }{r_2^4 }+\frac{12\xi ^{2}x_2 }{r_2^4 }+\frac{16\xi x_2^4 }{r_2^6 }-\frac{16\xi ^{2}x_2^3 }{r_2^6 }, \nonumber \\ G_{yyy} \left( {x,y;\xi ,\eta } \right)= & {} +\frac{3x_1 }{r_1^2 }-\frac{2x_1 ^{3}}{r_1^4 }-\frac{3x_2 }{r_2^2 }-\frac{2\xi }{r_2^2 }+\frac{2x_2^3 }{r_2^4 }+\frac{16\xi x_2^2 }{r_2^4 }-\frac{12\xi ^{2}x_2 }{r_2^4 }-\frac{16\xi x_2^4 }{r_2^6 }+\frac{16\xi ^{2}x_2^3 }{r_2^6 }, \nonumber \\ G_{yxy} \left( {x,y;\xi ,\eta } \right)= & {} \left( {y-\eta } \right) \left\{ {-\frac{1}{r_1^2 }+\frac{2x_1 ^{2}}{r_1^4 }+\frac{1}{r_2^2 }-\frac{2x_2^2 }{r_2^4 }-\frac{4\xi x_2 }{r_2^4 }+\frac{4\xi ^{2}}{r_2^4 }+\frac{16\xi x_2^3 }{r_2^6 }-\frac{16\xi ^{2}x_2^2 }{r_2^6 }} \right\} , \end{aligned}$$


$$\begin{aligned} x_1= & {} x-\xi ,\quad x_2 =x+\xi ,\quad y_1 =y-\eta , \nonumber \\ r^{2}= & {} x^{2}+y^{2},\quad r_1^2 =y_1^{2}+x_1 ^{2},\quad r_2^2 =y_1 ^{2}+x_2^2. \end{aligned}$$

Note that in global and local coordinate systems we have

$$\begin{aligned} x= & {} c+{x}'\cos \theta , \quad y=d+{x}'\sin \theta , \nonumber \\ \xi= & {} c+{\xi }'\cos \theta , \quad \eta =d+{\xi }'\sin \theta . \end{aligned}$$

Equation (A.2) can be expressed in local coordinate system as

$$\begin{aligned} x_1= & {} \left( {{x}'-{\xi }'} \right) \cos \theta ,\quad x_2 =2c+\left( {{x}'+{\xi }'} \right) \cos \theta ,\quad y_1 =\left( {{x}'-{\xi }'} \right) \sin \theta , \nonumber \\ r_1^2= & {} \left( {{x}'-{\xi }'} \right) ^{2},\quad r_2^2 =\left[ {2c+\left( {{x}'+{\xi }'} \right) \cos \theta } \right] ^{2}+\left[ {\left( {{x}'-{\xi }'} \right) \sin \theta } \right] ^{2}. \end{aligned}$$

The transformation matrix M is given by

$$\begin{aligned} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\left( {\cos \theta } \right) ^{3}}&{} {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\cos \theta \sin \left( {2\theta } \right) }&{} {\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\left( {\sin \theta } \right) ^{3}}&{} {\sin \theta \sin \left( {2\theta } \right) } \\ {-\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {-\left( {\sin \theta } \right) ^{3}}&{} {-\sin \theta \sin \left( {2\theta } \right) }&{} {\left( {\cos \theta } \right) ^{3}}&{} {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\cos \theta \sin \left( {2\theta } \right) } \\ {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\left( {\cos \theta } \right) ^{3}}&{} {-\cos \theta \sin \left( {2\theta } \right) }&{} {\left( {\sin \theta } \right) ^{3}}&{} {\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {-\sin \theta \sin \left( {2\theta } \right) } \\ {-\left( {\sin \theta } \right) ^{3}}&{} {-\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\sin \theta \sin \left( {2\theta } \right) }&{} {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\left( {\cos \theta } \right) ^{3}}&{} {-\cos \theta \sin \left( {2\theta } \right) } \\ {-\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\cos \theta \cos \left( {2\theta } \right) }&{} {-\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {\sin \theta \cos \left( {2\theta } \right) } \\ {\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {-\cos \theta \left( {\sin \theta } \right) ^{2}}&{} {-\sin \theta \cos \left( {2\theta } \right) }&{} {-\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\sin \theta \left( {\cos \theta } \right) ^{2}}&{} {\cos \theta \cos \left( {2\theta } \right) } \\ \end{array} }} \right] . \end{aligned}$$

Appendix B

The regular term \({K}'\left( {t;s} \right) \) of singular integral Eq. (19) is

$$\begin{aligned} {K}'\left( {t;s} \right)= & {} -\frac{\left( {C\cos \left( {2\theta } \right) +D\cos \left( {4\theta } \right) +E} \right) }{A}, \end{aligned}$$
$$\begin{aligned} A= & {} \left( {(1+s)^{2}+(1+t)^{2}+2m^{2}+2m\left( {2+s+t} \right) +2(1+m+s)(1+m+t)\cos \left( {2\theta } \right) } \right) ^{3}, \nonumber \\ C= & {} (1+m+s)(1+m+t)\nonumber \\&\left( {\begin{array}{l} 15+15m^{3}-2s^{3}+31t+20t^{2}+6t^{3}+s^{2}\left( {3+9t} \right) +m^{2}\left( {45+14s+31t} \right) \\ +2s\left( {7+11t+t^{2}} \right) +m\left( {45+3s^{2}+62t+20t^{2}+s\left( {28+22t} \right) } \right) \\ \end{array}} \right) , \nonumber \\ D= & {} (1+m+s)\nonumber \\&\left( {\begin{array}{l} 6+6m^{4}-s^{4}+16t+15t^{2}+7t^{3}+t^{4}+8m^{3}\left( {3+s+2t} \right) +s^{3}\left( {-1+3t} \right) \\ +s^{2}\left( {3+9t} \right) +s\left( {8+18t+9t^{2}+3t^{3}} \right) +3m^{2}\left( {12+s^{2}+16t+5t^{2}+s\left( {8+6t} \right) } \right) \\ +m\left( {24-s^{3}+48t+30t^{2}+7t^{3}+s^{2}\left( {6+9t} \right) +3s\left( {8+12t+3t^{2}} \right) } \right) \\ \end{array}} \right) ,\nonumber \\ E= & {} (1+m+t)^{2}\left( (1+m+t)\left( {10+10m^{2}+18s+9s^{2}+2t+t^{2}+2m\left( {10+9s+t} \right) } \right) \nonumber \right. \\&\left. +(1+m+s)^{3}\cos \left( {6\theta } \right) \right) , \end{aligned}$$


$$\begin{aligned} m=\frac{\delta }{l},\quad s=\frac{{\xi }'}{l}, \quad t=\frac{{x}'}{l}, \quad s,t\in \left[ {{-}\,1,1} \right] . \end{aligned}$$

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Zhang, Y., Ma, L. Surface Zener–Stroh crack model to slip band due to contact. Arch Appl Mech 90, 221–234 (2020).

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  • Slip band
  • Zener–Stroh crack
  • Distributed dislocations
  • Fracture toughness
  • Singular integral equation