Several Defects in a Hollow Cylinder Coated by a Functionally Graded Material (FGM) Subjected to Torsional Loading

Abstract

The present paper provides the Saint-Venant torsional analysis of an infinite hollow cylinder coated by a functionally graded material. The hollow cylinder contains multiple cracks and holes with arbitrary patterns. Pure mode III stress intensity factors are studied in this paper. The shear modulus of the through-radius functionally graded coating is supposed to vary according to an exponential function. First, the displacement and stress components of the hollow cylinder with its functionally graded coating weakened by a Volterra-type screw dislocation are found in terms of the dislocation density through Fourier transform. To model multiple arbitrarily shaped defects in the isotropic hollow cylinder, the dislocations are distributed on the boundaries of the defects. The distribution of the dislocations leads to a set of integral equations with a singularity of the Cauchy type. The solution of the integral equations using a numerical procedure leads to computing the stress intensity factors at the crack tips, the tangential hoop stresses around the boundary of the holes, and torsional rigidities of the cracked domain. The numerical results are presented to demonstrate the influences of critical parameters on the fracture behavior of a cracked hollow cylinder with a functionally graded coating. Our results show that the stress intensity factors are dependent on the material properties of the coating, revealing that the use of a suitable functionally graded coating may decrease the stress intensity factors.

Appendices

Appendix 1

$$ \begin{aligned} k_{{ij}} \left( {s,t} \right) & = \frac{{\mu _{0} }}{{2\pi r_{i} \left( s \right)}}\mathop \sum \limits_{{n = 1}}^{\infty } \Psi _{n} \Phi _{n} \left( {\gamma _{n} - 1} \right)\left( {\frac{{r_{i} \left( s \right)}}{{r_{j} \left( t \right)}}} \right)^{n} \left( {\cos \varphi _{i} \left( s \right)\cos \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right. \\ & \quad - \left. {\sin \varphi _{i} \sin \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right) + \cos \varphi _{i} \left( s \right)\tau _{{\theta z}}^{{{\rm{asy}},n}} \left( {r_{i} \left( s \right),\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - \sin \varphi _{i} \left( s \right)\tau _{{rz}}^{{asy,n}} \left( {r_{i} \left( s \right),\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad + \frac{{b_{z} \mu _{0} }}{{4\pi r_{i} \left( s \right)}}\left\{ {\cos \varphi _{i} \left( s \right)\mathop \sum \limits_{{m = 0}}^{\infty } \mathop \sum \limits_{{i = 0}}^{m} \mathop \sum \limits_{{j = 0}}^{\infty } \Pi _{{ij}}^{m} \left[ {U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)\kappa _{{23}} }}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right.} \right. \\ & \quad - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + U_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad + C_{{{\rm{eq}}}} \left( {U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)}}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{23}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad + \left. {U_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right) \\ & \quad + \left( {C_{{{\rm{eq}}}} - 1} \right)\left( {U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad + \left. {\left. {U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{23}} }}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right] \\ & \quad - \sin \varphi _{i} \left( s \right)\mathop \sum \limits_{{m = 0}}^{\infty } \mathop \sum \limits_{{i = 0}}^{m} \mathop \sum \limits_{{j = 0}}^{\infty } \Pi _{{ij}}^{m} \left[ {P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)\kappa _{{23}} }}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + P_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + C_{{{\rm{eq}}}} \left( {P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{23}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)}}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + P_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - \left. {P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right) + \left( {C_{{{\rm{eq}}}} - 1} \right)\left( {P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - \left. {\left. {\left. {P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right]} \right\} + r_{i} \left( s \right)\alpha \mu _{0} \cos \varphi _{i} \left( s \right) \\ & \quad + \frac{1}{{2D_{0} }}\mu _{0}^{2} r_{i} \left( s \right)\left( {R_{2}^{2} - r_{j} \left( t \right)^{2} } \right)\cos \left( {\phi _{i} \left( s \right)} \right),\quad 0 < r_{i} \left( s \right) < r_{j} \\ \end{aligned} $$

$$\begin{aligned} k_{{ij}} \left( {s,t} \right) & = \frac{{\mu _{0} }}{{2\pi r_{i} \left( s \right)}}\left[ {\mathop \sum \limits_{{n = 1}}^{\infty } \Psi _{n} \Lambda _{n} \left( {\frac{{r_{i} \left( s \right)}}{{R_{1} }}} \right)^{n} \left( {\cos \varphi _{i} \left( s \right)\cos \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right.} \right. \\ & \quad - \left. {\sin \varphi _{i} \left( s \right)\sin \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right) - \mathop \sum \limits_{{n = 1}}^{\infty } \left( {\frac{{r_{j} \left( t \right)}}{{r_{i} \left( s \right)}}} \right)^{n} \left( {\cos \varphi _{i} \left( s \right)\cos \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right. \\ & \quad - \left. {\sin \varphi _{i} \left( s \right)\sin \left( {n\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right] + \frac{{b_{z} \mu _{0} }}{{4\pi r_{i} \left( s \right)}}\cos \varphi _{i} \left( s \right)\left[ {U_{{00}}^{0} \left( {\frac{{r_{j} \left( t \right)}}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - 1} \right] \\ & \quad + \cos \varphi _{i} \left( s \right)\tau _{{\theta z}}^{{{\rm{asy}},n}} \left( {r_{i} \left( s \right),\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right) - \sin \varphi _{i} \left( s \right)\tau _{{rz}}^{{{\rm{asy}},n}} \left( {r_{i} \left( s \right),\left( {\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right) \\ & \quad + \frac{{b_{z} \mu _{0} }}{{4\pi r_{i} \left( s \right)}}\left[ {\cos \varphi _{i} \left( s \right)\mathop \sum \limits_{{m = 0}}^{\infty } \mathop \sum \limits_{{i = 0}}^{m} \mathop \sum \limits_{{j = 0}}^{\infty } \Pi _{{ij}}^{m} \left[ {U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)\kappa _{{23}} }}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right.} \right. \\ & \quad - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + U_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + C_{{{\rm{eq}}}} \left( {U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)}}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{12}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + U_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad - \left. {U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right) + \left( {C_{{{\rm{eq}}}} - 1} \right)\left( {U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - \left. {\left. {U_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{23}} }}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - U_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right] \\ & \quad - \sin \varphi _{i} \left( s \right)\mathop \sum \limits_{{m = 0}}^{\infty } \mathop \sum \limits_{{i = 0}}^{m} \mathop \sum \limits_{{j = 0}}^{\infty } \Pi _{{ij}}^{m} \left[ {P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)\kappa _{{23}} }}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad + P_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} \kappa _{{23}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + C_{{{\rm{eq}}}} \left( {P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{12}} }}{{r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)r_{j} \left( t \right)}}{{R_{2}^{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + P_{{ij}}^{m} \left( {\frac{{r_{j} \left( t \right)\kappa _{{12}} }}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) - P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)r_{j} \left( t \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) \\ & \quad + \left( {C_{{{\rm{eq}}}} - 1} \right)\left( {P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} \kappa _{{23}} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} } \right) - P_{{ij}}^{m} \left( {\frac{{R_{1}^{2} }}{{r_{i} \left( s \right)R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right) + P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)}}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right. \\ & \quad - \left. {\left. {\left. {P_{{ij}}^{m} \left( {\frac{{r_{i} \left( s \right)\kappa _{{23}} }}{{R_{2} }},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)} \right)} \right]} \right] + r_{i} \left( s \right)\alpha \mu _{0} \cos \varphi _{i} \left( s \right) + \frac{{b_{z} \mu _{0} }}{{4\pi r_{i} \left( s \right)}}P_{{00}}^{0} \left( {\frac{{r_{j} \left( t \right)}}{{r_{i} \left( s \right)}},\theta _{i} \left( s \right) - \theta _{j} \left( t \right)} \right)\sin \varphi _{i} \left( s \right) \\ & \quad + \frac{1}{{2D_{0} }}\mu _{0}^{2} r_{i} \left( s \right)\left( {R_{2}^{2} - r_{j} \left( t \right)^{2} } \right)\cos \left( {\phi _{i} \left( s \right)} \right),\quad r_{j} \left( t \right) < r_{i} \left( s \right) < R_{2} \\ \end{aligned} $$

$$ Q_{i} \left( s \right) = - \frac{T}{{D_{0} }}\mu_{0} r_{i} \cos \varphi_{i} $$

Appendix 2

Proof of Square-Root Singularity in Dislocation Approach

The kernel of Eq. (35) can be separated into the singular part and the regular part. The integral equation of the problem can be rewritten as:

$$ Q_{i} \left( {r_{i} \left( s \right),\theta_{i} \left( s \right)} \right) = \mathop \sum \limits_{j = 1}^{N} \mathop \int \limits_{ - 1}^{1} \left( {\frac{F\left( t \right)}{{s - t}} + F_{{{\rm{ns}}}} \left( t \right)} \right)b_{{{\rm{zj}}}} \left( t \right)dt. $$

The singular part of the above equation is defined as:

$$ I = \mathop \int \limits_{ - 1}^{1} b_{z} \left( t \right)\frac{F\left( t \right)}{{s - t}}dt $$

By expanding $F\left( t \right)$ in the form of Chebyshev polynomials, we have:

$$ F\left( t \right) = \mathop \sum \limits_{n = 0}^{\infty } B_{n} T_{n} \left( t \right) $$

Use the properties of Chebyshev polynomials as:

$$ \frac{1}{\pi }\mathop \int \limits_{ - 1}^{1} \frac{{T_{n} \left( t \right)dt}}{{\left( {t - s} \right)\sqrt {1 - t^{2} } }} = \left\{ {\begin{array}{*{20}c} 0 & {n = 0} & { - 1 < s < 1} \\ {U_{n - 1} \left( s \right) } & {n = 1,2, \ldots } & { - 1 < s < 1 } \\ { - \frac{\left| s \right|}{{s\sqrt {s^{2} - 1} }}\left[ {s - \frac{{\left| s \right|\sqrt {s^{2} - 1} }}{s}} \right]^{n} } & {n = 0,1,2, \ldots } & {\left| s \right| > 1 } \\ \end{array} } \right. $$

It should be mentioned that the singularity around the crack tips is evaluated for $\left| s \right| > 1$. By making use of Eq. (36a) and the above equation, we obtain:

$$ I = \mathop \sum \limits_{n = 0}^{\infty } B_{n} \mathop \int \limits_{ - 1}^{1} \frac{{g_{z} \left( t \right)T_{n} \left( t \right)}}{{\left( {s - t} \right)\sqrt {1 - t^{2} } }}dt = \pi \frac{\left| s \right|}{{s\sqrt {s^{2} - 1} }}\mathop \sum \limits_{n = 0}^{\infty } B_{n} \left[ {s - \frac{{\left| s \right|\sqrt {s^{2} - 1} }}{s}} \right]^{n} $$

It can be realized that the stress field around the embedded cracks shows square-root singularity for $s \to \pm 1$.