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IMA Journal of Applied Mathematics
, Volume 83 (1) – Feb 1, 2018

17 pages

/lp/ou_press/transient-analysis-for-torsional-impact-of-multiple-axisymmetric-TPNuPUBChr

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0272-4960
- eISSN
- 1464-3634
- D.O.I.
- 10.1093/imamat/hxx038
- Publisher site
- See Article on Publisher Site

Abstract The problem of determining the dynamic stress intensity factors (DSIFs) in a medium made by functionally graded orthotropic materials weakened by multiple axisymmetric cracks under torsional impact loading is investigated. It is assumed that the mass density and the shear modulus in the two principal directions of the functionally graded material (FGM) medium vary exponentially along the z-axis. The solution of a dynamic rotational Somigliana-type ring dislocation in an FGM orthotropic medium is obtained by using the Laplace and Hankel transforms. This solution is used to construct integral equations for a system of coaxial axisymmetric cracks, including annular and penny-shaped cracks. The integral equations are of Cauchy singular type, which are solved numerically to obtain the dislocation density on the faces of the cracks and the results are used to determine DSIFs for cracks. Numerical examples are provided to show the influences of material non-homogeneity and orthotropy as well as crack type on the DSIFs. 1. Introduction The study of fracture problems with various types of loading has always been an important branch in solid mechanics. The thermoelastic fracture problems of a cracked medium with an axisymmetric crack have been discussed extensively in the literature (Chen et al., 2004; Li et al., 2016, 2017,a,b). Static and dynamic analysis of axisymmetric crack in a medium is another important problem that has been treated by many researchers. This review focuses on the static and dynamic fracture mechanic at the mediums of the functionally graded materials (FGMs). In recent years, many attentions have been given to FGMs, whose mechanical properties vary continuously. Much attention has been paid to various theoretical and practical aspects of the mechanical behavior of FGMs, and there have been a considerable number of studies on the axisymmetric fracture mechanics of FGMs with axisymmetric crack. However, these studies have been mainly concerned with static loading at problems of the infinite FGM medium (Selvadurai, 2000; Huang et al., 2005; Chaudhur & Ray, 2013), non-homogeneous half space (Dhaliwal et al., 1992), interfacial FGM layer (Ozturk & Erdogan, 1995; Xuyue et al., 1996). For many engineering applications, the components made of FGMs could be subjected to impact loadings. Therefore, the dynamic fracture analysis in FGM medium weakened by multiple cracks is an important design consideration. Nowadays, the extensive dynamic fracture mechanics investigations are mainly concentrated on the problem of the cracked FGMs with a crack. In what follows, we review these papers. Ueda et al. (1983) reported the torsional impact response of a penny-shaped crack on a bimaterial interface. They determined the dynamic stress intensity factor and discussed its dependence on time and material constants. Li et al. (1999a, 1999b) considered the dynamic responses of an FGM and an orthotropic FGM with a penny-shaped crack subjected to torsional impact loading, respectively. In this study, they investigated the influence of the material non-homogeneity and the orthotropy on the dynamic stress intensity factor. Wang et al. (1999) investigated the dynamic response of FGMs with embedded multiple penny-shaped cracks by dividing the FGMs into some layers with different homogeneous properties. Using Laplace and Hankel transforms, the problem was reduced to a system of generalized singular integral equations which was solved via numerical inverse Laplace transform. Wang et al. (2000) reported a method for investigating the penny-shaped interface crack configuration in orthotropic multilayers under dynamic torsional loading. They used a similar method of solution explained in the reference Li et al. (1999b).,Li & Weng (2002), the elastodynamic response of a penny-shaped crack located in an FGM interlayer between two dissimilar homogeneous half spaces and subjected to a torsional impact loading is considered. In their work, the influences of the relative magnitudes of the adjoining material properties and the FGM interlayer thickness on the dynamic stress intensity factor were examined. Feng & Zou (2003) evaluated the transient stresses and dynamic stress intensity factor around a penny-shaped crack in a transversely isotropic FGM strip produced by torsional impact. This study was shown that the non-homogeneity and orthotropy of the strip have more significant effects on the fracture behavior than the strip’s highness. Several studies have also considered mediums such as beams with multiple cracks, which in those, vibration responses of cracked mediums are investigated (Zhao et al., 2016, 2017). According to the above review, the stress analysis of non-homogeneous medium under dynamic loading was mainly restricted only to a single axisymmetric crack. The aim of the present work is the dynamic fracture analysis of FGM orthotropic medium weakened by multiple axisymmetric cracks. In this article, we employ the distributed dislocation technique to the stress analysis a non-homogeneous medium with multiple axisymmetric cracks consist of annular and penny-shaped cracks. The paper is organized as follows. By using the Laplace and Hankel transforms, the solution of the axisymmetric rotational Somigliana ring dislocation in FGM orthotropic medium is given in Section 2. Section 3 presents the distributed dislocation to formulate and solve the Cauchy-type singular integral equations for the FGM orthotropic medium weakened by several axisymmetric cracks. In Section 4 several examples of cracks are solved to illustrate the influence of material non-homogeneity and orthotropy as well as crack type on the dynamic stress intensity factors (DSIFs). Concluding remarks are included in Section 5. 2. Formulation of problem Consider an FGM orthotropic medium with properties that vary as a function of variable z. Let the components of the displacement in the r, z and θ directions be labeled by ur, uz and uθ, respectively. Under conditions of anti-plane deformation, the only non-zero displacement component is uθ(r, z, t) being independent of θ. Consequently, the constitutive relationships are \begin{align} \tau_{r\theta}=\mu_{r}(z)\left(\frac{\partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r}\right) \tau_{\theta z}={\mu}_{z}(z)\frac{\partial u_{\theta}}{\partial z}, \end{align} (2.1)where μr(z) and μz(z) are the elastic shear modulus of elasticity of the FGM orthotropic medium in the r and z directions, respectively. By substituting Equation (2.1) into the equilibrium equation $$\frac {\partial t_{r\theta }}{\partial r}+\frac {\partial t_{\theta z}}{\partial z}+\frac {2}{r}t_{r\theta }=\rho (z)\frac {{\partial }^{2}u_\theta }{\partial t^{2}},$$ one can obtain \begin{align} \frac{\partial^{2}u_{\theta}}{\partial r^{2}}+\frac{1}{r}\frac{\partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r^{2}}+\frac{\mu^{\prime}_{z}(z)}{\mu_{r}(z)}\frac{\partial u_{\theta}}{\partial z}+\frac{\mu_{z}(z)}{\mu_{r}(z)}\frac{\partial^{2}u_{\theta}}{\partial z^{2}}=\frac{\rho(z)}{\mu_{r}(z)}\frac{\partial^{2}u_{\theta}}{\partial t^{2}}, \end{align} (2.2) where $${\mu }^{\prime }_{z}(z)$$ is the derivative of μz(z) with respect to the coordinate z, and ρ(z) is the mass density of the FGM orthotropic medium. For simplicity, we will assume that material properties are approximated by \begin{align} {\mu}_{r}(z)={({\mu}_{r})}_{0}e^{\alpha z},\quad{\mu}_{z}\left(z\right)={\left({\mu}_{z}\right)}_{0}e^{\alpha z},\quad\rho(z)=\rho_{0}e^{\alpha z} \end{align} (2.3) where $${\left ({\mu }_{r}\right )}_{0}$$ and $${\left ({\mu }_{z}\right )}_{0}$$ are the shear modulus at z = 0, and α is a constant (α > 0). The condition representing a Somigliana-type dynamic rotational ring dislocation located at r = ε, z = 0 in an FGM orthotropic medium with the cut of dislocation in radial direction starting from r = ε is (Hassani et al., 2017) \begin{align} u_\theta(r,0^{\mathrm{+}},t)\mathrm{-}u_\theta(r,0^{\mathrm{-}},t)\mathrm{=}\frac{\varepsilon}{r}b_\theta(t)H(r-\varepsilon), \end{align} (2.4) where $$\frac {\varepsilon }{r}b_{\theta }(t)$$ designates the magnitude of dislocation Burgers vector and H(.) is the Heaviside step-function. Furthermore, the continuity of traction vector on the cut of the dislocation implies that \begin{align} \tau_{\theta z}(r,0^{\mathrm{+}},t)\mathrm{=}\tau_{\theta z}(r,0^{\mathrm{-}},t). \end{align} (2.5) Application of the Laplace transformation to Equation (2.2), assuming that the medium is initially stationary and displacement field decays sufficiently quickly as $$\left |z\right |\to \infty ,$$ leads to \begin{align} \frac{{\partial }^{2}u^{*}_{\theta}}{\partial r^{2}}+\frac{1}{r}\frac{\partial u^{*}_{\theta}}{\partial r}-\left(\frac{\rho p^{2}}{{\mu}_{r}}+\frac{1}{r^{2}}\right)u^{*}_{\theta}+\frac{{\mu}^{\prime}_{z}}{{\mu}_{r}}\frac{\partial u^{*}_{\theta}}{\partial z}+\frac{{\mu}_{z}}{{\mu}_{r}}\frac{{\partial }^{2}u^{*}_{\theta}}{\partial z^{2}}=0, \end{align} (2.6) where $$u^{*}_{\theta }(r,z,p)={\mathcal {L}}[u_{\theta }(r,z,t);\ p]$$. We further use a Hankel transform of the first order to eliminate variable r, arriving at \begin{align} \frac{{\mu}_{z}}{{\mu}_{r}}\frac{{\partial }^{2}U_{\theta}}{\partial z^{2}}+\frac{{\mu}^{\prime}_{z}}{{\mu}_{r}}\frac{\partial U_{\theta}}{\partial z}-\left(\eta^{2}+\frac{\rho p^{2}}{{\mu}_{r}}\right)U_{\theta}=0, \end{align} (2.7) where $$U_{\theta }(\eta ,z,p)=H_{1}[u^{*}_{\theta }(r,z,p);\ \eta ].\ $$Substituting Equation (2.3) into Equation (2.7), the motion equation can be written as follows \begin{align} \frac{{({\mu}_{z})}_{0}}{{({\mu}_{r})}_{0}}\frac{{\partial }^{2}U_{\theta}}{\partial z^{2}}+\frac{{({\mu}_{z})}_{0}\alpha}{{({\mu}_{r})}_{0}}\frac{\partial U_{\theta}}{\partial z}-\left(\eta^{2}+\frac{\rho_{0}p^{2}}{{({\mu}_{r})}_{0}}\right)U_{\theta}=0. \end{align} (2.8) Letting ξ = λη and the orthotropic FGM coefficient $$\lambda =\sqrt {{({\mu }_{r})}_{0}/{({\mu }_{z})}_{0}}.$$ Equation (2.8) can be rewritten as \begin{align} \frac{{\partial }^{2}U_{\theta}}{\partial z^{2}}+\alpha\frac{\partial U_{\theta}}{\partial z}-\left(\xi^{2}+\frac{\rho_{0}p^{2}}{{\left({\mu}_{z}\right)}_{0}}\right)U_{\theta}=0. \end{align} (2.9) By defining \begin{align} y=e^{3\alpha z/4}U_{\theta},\qquad x=\xi\ e^{\alpha z}. \end{align} (2.10) The partial differential Equation (2.9) can be converted as \begin{align} x^{2}\frac{d^{2}y}{dx^{2}}+\frac{1}{2}x\frac{dy}{dx}-\beta^{2}y=0, \end{align} (2.11) where \begin{align} \beta=\sqrt{\frac{3}{16}+\frac{1}{\alpha^{2}}\left(\xi^{2}+\frac{\rho_{0}p^{2}}{{({\mu}_{z})}_{0}}\right)}. \end{align} (2.12) From the solution of the Euler Equation (2.11), the final solution of Equation (2.9) can be expressed as \begin{align} U_{\theta}(\eta,z,p)&=A(\eta,p)e^{\alpha m_{1}z},\quad z>0\nonumber\\ U_{\theta}(\eta,z,p)&=B(\eta,p)e^{\alpha m_{2}z},\quad z<0, \end{align} (2.13) where A(η, p) and B(η, p) are unknown coefficients and \begin{align} m_{1}=-\frac{1}{2}\left(1+\sqrt{\frac{1}{4}+4\beta^{2}}\right),\quad m_{2}=\frac{1}{2}\left(1-\sqrt{\frac{1}{4}+4\beta^{2}}\right). \end{align} (2.14) By taking the inverse Hankel transform of Equation (2.13), yields the Laplace transform of the displacement field \begin{align} u^{*}_{\theta}(r,z,p)&=\int^{\infty }_{0}{A(\eta,p)e^{\alpha m_{1}z}\eta J_{1}(r\eta)\text{ d}\eta},\ \ \ z>0\nonumber\\ u^{*}_{\theta}(r,z,p)&=\int^{\infty }_{0}{B(\eta,p)e^{\alpha m_{2}z}\eta J_{1}(r\eta) \text{ d}\eta},\ \ \ z<0, \end{align} (2.15) where J1(.) is the first kind of Bessel function of order 1. From the Equations (2.1) and (2.14), the Laplace transform of stress components $$\tau ^{\ast }_{r\theta }\ $$and $$\tau ^{\ast }_{\theta z}$$ can be obtained \begin{align} \tau^{\ast}_{r\theta}(r,z,p)&=-({\mu}_{r})_{0}e^{\alpha z}\int^{\infty}_{0}A(\eta,p)e^{m_{1}\alpha z}\eta^{2}J_{2}(r\eta) \ \text{d}\eta\quad z>0\nonumber\\ \tau^{\ast}_{r\theta}(r,z,p)&=-({\mu}_{r})_{0}e^{\alpha z}\int^{\infty}_{0}B(\eta,p)e^{m_{2}\alpha z}\eta^{2}J_{2}(r\eta) \ \text{d}\eta\quad z<0\nonumber\\ \tau^{\ast}_{\theta z}(r,z,p)&=\alpha({\mu}_{z})_{0}e^{\alpha z}\int^{\infty }_{0}{A(\eta,p)m_{1}e^{m_{1}\alpha z}\eta\ J_{1}(r\eta)\ \text{d}\eta}\quad z>0\nonumber\\ \tau^{\ast}_{\theta\ z}(r,z,p)&=\alpha {({\mu}_{z})}_{0}e^{\alpha z}\int^{\infty }_{0}{B(\eta,p)m_{2}e^{m_{2}\alpha z}\eta\ J_{1}(r\eta) \ \text{d}\eta}\quad z<0. \end{align} (2.16) The application of the Laplace transforms to conditions (2.4) and (2.5) results in \begin{align} u^{\ast}_{\theta}(r,0^{+},p)-u^{*}_{\theta}(r,0^{-},p)&=\frac{\varepsilon}{r}b^{*}_{\theta}(p)H(r-\varepsilon)\nonumber\\ \tau^{\ast}_{\theta z}(r,0^{+},p)&=\tau^{*}_{\theta z}(r,0^{-},p), \end{align} (2.17) where $$b^{*}_{\theta }(p)={\mathcal {L}}[b_{\theta }(t);\ p].\ $$ Application of conditions (2.17) to Equations (2.15) and (2.16) gives the unknown coefficients \begin{align} A(\eta,p)&=\left[\frac{m_{2}}{m_{2}-m_{1}}\right]\frac{\varepsilon b^{\ast}_{\theta}(p)}{\eta}J_{0}(\varepsilon\eta)\nonumber\\ B(\eta,p)&=\left[\frac{m_{1}}{m_{2}-m_{1}}\right]\frac{\varepsilon b^{\ast}_{\theta}(p)}{\eta}J_{0}(\varepsilon\eta). \end{align} (2.18) By substituting the coefficient (2.18) into the Laplace transform of stress components at Equation (2.16), we have \begin{align} \tau^{\ast}_{r\theta}(r,z,p)&=-\varepsilon b^{\ast}_{\theta}({\kern1pt}p){({\mu}_{r})}_{0}e^{\alpha z}\int^{\infty }_{0}{\frac{m_{2}}{m_{2}-m_{1}}e^{m_{1}\alpha z}\ \eta J_{0}(\varepsilon\eta)J_{2}(r\eta)\ \text{d}\eta}\quad z>0\nonumber\\ \tau^{\ast}_{r\theta}(r,z,p)&=-\varepsilon b^{\ast}_{\theta}({\kern1pt}p){({\mu}_{r})}_{0}e^{\alpha z}\int^{\infty }_{0}{\frac{m_{1}}{m_{2}-m_{1}}e^{m_{2}\alpha z}\eta J_{0}(\varepsilon\eta)J_{2}(r\eta)\ \text{d}\eta}\quad z<0\nonumber\\ \tau^{\ast}_{\theta z}(r,z,p)&=\alpha\varepsilon b^{\ast}_{\theta}({\kern1pt}p){({\mu}_{z})}_{0}e^{\alpha z}\int^{\infty }_{0}{\frac{m_{1}m_{2}}{m_{2}-m_{1}}e^{m_{1}\alpha z}J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta}\quad z>0\nonumber\\ \tau^{\ast}_{\theta z}(r,z,p)&=\alpha\varepsilon b^{\ast}_{\theta}({\kern1pt}p){({\mu}_{z})}_{0}e^{\alpha z}\int^{\infty }_{0}{\frac{m_{1}m_{2}}{m_{2}-m_{1}}e^{m_{2}\alpha z}J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta}\quad z<0. \end{align} (2.19) To examine the singular behavior of the Laplace transform of stress component $$t^{\ast }_{\theta z}(r,z,p)$$ at the dislocation location, we set z = 0 in the last equation in (19), Therefore we have \begin{align} \tau^{*}_{\theta z}(r,0,p)=\alpha\varepsilon b^{*}_{\theta}({\kern1pt}p){({\mu}_{z})}_{0}\int^{\infty }_{0}{\frac{m_{1}m_{2}}{m_{2}-m_{1}}J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta}\quad z<0. \end{align} (2.20) Since the integrand of above relation is continuous functions of η and also finite at η = 0, the singularity must occur as η tends to infinity. We must be note that $$\frac {m_{1}m_{2}}{m_{2}-m_{1}}\to -\frac {\lambda }{2a}\eta $$ as $$\eta \to \infty .$$ Equation (2.20) may be expressed as \begin{align} \tau^{*}_{\theta z}(r,0,p)&=\alpha\varepsilon b^{*}_{\theta}(p){({\mu}_{z})}_{0}\int^{\infty}_{0}{\left[\frac{m_{1}m_{2}}{m_{2}-m_{1}}+\frac{\lambda}{2a}\eta\right]J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta}\nonumber\\ &\quad-\frac{1}{2}\varepsilon b^{\ast}_{\theta}(p)\sqrt{{({\mu}_{r})}_{0}{({\mu}_{z})}_{0}}\int^{\infty}_{0}\eta J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta.\ \end{align} (2.21) The Equation (2.21) is rewritten as follows \begin{align} \tau^{*}_{\theta z}(r,0,p)=&\,\alpha\varepsilon b^{*}_{\theta}(p){({\mu}_{z})}_{0}\int^{\infty}_{0}{\left[\frac{m_{1}m_{2}}{m_{2}-m_{1}}+\frac{\lambda}{2\alpha}\eta\right]J_{0}(\varepsilon\eta)J_{1}(r\eta)\ \text{d}\eta}\nonumber\\ &-\frac{\sqrt{{({\mu}_{r})}_{0}{({\mu}_{z})}_{0}}b^{*}_{\theta}(p)}{\pi r}\begin{cases} \frac{r\varepsilon}{r^{2}-\varepsilon^{2}}{{{\boldsymbol{E}}} \left(\frac{\varepsilon^{2}}{r^{2}}\right)\ }&r>\varepsilon\\ {{\boldsymbol{K}} \left(\frac{r^{2}}{\varepsilon^{2}}\right)\ }+\frac{\varepsilon^{2}}{r^{2}-\varepsilon^{2}}{{{\boldsymbol{E}}} \left(\frac{r^{2}}{\varepsilon^{2}}\right)\ }&r<\varepsilon, \end{cases} \end{align} (2.22) where $${{\boldsymbol{K}}}(k)=\int ^{\pi/2}_{0}{dx/\sqrt {1-k^{2}{\sin}^{2}x}}$$ and $${{\boldsymbol{E}}}(k)=\int ^{\pi/2}_{0}{\sqrt {1-k^{2}{\sin}^{2}x}}\ \text {d}x$$ are the complete elliptic integrals of the first and second kind, respectively. From Equation (2.22), we may observe that the Laplace transform of stress component $$\tau ^{\ast }_{\theta z}(r,0,p)$$ exhibit the familiar Cauchy-type singularity at dislocation location, i.e. $$\tau ^{\ast }_{\theta z}\left (r,0,p\right )\sim \frac {1}{r-\varepsilon }\ {\rm as}\ \ r\to \varepsilon $$. This kind of singularity was previously reported by Pourseifi & Faal (2017) for an infinite isotropic cylinder with a climb and glide edge dislocations. 3. Axisymmetric crack formulation Let us implement the dislocation solutions, obtained in the Section 2, to analyze axisymmetric crack problems in an FGM orthotropic medium. In this technique, the dislocations are distributed in the locations of the crack and the stress fields are determined for the cracked medium. We assume that the FGM orthotropic medium is weakened by N1 penny-shaped cracks and N − N1 annular cracks, so that N denotes the number of all above-mentioned defects. As shown in Fig. 1, these defects are located horizontally and concentrically in the medium. The inner and outer radii of the annular cracks are aj and $$b_{j},\ \ j=1,2,\dots ,N-N_{1}.$$ Also, the radii of the penny-shaped cracks are $$c_{j},\ \ j=1,2,\dots ,N_{1}$$. These cracks can be presented by the following parametric equations \begin{align} r_{j}(s)&=L_{j}(1-s),-1\le s\le 1\quad \textrm{for penny-shaped cracks}\nonumber\\r_{j}(s)&=L_{j}s+0.5(b_{j}+a_{j}),-1\le s\le 1\ \quad \textrm{for annular cracks.} \end{align} (3.1) Fig. 1. View largeDownload slide An FGM orthotropic medium with axisymmetric cracks (penny-shaped crack and annular crack). Fig. 1. View largeDownload slide An FGM orthotropic medium with axisymmetric cracks (penny-shaped crack and annular crack). Noting that, \begin{align} L_{j}=0.5\begin{cases} c_{j}&\textrm{for penny-shaped cracks}\ (j=1,2,\dots,N_{1}) \\ b_{j}-a_{j}&\textrm{for annular cracks}\ (\,j=1,2,\dots,N-N_{1}). \end{cases} \end{align} (3.2) Suppose the Laplace transform of the dislocations with unknown density $$b^{\ast }_{\theta j}(q,p)$$ are distributed on the infinitesimal segment at the surfaces of the j-th concentric crack located at z = zj . Employing the principal of superposition the traction component τθz(ri(s), zj, p), $$i=1,2,\dots N$$ caused by dislocations distributed on the face of all cracks yields \begin{align} \tau^{*}_{\theta z}(r_{i}(s),z_{i},p)=\sum^{N}_{j=1}{L_{j}\int^{1}_{-1}{k_{ij}\left(s,q,p\right)b^{*}_{\theta j}\left(q,p\right)}\ \text{d}q},\qquad i=1,2,\dots,N. \end{align} (3.3) From Equation (2.19), the kernel of integral (3.3) become \begin{align} k_{ij}(s,q,p)=\begin{cases} \alpha\ r_{j}(q){({\mu}_{z})}_{0}e^{\alpha(z_{i}-z_{j})}\int^{\infty }_{0}{\frac{m_{1}m_{2}}{m_{2}-m_{1}}e^{m_{1}\alpha(z_{i}-z_{j})}J_{0}(r_{j}(q)\eta)J_{1}(r_{i}(s)\eta)\ \text{d}\eta}&z_{i}>z_{j}\\ \alpha\ r_{j}(q){({\mu}_{z})}_{0}e^{\alpha(z_{i}-z_{j})}\int^{\infty }_{0}{\frac{m_{1}m_{2}}{m_{2}-m_{1}}e^{m_{2}\alpha(z_{i}-z_{j})}J_{0}(r_{j}(q)\eta)J_{1}(r_{i}(s)\eta)\ \text{d}\eta}& z_{i}<z_{j}. \end{cases} \end{align} (3.4) By virtue of the Buckner’s principle the left side of Equation (3.3), after changing the sign,is the traction caused by external loading on the uncracked medium at the presumed surfaces of cracks. Using the definition of dislocation density function, we have \begin{align} u^{*,+}_{\theta j}(s,p)-u^{*,-}_{\theta j}(s,p)=\frac{L_{j}}{r_{i}(s)}\int^{s}_{-1}{r_{j}(q)\ b^{\ast}_{\theta j}(q,p)}\ \text{d}q,\qquad j=1,2,\dots,N. \end{align} (3.5) Fig. 2. View largeDownload slide Variation of normalized DSIFs at the inner tip of annular crack with normalized time for different non-homogeneity when a1 = 0.5b1 and λ = 0.5. Fig. 2. View largeDownload slide Variation of normalized DSIFs at the inner tip of annular crack with normalized time for different non-homogeneity when a1 = 0.5b1 and λ = 0.5. Table 1 Normalized $$k_{III}(t)/\tau _{0}\sqrt {c_{1}}$$ for a penny-shaped crack at the infinite FGM medium for $$t/t_{0}\to \infty $$ $$\frac {k_{III}}{\tau _{0}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.4989 0.5 0.1 0.4990 0.5000 1.0 0.5044 0.5049 2.0 0.5175 0.5177 3.0 0.5359 0.5356 4.0 0.5573 0.5564 $$\frac {k_{III}}{\tau _{0}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.4989 0.5 0.1 0.4990 0.5000 1.0 0.5044 0.5049 2.0 0.5175 0.5177 3.0 0.5359 0.5356 4.0 0.5573 0.5564 View Large Table 2 Normalized $$k_{III}(t)/\tau _{1}\sqrt {c_{1}}$$ for a penny-shaped crack at the infinite FGM medium for $$t/t_{0}\to \infty $$ $$\frac {k_{III}}{\tau _{1}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.4239 0.4244 0.1 0.4239 0.4244 1.0 0.4272 0.4277 2.0 0.4361 0.4365 3.0 0.4487 0.4459 4.0 0.4633 0.4632 $$\frac {k_{III}}{\tau _{1}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.4239 0.4244 0.1 0.4239 0.4244 1.0 0.4272 0.4277 2.0 0.4361 0.4365 3.0 0.4487 0.4459 4.0 0.4633 0.4632 View Large Table 3 Normalized $$k_{III}(t)/\tau _{2}\sqrt {c_{1}}$$ for a penny-shaped crack at the infinite FGM medium for $$t/t_{0}\to \infty $$ $$\frac {k_{III}}{\tau _{2}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.3744 0.3750 0.1 0.3744 0.3750 1.0 0.3769 0.3775 2.0 0.3835 0.3839 3.0 0.3927 0.3931 4.0 0.4035 0.4037 $$\frac {k_{III}}{\tau _{2}\sqrt {c_{1}}}$$ $$\alpha c_1$$ Present work Ozturk & Erdogan (1995) 0.0 0.3744 0.3750 0.1 0.3744 0.3750 1.0 0.3769 0.3775 2.0 0.3835 0.3839 3.0 0.3927 0.3931 4.0 0.4035 0.4037 View Large The Laplace transform of the displacement field is single-valued away from the penny-shaped and annular cracks. Thus, the Laplace transform of the dislocation density for the j-th crack of these kinds must be subjected to the closure requirement as \begin{align} \int^{1}_{-1}{r_{j}(q)\ b^{*}_{\theta j}(q,p)}\ \text{d}q=0,\quad\quad j=1,2,\dots,N. \end{align} (3.6) Fig. 3. View largeDownload slide Variation of normalized DSIFs at the outer tip of annular crack with normalized time for different non-homogeneity when a1 = 0.5b1 and λ = 0.5. Fig. 3. View largeDownload slide Variation of normalized DSIFs at the outer tip of annular crack with normalized time for different non-homogeneity when a1 = 0.5b1 and λ = 0.5. Fig. 4. View largeDownload slide Variation of normalized DSIFs of penny-shaped crack with normalized time for different non-homogeneity when λ = 0.5. Fig. 4. View largeDownload slide Variation of normalized DSIFs of penny-shaped crack with normalized time for different non-homogeneity when λ = 0.5. Fig. 5. View largeDownload slide Variation of normalized DSIFs at the inner tip of annular crack with normalized time for different orthotropy when a1 = 0.5b1 and αl = 0.5. Fig. 5. View largeDownload slide Variation of normalized DSIFs at the inner tip of annular crack with normalized time for different orthotropy when a1 = 0.5b1 and αl = 0.5. Fig. 6. View largeDownload slide Variation of normalized DSIFs at the outer tip of annular crack with normalized time for different orthotropy when $$\alpha_1 =0.5b_1$$ and $$\alpha l = 0.5$$. Fig. 6. View largeDownload slide Variation of normalized DSIFs at the outer tip of annular crack with normalized time for different orthotropy when $$\alpha_1 =0.5b_1$$ and $$\alpha l = 0.5$$. Fig. 7. View largeDownload slide Variation of normalized DSIFs of penny-shaped crack with normalized time for different orthotropy when $$\alpha l = 0.5$$. Fig. 7. View largeDownload slide Variation of normalized DSIFs of penny-shaped crack with normalized time for different orthotropy when $$\alpha l = 0.5$$. To obtain the Laplace transform of the dislocation density, the integral equation (3.3) and the closure equations for the penny-shaped and annular cracks are to be solved simultaneously. The stress field near the crack tips has a square-root singularity (Sneddon & Lowengrub, 1969), therefore by choosing that the embedded crack tips to be singular at q = −1, the Laplace transform of the dislocation densities for each kind of cracks are grouped as (Pourseifi & Faal, 2015) \begin{align} \begin{cases} b^{*}_{\theta j}(q,p)=\frac{g^{*}_{\theta j}(q,p)}{\sqrt{1-q^{2}}}& \textrm{for annular cracks} \\ b^{*}_{\theta j}(q,p)=g^{*}_{\theta j}(q,p)\sqrt{\frac{1-q}{1+q}}& \textrm{for penny-shaped crack.} \end{cases} \end{align} (3.7) The function $$g^{\ast }_{\theta j}(q,p)$$ in above equation is continuous in − 1 ≤ q ≤ 1. Substituting (3.9) into (3.3) and (3.8) and discretizing the domain,− 1 ≤ q ≤ 1, by m + 1 segments, the integral equations reduced to the following system of N × m linear algebraic equations (Hassani et al., 2017) \begin{align} \left[\begin{array}{cc} \begin{array}{cc} H_{11} & H_{12} \\ H_{21} & H_{22} \end{array} & \begin{array}{cc} \dots & H_{1N} \\ \dots & H_{2N} \end{array} \\ \begin{array}{cc} \vdots & \vdots \\ H_{N1} & H_{N2} \end{array} & \begin{array}{cc} \ddots & \vdots \\ \dots & H_{NN} \end{array} \end{array} \right]\left[ \begin{array}{c} g^{*}_{\theta 1}\left(q_{k},p\right) \\ g^{*}_{\theta 2}\left(q_{k},p\right) \\ \vdots \\ g^{*}_{\theta N}\left(q_{k},p\right) \end{array} \right]=\left[ \begin{array}{c} F_{1}\left(s_{l},p\right) \\ F_{2}\left(s_{l},p\right) \\ \vdots \\ F_{N}\left(s_{l},p\right) \end{array} \right]. \end{align} (3.8) Where the collocation points are \begin{align} s_{l}&={\cos \left(\frac{l}{m}\pi\right)\ },\quad l=1,2,\dots,m-1\nonumber\\ q_{k}&={\cos \left(\frac{2k-1}{2m}\pi\right)\ },\quad k=1,2,\dots,m. \end{align} (3.9) The components of this system are \begin{align} H_{ij}=&\,\left[ \begin{array}{cccc} A_{j1}k_{ij}\left(s_{1},q_{1}\right) & A_{j2}k_{ij}\left(s_{1},q_{2}\right) & \cdots & A_{jm}k_{ij}\left(s_{1},q_{m}\right)\\ A_{j1}k_{ij}\left(s_{2},q_{1}\right) & A_{j2}k_{ij}\left(s_{2},q_{2}\right) & \cdots & A_{jm}k_{ij}\left(s_{2},q_{m}\right)\\ \vdots & \vdots & \ddots & \vdots\\ A_{j1}k_{ij}\left(s_{m-1},q_{1}\right) & A_{j2}k_{ij}\left(s_{m-1},q_{2}\right) & \cdots & A_{jm}k_{ij}\left(s_{m-1},q_{m}\right)\\ A_{j1}B_{ij}\left(q_{1}\right) & A_{j2}B_{ij}(q_{2}) & \cdots & A_{jm}B_{ij}(q_{m})\end{array}\right]\\ g_{\theta j}(q_{k},p)=&\, {\left[g_{\theta j}(q_{1},p)\ \ g_{\theta j}(q_{2},p)\ \ \ g_{\theta j}(q_{3},p)\ \dots \ \ g_{\theta j}(q_{m},p)\right]}^{T},\ \ \ \ \ j=1,2,\dots,N\nonumber\\ F_{i}(s_{l},p)=&\,{\left[\tau^{\ast}_{\theta z}(r_{i}(s_{1}),z_{i},p)\ \tau^{\ast}_{\theta z}(r_{i}(s_{2}),z_{i},p)\dots \tau^{\ast}_{\theta z}(r_{i}(s_{m-1}),z_{i},p)\ 0\right]}^{T},\ \ i=1,\dots,N\nonumber. \end{align} (3.10) The superscript T stands for the transpose of a vector and Ajd and Bij(q) are defined as follows \begin{align} \ A_{jd}&=\frac{\pi}{m}\qquad j=1,2,\ \dots,\ N,\ \ \ \ \ \ d=1,2,\dots m\nonumber\\ B_{ij}(q)&=\delta_{ij}r_{i}(q)L_{i}\qquad i=1,2,\ \dots,\ N ). \end{align} (3.11) The Laplace transform of the DSIFs for the penny-shaped and annular cracks in the FGM orthotropic medium are \begin{align} \begin{cases} k^{*}_{III,j}(p)=\frac{{({\mu}_{z})}_{0}e^{\alpha z_{j}}\sqrt{L_{j}}}{2}\ g^{*}_{\theta j}(-1,p)& \textrm{for penny-shaped cracks} \\ k^{*}_{IIIL,j}(p)=\frac{{({\mu}_{z})}_{0}e^{\alpha z_{j}}\sqrt{L_{j}}}{2}g^{\ast}_{\theta j}(-1,p) &\\ k^{*}_{IIIR,j}(p)=-\frac{{({\mu}_{z})}_{0}e^{\alpha z_{j}}\sqrt{L_{j}}}{2}g^{\ast}_{\theta j}(+1,p) & \textrm{for annular cracks.} \end{cases} \end{align} (3.12) The DSIFs in the time domain can be obtained by means of the Stehfest’s method, see e.g. Vafa et al. (2015). Fig. 8. View largeDownload slide Variation of normalized DSIFs at the inner tips of two annular crackswith normalized time: a1 = 0.5b1 and αl = 0.5, λ = 2. Fig. 8. View largeDownload slide Variation of normalized DSIFs at the inner tips of two annular crackswith normalized time: a1 = 0.5b1 and αl = 0.5, λ = 2. Fig. 9. View largeDownload slide Variation of normalized DSIFs at the outer tips of two annular crackswith normalized time: α1 = 0.5b1 and al = 0.5, λ = 2. Fig. 9. View largeDownload slide Variation of normalized DSIFs at the outer tips of two annular crackswith normalized time: α1 = 0.5b1 and al = 0.5, λ = 2. Fig. 10. View largeDownload slide Variation of normalized DSIFs of two penny-shaped crackswith normalized time: αl = 0.5, λ = 2. Fig. 10. View largeDownload slide Variation of normalized DSIFs of two penny-shaped crackswith normalized time: αl = 0.5, λ = 2. 4. Numerical examples In this section several examples are solved to demonstrate the applicability of the distributed dislocation technique. The analysis developed in preceding section, allows the consideration of an FGM orthotropic medium with multiple axisymmetric cracks. These defects may include penny-shaped and annular cracks, as shown in Fig. 1. In all Examples, the dynamic shear stress $$\tau _{\theta z}=\frac {\tau _{0}r}{l}H(t)$$ is applied on the surface of the crack except for Example 1, wherein the dynamic traction resulting of torsion on the surface of the crack are considered as $$\tau _{\theta z}=\left (\tau _{0}H(t),\tau _{1}\frac {r}{l}H(t),\tau _{2}{\left (\frac {r}{l}\right )}^{2}H(t)\right ),$$ in which τ0, τ1 and τ2 are constant twisting loads, H(.) is the Heaviside unit step function and also r and l are the radial general coordinate and the length of the crack, respectively. The quantities of interest are the dimensionless stress intensity factors, kIII(t)/k0, where we take $$k_{0}=\tau _{0}\sqrt {l}$$ . The results of examples are represented verses normalized time t/t0, where t0 = l/(cz)0 and $${(c_{z})}_{0}=\sqrt {{({\mu }_{z})}_{0}/\rho _{0}}$$. Example 1 (An infinite FGM medium with a penny-shaped crack). In order to demonstrate and verify the solution of the dislocation method given in this study with the published results, for our first example we consider the problem of an FGM infinite domain (λ = 1) containing a penny-shaped crack with radius c1 under loading conditions that mentioned above. The dimensionless DSIFs of the singular crack tip for $$t/t_{0}\to \infty $$ (static value) are tabulated in Tables 1–3. The numerical results are compared with those of Ozturk & Erdogan (1995) to establish their accuracy. A good agreement is seen in the results. Example 2 (An infinite FGM orthotropic medium with an axisymmetric crack). The second example deals with an FGM orthotropic medium weakened by an axisymmetric crack with length l. In this problem, variations of the orthotropic FGM coefficient and the non-homogeneity constant are investigated on the normalized DSIFs. The example split into two cases. In the first case, an annular crack with length 2l = (b1 − a1), where a1 and b1 are the inner and outer radii of the annular crack, is studied. In the second case, a penny-shaped crack with length l = c1 is investigated, in which c1 is the radius of the penny-shaped crack. The normalized DSIFs versus t/t0 for different non-homogeneity constant are depicted in Figs 2–4. From these figures, a general feature of the curves is that the DSIFs rise rapidly and reach a peak, then oscillate about their static values with decreasing magnification. Also, it can be seen that for definite values of the orthotropic FGM coefficient, DSIFs are increased with growing non-homogeneity constant. Note that for definite values of the orthotropic FGM coefficient and the non-homogeneity constant, the normalized DSIFs for annular crack is higher than the penny shape crack. Next, let’s study the influence of the orthotropic FGM coefficient on normalized DSIFs.The results for an annular crack and a penny-shaped crack are shown in Figs 5–7. As demonstrated in Figs 5– 7, for fixed values of non-homogeneity constant the normalized DSIFs are decreased with increasing the orthotropic FGM coefficient. Example 3 (An infinite FGM orthotropic medium weakened by two concentric cracks). In another example, an FGM orthotropic medium containing two coplanar concentric interacting cracks is analyzed. The example split into two cases. In the first case, an FGM orthotropic medium containing two concentric annular cracks with length 2l = (b1 − a1) = (b2 − a2) is evaluated. The cracks are located at z1/l = 1and z2/l = 2. The normalized DSIFs of two interacting annular cracks for value of the orthotropic FGM coefficient λ = 2 and the non-homogeneity parameter al = 0.5, is depicted in Figs 8–9. In the second case, two penny-shaped cracks with length l = c1 = c2 in an FGM orthotropic medium are considered. The location of cracks is assumed to be similar to previous case. The effect of interaction between two penny-shaped cracks on the DSIFs demonstrated in Fig. 10. It is interesting to note that, in the non-homogeneous medium, in the stiffer portion of the materials, the crack surface displacement is smaller than that of the less stiff portion of the medium. Therefore, as expected, for the crack which is located in a stiffer zone, the stress intensity factor is higher than the other crack. 5. Concluding remarks In this paper, the multiple axisymmetric cracks problem in functionally graded orthotropic medium under torsional impact loading is investigated. A solution for the stress field caused by the dynamic rotational Somigliana ring dislocation in an FGM orthotropic medium is first obtained. Next, using the distributed dislocation technique, the problem was formulated for an FGM orthotropic medium with a system of coaxial axisymmetric cracks. The unknown dislocation density on the surfaces of the cracks was obtainable by solving a set of integral equations of Cauchy singular type. To study the effect of the nonhomogeneity and orthotropy as well as crack type, DSIFs are obtained for some examples. The following key points are observed: The non-homogeneity parameter and orthotropy have quite a considerable influence on the DSIFs. The peak value of the DSIFs with increasing the non-homogeneity parameter and orthotropy, increases and decreases, respectively. The DSIFs, for definite values of orthotropy, increase with growing in the values of the non-homogeneity. The DSIFs for fixed values of non-homogeneity reduce with increasing in the values of the orthotropy. For the cracks located in a stiffer portion, the DSIFs are higher than the other cracks. Acknowledgements The authors would like to acknowledge Islamic Azad University, Nazar Abad Center (Iran), for financial support of this research. References Chen, W. Q., Ding, H. J. & Ling, D. S. ( 2004) Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. Int. J. Solids and Struct. , 41, 69-- 83. Google Scholar CrossRef Search ADS Chaudhur, P. & Ray, S. ( 2013) Penny-shaped crack in a nonhomogeneous solid under torsion. J. Indian Inst. Sci. , 74, 727. Dhaliwal, R. S., Saxena, H. S. & Rokne, J. G. ( 1992) Penny-shaped crack in a non-homogeneous isotropic elastic half-space under axisymmetric torsion. Indian J. Pure Appl. Mathematics , 23, 887-- 887. Feng, W. J. & Zou, Z. Z. 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Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Applied Mathematics – Oxford University Press

**Published: ** Feb 1, 2018

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