CRACK TIP STRESS FIELDS FOR ANISOTROPIC MATERIALS WITH CUBIC SYMMETRY

Transcription

1 CRACK TIP TRE FIELD FOR ANIOTROPIC MATERIAL WITH CUBIC YMMETRY D.E. Lempidaki, N.P. O Dowd, E.P. Buo Department of Mechanical Engineering, Imperial College London, outh Kenington Campu, London, W7 AZ United Kingdom. {d.lempidaki, n.odowd, e.buo}@imperial.ac.uk Abtract Fracture mechanic of linear elatic material i generally baed on the K field which ha been derived for iotropic material. Many application require the ue of advanced material, which are often aniotropic and thu the iotropic elatic K-field i not applicable. In thi work, a harp crack lying in a homogenou, aniotropic material with cubic ymmetry i tudied and crack tip tre field are preented. It i hown that the crack tip field depend on material propertie through the aniotropy factor, ρ. The tre field are applicable for both plane tre and plane train condition, though the definition of ρ i different in each cae. The theoretical K field obtained ha been compared to reult from finite element tudie and excellent agreement ha been obtained. Introduction The majority of olution for crack tip field addre the problem of a crack within an iotropic material and are baed on the iotropic K-field []. In thi work, crack tip field for material with cubic ymmetry only are examined. The motivation for the work arie from the increaing ue of ingle crytal nickel alloy (which have elatic cubic ymmetry) in ga turbine blade. In [], ih et al. invetigated the tre field in the vicinity of a harp crack lying within an aniotropic olid. Following the approach preented in [3] the elatic crack tip field were obtained and it wa hown that the quare root crack tip ingularity i preent in the aniotropic cae. More recently, tre field for interface crack in linear elatic aniotropic bimaterial have been examined in [], [5]. Fourier tranform were ued in [], [5] to obtain the full crack tip field of a finite crack lying along the interface between aniotropic elatic media. More recently, crack tip field around kinked crack in aniotropic elatic olid were obtained [6], [7]. In [6], the troh formalim i ued to derive crack tip field, in aniotropic elatic olid, including elatic T-tree and other coefficient of the higher-order term. Numerical reult for variou kink angle and mode mixitie, were preented. The problem of an inclined crack in an orthotropic medium wa tudied in [8] and fully analytical olution for crack tip field in orthotropic material have been obtained uing complex potential. One of the drawback of thee earlier work i that the precie form of the crack tip tre ditribution ha not been provided, o analytical tudie of crack in aniotropic

2 material cannot be baed directly on thee work. Here, our attention i directed to aniotropic material with cubic ymmetry. The approach adopted in [] i ued to determine the tre field in the vicinity of a harp crack in uch a material. olution are provided for Mode I loading. The emi-analytical reult are compared to thoe obtained from a finite element analyi of a crack in an infinite plate under tenile loading. Crack Tip tre Field The generalied Hooke' law for aniotropic material can be expreed a: ε ij = ijkl kl, ij Cijklε kl =, () where ε ij and ij are the coefficient of the tre and train tenor repectively and iklj and C ijkl are the coefficient of the fourth order tenor (the compliance and tiffne matrix repectively) which contain the material contant for an elatic body. A material with cubic ymmetry ha four-fold rotational ymmetry. For uch a material there are three independent material propertie and Eq. ha the following matrix form: ε ε ε 33 = ε 3 ε 3 ε where [] = ; E (3) ν = ; E [] = G Here, axe are deignated a [], [] and [] correponding to x, y and z. In Eq. (3), E [] repreent the tenile modulu in the [] direction for the cubic material, G [] the hear modulu in the [] direction and ν i the ratio between normal, ε, and tranvere train, ε (Poion ratio). The plane problem for an infinitely harp crack i illutrated in Fig.. Auming that tre gradient in the out of plane (z) direction are negligible, the out of plane hear tre, xz and yz are zero and the tree depend only on x and y. For the plane tre/plane train problem examined here, Eq. () can then be rewritten a, [] () ε ε γ xx yy xy = where γ xy = ε xy and xx yy xy, ()

3 ij plane tre = 3 ij i j3 (5) ij plane train 33 ([]) FIGURE. Plane crack problem. ([]) The general aniotropic problem in two dimenion ha been tudied in [9]. Here, we pecialie to the cae of material having cubic ymmetry. A dicued in [], the governing differential equation for the plane problem of a cubic material can be written in term of a tre function, U(x, y), uch that, U U + ( + ) + =. (6) x x y The characteritic equation of the differential equation, Eq. (6), i then µ + + ) µ + (7) ( = and the problem reduce to finding the root of Eq. 7. ubtituting the value of ij from Eq. 3 and leaving out the ubcript [] for implicity, one obtain for a cubic material under plane tre condition, E µ + ν µ + =. G By introducing a factor ρ [], + 66 ρ = (9) and defining ij a in Eq. 5, Eq. 8 can be written for plane tre or plane train a, µ + (ρ) µ + =. () Equation ha four ditinct complex root, which may be written a, (8)

4 u, = x + iy u = x + iy, u =, u = u () where u i indicate the conjugate of u i. 3 u Having determined the root of the characteritic equation, crack tip tree may be determined for Mode I loading (following [9]) a, K = I uu u u Re πr u u coθ + u inθ coθ + u inθ xx, K = I u u Re πr u u coθ + u inθ coθ + u inθ, () yy = K I uu Re πr u u inθ inθ xy. coθ + u coθ + u In Eq. 3, Re repreent the real part of the complex number and K I i the tre intenity factor under Mode I loading. It i een in Eq. that a quare root ingularity i till preent for the cae of a cubic aniotropic elatic body. However, the angular function in Eq. depend on the material propertie through the characteritic equation, Eq.. The angular function in Eq. cannot be plotted directly a they depend implicitly on ρ through Eq.. Therefore, the characteritic equation, Eq., i firt olved analytically for a given value of ρ and the Real part of the olution in Eq. determined numerically (in thi work the oftware package Mathematica [] wa ued for thi purpoe). In thi way, a emi-analytical olution for the crack tip field i obtained. Note that the angular ditribution i the ame for plane tre and train, though the value of ρ will depend on whether plane tre or plane train condition are aumed. u REULT The angular ditribution for Mode I loading are hown in Fig. a a function of the crack tip angle θ. Finite element (FE) olution are alo provided in thi figure, which will be dicued in the next ection. Reult are provided for ρ value varying from.5 to.. It can be een in Fig. (b) and (c) that the hoop and hear tre, θθ and r θ, are relatively independent of the aniotropy parameter ρ, in contrat to rr which i trongly dependent on ρ a hown in Fig. (a) (Tabulated olution for variou value of ρ and for Mode I and Mode II are given in []). For the iotropic cae, correponding to ρ =, Eq. ha equal root and the olution obtained from Eq. and ocillate. Therefore, the dotted line for ρ = in Fig. repreent the iotropic Mode I K-field []. (Note that for ρ =.999 there wa no ocillation of the emi-analytical olution and the reult i inditinguihable from the iotropic K-field.)

5 FIGURE. Analytical olution (line) and finite element olution (ymbol) for angular function of a cubic material under mode I loading. Uing Eq. and, the angular variation of the tree have been plotted for variou value of ρ and compared with the reult from finite element calculation. ome typical value of ρ are aluminium: ρ =.7, copper:.3, iron:. []. For the ingle crytal nickel alloy, CMX and CM86, ρ =. [3] and. [], repectively.

6 Finite Element Analyi The emi-analytical crack tip field obtained in the previou ection are compared with finite element (FE) olution. The FE calculation were conducted, uing the finite element code ABAQU [5], on a centre cracked plate loaded in tenion (ee Fig. 3). The crack length to pecimen width ratio a/w =., which eentially correpond to a crack in an infinite plate. The material wa modelled a being linear elatic. [] a [] W FIGURE 3. Loading configuration of a Mode I pecimen The FE meh ued for the analyi i hown in Fig.. A full meh i employed to allow a range of mixed mode configuration to be examined in future work. In the figure three part compoing the meh, are hown. ection (b) fit into the rectangular white area of (a) and ection (c) fit into the white circle of (b). The current analyi wa conducted under plane tre condition uing four node bilinear, reduced integration element. The meh ued i compoed of about 7 element, with the mallet ize element being on the order of -5 of the crack length. FIGURE. Meh ued for the FE analyi. Part (b) fit into the rectangular white area of (a) and part (c) fit into the white circle of part (b). In Fig., the comparion between the emi-analytical olution and the FE analyi i hown. The open ymbol repreent the FE olution and the line are the emi-analytical olution. The FE tree have been obtained at a ditance r/a = -, where the K-field

7 i expected to be dominant. It can be een that excellent agreement i obtained between the FE and the analytical olution. Concluion In thi paper the precie form of the crack tip tre ditribution in a cubic aniotropic material have been obtained under plane tre/train condition. The olution obtained are emi-analytical: the complex root of the characteritic equation are firt determined for a given value of the aniotropy parameter ρ, and the crack tip tre field are then obtained a a function of thee root. The crack field thu depend on the material propertie only through the parameter ρ. It i found that the radial tre component rr depend trongly on ρ wherea the other two in-plane component, θθ and rθ are almot independent of ρ. Finite element calculation have been conducted on a large plate containing a harp crack and excellent agreement i obtained between the emi-analytical and FE reult. Reference. Irwin, G. R., J. Appl. Mech., vol., 36 36, 957. ih, G.C., Pari, P.C. and Irwin, G.R., Int. J. Fract. Mech., vol., 89 3, Lekhnitkii,.G., Theory of Elaticity of an Aniotropic Body, Holden-Day, Francico, UA, 963. Baani, J.L. and Qu, J., J. Mech. Phy. olid, vol. 37, 35 53, Qu, J. and Baani, J.L., ibid, 7 33, Yang,. and Yuan, F.G., Int. J. olid truct., vol. 37, , 7. Yang,. and Yuan, F.G., Int. J. Fracture, vol., 9 39, 8. Nobile, L. Piva, A. and Viola, E., Eng. Fract. Mec., vol. 7, 59 56, 9. ih, G.C. and Liebowitz, H., In: Fracture: An Advanced Treatie, edited Liebowitz, vol., Academic Pre, New York, 968, Lempidaki, D.E. O Dowd, N.P. and Buo, E.P., work in preparation,. Wolfram Reearch, Inc. Mathematica, uo, Z., J. Appl. Mech., vol. 57, 67 63, 99 an by H. 3. Denni, R.J. Mechanitic Modelling of Deformation and Void Growth Behaviour in uperalloy ingle Crytal. PhD Thei, Imperial College London,. Regino, G.M. A Multicale Contititive Approach to Model the Mechanical Behaviour of Inhomogeneou ingle Crytal uperalloy, PhD Thei, Imperial College London, work in preparation, 5. Hibbitt, Karlon and orenen Ltd. ABAQU, Verion 6..