Research Article Oscillatory Singularity Behaviors Near Interface Crack Tip for Mode II of Orthotropic Bimaterial

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1 pplied Mathematics Volume, rticle ID, 9 pages Research rticle Oscillatory Singularity ehaviors Near Interface rack Tip for Mode II of Orthotropic imaterial Xiaomei Yang, Weiyang Yang, Junlin Li, and Xuexia Zhang School of pplied Science, Taiyuan University of Science and Technology, Taiyuan, hina orrespondence should be addressed to Xiaomei Yang; Received December ; Revised May ; ccepted May cademic Editor: Ray K. L. Su opyright Xiaomei Yang et al. This is an open access article distributed under the reative ommons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The fracture behaviors near the interface crack tip for mode II of orthotropic bimaterial are discussed. The oscillatory singularity fields are researched. The stress functions are chosen which contain twelve undetermined coefficients and an unknown singularity exponent. ased on the boundary conditions and linear independence, the system of twelve nonhomogeneous linear equations is derived. ccording to the condition for the system of nonhomogeneous linear equations which has a solution, the singularity exponent is determined. Total coefficients are found by means of successive elimination of the unknowns. The theoretical formulae of stress intensity factors and analytic solutions of stress field near the interface crack tip are obtained. The crack tip field is shown by figures.. Introduction Many researchers have studied the singularity behavior near interface crack tip for isotropic, orthotropic, or anisotropic bimaterial. The method of eigenfunction expansions is used by Williams []. The complex function method is developed by Rice and Sih []. Erdogan []andengland[], present and research various interface crack problems. These academic authorities play a leading role in fracture mechanics for interface crack, and they have made the tremendous contribution. The subsequent papers are published one after another over several decades [ ]. In this paper, the solution method is proposed to research the singularity behavior near interface crack tip for mode II of orthotropic bimaterial. It can be seen that the following differences exist between this method and previous methods by comparison. () The stress function contains twelve undetermined coefficients, rather than eight. () The system of twelve nonhomogeneous linear equations is deduced based on the boundary conditions, rather than the system of eight homogeneous linear equations. () The characteristic equation is found by using the condition for the system of nonhomogeneous linear equations that possess a solution, rather than being basedontheconditionforthesystemofhomogeneous linear equations which has a nontrivial solution. () In order to determine total coefficients, we only need to solve the system of nonhomogeneous linear equations, rather than to solve the system of homogeneous linearequationsfirst,andthenusetheloadconditions at infinity. () The stress intensity factors are defined by right-hand limit and left-hand limit rather than by limit. () The oscillatory singularity fields near interface crack tip of three orthotropic bimaterial are illustrated by the help of two tables and seven figures.. Mechanical Model The plane y>is the upper orthotropic material (j = ), and its elastic constants are E,E, ] and μ.theplane y<is the lower orthotropic material (j = ),anditselastic constants are E,E, ],andμ. y=, x < a is the crack surfaces, and y=, x > a isthebondedinterfaceoftwo dissimilar materials.

2 pplied Mathematics τ y E,E,,μ y h / Upper material h / a O r a θ M x Inter -face rack O rack Inter -face x E,E,,μ Lower material h / h / z (a) Geometric state τ (b) Shear load Figure : Interface crack for mode II of orthotropic bimaterial. The stress functions U j (x, y), (j =, ) satisfy the governing equations [ ]:. Stress Function The relationships between the stresses and the stress functions can be obtained as (b ) j U j x +[(b ) j +(b ) j ] U j x y +(b ) j U j y =, (j =, ). () (σ x ) j = U j y = k= Re [( jk,λ +i jk,λ ) i( jk,λ +i jk,λ )] The boundary conditions of the interface crack for mode II are as follows: y=, x <a:(σ y ) =(σ y ) =, (τ xy ) =(τ xy ) =, y=, x >a:(σ y ) =(σ y ), (τ xy ) =(τ xy ), () (σ y ) j = U j x = k= ( β jk )U jk,λ (z jk )}, Re [( jk,λ +i jk,λ ) i ( jk,λ +i jk,λ )] U jk,λ (z jk )}, (a) (b) (u) = (u), (V) = (V), () y + : (σ y) =(σ y ) =, (τ xy ) =(τ xy ) =τ. () (τ xy ) j = U j x y = k= Re [( jk,λ +i jk,λ ) i( jk,λ +i jk,λ )] The geometric and load conditions for the investigated problem are given as shown in Figure. (iβ jk )U jk,λ (z jk )}, (j =, ) (c)

3 pplied Mathematics in which the stress functions U j of right side of (a) (c) contain twelve undetermined coefficients jk,λ, jk,λ, jk,λ (j,k=,)and an unknown singularity exponent λ. onsidering the boundary conditions (), (), and (), the functions U jk,λ (z jk ) of right side of (a) (c) canbechosen as λ U jk,λ (z jk )=τ( a zjk ) in which z λ jk =τ, (j, k =, ). (zjk λ a ) () z jk =x+s jk y=x+iβ jk y, (j,k=,) () z jk a=x a+iβ jk y=r(cos θ+iβ jk sin θ), (j,k=,). From (a) (c) and (), an unknown number λ is called singularity exponent, and sometimes it is also called eigenvalue.. Singularity Exponent If the singularity exponent λ is the complex number, the stress near the interface crack tip shows oscillatory singularity. We let [ ] () λ= +iε. (9) Substituting (9)into(),itcanbefoundas (/)+iε U jk,ε (z jk )=τ( a zjk ) z iε jk =τ, (j, k =, ). (zjk (/) iε a ) () Equations () and() aresubstitutedinto(), and we have επ τx U jk,ε (z jk ) y=, x <a =e( )j a x ( )j [sin (ε ln a x ) x i cos (ε ln a x )]. x () at the crack surfaces. Substituting (), (b), and (c)intotheboundarycondition (), by the help of the functions cos(ε ln(( a x )/x)) and sin(ε ln(( a x )/x)) which are linearly independent, the four homogeneous linear equations can be deduced as,ε +,ε,ε,ε +,ε +,ε =, (a),ε,ε +,ε +,ε,ε,ε =, β,ε β,ε β,ε β,ε =, (b) (c) β,ε +β,ε +β,ε +β,ε =. (d) Similarly, we can find other eight linear equations (e) to (l) which are omitted here. y (τ xy ) = (τ xy ) = τ in condition (), two nonhomogeneous linear equations are contained in the eight linear equations. The system of nonhomogeneous linear equations (a) to (l) is solved by eliminating the unknowns. To let the rank of the coefficient matrix λ for the system (a) (l) and the rank of the augmented matrix λ =( λ,b II ) be equal,therankmustbe;thatis,rank( λ )=rank( λ )=. For this reason [], the number ε must satisfy the following characteristic equation: (e f )f g h ε = () in which bimaterial parameters are given as e =, f μ μ = β β ] β β ], E E (a) g = β +β E + β +β E, h = β +β E β β + β +β E β β. (b) The singularity exponent λ of (9) isacomplexnumber, so ε =in (), and e =,f =,g >,h >in (a) and (b) for determinate bimaterial. To the equation () has a solution, the discrimination rule can be deduced as ( ) D : if e and f in (a)satisfye >and f >, then thenegativerootof()isthebielasticconstant ε. Substituting ε solved by ()into(9), the complex singularity exponent λ canbeobtained. The system of eight equations which remained through sequence elimination is solved by means of the inverse sequence backsubstitution. We found all coefficients as follows: jk,ε + jk,ε = ( ) k f +β jk g ε, (a) (β j β j )f jk,ε jk,ε = ( ) k f β jk g ε (β j β j )f, (j, k =, ). (b) in which k=, k =; k=, k =.y(a) and(b), the stress functions U j of the right side of (a) (c) contain practically eight coefficients.

4 pplied Mathematics. Stress Intensity Factors onsidering the stress expressions (b)and(c)and also the load condition in (), the stress intensity factors are defined as (k + ) j = lim Re (π z jk a + z jk a )(/) iε k= [( jk,ε + jk,ε ) i ( jk,ε jk,ε )] (iβ jk )U jk,ε (z jk )}, (k ) j = lim Re (π z jk a z jk a )(/) iε k= k = k+ k = [( jk,ε + jk,ε ) i ( jk,ε jk,ε )] ( ) j (ε i) (iβ jk ) (a) U jk,ε (z jk ) }, (j =, ), (b) j= (k + ) j j= (k + ) j = lim Re (π z k= jk a + z jk a )(/) iε (k ) j, (c) [( jk,ε + jk,ε ) i( jk,ε jk,ε )] U jk,ε (z jk )}, (k ) j = lim Re (π z jk a z jk a )(/) iε k= k = k+ +k (a) [( jk,ε + jk,ε ) i( jk,ε jk,ε )] ( ) j (ε i) U jk,ε (z jk )}, = j= (k + ) j + j= (j =, ), (b) (k ) j, (c) k=k +ik. () In order to express the change process of z jk a completely, it is necessary that the factor of (): e ( )jεπ ( ) j [sin(ε ln(( a x )/x)) i cos(ε ln(( a x )/x))] = ( ) j (ε i),(ε ) is substituted into the expressions (b)and(b). t the same time, the minus is used in (c)becausetheloadconditionis given in (). Substituting (), (), (), (a), and (b)into(a) (c), (a) (c), and (), the stress intensity factors of interface crack for mode II can be deduced as k =τ(πa) / p (ε), k = τ(πa) / g ε q (ε), f (9) k=k +ik in which p (ε) = [+(+ε) chεπ] cos (ε ln πa) [+( ε) chεπ] sin (ε ln πa)}, q (ε) = [ (+ε) chεπ] cos (ε ln πa) [ ( ε) chεπ] sin (ε ln πa)}.. Stress From (), (), and (),itcanbefoundasfollowsthat lim [π (z z jk a jk a)] (/) iε U jk,ε (z jk )} = τ(πa) (/) iε, in the vicinity of the crack tip. y (), it can be known that U jk,ε (z jk )= τ(πa)(/) iε (π) (/) iε (z jk a) (/) iε, (z jk a; j, k =, ). (a) (b) (j, k =, ) () () Substituting (), (a), (b), (9), and ()into(a) (c), the stresses in the vicinity of the interface crack tip (z jk a, r ; j, k =, ) formodeiioforthotropicbimaterial can be expressed as follows: (σ x ) j = k (πr) / (β j β j )p(ε) [cos ( π εln πa) cos (ε ln πr) sin ( π εln πa) sin (ε ln πr)]

5 pplied Mathematics + β j [ Re [ (cos θ+iβ j sin θ) (/) iε β j ] (cos θ+iβ j sin θ) (/) iε ] [cos ( π +εln πa) cos (ε ln πr) + sin ( π +εln πa) sin (ε ln πr)] β j [ Im [ (cos θ+iβ j sin θ) (/) iε k (πr) / β j (cos θ+iβ j sin θ) (/) iε } ]} β jβ j (β j β j )q(ε) [cos ( π +εln πa) cos (ε ln πr) + sin ( π +εln πa) sin (ε ln πr)] β j [ Re [ (cos θ+iβ j sin θ) (/) iε β j ] (cos θ+iβ j sin θ) (/) iε ] +[cos ( π εln πa) cos (ε ln πr) sin ( π εln πa) sin (ε ln πr)] sin ( π εln πa) sin (ε ln πr)] [ Re [ (cos θ+iβ j sin θ) (/) iε ] (cos θ+iβ j sin θ) (/) iε ] [cos ( π +εln πa) cos (ε ln πr) + sin ( π +εln πa) sin (ε ln πr)] [ Im [ (cos θ+iβ j sin θ) (/) iε + (cos θ+iβ j sin θ) (/) iε } k (πr) / (β j β j )q(ε) [cos ( π +εln πa) cos (ε ln πr) + sin ( π +εln πa) sin (ε ln πr)] β j [ Re [ (cos θ+iβ j sin θ) (/) iε β j ] (cos θ+iβ j sin θ) (/) iε ] +[cos ( π εln πa) cos (ε ln πr) (σ y ) j = β j [ Im [ (cos θ+iβ j sin θ) (/) iε k (πr) / β j (cos θ+iβ j sin θ) (/) iε }, ]} (β j β j )p(ε) [cos ( π εln πa) cos (ε ln πr) (a) (τ xy ) j = sin ( π εln πa) sin (ε ln πr)] β j [ Im [ (cos θ+iβ j sin θ) (/) iε β j (cos θ+iβ j sin θ) (/) iε }, ]} (b) k (πr) / (β j β j )p(ε)

6 pplied Mathematics [cos ( π +εln πa) cos (ε ln πr) + + sin ( π +εln πa) sin (ε ln πr)] β j [ Re [ (cos θ+iβ j sin θ) (/) iε β j ] (cos θ+iβ j sin θ) (/) iε ] +[cos ( π εln πa) cos (ε ln πr) sin ( π εln πa) sin (ε ln πr)] β j [ Im [ (cos θ+iβ j sin θ) (/) iε k (πr) / β j (cos θ+iβ j sin θ) (/) iε } β jβ j (β j β j )q(ε) [cos ( π εln πa) cos (ε ln πr) sin ( π εln πa) sin (ε ln πr)] [ Re [ (cos θ+iβ j sin θ) (/) iε ] (cos θ+iβ j sin θ) (/) iε ] [cos ( π +εln πa) cos (ε ln πr) + sin ( π +εln πa) sin (ε ln πr)] [ Im [ (cos θ+iβ j sin θ) (/) iε (cos θ+iβ j sin θ) (/) iε } ]} (c) in which p(ε) and q(ε) are given by (a)and(b). Table : Mechanical properties of each orthotropic material. Materials E j / GPa E j / GPa ] j μ j / GPa Δ j β j β j j= j= j= j= j= j= Table : Mechanical properties of three orthotropic bimaterials. imaterial e f g h ε λ i i i Note that (a) (c) of the stress field contain the following terms: (σ x ) j ((σ y ) (ε ln πr) j ) (cos ). () / (πr) sin (ε ln πr) (τ xy ) j The right side of () tends to change the sign rapidly at the crack tip (r ), and therefore the stresses possess an oscillatory character.. Oscillatory Field The test and calculus results [, ] for the mechanicals properties of three orthotropic bimaterial are shown in Tables and. From (9), the normalized stress intensity factors k /τ and k /τ depend on the length of crack a and the bielastic constant ε. The variations of the stress intensity factors versus crack length are plotted in Figure. Thevariationsofthe stress intensity factors with bielastic constant are illustrated in Figure. The minute variations of factor k /τ can be observed by the inner small figure of Figure. The factors k /τ and k /τ increase almost linearly when the length a or the constant ε increasesasshowninfigures and. y (a) (c), the normalized stresses σ x /τ, σ y /τ, and τ xy /τ depend on the polar angle θ and the polar radius r. The variations of the normalized stresses for the above three bimaterials with respect to polar angle θ are plotted in Figure. Figure shows that three stresses have alternately the increase and decrease with θ increase, their maximum values can be always reached at both sides of the crack. Figures and show the variations of the normalized stresses for the above three bimaterials with respect to polar radius r for θ = ± and θ = ±.Twofiguresmean that the stresses σ x /τ, σ y /τ, andτ xy /τ are the monotonic decreasing functions of on the two half-planes.

7 pplied Mathematics. k i /τ... k /τ k /τ ε =. ε =. ε =.9 Figure : Normalized stress intensity factors as a function of a.. a σ x /τ =. =. θ (deg) σy/τ τ xy /τ =. θ (deg) θ (deg) Figure : Normalized stresses versus θ for =.. k i /τ.. k /τ k /τ k /τ ε(a = ) σ x /τ θ=± σy/τ θ=± τ xy /τ θ=±.... ε(a = ).. Figure : Normalized stress intensity factors as a function of ε when a=. Figures and show the variations of the normalized logarithmic stresses with respect to the polar radius r for θ= ± and θ=±. It can be found that the stress distribution in Figures and is not straight line and does not parallel to each other. Such distributions mean oscillatory singularity state.. onclusion From the above derivation, the following results are very significant. () New stress functions are chosen. () The system of twelve nonhomogeneous linear equations is derived Figure : Normalized stresses versus r for θ=±. () The characteristic equation can be given. () y the help of the distinction rule ( ), the complex singularity exponent λ can be found. () Total coefficients are determined. () For the oscillatory field, the theoretical formulae of stress intensity factors and the analytic solutions for stresses near interface crack tip of mode II are

8 pplied Mathematics 9 9 σ x /τ θ=± σy/τ θ=± τ xy /τ θ=± log( σ x /τ) θ=± log( σ y /τ) θ=± log( τ xy /τ) θ=± Figure : Normalized stresses versus r for θ=± Figure : Normalized stresses versus r for θ=±. log( σ x /τ) 9 θ=± log( σ y /τ)..... θ=± log( τ xy /τ) 9 θ=± Figure : Normalized stresses versus r for θ=±. yielded, respectively. The stress intensity factors and the stresses show mixed crack characteristics. () The crack tip fields for oscillatory singularities are illustrated in Figures to. cknowledgment This work was supported by the Natural Science Foundation of Shanxi province (no. -), and the doctoral fund of Taiyuan University of Science and Technology (no. ). References [] M. L. Williams, The stresses around a fault or crack in dissimilar media, ulletin of the Seismological Society of merica,vol. 9,no.,pp.99,99. [] J. R. Rice and G.. Sih, Plane problems of cracks in dissimilar media, pplied Mechanics,vol.,no.,pp., 9. [] F. Erdogan, Stress distribution in bonded dissimilar materials with cracks, pplied Mechanics, vol.,no.,pp., 9. []. H. England, crack between dissimilar media, pplied Mechanics,vol.,no.,pp.,9. [] K. Y. Lin and J. W. Mar, Finite element analysis of stress intensity factors for cracks at a bi-material interface, International Fracture,vol.,no.,pp.,9. [] Z. Suo, Singularities, interfaces and cracks in dissimilar anisotropic media, Proceedings of the Royal Society of London Series,vol.,no.,pp.,99. [] J. hang and J.-Q. Xu, The singular stress field and stress intensity factors of a crack terminating at a bimaterial interface, International Mechanical Sciences, vol.9,no.,pp. 9,. [] D. Ying and J. Xing, Researches on stress singulality of interface end and distributive law of interface stress, Science in hina, G Series,vol.,no.,pp.,(hinese). [9] X. S. Zhang, central crack at the interface between two different orthotropic media for the mode I and mode II, Engineering Fracture Mechanics,vol.,no.,pp.,99. [] Z. Suo and J. W. Hutchinson, Interface crack between two elastic layers, International Fracture, vol., no., pp., 99. [] H. Gao, M. bbudi, and D. M. arnett, Interfacial crack-tip field in anisotropic elastic solids, the Mechanics and Physics of Solids, vol., no., pp. 9, 99.

9 pplied Mathematics 9 [] F. Erdogan and. Wu, Interface crack problems in layered orthotropic materials, the Mechanics and Physics of Solids,vol.,no.,pp.9 9,99. [] K.-P. Ma and.-t. Liu, Semi-weight function method on computation of stress intensity factors in dissimilar materials, pplied Mathematics and Mechanics, vol., no., pp.,. [] L. Marsavina and T. Sadowski, Stress intensity factors for an interface kinked crack in a bi-material plate loaded normal to the interface, International Fracture, vol., no., pp.,. [] Y. hen, P. Z. Qiao, H. D. Jiang, and Q. W. Ren, Review on experimental methods and fracture models for bimaterial interfaces, dvances in Mechanics, vol., no., pp., (hinese). [] G.. Sih and E. P. hen, racks in omposite Materials, vol. of Mechanics of Fracture, Martinus Nijhoff, The Hague, The Netherlands, 9. [] G.. Sih and H. Liebowitz, Mathematical theories of brittle fracture, in Fracture, H.Liebowitz,Ed.,vol.,pp.9, cademic Press, New York, NY, US, 9. [] H. T. orten, Fracture mechanics of composites, in Fracture, H. Liebowitz, Ed., vol., pp. 9, cademic Press, New York, NY, US, 9. [9] S. G. Lekhnitskii, TheoryofElasticityofannisotropicElastic ody, Science Press, eijing, hina, 9. [] W.-y. Yang, J.-l. Li, and X.-x. Zhang, Method of a omplex variable for fracture in omposite Materials, Science Press, eijing, hina,. [] Department of mathematics, eijing University, dvanced lgebra, dvanced Education Press, eijing, hina,. [] S. Q. Zhang and W.-Y. Yang, Prediction of mode I crack propagation direction in carbon-fiber reinforced composite plate, pplied Mathematics and Mechanics, vol., no., pp.,. [] J.-L.Li,G.-L.Feng,.-.hen,andJ.Zhang, pplicationof complex method in the fracture problem of bimaterial, Journal of North University of hina, vol.,no.,pp., (hinese). [] J.-l. Li and X.-l. Wang, Interface end stress field of antiplane of orthotropic bimaterials, pplied Mathematics and Mechanics, vol., no. 9, pp. 9, 9.

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