JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 6 Mode III Stress Singularity Analysis of Isotropic and Orthotropic Bi-material near the Interface End Junlin Li, Jing Zhao, Wenting Zhao, Caixia Ren, and Shuaishuai Hu School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, China Email: liunlin976@yahoo.com.cn, reputation5, zwt_0356, zrcx, hsshuai} @63.com Abstract The stress singularity near the Mode III interface end of isotropic and orthotropic bi-material was studied. Based on arbitrary angle boundary conditions, by solving a class of harmonic equations we discussed the root of the characteristic equation with the help of complex function method of material fracture. The expression and variation of the singularity index of the tip of interfacial crack and the symmetric interface end is obtained as well. The analytical expressions of the stress and displacements fields and the stress intensity factor around the interfacial crack are derived. Index Terms isotropic and orthotropic, bi-material, Mode III interface end, symmetric, asymmetric, singularity index, stress field I. INTRODUCTION Several methods of singularity analysis are available for the class of problems under consideration. The interfacial crack of an isotropic bi-material was studied, For Mode I, II, the stress field has oscillatory singularity. For Mode III the stress has singularity of r around the tip of interfacial crack without oscillatory []. An isotropic bi-material interface end was studied by the Mellin transform method and the stress field around the interface end shows the singularity []. However, based on the elastic equations and series expansion, the paper presents the general solution of anti-plane interface end. The research showed the relationship of singularity index and interface end angular [3]. A technique, applied to the bi-material wedge by Bogy [4] and others, employs a Mellin transform in the literature this last method is the only one to provide conditions for a logarithmic stress singularity. Used separation of variables on the Airy stress function, the usual determinant conditions for singularities of the form Or ( as r 0 are established and further conditions are derived for singularities of the form Or ( ln r as r 0. The order of the determinant involved in these conditions depends upon the number of materials comprising the wedge. Two systematic methods of expanding the determinant for the N material wedge are presented [5]. Orthotropic bi-materials anti-plane interface end of flap lap was studied by constructing new stress function and using composite complex function method of material fracture. The expression of stress fields, displacements fields and stress intensity factor around flat lap interface end are derived by solving a class of generalized bi-harmonic equations [6]. Two systems of non-homogeneous linear equations with 8 unknowns are obtained. By solving the above systems of non-homogeneous linear equations, the two real stress singularity exponents can be determined when the double material parameters meet certain conditions. The expression of the stress function and all coefficients are obtained based on the uniqueness theorem of limit [7]. In this article, by enoying the composite complex function method of material fracture, the singularity index of the tip of interfacial crack and the symmetric interface end were determined as well. The analytical expressions of the stress and displacement fields and the stress intensity factor around the interfacial crack are derived by translating the governing equation into the generalized harmonic equations. The corresponding results of asymmetric interface end can be obtained similarly. This paper is organized as follows. Section II reviews the Mechanical model. Section III gives the Stress function and then Section IV discusses the symmetric and asymmetric interface end, the analytical expressions of the stress and displacement fields are derived. Section V presents the conclusions. II. MECHANICAL MODEL Mode III interface end with arbitrary angle is shown in Fig.. The plane is bonded at x > 0 and y. The part of the plane with y > 0 is an isotropic material. Its material parameters are ( G3 ( G 3 µ. The part of the plane with y < 0 is an orthotropic material. The material parameters are( G 3 and ( G. 3 0 ACADEMY PUBLISHER doi:0.4304/cp.6.8.6-67
6 JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 Equation (4 has a couple of conugation imaginary roots. Choose the roots of the imaginary part to be greater than zero as follows: s iβ,(,. β ( Q ( Q 55, β (5 44 Figure. The model of an isotropic and orthotropic bi-material Mode III interface end with arbitrary angle From the theory of elasticity, the governing equations can be expressed as w w x y w w ( Q55 ( Q 44 x y Where (, ( w are the displacements. The expressions of boundary conditions for polar coordinates are, z z, w w, z (, z Based on the theory of elasticity, the formulae of stress components are given as w µ w, ( Q55 r r w z µ r, w z ( Q44 (3 r Where the displacements u v 0,,(,. ( G3 ( G 3 µ, ( Q ( G 44 3 µ,( G3 and w w ( x, y, r and are polar coordinates defined in Fig. 3, G are shear moduli III. STRESS FUNCTION ( Q ( G. 55 3 Assume the displacement w w ( x s y, (,, where w w ( x s y is an arbitrary complex function for a given.substituting w w ( x s y into (, we can obtain the characteristic equations s ( Q55 ( Q44 s (4 By the complex function method in the z plane, the governing equations can be translating into the harmonic equations w w x y (6 w w x y From (,the special stress functions are chosen as z z z z w(, r a b,,. i Where (7 z x sy r(cos ssin z x sy r(cos ssin, Let (, w r in (7 satisfy (.Substituting (7 and (8 into (3,we can get w µ r ( cos s sin µ r a ( cos s sin ( cos s sin 55 ( cos s sin w ( Q r b i ( Q r ( cos s sin ( cos s sin ( cos s sin b i 55 a ( cos s sin (8 (9 (0 0 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 63 w z µ r z ( cos sin ( sin cos µ r s s ( a ( cos ssin ( sin scos ( cos s sin ( sin s cos b ( cos ssin ( sin scos 44 i w ( Q r ( Q r ( cos s sin ( sin s cos 44 a ( ( cos ssin ( sin scos ( cos s sin ( sin s cos b i z r cos s sin,we define ( cos ssin ( sin scos As (7 and (8 have ( i cos s sin ρ e φ,where ρ cos β sin, tanφ β tan (, Substituting the boundary conditions into (9, (0 and (, ( respectively, we can obtain the fourth-order homogeneous linear system a, a, b, b as follows about a a µ b ( Q44 β b sin( a cos( b (3 ( sincosφ βcossinφ a ( βcoscosφ sinsinφ b Simplify (3, we can obtain cos( ( Q55 ( Q44 sin cosφ ( Q cos sinφ 55 ( Q ( Q 55 sin ( µ coscosφ 44 µ sin sinφ ] (4 In order to make the homogeneous linear system have nonzero solutions, the determinant of the coefficient must be zero. Form the theory of trigonometric functions, by introducing the angle ϕ sinϕ ( Q ( Q sin 55 44 55 55 44 ( Q cos ( Q ( Q sin We have the characteristic equation as follows: ( µ ( Q55 ( Q44 sin φ ϕ ( µ ( Q55 ( Q44 sin φ ϕ (5 (6 A. Interficial crack Mode III interfacial crack is shown in Fig.. Figure. The model of an isotropic and orthotropic bi-material Mode III interfacial crack Where, φ, arctan( β tan ϕ, Substituting these into the characteristic equation (6 we.thus sin k. ( k,, (7 If k is an even number, n,( n,, and k is an odd number, n,( n,, Then, For Mode III the stress has singularity of r around the tip of interfacial crack without oscillatory. have B. Symmetric right-angled Interface end Mode III interface end with right angle is shown in Fig. 3. cosϕ ( Q cos 55 ( Q cos ( Q ( Q sin 55 55 44 Figure 3. The model of an isotropic and orthotropic bi-material Mode III right-angled interface crack 0 ACADEMY PUBLISHER
64 JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 Where,, φ arctan( β tan, ϕ Substituting these into the characteristic equation (6 we obtain k. ( k,,, (8 From the analytical expressions of the stress field, we can obtain the stress fields have no singularity in this case. C. Symmetric convex and concave angled Interface end Mode III with symmetric convex and concave angle are shown in Fig. 4 and Fig. 5. Figure 4. The model of an isotropic and orthotropic bi-material Mode III convex angled interface crack Figure 5. The model of an isotropic and orthotropic bi-material Mode III concave angled interface crack Transpose and simplify (6, we have sin φ ϕ µ sin φ ϕ µ ( Q55 ( Q44 ( Q ( Q 55 44 (9 and discuss the following three cases (a µ ( Q55 ( Q44 > 0, considering in one period, the angle ( φ ϕ and ( φ ϕ are both in x-axis upper half plane or both in x-axis lower half plane. (b µ ( Q55 ( Q44, k ϕ, k,, φ µ Q Q < 0, considering in one period, (c ( 55 ( 44 the angle ( φ ϕ and φ ϕ are in x-axis upper half plane and x-axis lower half plane respectively. We discuss the former case (a for example 0 < ( φ ϕ <. 0 < ( φ ϕ < the solution is < < < ( <. < ( < the solution is < < φ ϕ φ ϕ 0 < ( φ ϕ < 3. < ( φ ϕ < 3 the solution is < < 4 4 Summarization above results, we have obtained 0< <, > 0. < <, < < 0. (0 ( ( (3 Thus, the stress fields of isotropic and orthotropic bimaterial Mode III symmetric convex angled interface end has no singularity, case when symmetric concave angled interface end the singularity still exist. From (6, taking two cases into consideration, the variation of the singularity index of symmetric concave angled interface end are presented. The results equates with (0. ( Case when the symmetric concave angle are 3 4 (data, 4 5 (data and 5 6 (data3 respectively, from Ref.[9], β.863 is chosen, the variation of singularity index with the ratio of material parameters µ ( Q55 ( Q44 is shown in Fig. 6, Fig. 7 and Fig. 8 respectively. Figure 6. Variation chart of with µ ( Q ( Q (data 55 44 0 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 65 Figure 7. Variation chart of with µ ( Q ( Q (data 55 44 Figure 0. Variation chart of with (data Figure 8. Variation chart of with µ ( Q ( Q (data 3 55 44 ( Based on Ref.[9],case when the ratio of material parameters are µ ( Q ( Q.756(data, 55 44 55 44 µ ( Q ( Q 4.360(data and µ ( Q ( Q 4.6634 55 44 (data3 respectively, β.863 is chosen, the variation of singularity index with the symmetric concave angle < < is shown in the Fig. 9, Fig. 0 and ( Fig.. That is, from Fig. 6, Fig. 7 and Fig. 8, case when the symmetric concave angle is determined, the stress fields singularity of the symmetric concave angled interface end is increscent with the increase of µ ( Q55 ( Q44. From Fig. 9, Fig. 0 and Fig., case when the ratio of material parameters is determined, the stress fields singularity of the symmetric concave angled interface end increases with the concave angle increases. Figure. Variation chart of with (data 3 D. Asymmetric Interface end From (6, singularity index, stress and displacements field and the stress intensity factor of Mode III isotropic and orthotropic bi-material asymmetric interface end such as the flat lap and others can be obtained similarly. An isotropic and orthotropic Mode III composite bimaterial of the flat lap interface end is shown in Fig. Figure. The model of an isotropic and orthotropic bi-material Mode III Flat lap interface crack Figure 9. Variation chart of with (data Where,, φ arctan( βtan(, ϕ Substituting these into the characteristic equation (6 we obtain ( Q55 ( Q44 µ arcsin (4 ( ( Q55 ( Q44 µ Thus, the variation of singularity index with the growth of µ ( Q55 ( Q44 is shown in the Fig. 3. 0 ACADEMY PUBLISHER
66 JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 Figure 3. Variation chart of with µ ( Q ( Q 55 44 As is shown in Fig. 3, this kind of problem has one singularity which is power singularity. The singularity index tends to as µ ( Q55 ( Q44 increases IV. STRESS AND DISPLACEMENT FIELDS AND THE STRESS INTENSITY FACTOR Taking the case of interfacial crack, the Mode III stress fields of isotropic and orthotropic bi-material are shown as follows, and other cases can be obtained similarly. From (6, if k is an even number, b and k is an odd number, a. τ µ cos a µ r sin b τ Q55 ρ cosφ a φ ( Q55 r ρ sin b z µ sin a µ r cos b z ( Q44 sin a β φ ( Q44 r ρ cos (5 β φ cos b The displacement fields are shown as w r, r cos a r sin b w( r, r ρcosφ a φ (6 r ρ sin b Define the stress intensity factor, τ z III r 0 lim r τ µ b, Q55 ρ b, III τ sin r III φ τ sin r III τ z cos r ( Q44 III Q ρ r 55 β φ β φ cos cos (, III r w r sin µ w, III r φ sin ( r ( Q (7 (8 55 V. CONCLUTIONS By the complex function method in the z plane, the stress Singularity near the Mode III symmetric and asymmetric interface end of isotropic and orthotropic bimaterial is studied. The results are as follows:(the stress has singularity of around the tip of interfacial r crack without oscillatory.(the singularity index of symmetric right-angled and convex-angled interface end is greater than zero, so the stress fields have no singularity in this case.(3 The value range of singularity index of symmetric concave angled interface end is from 0 to -. (4The corresponding conclusions of asymmetric interface end such as the flat lap and others can be obtained similarly.(5 Taking the case of interfacial crack, the analytical expressions of the stress and displacements fields and the stress intensity factor are derived. ACNOWLEDGMENT This work is supported by the Natural Science Foundation of Shanxi Province of China (No. 0070008; Ph.D. Science Foundation of Taiyuan University of Science and Technology (No. 008009. REFERENCES [] M. L. Williams, The Stresses around a Fault or Crack in Dissimilar Media, Bulletin of the Seismological Society of America, vol. 49, pp. 09-04, 959. 0 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 67 [] J. L. Bassani, F. Erdogan. Stress intensity factors in bonded half planes containing inclined cracks and subected to antiplane shear loading, Int. J. Fract, vol. 5, pp. 45-58, 979 [3] Bailin Zheng, Ying Dai, Xing Ji, and Yuangong Wang. The stress singularity analysis of an anti-plane problem near the interface end (in Chinese, Chinese Journal of Applied Mechanics, vol. 6, pp. -6, 999. [4] D. B. Bogy, Edge bonded dissimilar elastic wedges under normal and shear loading, J. Appl. Mech, vol. 35, pp.46-54, 968. [5] J. P. Dempsey, G. B. Sinclair, On the stress singularities in the plane elasticity of the composite wedge, Journal of Elasticity, vol. 9, pp. 373-39, 979. [6] Jinquan Xu, Interface Mechanics, Science Press, Beiing, pp. 0-07, 006. [7] Xuexia Zhang, Xiaochao Cui, Weiyang Yang and Junlin Li, Crack-tip field on mode II interface crack of double dissimilar orthotropic composite materials, Appl.Math.Mech.-Engl Ed, vol. 30, pp. 489-504, 009. [8] Ying Dai, Xing Ji, Analysis of Stress singularity and distribution of interface end(in Chinese, Science in China(Series G: Physics, Mechanics & Astronomy, vol. 37, pp. 535-543, 007. [9] Zhenming Wang, Mechanics and structural mechanics of composite materials, Beiing: Machinery Industry Press, vol., pp.9-7, 00. [0] C. C. Ma, B. L. Hour. Analysis of dissimilar anisotropic wedges subected to anti-plane shear deformation, International Journal of Solids, Vol.5, pp.95 350, 989. [] Junlin Li, Xiaoli Wang, Interface end stress field of antiplane of orthotropic bi-materials, Appl.Math.Mech.- Engl.Ed, Vol.30, pp.078-084, 009. [] D. B. Bogy, Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading, J. appl. Mech., Vol 35, pp.460-466, 968. [3] M. H. argarnovin, A. R. Shahani, S. J. Faribo, Analysis of an isotropic finite wedge under anti-plane deformation., Int J Solids Struct, vol. 34, pp. 3-8, 997. [4] T. C. T. Ting, Explicit solution and incariance of the singularities at an interface crack in anisotropic composites, Int J Solids Struct, vol. pp. 965-983, 986. [5] A. R. Shahani, A note on the paper Analysis of perfectly bonded wedges and bonded wedges with an interfacial crack under antiplane shear loading, Int J Solids Struct, vol. 38, pp. 504-5043, 00. [6] A. R. Shahani, Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials, Arch Appl Mech, vol. 75, pp. 405-4, 006. Li Junlin, Male, born in 963, Professor, He received his bachelor's degree from Shanxi Normal University in 984 and received his master s degree from Hunan University in 997 respectively. In 007, He graduated from School of Aeronautic Science and Technology, Beiing University of Aeronautics and Astronautics, obtained Ph.D. degree. And now, he works in Taiyuan University of Science and Technology as a professor, Mater supervisor and the director of Finance Department. His research interests include Interface crack fracture of composite materials, Theory and application of partial differential equations et al. He has published 3 books: []. Complex function method in Composite Material Fracture (Science Press, 005. []. Methods of Mathematical modeling and Contest (Weapon Industry Press, 006. [3].Probability Statistics and Modeling (Weapon Industry Press, 007, and lots of research papers in ournals and international conferences: [].Stress Field near Interface Crack Tip of Double Dissimilar Orthotropic Composite Materials (Applied Mathematics and Mechanics ISSN:053-487,008.8. [].Interface End Stress Field of Anti-plane of Orthotropic Bi-materials (Applied Mathematics and Mechanics ISSN:053-487,009.9. [3].An Analysis of Fracture Problems of Orthotropic Composite Plate under Bending and Twisting (ey Engineering Materials,006.3 Current research interests: Interface crack fracture of composite materials et al. Prof. Li is a chief scientific leader of Applied Mathematics, Mathematical Society executive director of the Shanxi Province. Zhao Jing is currently a master candidate at School of Applied Science Taiyuan University of Science and Technology, Taiyuan, China. Her research interests include Interface crack fracture of composite materials, Theory and application of partial differential equations et al. Zhao Wenting is currently a master candidate at School of Applied Science Taiyuan University of Science and Technology, Taiyuan, China. Her research interests include Interface crack fracture of composite materials et al. Ren Caixia is currently a master candidate at School of Applied Science Taiyuan University of Science and Technology, Taiyuan, China. Her research interests include Theory and application of partial differential equations et al. Hu Shuaishuai is currently a master candidate at School of Applied Science Taiyuan University of Science and Technology, Taiyuan, China. His research interests include Interface crack fracture of composite materials. 0 ACADEMY PUBLISHER