- Email: [email protected]
Multiple cylindrical interface cracks in FGM coated cylinders under torsional transient loading
Accepted Manuscript Multiple Cylindrical Interface Cracks in FGM Coated Cylinders under Torsional Transient Loading M. Noroozi, A. Ghassemi, A. Atrian, M. Vahabi PII: DOI: Reference:
S0167-8442(18)30108-3 https://doi.org/10.1016/j.tafmec.2018.08.015 TAFMEC 2094
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
4 March 2018 11 August 2018 27 August 2018
Please cite this article as: M. Noroozi, A. Ghassemi, A. Atrian, M. Vahabi, Multiple Cylindrical Interface Cracks in FGM Coated Cylinders under Torsional Transient Loading, Theoretical and Applied Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.tafmec.2018.08.015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multiple Cylindrical Interface Cracks in FGM Coated Cylinders under Torsional Transient Loading M. Noroozi1,2, A. Ghassemi 1,2*, A. Atrian1,2, M. Vahabi1,2 1
Modern manufacturing technologies research center, Najafabad Branch , Islamic Azad University, Najafabad, Iran 2 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad , Iran [email protected] , *[email protected], [email protected] , [email protected]
Abstract In the present study, the problem of multiple cylindrical interface cracks between a homogeneous circular cylinder and its functionally graded materials (FGM) coating under torsional impact loading is investigated. First, by using the Laplace and complex Fourier transforms, the solution for Somigliana-type dynamic rotational ring dislocation in a cylinder and its coating is obtained. Next, the distributed dislocation technique is employed to derive a set of Cauchy singular integral equations for multiple cylindrical interface cracks situated between the isotropic cylinder and its FGM coating. These integral equations are solved by Erdogan’s collocation method; the dislocation densities on the faces of the cracks are obtained, and the results are used to determine dynamic stress intensity factors (DSIFs). Several examples are provided to study the influences of material nonhomogeneity constant, the FGM layer thickness and crack geometry configuration on the DSIFs at the tips of cracks and, the interaction between the cracks. Keywords:
Dynamic Stress Intensity FGM Coating; Dislocation Density.
Factors;
Transient;
Cylindrical Interface Cracks;
1. Introduction In recent years, functionally graded materials (FGMs) as coating and interfacial zones have been widely applied in structures exposed to harsh environments and severe thermal shocks. FGMs coating and FGMs interlayer help reduce mechanically- and thermally-induced stresses caused by material property mismatch; they also improve bonding strength. Typical applications of FGMs comprise thermal barrier coatings (TBCs) of high temperature components in gas turbines and as inter-layers in microelectronic and optoelectronic components. Therefore, fracture analysis of an FGMs coating-substrate system and an FGMs interlayer is an important design consideration and has attracted the attention of many researchers [1-3]. Delale and Erdogan identified the stress intensity factors for a crack situated in the interfacial nonhomogeneous layer between two dissimilar elastic half-planes under tension [4]. Ozturk and Erdogan determined axisymmetric solutions for a pennyshaped crack in an interfacial nonhomogeneous layer between two dissimilar elastic halfspaces under torsion [5] and under tension [6]. The axially symmetric problem of a cylindrical crack in the nonhomogeneous region of two coaxial elastic cylinder under ModeIII loading was studied by Han and Wang [7]. Choi et al. investigated the embedded collinear cracks in a layered half plane with a graded nonhomogeneous interfacial zone under static mechanical load [8]. Shbeeb and Binienda analyzed an interface crack in an FGM strip 1
sandwiched between two homogeneous layers of finite thickness and determined the stress intensity factors and strain energy release rate [9]. The axisymmetric problem for a cylindrical crack in an interfacial zone between a circular elastic cylinder and an infinite elastic medium under Mode I loading was investigated by Itou and Shima [10]. Itou directed a study of a crack in an FGM layer between two homogeneous half planes. In his paper, the integral equation was solved using the Schmidt method, and a stress intensity factor was calculated [11]. An analytical model for fracture analysis of an FGM interfacial zone with a Griffith crack under the anti-plane shearing load was developed by Wang et al. [12]. Dhaliwal et al. analyzed the problem of a cylindrical interface crack between two dissimilar nonhomogeneous coaxial finite elastic cylinders under axially symmetric longitudinal shear stress. They identified the stress intensity factor using dual series equations [13]. Jin and Batra [14] investigated the interface cracking between ceramic and/or FGM coatings and a substrate under anti-plane shear. The interface crack problem between the functionally graded ceramic coating and the homogeneous substrate was studied by Chen and Erdogan [15]. In this study, the mixed mode crack problem was formulated for a crack subjected to surface tractions. Huang et al. developed a brand-new model for the approximate analysis of FGMs, the properties of which may vary arbitrarily, and solved the problem of a crack in an FGM coating bonded to a homogeneous substrate under a static anti-plane shearing load [16]. The crack problem of an FGM coating-substrate structure with an internal or edge crack perpendicular to the interface was investigated under an in-plane load by Guo et al. [17]. A multi-layered model for the crack problem of FGMs under plane stress state deformation with arbitrarily varying Young’s modulus was proposed by Huang et al. [18]. A study by Chen and Chue [19] dealt with the anti-plane problems of two bonded FGM strips weakened by an internal crack normal to the interface. Asadi et al. [20] obtained the stress fields for an orthotropic strip with defects and imperfect FGM coating using the Volterra screw dislocation. Cheng et al. [21] studied the plane elasticity problem of two bonded dissimilar FGM strips containing an interface crack in which material properties varied arbitrarily. The problem of a tubular interface crack in a bi-layered cylindrical composite of finite thickness under torsion was studied by Li et al. [22]. The fracture analysis on a layered cylinder consisting of an inner and an outer functionally graded elastic circular tube with a cylinder interface crack under static torsion was performed by Shi [23]. In many engineering applications, nonhomogeneous structures may be subjected to dynamic loadings. In relation to the dynamic crack problem, Ueda et al. [24] 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. Itou obtained the dynamic stresses of a crack in a nonhomogeneous interfacial layer between two elastic halfplanes under tension [25]. The problem of a cylindrical crack located in an FGM interlayer between two coaxial dissimilar homogeneous cylinders and subjected to a torsional impact loading was examined by Li et al. [26]. Li and Weng [27] investigated the dynamic stress intensity factor of a cylindrical interface crack located between two coaxial dissimilar homogeneous cylinders that are bonded with an FGM interlayer and subjected to a torsional impact loading. Later, they considered 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 [28]. The transient response analysis of an FGM coating2
substrate system with an internal or edge crack perpendicular to the interface under an inplane impact load was carried out by Guo et al. [29]. The problem of a cylindrical interface crack located between an FGM interlayer and its external homogeneous cylinder and subjected to a torsional impact loading was considered by Feng et al. [30]. The main objective of the present study was to provide a theoretical analysis of the dynamic fracture behavior of multiple cylindrical interface cracks between a homogeneous cylinder and its FGM coating subjected to torsional transient loading. The technique necessitates the solution of the Somigliana-type dynamic rotational ring dislocation in the region. Laplace and complex Fourier transforms were employed to obtain transformed displacement and stress fields. The dislocation solution and Buckner’s principle were employed in the construction of integral equations for the homogeneous circular cylinder and its FGM coating weakened by cylindrical interface cracks. The integral equations were of the Cauchy singular types and solved numerically to determine the density of the dislocation on a crack’s faces. The numerical examples are given to show the effects thickness of the FGM layer, the material properties, and the interaction between the cracks upon the fracture behavior. 2. Formulation of the Problem In this study, an isotropic circular cylinder with its radius coated by an FGM layer with a thickness of . The circular cylindrical coordinate was selected such that the -axis was along the longitudinal direction and the origin was located at the center of the cross-section. For the problem in question, there is only anti-plane deformation being independent of , and two other elastic displacements, and , oriented in the and axes vanish. Consequently, the constitutive equations read as
(1)
where is the shear modulus of elasticity of the isotropic cylinder, and modulus in the FGM coating. By using Eq. (1), the equilibrium equation, each region leads to
is the shear , for
(2)
where and are the materials mass density of the isotropic cylinder and FGM coating, respectively. We will assume that material properties of the FGM are approximated by (3)
3
where , and are the material constants of FGM. Substituting Eq. (3) into Eq. (2), the equation of motion can be rewritten as
(4)
The values and are the reciprocal of the shear wave velocity in the isotropic cylinder and FGM coating, respectively. Let a Somigliana-type dynamic rotational ring dislocation with Burgers vector be situated at with dislocation cut in axial direction . The stress and displacement fields for the dislocation for the current problem can be expressed as follows (5)
where is the Heaviside step-function. The traction-free condition on the curved surface of FGM coating is (6)
Applying the standard Laplace transform and letting (7)
Eq. (4) can be converted into
(8)
To obtain the solution of the spatial differential Eq. (8), the complex Fourier transform is introduced as follows (9)
The application of Eq. (9) to Eq. (8) yields (10)
4
The finite solutions to Eq. (10) as
are readily known
(11)
where and and the coefficients and are unknown functions. Also, In Eq. (10), and are the modified Bessel functions of the first and second kinds, respectively. From Eq. (9) and Eq. (10), we can obtain the displacements in the Laplace domain as follow
(12)
Subsequently, the shear stresses obtained from Eq. (1) and Eq. (12).
and
in the Laplace transform domain can be
(13)
The application of the Laplace transforms to conditions (4) and (5) results in (14)
Applying the boundary condition in Eq. (14) to the Eq. (12) and Eq. (13) leads to
(15)
wherein 5
(16)
Substituting Eq. (15) into Eq. (13), and after a lengthy analysis, the stress components yields
(17)
where
(18)
The integrals in the Laplace transform of the stress component , do not converge for the points in the vicinity of the dislocation . To circumvent this difficulty, an asymptotic analysis of the integrands is carried out as , leading to
(19)
From Eq. (19), it may be observed that the Laplace transform of the stress component exhibits the familiar Cauchy-type singularity at the dislocation location, i.e. . 6
3. Formulation of Multiple Cylindrical Interface Cracks To analyze the cylindrical interface crack problems in an isotropic cylinder with FGM coating, the dislocation solutions obtained in Section 2 can be implemented. With this technique, the dislocations are distributed in the locations of the crack, and the stress fields are determined for the cracked medium. Consider an isotropic cylinder bonded to an FGM coating weakened by an cylindrical interface crack. The geometry of an interfacial crack may be described in parametric form, as
(20)
where and are the coordinates of the upper and lower tips and half-length of the j-th crack, respectively. Suppose the Laplace transform of the dislocations with unknown density are distributed on the infinitesimal segment at the surfaces of the j-th interface crack located at Employing the principal of superposition, the traction component caused by the dislocations distributed on the surface of all cracks is (21)
From Eq. (16), the kernel of integral (25) become
(22)
The kernel in Eq. (17) represents a Cauchy-type singularity for means of the Taylor series expansion, we have
as
. Therefore, by
(23)
Using the definition of dislocation density function, the equation for the crack opening displacement across the j-th crack yields (24)
7
The single-valuedness of the displacement field out of the embedded cracks, i.e. the cylindrical interface cracks, implies that their related dislocation densities are imposed on the closure requirement as (25)
To obtain the dislocation density, the Cauchy singular integral (25) and the closure equation (28) are to be solved simultaneously. The stress field near the interfacial crack tips has a square-root singularity [31]; therefore, the Laplace transform of the dislocation densities on the surface of embedded cracks is grouped as [32]
(26)
The substitution of Eq. (26) into Eqs. (21) and (25) and the application of the numerical technique devised by Erdogan et al. [33] result in , . The DSIFs in the Laplace transform domain for a cylindrical interface crack are defined as
(27)
The substitution of Eqs. (23) and (26) into (21) and the resultant equations into (27) yield the Laplace transform of the DSIFs at the crack tips for the j-th crack
(28)
The dynamic stress intensity factor in the time domain can be obtained by the Stehfest’s method, e.g., see Vafa et al. [34]. 4. Numerical Examples The theoretical analysis developed in the preceding section was used to investigate the effects of pertinent parameters upon the DSIFs of the multiple cylindrical interface cracks. Several examples were solved to demonstrate the applicability of the distributed dislocation technique. Variations in the interface crack length, material nonhomogeneity constant, FGM layer thickness, and the interaction between the cracks on the DSIFs were investigated. In all the proceeding examples, the radius of the isotropic cylinder was . Also, the dynamic shear stress was applied on the surface of the cracks, in which is the constant twisting load. The DSIFs were normalized as where is taken and is the half length of the crack. The results of the examples are represented versus non-dimensional time
, where
8
Example 1: An infinite medium with a cylindrical crack The solution of the dislocation method given in this manuscript was demonstrated and verified with the published results. For the first example, the problem of an isotropic infinite medium ( ) containing a cylindrical crack under statically torsional loading ( ) was considered. Fig. 2 shows the variations in the normalized stress intensity factor versus the normalized crack length for the crack tip. As can be seen in Fig. 1, the results obtained from [35] validate the results of this study.
4
Present Study Refference [32]
3.5
=0 1=2 1=2
k III (t) / k0
3
2.5
2
1.5
1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a/l
Fig. 1. Normalized
at
for infinite isotropic medium with cylindrical crack
Example 2: An isotropic cylinder with FGM coating weakened by a cylindrical interface crack The second example dealt with a homogeneous cylinder with radius bound to a nonhomogeneous layer with a thickness of , which was weakened by a cylindrical crack at the interface. Considering the crack symmetrically relative to the , we have , and then they are written as The normalized DSIFs versus normalized time were calculated, and the results are shown graphically in Figs. 2–4. From these figures, it is clearly seen that generally, the DSIFs had a main peak value appearing at about , and after reaching the main peak, those values diminished gradually and tended to the static values. Fig. 2 shows the influence of the interfacial crack length on the normalized DSIFs with at and As expected, normalized DSIFs increased with increases to the interface crack length
9
3.5
l/a=0.5 l/a=1.0 l/a=1.5 l/a=2.0
3
b/a=2 =1
k III(t) / k0
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t / l
Fig. 2. The variation of normalized DSIFs with crack length
The effect of the FGM thickness, i.e. , on the variation of the normalized DSIFs is shown in Fig. 3. The other parameters were chosen as and , indicating that the normalized DSIFs decreased with increases in ; that is, a thicker FGM coating was beneficial in reducing DSIFs.
3.5
b/a=1.15 b/a=1.25 b/a=1.50 b/a=2.0
3
l/a=1 =1
k III (t) / k0
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t / l
Fig. 3. The effect of the FGM coating thickness on the normalized DSIFs
The normalized DSIFs versus for different values of the nonhomogeneity constants of the FGM coating is depicted in Fig. 4, while and . As it might be observed,
10
for definite values of constants.
and
, DSIFs are reduced with growing nonhomogeneity
2.5
=0.5 =1.5 =2.5 =5.0
l/a=1 b/a=2
k III (t) / k0
2
1.5
1 0
2
4
6
8
10
12
c1 t / l
Fig. 4. The effect of the nonhomogeneity constant on the normalized DSIFs
Example 3: An isotropic cylindrical interface cracks
cylinder
with
FGM
coating
weakened
by
two
In the last example, an isotropic cylinder with FGM coating containing two concentric cylindrical interface cracks was analyzed. In this problem, the interaction between the two cylindrical interfacial cracks was studied. The cracks were symmetrically located relative to the , where the distance separating the two inner crack tips was The up and down cracks relative to the were designated cracks 1 and 2, respectively. They were identical, each having a crack length of Therefore, we have and The graphs of those versus are plotted in Figs. 5-6, respectively. Figs. 5-6 illustrate the effect of the interaction between the two cylindrical interfacial cracks for different values of spacing , where other factors were selected as , and . In comparison with the previous example, it was observed that the interaction between cracks enhanced the SIFs of the crack tips, which were more pronounced at the lower tip of the upper crack (crack number one) and the upper tip of the lower crack (crack number two), because these are closer to each other. Also, as approaches infinity, the values of DSIFs are set to the values of a crack.
11
10
d/a=0.01 d/a=0.10 d/a=1.00 d/a=Infinity
9
k III,l1 (t) / k0 , k III,u2 (t) / k0
8
l/a=1 b/a=2 =1
7 6 5 4 3 2 1 0 0
2
4
6
8
10
12
c1 t / l
Fig. 5. Interaction between two concentric cylindrical interface cracks
3
d/a=0.01 d/a=0.10 d/a=1.00 d/a=Infinity
k III,u1 (t) / k0 , k III,l2 (t) / k0
2.5
l/a=1 b/a=2 =1
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t/l
Fig. 6. Interaction between two concentric cylindrical interface cracks
5. Concluding remarks In this paper, the fracture behavior of multiple cylindrical interface cracks between a homogeneous circular cylinder and its FGM coating subjected to torsional transient loading using the distributed dislocation technique was studied. The problem was reduced to a set of singular integral equations with a Cauchy-type singular kernel by means of Laplace and complex Fourier transforms. The numerical solutions to integral equations result in the dislocation density function on a crack surface, thereby determining the SIFs for cracks. These integral equations were solved with the numerical method, and the SIFs at the interfacial crack tips were calculated. To study the effect of the nonhomogeneity constant, the FGM layer thickness and crack length as well as the interaction between cracks and DSIFs 12
were obtained for some examples. From the numerical results, the following key points were observed: (i) The DSIFs rise rapidly, and after reaching the main peak, they diminish gradually and tend to the static values. (ii) The DSIFs increase with increases in interface crack length. (iii) The magnitude of the DSIFs decreases with increases in the material gradient and FGM coating thickness. (iv) The interaction between cracks is an important factor affecting the DSIFs of crack tips. References [1] Karmini, M., Atrian, A., Ghassemi, A., and Vahabi, M., 2017. Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks. Theoretical and Applied Fracture Mechanics 90, pp.110–121. [2] Noroozi, M., Atrian, A., Ghassemi, A., and Vahabi, M., 2018. Torsion analysis of infinite hollow cylinders of functionally graded materials weakened by multiple axisymmetric cracks. Theoretical and Applied Fracture Mechanics, 96, pp.811–819. [3] Atrian, A., Jafari Fesharaki, J., and Nourbakhsh, S.H., 2015. Thermo-electromechanical behavior of functionally graded piezoelectric hollow cylinder under non-axisymmetric loads. Applied Mathematics and Mechaics (English Edition), 35(7), pp.939–954. [4] Delale, F. and Erdogan, F., 1988. On the mechanical modeling of the interfacial region in bonded half-planes. Journal of Applied Mechanics, 55(2), pp.317-324. [5] Ozturk, M. and Erdogan, F., 1995. An axisymmetric crack in bonded materials with a nonhomogeneous interfacial zone under torsion. Journal of Applied Mechanics, 62(1), pp.116-125. [6] Ozturk, M. and Erdogan, F., 1996. Axsiymmetric crack problem in bonded materials with a graded interfacial region. International Journal of Solids and Structures, 33(2), pp.193-219. [7] Xue-Li, H. and Duo, W., 1996. The crack problem of a fiber-matrix composite with a nonhomogeneous interfacial zone under torsional loading—Part I. A cylindrical crack in the interfacial zone. Engineering fracture mechanics, 54(1), pp.63-69. [8] Choi, H.J., Lee, K.Y. and Jin, T.E., 1998. Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone–Part I: Mechanical response. International Journal of Fracture, 94(2), pp.103-122. [9] Shbeeb, N.I. and Binienda, W.K., 1999. Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness. Engineering Fracture Mechanics, 64(6), pp.693-720. [10] Itou, S. and Shima, Y., 1999. Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials. International journal of solids and structures, 36(5), pp.697-709. [11] Itou, S., 2001. Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes. International Journal of Fracture, 110(2), pp.123-135. [12] Wang, Y.S., Huang, G.Y. and Dross, D., 2003. On the mechanical modeling of functionally graded interfacial zone with a Griffith crack: anti-plane deformation. Journal of Applied Mechanics, 70(5), pp.676-680. [13] Dhaliwal, R.S., Saxena, H.S., He, W. and Rokne, J.G., 1992. Stress intensity factor for the cylindrical interface crack between nonhomogeneous coaxial finite elastic cylinders. Engineering fracture mechanics, 43(6), pp.1039-1051. [14] Jin, Z.H. and Batra, R.C., 1996. Interface cracking between functionally graded coatings and a substrate under antiplane shear. International Journal of Engineering Science, 34(15), pp.1705-1716.
13
[15] Chen, Y.F. and Erdogan, F., 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids, 44(5), pp.771787. [16] Huang, G.Y., Wang, Y.S. and Gross, D., 2002. Fracture analysis of functionally graded coatings: antiplane deformation. European Journal of Mechanics-A/Solids, 21(3), pp.391-400. [17] Guo, L.C., Wu, L.Z., Ma, L. and Zeng, T., 2004. Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface-Part I: Static problem. International Journal of fracture, 127(1), pp.21-38. [18] Huang, G.Y., Wang, Y.S. and Yu, S.W., 2005. A new model for fracture analysis of functionally graded coatings under plane deformation. Mechanics of materials, 37(4), pp.507-516. [19] Chen, Y.J. and Chue, C.H., 2009. Mode III crack problems of two bonded functionally graded strips with internal cracks. International Journal of Solids and Structures, 46(2), pp.331-343. [20] Asadi, E., Fariborz, S.J. and Fotuhi, A.R., 2012. Anti-plane analysis of orthotropic strips with defects and imperfect FGM coating. European Journal of Mechanics-A/Solids, 34, pp.12-20. [21] Cheng, Z., Gao, D. and Zhong, Z., 2012. Interface crack of two dissimilar bonded functionally graded strips with arbitrary distributed properties under plane deformations. International Journal of Mechanical Sciences, 54(1), pp.287-293. [22] Li, Y.D., Zhao, H. and Xiong, T., 2013. The cylindrical interface crack in a layered tubular composite of finite thickness under torsion. European Journal of Mechanics-A/Solids, 39, pp.113-119. [23] Shi, P., 2016. Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion. Mechanics Based Design of Structures and Machines, 44(3), pp.250-269. [24] Ueda, S., Shindo, Y. and Atsumi, A., 1983. Torsional impact response of a penny-shaped crack lying on a bimaterial interface. Engineering Fracture Mechanics, 18(5), pp.1059-1066. [25] Itou, S., 2001. Transient dynamic stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes. International journal of solids and structures, 38(20), pp.3631-3645. [26] Li, C., Weng, G.J. and Duan, Z., 2001. Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. International Journal of solids and Structures, 38(42), pp.7473-7485. [27] Li, C. and Weng, G.J., 2001. Dynamic stress intensity factor of a cylindrical interface crack with a functionally graded interlayer. Mechanics of Materials, 33(6), pp.325-333. [28] Li, C. and Weng, G.J., 2002. Dynamic fracture analysis for a penny-shaped crack in an FGM interlayer between dissimilar half spaces. Mathematics and Mechanics of Solids, 7(2), pp.149-163. [29] Guo, L.C., Wu, L.Z., Zeng, T. and Ma, L., 2004. Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface-Part II: Transient problem. International Journal of Fracture, 127(1), pp.39-59. [30] Feng, W.J., Su, R.K.L. and Jiang, Z.Q., 2005. Torsional impact response of a cylindrical interface crack between a functionally graded interlayer and a homogeneous cylinder. Composite structures, 68(2), pp.203-209. [31] Lowengrub, M., 1969. Crack Problems in the Classical Theory of Elasticity. John Wiley & Sons. [32] Vafa, J.P. and Fariborz, S.J., 2016. Transient analysis of multiply interacting cracks in orthotropic layers. European Journal of Mechanics-A/Solids, 60, pp.254-276. [33] Erdogan, F., Gupta, G.D.A. and Cook, T.S., 1973. Numerical solution of singular integral equations. In Methods of analysis and solutions of crack problems (pp. 368-425). Springer Netherlands. [34] Vafa, J.P., Baghestani, A.M. and Fariborz, S.J., 2015. Transient screw dislocation in exponentially graded FG layers. Archive of Applied Mechanics, 85(1), pp.1-11.
14
[35] Demir, I. and Khraishi, T.A., 2005. The torsional dislocation loop and mode III cylindrical crack. Journal of Mechanics, 21(2), pp.109-116.
15
We solve problem of Somigliana-type dynamic rotational dislocation in FGM coated cylinder. We use the dislocation cylindrical interface cracks.
distribution
technique
for
analysis
Nonhomogeneity constant has an efficient effect on DSIFs. FGM layer thickness and crack length have also a crucial effect on DSIFs.
16
of
multiple
S0167-8442(18)30108-3 https://doi.org/10.1016/j.tafmec.2018.08.015 TAFMEC 2094
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
4 March 2018 11 August 2018 27 August 2018
Please cite this article as: M. Noroozi, A. Ghassemi, A. Atrian, M. Vahabi, Multiple Cylindrical Interface Cracks in FGM Coated Cylinders under Torsional Transient Loading, Theoretical and Applied Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.tafmec.2018.08.015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multiple Cylindrical Interface Cracks in FGM Coated Cylinders under Torsional Transient Loading M. Noroozi1,2, A. Ghassemi 1,2*, A. Atrian1,2, M. Vahabi1,2 1
Modern manufacturing technologies research center, Najafabad Branch , Islamic Azad University, Najafabad, Iran 2 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad , Iran [email protected] , *[email protected], [email protected] , [email protected]
Abstract In the present study, the problem of multiple cylindrical interface cracks between a homogeneous circular cylinder and its functionally graded materials (FGM) coating under torsional impact loading is investigated. First, by using the Laplace and complex Fourier transforms, the solution for Somigliana-type dynamic rotational ring dislocation in a cylinder and its coating is obtained. Next, the distributed dislocation technique is employed to derive a set of Cauchy singular integral equations for multiple cylindrical interface cracks situated between the isotropic cylinder and its FGM coating. These integral equations are solved by Erdogan’s collocation method; the dislocation densities on the faces of the cracks are obtained, and the results are used to determine dynamic stress intensity factors (DSIFs). Several examples are provided to study the influences of material nonhomogeneity constant, the FGM layer thickness and crack geometry configuration on the DSIFs at the tips of cracks and, the interaction between the cracks. Keywords:
Dynamic Stress Intensity FGM Coating; Dislocation Density.
Factors;
Transient;
Cylindrical Interface Cracks;
1. Introduction In recent years, functionally graded materials (FGMs) as coating and interfacial zones have been widely applied in structures exposed to harsh environments and severe thermal shocks. FGMs coating and FGMs interlayer help reduce mechanically- and thermally-induced stresses caused by material property mismatch; they also improve bonding strength. Typical applications of FGMs comprise thermal barrier coatings (TBCs) of high temperature components in gas turbines and as inter-layers in microelectronic and optoelectronic components. Therefore, fracture analysis of an FGMs coating-substrate system and an FGMs interlayer is an important design consideration and has attracted the attention of many researchers [1-3]. Delale and Erdogan identified the stress intensity factors for a crack situated in the interfacial nonhomogeneous layer between two dissimilar elastic half-planes under tension [4]. Ozturk and Erdogan determined axisymmetric solutions for a pennyshaped crack in an interfacial nonhomogeneous layer between two dissimilar elastic halfspaces under torsion [5] and under tension [6]. The axially symmetric problem of a cylindrical crack in the nonhomogeneous region of two coaxial elastic cylinder under ModeIII loading was studied by Han and Wang [7]. Choi et al. investigated the embedded collinear cracks in a layered half plane with a graded nonhomogeneous interfacial zone under static mechanical load [8]. Shbeeb and Binienda analyzed an interface crack in an FGM strip 1
sandwiched between two homogeneous layers of finite thickness and determined the stress intensity factors and strain energy release rate [9]. The axisymmetric problem for a cylindrical crack in an interfacial zone between a circular elastic cylinder and an infinite elastic medium under Mode I loading was investigated by Itou and Shima [10]. Itou directed a study of a crack in an FGM layer between two homogeneous half planes. In his paper, the integral equation was solved using the Schmidt method, and a stress intensity factor was calculated [11]. An analytical model for fracture analysis of an FGM interfacial zone with a Griffith crack under the anti-plane shearing load was developed by Wang et al. [12]. Dhaliwal et al. analyzed the problem of a cylindrical interface crack between two dissimilar nonhomogeneous coaxial finite elastic cylinders under axially symmetric longitudinal shear stress. They identified the stress intensity factor using dual series equations [13]. Jin and Batra [14] investigated the interface cracking between ceramic and/or FGM coatings and a substrate under anti-plane shear. The interface crack problem between the functionally graded ceramic coating and the homogeneous substrate was studied by Chen and Erdogan [15]. In this study, the mixed mode crack problem was formulated for a crack subjected to surface tractions. Huang et al. developed a brand-new model for the approximate analysis of FGMs, the properties of which may vary arbitrarily, and solved the problem of a crack in an FGM coating bonded to a homogeneous substrate under a static anti-plane shearing load [16]. The crack problem of an FGM coating-substrate structure with an internal or edge crack perpendicular to the interface was investigated under an in-plane load by Guo et al. [17]. A multi-layered model for the crack problem of FGMs under plane stress state deformation with arbitrarily varying Young’s modulus was proposed by Huang et al. [18]. A study by Chen and Chue [19] dealt with the anti-plane problems of two bonded FGM strips weakened by an internal crack normal to the interface. Asadi et al. [20] obtained the stress fields for an orthotropic strip with defects and imperfect FGM coating using the Volterra screw dislocation. Cheng et al. [21] studied the plane elasticity problem of two bonded dissimilar FGM strips containing an interface crack in which material properties varied arbitrarily. The problem of a tubular interface crack in a bi-layered cylindrical composite of finite thickness under torsion was studied by Li et al. [22]. The fracture analysis on a layered cylinder consisting of an inner and an outer functionally graded elastic circular tube with a cylinder interface crack under static torsion was performed by Shi [23]. In many engineering applications, nonhomogeneous structures may be subjected to dynamic loadings. In relation to the dynamic crack problem, Ueda et al. [24] 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. Itou obtained the dynamic stresses of a crack in a nonhomogeneous interfacial layer between two elastic halfplanes under tension [25]. The problem of a cylindrical crack located in an FGM interlayer between two coaxial dissimilar homogeneous cylinders and subjected to a torsional impact loading was examined by Li et al. [26]. Li and Weng [27] investigated the dynamic stress intensity factor of a cylindrical interface crack located between two coaxial dissimilar homogeneous cylinders that are bonded with an FGM interlayer and subjected to a torsional impact loading. Later, they considered 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 [28]. The transient response analysis of an FGM coating2
substrate system with an internal or edge crack perpendicular to the interface under an inplane impact load was carried out by Guo et al. [29]. The problem of a cylindrical interface crack located between an FGM interlayer and its external homogeneous cylinder and subjected to a torsional impact loading was considered by Feng et al. [30]. The main objective of the present study was to provide a theoretical analysis of the dynamic fracture behavior of multiple cylindrical interface cracks between a homogeneous cylinder and its FGM coating subjected to torsional transient loading. The technique necessitates the solution of the Somigliana-type dynamic rotational ring dislocation in the region. Laplace and complex Fourier transforms were employed to obtain transformed displacement and stress fields. The dislocation solution and Buckner’s principle were employed in the construction of integral equations for the homogeneous circular cylinder and its FGM coating weakened by cylindrical interface cracks. The integral equations were of the Cauchy singular types and solved numerically to determine the density of the dislocation on a crack’s faces. The numerical examples are given to show the effects thickness of the FGM layer, the material properties, and the interaction between the cracks upon the fracture behavior. 2. Formulation of the Problem In this study, an isotropic circular cylinder with its radius coated by an FGM layer with a thickness of . The circular cylindrical coordinate was selected such that the -axis was along the longitudinal direction and the origin was located at the center of the cross-section. For the problem in question, there is only anti-plane deformation being independent of , and two other elastic displacements, and , oriented in the and axes vanish. Consequently, the constitutive equations read as
(1)
where is the shear modulus of elasticity of the isotropic cylinder, and modulus in the FGM coating. By using Eq. (1), the equilibrium equation, each region leads to
is the shear , for
(2)
where and are the materials mass density of the isotropic cylinder and FGM coating, respectively. We will assume that material properties of the FGM are approximated by (3)
3
where , and are the material constants of FGM. Substituting Eq. (3) into Eq. (2), the equation of motion can be rewritten as
(4)
The values and are the reciprocal of the shear wave velocity in the isotropic cylinder and FGM coating, respectively. Let a Somigliana-type dynamic rotational ring dislocation with Burgers vector be situated at with dislocation cut in axial direction . The stress and displacement fields for the dislocation for the current problem can be expressed as follows (5)
where is the Heaviside step-function. The traction-free condition on the curved surface of FGM coating is (6)
Applying the standard Laplace transform and letting (7)
Eq. (4) can be converted into
(8)
To obtain the solution of the spatial differential Eq. (8), the complex Fourier transform is introduced as follows (9)
The application of Eq. (9) to Eq. (8) yields (10)
4
The finite solutions to Eq. (10) as
are readily known
(11)
where and and the coefficients and are unknown functions. Also, In Eq. (10), and are the modified Bessel functions of the first and second kinds, respectively. From Eq. (9) and Eq. (10), we can obtain the displacements in the Laplace domain as follow
(12)
Subsequently, the shear stresses obtained from Eq. (1) and Eq. (12).
and
in the Laplace transform domain can be
(13)
The application of the Laplace transforms to conditions (4) and (5) results in (14)
Applying the boundary condition in Eq. (14) to the Eq. (12) and Eq. (13) leads to
(15)
wherein 5
(16)
Substituting Eq. (15) into Eq. (13), and after a lengthy analysis, the stress components yields
(17)
where
(18)
The integrals in the Laplace transform of the stress component , do not converge for the points in the vicinity of the dislocation . To circumvent this difficulty, an asymptotic analysis of the integrands is carried out as , leading to
(19)
From Eq. (19), it may be observed that the Laplace transform of the stress component exhibits the familiar Cauchy-type singularity at the dislocation location, i.e. . 6
3. Formulation of Multiple Cylindrical Interface Cracks To analyze the cylindrical interface crack problems in an isotropic cylinder with FGM coating, the dislocation solutions obtained in Section 2 can be implemented. With this technique, the dislocations are distributed in the locations of the crack, and the stress fields are determined for the cracked medium. Consider an isotropic cylinder bonded to an FGM coating weakened by an cylindrical interface crack. The geometry of an interfacial crack may be described in parametric form, as
(20)
where and are the coordinates of the upper and lower tips and half-length of the j-th crack, respectively. Suppose the Laplace transform of the dislocations with unknown density are distributed on the infinitesimal segment at the surfaces of the j-th interface crack located at Employing the principal of superposition, the traction component caused by the dislocations distributed on the surface of all cracks is (21)
From Eq. (16), the kernel of integral (25) become
(22)
The kernel in Eq. (17) represents a Cauchy-type singularity for means of the Taylor series expansion, we have
as
. Therefore, by
(23)
Using the definition of dislocation density function, the equation for the crack opening displacement across the j-th crack yields (24)
7
The single-valuedness of the displacement field out of the embedded cracks, i.e. the cylindrical interface cracks, implies that their related dislocation densities are imposed on the closure requirement as (25)
To obtain the dislocation density, the Cauchy singular integral (25) and the closure equation (28) are to be solved simultaneously. The stress field near the interfacial crack tips has a square-root singularity [31]; therefore, the Laplace transform of the dislocation densities on the surface of embedded cracks is grouped as [32]
(26)
The substitution of Eq. (26) into Eqs. (21) and (25) and the application of the numerical technique devised by Erdogan et al. [33] result in , . The DSIFs in the Laplace transform domain for a cylindrical interface crack are defined as
(27)
The substitution of Eqs. (23) and (26) into (21) and the resultant equations into (27) yield the Laplace transform of the DSIFs at the crack tips for the j-th crack
(28)
The dynamic stress intensity factor in the time domain can be obtained by the Stehfest’s method, e.g., see Vafa et al. [34]. 4. Numerical Examples The theoretical analysis developed in the preceding section was used to investigate the effects of pertinent parameters upon the DSIFs of the multiple cylindrical interface cracks. Several examples were solved to demonstrate the applicability of the distributed dislocation technique. Variations in the interface crack length, material nonhomogeneity constant, FGM layer thickness, and the interaction between the cracks on the DSIFs were investigated. In all the proceeding examples, the radius of the isotropic cylinder was . Also, the dynamic shear stress was applied on the surface of the cracks, in which is the constant twisting load. The DSIFs were normalized as where is taken and is the half length of the crack. The results of the examples are represented versus non-dimensional time
, where
8
Example 1: An infinite medium with a cylindrical crack The solution of the dislocation method given in this manuscript was demonstrated and verified with the published results. For the first example, the problem of an isotropic infinite medium ( ) containing a cylindrical crack under statically torsional loading ( ) was considered. Fig. 2 shows the variations in the normalized stress intensity factor versus the normalized crack length for the crack tip. As can be seen in Fig. 1, the results obtained from [35] validate the results of this study.
4
Present Study Refference [32]
3.5
=0 1=2 1=2
k III (t) / k0
3
2.5
2
1.5
1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a/l
Fig. 1. Normalized
at
for infinite isotropic medium with cylindrical crack
Example 2: An isotropic cylinder with FGM coating weakened by a cylindrical interface crack The second example dealt with a homogeneous cylinder with radius bound to a nonhomogeneous layer with a thickness of , which was weakened by a cylindrical crack at the interface. Considering the crack symmetrically relative to the , we have , and then they are written as The normalized DSIFs versus normalized time were calculated, and the results are shown graphically in Figs. 2–4. From these figures, it is clearly seen that generally, the DSIFs had a main peak value appearing at about , and after reaching the main peak, those values diminished gradually and tended to the static values. Fig. 2 shows the influence of the interfacial crack length on the normalized DSIFs with at and As expected, normalized DSIFs increased with increases to the interface crack length
9
3.5
l/a=0.5 l/a=1.0 l/a=1.5 l/a=2.0
3
b/a=2 =1
k III(t) / k0
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t / l
Fig. 2. The variation of normalized DSIFs with crack length
The effect of the FGM thickness, i.e. , on the variation of the normalized DSIFs is shown in Fig. 3. The other parameters were chosen as and , indicating that the normalized DSIFs decreased with increases in ; that is, a thicker FGM coating was beneficial in reducing DSIFs.
3.5
b/a=1.15 b/a=1.25 b/a=1.50 b/a=2.0
3
l/a=1 =1
k III (t) / k0
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t / l
Fig. 3. The effect of the FGM coating thickness on the normalized DSIFs
The normalized DSIFs versus for different values of the nonhomogeneity constants of the FGM coating is depicted in Fig. 4, while and . As it might be observed,
10
for definite values of constants.
and
, DSIFs are reduced with growing nonhomogeneity
2.5
=0.5 =1.5 =2.5 =5.0
l/a=1 b/a=2
k III (t) / k0
2
1.5
1 0
2
4
6
8
10
12
c1 t / l
Fig. 4. The effect of the nonhomogeneity constant on the normalized DSIFs
Example 3: An isotropic cylindrical interface cracks
cylinder
with
FGM
coating
weakened
by
two
In the last example, an isotropic cylinder with FGM coating containing two concentric cylindrical interface cracks was analyzed. In this problem, the interaction between the two cylindrical interfacial cracks was studied. The cracks were symmetrically located relative to the , where the distance separating the two inner crack tips was The up and down cracks relative to the were designated cracks 1 and 2, respectively. They were identical, each having a crack length of Therefore, we have and The graphs of those versus are plotted in Figs. 5-6, respectively. Figs. 5-6 illustrate the effect of the interaction between the two cylindrical interfacial cracks for different values of spacing , where other factors were selected as , and . In comparison with the previous example, it was observed that the interaction between cracks enhanced the SIFs of the crack tips, which were more pronounced at the lower tip of the upper crack (crack number one) and the upper tip of the lower crack (crack number two), because these are closer to each other. Also, as approaches infinity, the values of DSIFs are set to the values of a crack.
11
10
d/a=0.01 d/a=0.10 d/a=1.00 d/a=Infinity
9
k III,l1 (t) / k0 , k III,u2 (t) / k0
8
l/a=1 b/a=2 =1
7 6 5 4 3 2 1 0 0
2
4
6
8
10
12
c1 t / l
Fig. 5. Interaction between two concentric cylindrical interface cracks
3
d/a=0.01 d/a=0.10 d/a=1.00 d/a=Infinity
k III,u1 (t) / k0 , k III,l2 (t) / k0
2.5
l/a=1 b/a=2 =1
2
1.5
1
0.5
0 0
2
4
6
8
10
12
c1 t/l
Fig. 6. Interaction between two concentric cylindrical interface cracks
5. Concluding remarks In this paper, the fracture behavior of multiple cylindrical interface cracks between a homogeneous circular cylinder and its FGM coating subjected to torsional transient loading using the distributed dislocation technique was studied. The problem was reduced to a set of singular integral equations with a Cauchy-type singular kernel by means of Laplace and complex Fourier transforms. The numerical solutions to integral equations result in the dislocation density function on a crack surface, thereby determining the SIFs for cracks. These integral equations were solved with the numerical method, and the SIFs at the interfacial crack tips were calculated. To study the effect of the nonhomogeneity constant, the FGM layer thickness and crack length as well as the interaction between cracks and DSIFs 12
were obtained for some examples. From the numerical results, the following key points were observed: (i) The DSIFs rise rapidly, and after reaching the main peak, they diminish gradually and tend to the static values. (ii) The DSIFs increase with increases in interface crack length. (iii) The magnitude of the DSIFs decreases with increases in the material gradient and FGM coating thickness. (iv) The interaction between cracks is an important factor affecting the DSIFs of crack tips. References [1] Karmini, M., Atrian, A., Ghassemi, A., and Vahabi, M., 2017. Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks. Theoretical and Applied Fracture Mechanics 90, pp.110–121. [2] Noroozi, M., Atrian, A., Ghassemi, A., and Vahabi, M., 2018. Torsion analysis of infinite hollow cylinders of functionally graded materials weakened by multiple axisymmetric cracks. Theoretical and Applied Fracture Mechanics, 96, pp.811–819. [3] Atrian, A., Jafari Fesharaki, J., and Nourbakhsh, S.H., 2015. Thermo-electromechanical behavior of functionally graded piezoelectric hollow cylinder under non-axisymmetric loads. Applied Mathematics and Mechaics (English Edition), 35(7), pp.939–954. [4] Delale, F. and Erdogan, F., 1988. On the mechanical modeling of the interfacial region in bonded half-planes. Journal of Applied Mechanics, 55(2), pp.317-324. [5] Ozturk, M. and Erdogan, F., 1995. An axisymmetric crack in bonded materials with a nonhomogeneous interfacial zone under torsion. Journal of Applied Mechanics, 62(1), pp.116-125. [6] Ozturk, M. and Erdogan, F., 1996. Axsiymmetric crack problem in bonded materials with a graded interfacial region. International Journal of Solids and Structures, 33(2), pp.193-219. [7] Xue-Li, H. and Duo, W., 1996. The crack problem of a fiber-matrix composite with a nonhomogeneous interfacial zone under torsional loading—Part I. A cylindrical crack in the interfacial zone. Engineering fracture mechanics, 54(1), pp.63-69. [8] Choi, H.J., Lee, K.Y. and Jin, T.E., 1998. Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone–Part I: Mechanical response. International Journal of Fracture, 94(2), pp.103-122. [9] Shbeeb, N.I. and Binienda, W.K., 1999. Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness. Engineering Fracture Mechanics, 64(6), pp.693-720. [10] Itou, S. and Shima, Y., 1999. Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials. International journal of solids and structures, 36(5), pp.697-709. [11] Itou, S., 2001. Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes. International Journal of Fracture, 110(2), pp.123-135. [12] Wang, Y.S., Huang, G.Y. and Dross, D., 2003. On the mechanical modeling of functionally graded interfacial zone with a Griffith crack: anti-plane deformation. Journal of Applied Mechanics, 70(5), pp.676-680. [13] Dhaliwal, R.S., Saxena, H.S., He, W. and Rokne, J.G., 1992. Stress intensity factor for the cylindrical interface crack between nonhomogeneous coaxial finite elastic cylinders. Engineering fracture mechanics, 43(6), pp.1039-1051. [14] Jin, Z.H. and Batra, R.C., 1996. Interface cracking between functionally graded coatings and a substrate under antiplane shear. International Journal of Engineering Science, 34(15), pp.1705-1716.
13
[15] Chen, Y.F. and Erdogan, F., 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids, 44(5), pp.771787. [16] Huang, G.Y., Wang, Y.S. and Gross, D., 2002. Fracture analysis of functionally graded coatings: antiplane deformation. European Journal of Mechanics-A/Solids, 21(3), pp.391-400. [17] Guo, L.C., Wu, L.Z., Ma, L. and Zeng, T., 2004. Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface-Part I: Static problem. International Journal of fracture, 127(1), pp.21-38. [18] Huang, G.Y., Wang, Y.S. and Yu, S.W., 2005. A new model for fracture analysis of functionally graded coatings under plane deformation. Mechanics of materials, 37(4), pp.507-516. [19] Chen, Y.J. and Chue, C.H., 2009. Mode III crack problems of two bonded functionally graded strips with internal cracks. International Journal of Solids and Structures, 46(2), pp.331-343. [20] Asadi, E., Fariborz, S.J. and Fotuhi, A.R., 2012. Anti-plane analysis of orthotropic strips with defects and imperfect FGM coating. European Journal of Mechanics-A/Solids, 34, pp.12-20. [21] Cheng, Z., Gao, D. and Zhong, Z., 2012. Interface crack of two dissimilar bonded functionally graded strips with arbitrary distributed properties under plane deformations. International Journal of Mechanical Sciences, 54(1), pp.287-293. [22] Li, Y.D., Zhao, H. and Xiong, T., 2013. The cylindrical interface crack in a layered tubular composite of finite thickness under torsion. European Journal of Mechanics-A/Solids, 39, pp.113-119. [23] Shi, P., 2016. Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion. Mechanics Based Design of Structures and Machines, 44(3), pp.250-269. [24] Ueda, S., Shindo, Y. and Atsumi, A., 1983. Torsional impact response of a penny-shaped crack lying on a bimaterial interface. Engineering Fracture Mechanics, 18(5), pp.1059-1066. [25] Itou, S., 2001. Transient dynamic stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes. International journal of solids and structures, 38(20), pp.3631-3645. [26] Li, C., Weng, G.J. and Duan, Z., 2001. Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. International Journal of solids and Structures, 38(42), pp.7473-7485. [27] Li, C. and Weng, G.J., 2001. Dynamic stress intensity factor of a cylindrical interface crack with a functionally graded interlayer. Mechanics of Materials, 33(6), pp.325-333. [28] Li, C. and Weng, G.J., 2002. Dynamic fracture analysis for a penny-shaped crack in an FGM interlayer between dissimilar half spaces. Mathematics and Mechanics of Solids, 7(2), pp.149-163. [29] Guo, L.C., Wu, L.Z., Zeng, T. and Ma, L., 2004. Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface-Part II: Transient problem. International Journal of Fracture, 127(1), pp.39-59. [30] Feng, W.J., Su, R.K.L. and Jiang, Z.Q., 2005. Torsional impact response of a cylindrical interface crack between a functionally graded interlayer and a homogeneous cylinder. Composite structures, 68(2), pp.203-209. [31] Lowengrub, M., 1969. Crack Problems in the Classical Theory of Elasticity. John Wiley & Sons. [32] Vafa, J.P. and Fariborz, S.J., 2016. Transient analysis of multiply interacting cracks in orthotropic layers. European Journal of Mechanics-A/Solids, 60, pp.254-276. [33] Erdogan, F., Gupta, G.D.A. and Cook, T.S., 1973. Numerical solution of singular integral equations. In Methods of analysis and solutions of crack problems (pp. 368-425). Springer Netherlands. [34] Vafa, J.P., Baghestani, A.M. and Fariborz, S.J., 2015. Transient screw dislocation in exponentially graded FG layers. Archive of Applied Mechanics, 85(1), pp.1-11.
14
[35] Demir, I. and Khraishi, T.A., 2005. The torsional dislocation loop and mode III cylindrical crack. Journal of Mechanics, 21(2), pp.109-116.
15
We solve problem of Somigliana-type dynamic rotational dislocation in FGM coated cylinder. We use the dislocation cylindrical interface cracks.
distribution
technique
for
analysis
Nonhomogeneity constant has an efficient effect on DSIFs. FGM layer thickness and crack length have also a crucial effect on DSIFs.
16
of
multiple