An internal crack problem in an infinite transversely isotropic elastic layer

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1 Int. J. Adv. Appl. Math. and Mech. 3() (05) 6 70 (ISSN: ) Journal homepage: International Journal of Advances in Applied Mathematics and Mechanics An internal crack prolem in an infinite transversely isotropic elastic layer Research Article Rajesh Patra a, S. P. Barik,, P. K. Chaudhuri c a Department of Mathematics, Hooghly Engineering & Technology College, Vivekananda Road, Hooghly, India Department of Mathematics, Goardanga Hindu College, 4-Parganas (N), India c Department of Applied Mathematics, University of Calcutta, 9, A. P. C. Road, Kolkata , India Received 04 June 05; accepted (in revised version) 0 August 05 Astract: MSC: This paper is concerned with an internal crack prolem in an infinite transversely isotropic elastic layer. The crack is opened y an internal uniform pressure p 0 along its surface. The layer surfaces are supposed to e acted on y symmetrically applied concentrated forces of magnitude P with respect to the centre of the crack. The applied concentrated force may e compressive or tensile in nature. The prolem is solved y using integral transform tech- nique. The solution of the prolem has een reduced to the solution of a Cauchy type singular integral equation, which requires numerical treatment. The integral equation is solved numerically y using Gauss-Cheychev and Gauss-Laguerre quadratures. The stress-intensity factors and the crack opening displacements are determined and the effects of anisotropy on oth cases are shown graphically. 74E0 74R0 Keywords: Transversely isotropic medium Fourier integral transform Singular integral equation Stress-intensity factor 05 The Author(s). This is an open access article under the CC BY-NC-ND license ( Introduction The study relating the ehaviour of elastic material under applied load needs special attention and care when the elastic ody develops a crack in it. It is ovious that the presence of a crack in a structure not only affects the stress distriution ut also drastically reduces the life span of the structure.propagation of elastic disturance in a solid is also distured y the presence of a crack. But cracks are present essentially in all structural materials, either as natural defects or as a result of farication processes. Stress distriution in a ody which develops a crack in it is entirely different from that in a ody without a crack. In literature, considerale effort has een devoted to the study of cracks in solids, due to their applications in industry in general and in farication of electronic components in particular. Presence of a crack in a solid significantly affects its response to the applied load. Stress distriutions in the solid with a crack are studied in two regions: the region in the neighourhood of crack, called the near field region and the region faraway from the crack, called the far field region. Study of stress distriution in the near field region is very important. Stress intensity factor, crack energy etc. are some of the quantities responsile for spreading of a crack. For a solid with a crack in it loaded mechanically or thermally determination of stress intensity factor (SIF) ecomes a very important topic in fracture mechanics. The SIF is a parameter that gives a measure of stress concentration around cracks and defects in a solid. SIF needs to e understood if we are to design fracture tolerant materials used in ridges, uildings, aircraft,or even ells. Polishing just won t do if we detect crack. Typically for most materials if a crack can e seen it is very close to the critical stress state predicted y the SIF. Corresponding author. address: patra_rajesh@yahoo.co.in (Rajesh Patra), sparik@gmail.com (S. P. Barik), pranay_chaudhuri@yahoo.co.in (P. K. Chaudhuri)

2 Rajesh Patra et al. / Int. J. Adv. Appl. Math. and Mech. 3() (05) Over the last few decades anisotropic materials have een increasingly used. There are materials which have natural anisotropy such as zinc, magnesium, sapphire, wood, some rocks and crystals, and also there are artificially manufactured materials such as fire-reinforced composite materials, which exhiit anisotropic character. The advantage of composite materials over the traditional materials lies on their valuale strength, elastic and other properties []. A reinforced material may e regarded to some order of approximation, as homogeneous and anisotropic elastic medium having a certain kind of elastic symmetry depending on the symmetry of reinforcement. Some glass fire reinforced plastics may e regarded as transversely isotropic. Thus solid mechanics prolems should not e restricted to the isotropic medium only. Increasing use of anisotropic media demands that the study should e extended to anisotropic medium also. Crack prolem in isotropic elastic medium have een extensively studied in literature following classical theory. A comprehensive list of work on crack prolems y earlier investigators has een provided in Chaudhuri and Ray [], Barik et. al.[3], Ozturk and Erdogan [4], Zhou and Hanson [5], Farikant [6], Dag and Erdogan [7], Sherief and El- Mahray [8], Chen et. al. [9], Lee [0], Brock[]etc. The present investigation aims at studying an internal crack prolem in infinite transversely isotropic elastic layer. Following the integral transform technique the prolem has een reduced to a prolem of Cauchy type singular integral equation, which has een solved numerically. Numerical computations have een done to assess the effects of anisotropy considered in the prolem on various sujects of interest and the results have een shown graphically.. Formulation of the prolem We consider an infinite transversely isotropic layer of thickness h weakened y an internal crack of length, which is opened y an uniform internal pressure p 0 along its surface. The layer is sujected to two different types of loadings on its surfaces in a direction perpendicular to its length a symmetric pair of compressive concentrated normal loads P, () a symmetric pair of tensile concentrated normal loads P (Figs. ()-()). It is assumed that the effect of the gravity force is neglected. The prolem is formulated in cartesian co-ordinate system (x, y) in which crack lies along x-axis with origin at the centre of the crack. Due to symmetry of the crack location with respect to the layer and also of the applied load with respect to the crack, it is sufficient to consider solution of the prolem in the region 0 x < and 0 y h. The strain displacement relations, linear stress-strain relations and equations of equilirium Concentrated compressive load condition () Concentrated tensile load condition Fig.. Geometry of the prolem (Ding et. al. 006) are, respectively, given y ε x = u x, ε y = v y, γ x y = u y + v x ; () σ x = C ε x +C ε y, σ y = C ε x +C ε y, τ x y = (C C )γ x y () σ x x + τ x y y = 0 (3) τ x y x + σ y y = 0 (4) The present prolem is mathematically equivalent to the solution of the following equations and oundary conditions:

3 64 An internal crack prolem in an infinite transversely isotropic elastic layer Equilirium equations: C u x + (C C ) u y + (C +C ) v x y = 0 (5) (C C ) v x +C v y + (C +C ) u x y = 0 (6) The oundary conditions: τ x y (x,0) = 0, < x < (7) τ x y (x,h) = 0, < x < (8) [ P σ y (x,h) = δ(x a) + P ] δ(x + a), < x < (9) { f (x), < x < ; x [v(x,0)] = 0, < x <. (0) σ y (x,0) = p 0, x () where u and v are the x and y components of the displacement vectors; σ x,σ y,τ x y are the normal and sharing stress components; C, C are transversely isotropic elastic coefficients; f (x) is an unknown function and δ(x) is the Dirac delta function. In Eq. (9) positive sign indicates tensile force while negative sign corresponds to compressive force. 3. Method of solution To solve the partial differential Eqs. (5) and (6), the Fourier transform is applied to the equations with respect to the variale x. The equations in the transformed domain can e written as (C C ) d u d y C ξ u iξ (C +C ) d v d y = 0 () C d v d y ξ (C C )v iξ (C +C ) du d y = 0 (3) where u and v are the Fourier transforms of u and v, respectively defined y (u, v) = (u, v)e iξx d x. Now, elimination of u from the Eqs. () and (3) yields a differential equation in v; d 4 v d y 4 d v ξ d y + ξ4 v = 0. (4) Eq. (4) is an ordinary differential equation in independent variale y, its general solution can e otained as v(ξ, y) = (A + B y)e ξy + (C + D y)e ξy (5) where A,B,C and D are the unknown constants. Eliminating derivatives of u from Eqs. ()-(3) and using Eq. (5) we have u(ξ, y) = [ { ξ(a + B y) + 3C C B}e ξy + {ξ(c + D y) + 3C ] C D}e ξy (6) iξ C +C C +C

4 Rajesh Patra et al. / Int. J. Adv. Appl. Math. and Mech. 3() (05) where i =. Application of Fourier inversion formula on the Eq. (5) and (6) we get v(x, y) = {(A + B y)e ξy + (C + D y)e ξy }e iξx dξ (7) π u(x, y) = i [ {ξ(a + B y) 3C C B}e ξy {ξ(c + D y) + 3C ] C D}e ξy e iξx dξ (8) π ξ C +C C +C Comination of Eqs. (), (), (7) and (8) yields σ x (x, y) = π σ y (x, y) = π [{ξ(c C )e ξy }A + {ξy(c C ) + C C +C 3C }e ξy B C +C + { ξ(c C )e ξy }C + { ξy(c C ) + C C +C 3C }e ξy D]e iξx dξ (9) C +C [{ ξ(c C )e ξy }A + { ξy(c C ) + C C +C +C }e ξy B C +C + {ξ(c C )e ξy }C + {ξy(c C ) + C C +C +C }e ξy D]e iξx dξ (0) C +C τ x y (x, y) = C C i π [{ ξe ξy }A + { ξy + 4C }e ξy B + { ξe ξy }C C +C + { ξy 4C }e ξy D]e iξx dξ. () C +C Utilizing of the oundary conditions (7)-(0), the unknown constants A, B,C and D can e found out and sustitution of these values into the equation () will lead to the following singular integral equation: [ ] f (t) π t x + k (x, t)]d t = C C C [ p 0 ± P π k (x), ( < x < ) () where ( + ξh + ξ h )e ξh e 4ξh k (x, t) = 0 + 4ξhe ξh e 4ξh si nξ(t x)dξ (3) and e ξh [ + ξh + ( + ξh)e ξh ] k (x) = 0 + 4ξhe ξh e 4ξh [cosξ(x a) + cosξ(x + a)]dξ. (4) The kernels k (x, t) and k (x) are ounded and continuous in the closed interval x. The integral equation must e solved under the following single-valuedness condition f (t)d t = 0. (5) Before further proceeding it will e convenient to introduce non-dimensional variales r and s y rescaling all lengths in the prolems y length scale : x = r, t = s (6) p 0 f (t) = f (s) = φ(s),ω = ξ. (7) C C In terms of non-dimensional variales the integral Eq. () and single valuedness condition (5) ecome π [ s r + k (r, s)]φ(s)d s = C [ ± Q C +C π k (r )], ( < r < ) (8) and φ(s)d s = 0 (9) where k (r, s) = [( + ξh + ξ h )e ξh e 4ξh ] + 4ξhe ξh e 4ξh si nω(s r ) dω, (30) 0 k (r ) = 0 e ξh [ + ξh + ( + ξh)e ξh ] + 4ξhe ξh e 4ξh {cosω(r a ) + cosω(r + a )} dω, (3) a = a and Q is the load ratio defined as: Q = P. (3) p 0

5 66 An internal crack prolem in an infinite transversely isotropic elastic layer 4. Solution of integral equations The singular integral Eq. (8) is a Cauchy-type singular integral equation for an unknown function φ(s). For the evaluation of displacement and stress components it is necessary to solve the integral Eq. (8) for the unknown function φ(s). Expressing now the solution of equation (8) in the form φ(s) = ψ(s), ( < s < ) (33) s where ψ(s) is a regular and ounded unknown function and using the Gauss-Cheyshev formula [] to evaluate the integral Eq. (8), we otain [ N [ ] ] + k N s k r (r j, s k ) ψ(s k ) = C [ ± Q j C +C π k (r j )], j =,,..., N (34) and N k= N ψ(s k ) = 0 (35) k= where s k and r j are given y ( ) k s k = cos N π,(k =,,3,..., N ) (36) ( ) j π r j = cos,(j =,,3,..., N ). (37) N j π We oserve that corresponding to (N ) collocation points x j = cos( (N + ) ), j =,,..,(N ) the Eqs. (34) and (35) represent a set of N linear equations in N unknowns ψ(s ),ψ(s ),...,ψ(s N ). This linear algeraic system of equations are solved numerically y utilizing Gaussian elimination method. 5. Determination of stress intensity factor Presence of a crack in a solid significantly affects the stress distriution compared to that when there is no crack. While the stress distriution in a solid with a crack in the region far away from the crack is not much distured, the stresses in the neighourhood of the crack tip assumes a very high magnitude. In order to predict whether the crack has a tendency to expand further, the stress intensity factor (SIF), a quantity of physical interest, has een defined in fracture mechanics. The load at which failure occurs is referred to as the fracture strength. The stress intensity factor is defined as k() = lim r (r )σ y (r,0). (38) Use of the Eqs. (7) and () and after some manipulation, the expression for k() is otained as k() p 0 = C +C C ψ() (39) where ψ() can e found out from ψ(s k ), (k =,,3,..., N ) using the interpolation formulas given y [3]. Following the method as in [4] we otain the crack surface displacement in the form x v(x,0) = f (t)d t, ( < x < ). (40) Using the Eqs. (6) and (7) into the Eq. (40), we can express the dimensionless normal displacement as v (r,0) = v(x,0)(c C ) r = φ(s)d s, ( < r < ) (4) p 0 which can e otained numerically, using Simpson s integration formula and the appropriate interpolation formula. 3 It has een verified that the results of [5] follow directly [with the exception of one / two printing errors / omissions] from our results y using the transformations: C = C = λ + µ,c = λ,c 33 = µ. 6. Numerical results and discussions: The present study is related to the study of an internal crack prolem in an infinite transversely isotropic elastic layer. In our present discussion we have considered the transversely isotropic material as Sapphire, Magnesium and Graphite to illustrate theoretical results. The numerical values of the elastic coefficients for the materials are listed in Tale [6].

6 Rajesh Patra et al. / Int. J. Adv. Appl. Math. and Mech. 3() (05) Tale. Basic data for three transversely isotropic materials. Quantity Unit Sapphire Magnesium Graphite c N m c N m () Fig.. Variation of normalized stress-intensity factor k () for various values of Q in the case of compressive concentrated forces (a/ = 0.50). () Variation of normalized stress-intensity factor k () for various values of Q in the case of tensile concentrated forces (a/ = 0.50). Utilizing the input parameters and following the standard numerical method descried in section 4 we compute the normal displacement component and the stress intensity factor and are shown graphically. Considering the elastic material of the layer as Sapphire, the variation of normalized stress intensity factor with are shown in Figs. -() h for oth the cases of two symmetric transverse pair of compressive and tensile concentrated forces. It is oserved from Fig. that for compressive concentrated forces the normalized stress-intensity factor k () decreases with the increase of the load ratio Q, and the increase of k () is quite significant for smaller values of Q. It is also oserved from Fig. that the load ratio Q has not much effect on the stress intensity factor k () when the crack length is sufficiently small. Fig. () represents the variations of k () with crack length under tensile nature of the concentrated force. Contrary to the previous case it is oserved that k () increases with Q. For small crack length, the ehaviour of k () is similar to the case of compressive concentrated load. Figs. 3-3() display the variation of normalized stress-intensity factor k () for different position of loading. () Fig. 3. Variation of normalized stress intensity factor k () for different values a/ in the case of compressive concentrated forces (Q = 0.5). () Variation of normalized stress intensity factor k () for different values of a/ in the case of tensile concentrated forces (Q = 0.5).

7 68 An internal crack prolem in an infinite transversely isotropic elastic layer () Fig. 4. Effect of anisotropy on normalized stress intensity factor k () in the case of compressive concentrated forces (a/ = 0.50,Q = 0.50). () Effect of anisotropy on normalized stress intensity factor k () in the case of tensile concentrated forces (a/ = 0.50,Q = 0.50). () Fig. 5. Variation of normalized crack surface displacement v (r,0) for various values of Q in the case of compressive concentrated forces (a/ = 0.5,/h = ). () Variation of normalized crack surface displacement v (r,0) for various values of Q in the case of tensile concentrated forces (a/ = 0.5,/h = ). () Fig. 6. Variation of normalized crack surface displacement v (r,0) for various values of a/ in the case of compressive concentrated forces (Q = 0.5,/h = ). () Variation of normalized crack surface displacement v (r,0) for various values of a/ in the case of tensile concentrated forces (Q = 0.5,/h = ).

8 Rajesh Patra et al. / Int. J. Adv. Appl. Math. and Mech. 3() (05) () Fig. 7. Effect of anisotropy on normalized crack surface displacement v (r,0) in the case of compressive concentrated forces (a/ = 0.50,Q = 0.5,/h = ). () Effect of anisotropy on normalized crack surface displacement v (r,0) in the case of tensile concentrated forces (a/ = 0.50,Q = 0.5, /h = ). Fig. 8. Transition of normalized crack surface displacement v (r,0) from anisotropic materials to isotropic one ( as C λ and C λ + µ ) in the case tensile forces It is noted that in the case of compressive concentrated forces, k () increases with increasing a, ut it decreases in the case of tensile concentrated forces. In Figs. 4-4() normalized stress intensity factor experiences the effect of anisotropy in the cases of two symmetric transverse pair of compressive and tensile concentrated forces for load ratio Q = 0.5. Figs. 5-5() depict the variation of normalized crack surface displacement v (r,0) with r for different values h of load ratio Q. It is clear from Fig. 5 that for two symmetric transverse pair of compressive concentrated forces the normalized crack surface displacement v (r,0) decreases as load ratio Q increases, ut for tensile concentrated forces the normalized crack surface displacement decreases as load ratio Q also decreases. For oth the cases of compressive and tensile concentrated forces the graphs show that the normalized crack surface displacement is symmetrical with respect to origin. Figs. 6-6() illustrate the role of the point of application of loading on the normalized crack surface displacement for a particular load ratio Q = 0.5 and =.00. It is oserved in Fig. 6 that for compressive concentrated h loading the normalized crack surface displacement increases with the increased values of a ut ehaviour is just opposite (Fig. 6()) for tensile concentrated loading. The effect of anisotropy on normalized crack surface displace-

9 70 An internal crack prolem in an infinite transversely isotropic elastic layer ment v (r,0) is oserved in Figs. 7-7() for oth the cases of compressive and tensile concentrated forces as a 0, =.00 and load ratio Q = 0.5. As expected it is oserved from Figs. 5-7,that the normalized crack surface displacement v (r,0) assumes its maximum magnitude near the origin. h It is to e noted that all physical quantities like stress intensity factor, crack surface displacement etc. in isotropic medium are the degenerate case of corresponding results otained for transversely isotropic media. For instance, Fig. 8 exhiits that variation of crack surface displacement in our discussion approaches the corresponding variation of crack surface displacement in Copper (with λ= 8 GPa; µ= 4 GPa), an isotropic medium. An important oservation may e availale from Fig.. Under the compressive load condition we find that with the increase of Q, there are critical crack lengths for which the stress intensity factor has zero value i,e. for which the stress magnitude σ y is finite at the crack tips. For example, from Fig., if Q = 0.9, the critical crack length is approximately 3.5. References [] S.G Lekhnitskii, Theory of Elastic of an Anisotropic Body, Mir Pulication, Moscow, 98. [] P.K Chaudhuri, S. Ray,Sudden twisting of a penny-shaped crack in a nonhomogeneous elastic medium,journal of Mathematical and Physical Science 9 (995) 07. [3] S.P Barik, M. Kanoria, P.K Chaudhuri,Stress distriution in the neighourhood of an external crack in a transversely isotropic medium, Indian Journal of Theoretical Physics 55 (007) [4] M.Ozturk, F. Erdogan, The axisymmetric crack prolem in a nonhomogeneous medium,journal of Applied Mechanics 60 (993) [5] Y. Zhou, M.T Hanson, A circular crack system in an infinite elastic medium under aritrary normal loads,journal of Applied Mechanics 6 (994) [6] V.I Farikant, Interaction of an aritrary force with a flexile punch or with an external circular crack,international Journal of Engineering Science 34 (996) [7] S. Dag, F. Erdogan, A surface crack in a graded medium under general loading conditions, Journal of Applied Mechanics 69 (00) [8] H.H Sherief, N.M El-Mahray, An internal penny-shaped crack in an infinite thermoelastic solid,journal of Thermal Stresses 6 (003) [9] W.Q Chen, H.J Ding, D.S Ling, Transient field of a transversely isotropic elastic medium containing a pennyshaped : exact fundamental solution, International Journal of Solids and Structures 4 (004) [0] D.S Lee, The prolem of internal cracks in an infinite strip having a circular hole,acta Mechanica 69 (004) 0 0. [] L.M. Brock, The coupled thermoelastic transversely isotropic imaterial:interface crack extension, Journal of Applied Mechanics 7 (005) [] F. Erdogan, G.D Gupta, On the numerical solution of singular integral equations, Quaterly Applied Mathematics 9 (97) [3] S. Krenk, A note on the use of the interpolation polynomial for solution of singular integral equations, Quaterly Applied Mathematics 3 (975) [4] G.D Gupta, F. Erdogan, The prolem of edge cracks in an infinite strip, Journal of Applied Mechanics 4 (974) [5] A. Birinci, F. Birinci, F.L Cakiroglu, R. Erdol, An internal crack prolem for an infinite elastic layer, Archive of Applied Mechanics 80 (00) [6] S.P Barik, M. Kanoria, P.K Chaudhuri, Effect of anisotropy on thermoelastic contact prolem,applied Mathematics and Mechanics 9 (008) [7] C.C. Ma, J.J. Luo, Plane solution of interface cracks in anisotropic dissimilar media, Journal of Engineering Mechaics (996) [8] H. Ding, W. Chen, L. Zhang, Elasticity of Transversely Isotropic Materials, Springer, Netherlands, 006.