AN ANALYSIS OF 3D FINITE CRACKED BODIES

ACTA MECHANICA SOLIDA SINICA, Val. 14, No. 1, March 2001ISSN 0894 - 9166Published by HUST, Wuhan, ChinaAN ANALYSIS OF 3D FINITE CRACKED BODIESOTao WeimingGuo Yimu(Department of Mechanics，Thejiang Unirersily, Hangzhou 310027, China)Cao Zhiyuan(Department of Mechanics , Tongji University, Shanghai 200092, China)ABSTRACT The new type traction boundary integral equations developed by Hu and with no hyper-singular integral are applied to analysis of 3D finite cracked bodies. A numerical algorithm for general3D problens and a semi-analytica one for axisymmetric problemns are presented. Some examples ofthick plates and eylindrical columns including penny shaped crack(s)，and rectangular plates includingan elptial crnack nommal to the suface are analyzed. The comparison between present resuts andthose in literature shows the high accuracy and efectiveness of the present method.KEY wORDS principal integral, cruck, stress intensity factor, traction boundary integral equation,three -dimensional finite bodyI . INTRODUCTIONAll crack problems are strictly 3D ones, and the objects found in engineering (such as ma-chinery components, structural members) are always finite. So a 3D analysis of finite crackedbodies is of significance to both theory and practice. In literature and handbooksitl there are alot of stress intensity factors (SIF) results of such problems. Most of the results are calculatedby such methods as the method of integral transformation, method of weighted function,boundary alolationo method, the dual-integral equation method and finite element method. Theboundary integral equation method (BIEM or BEM) is a significant technique for engineeringand theoretical analysis and has made great progress in the recent 20 years. In comparison to .‘domain-type' methods, such as FEM and FDM, it is reduced by one dimension and can rea-sonably simplify numerical computation for the problem concerned. And it is considered thatBIEM is superior to FEM for crack problems and/or those with unbounded domain.For crack problems, however, typical BIEM will lead to a set of singular linear algebraicequations. We can combine it with the multi domain method to overcome the difficulty. Gener-ally, the traction BIEM is fairly direct and simple in form for crack problems, and is frequentlyused4- 8]. Unfortunately, these equations derived from Rizzo-type BIEs by partial derivationcontain hypersingular integrals, which will meet with computational dfcultie.. Hu10] formu-lated principal-type traction BIEs for the 3D elasto- static problem by employing two-state con-servation law. These equations only include regular or principal integrals without any hyper-sin-gularities. These BIEs are applied to 3D finite cracked中国煤化工nding numericaland semi-analytical method is proposed. Examples ofTHCN M H Gcal columns with①Project supprted by the National Natural Science Foundation of China (No. 19972060) and the Foundation of KeyL aboratory of the Ministry of Education， Tongji Univcrsity.Received 10 Oetober 2000..ACTA MECHA.NICA SOLIDA SINICA2001penny-shaped crack(s). and rectangular plates with an elliptical crack normal to the surface areanalyzed.I. FUNDAMENTAL EQUATIONSHu directly formulated principal-type traction BIEs for 3D elasto-static problems by em-ploying the two-state conservation law-10]专p(白)= Dwm[。[[enerpoiup,n,- U.dp]dr， (ξ∈s, Sis smooth) (1)where p,，1; are traction and displacement components; σj放， ui; are stress and displacementfundamental solutions; S, n; stand for the object 's boundary and its outward normal cosine andD;jk for elastic tensor, ejke is a 3rd-order sign tensor.We can sce that in Eq. (1) there is no hypersingular integral, so no special difficulties willbe met in numerical computation. It is very significant and useful for an analysis of 3D crackproblems.The total solution o[ a crack problem can be decomposed into two parts under the assump-tion of linear elasticity and with no contact between crack faces. One part is relevant to the cor-responding problem without a crack, which makes no convolution to stress intensity faclorswhile the other is relevant to one with a crack. We need only pay attention to the latter. Gen-erally, the boundary conditions at crack faces can be expressed by a set of distribution forcesacting on the upper and lower faces, andp;|r' =- p;Ir = p;(i)，t ∈F+ (or I-)(2)where r+ ,r stand for the upper and lower faces of the crack, and their outward normals arerelated byn;Ir=-n;Ir~(3)Applying Eq. (1) to a cracked object, we can obtain the following integral equationsp;(ξ) = Djxmn/。enaernp imeSup.n,dr +Diaxnj[erxerpipup.m.- Up,IPp]dr"，(∈ r*)(4a)立p;() = Dikmj,enerupmAup.mudr. +Dxn.ermpiup, - Ux,dPp]dr， (ξ∈S)(4b)where S denotes the outward boundary, and Ou:= ui|r*- u;Ir~ the CODs. .If the surface S is in free condition, Eqs. (4) can be simplified tcp,(E)= Dxaij。esterpxeup.n.dr, .(ξ∈B)(5)where B= I+ + S, andp;(e) ={p(ξ)， (er"); a(自= jsu;, (ξ∈r").; n;(ξ) =n;， (ξ∈r+)l0,(ξ∈S)( u;,中国煤化工n，(ξ∈S)If there are m cracks, and Tt denotes the upMYHCNMH G, thenI*= rtin the above-mentioned equations..Vol. 14, No.1Tao Weiming et al.: An Analysis of 3D Finite Cracked Bodies●43●I. NUMERICAL ALGORITHMIt is very difficult to find the exact solutions of BIEs (4) and (5), so we normally do sowith numerical methods. A numerical process for an infinite medium with a planar crack locat-ed at x10x2 plane is described in Ref. [11 ]. Numerical processes for general 3D and axisym-metric problems of cracks or a crack system will be presented in this paper.We firstly subdivide the crack upper face and the body' s surface into M elements, i.e. B当= B。，and assume the displacements ase-1u;(x) =pH(i)ij(6)1，(z∈B,)where$%r(z) = g;(t)H*(i)， H(F) =\0，(i乐B)ring element/B.B.袁R.gofqcrack frontR' nodal 1ineelement(a) General 3D boundary element(b) Ring elementFig.1 Boundary elernent shape.Substituting Eq. (6) into Eq. (5), and ltting ξ cover each clement's central coordinate王|a(d= 1,2,.,M), we can obtain a set of linear algebraic equations relevant to displacementsp= p:(∈|a)=A等w(7)=1A腊= Digilenera}。 oxl[gp),(2)n,(1)- 8p1,.(ξlu)n,(ξlu)]dr +Br(a)n(a)exexPmeUamgdl - eneing oxwu(i)n(t)m(x)dl} (8a)JaB(for singular elements in which ξlu=zle)AT = Di;kiexerujJ。OpkBpr,(&)n(i)dI- exterin」,,σxgpr(f)n,(f)m,(z)dl} (8b)(for normal elerments in which ξla≠il。)The shape function g;r(x ) should include‘ square root’characteristic at the crack front.For gencral 3D problems (Fig. 1(a)), we assume g;(i)= g(z)8;. Andg(2) = 1.5), the effect ofVol. 14, No. 1Tao Weiming et al. : An Analysis of 3D Finite Cracked Bodies45 .2.4k=h.20具■Present: Ref. [12]1.8|51.5.2-T iHH,tThz0.00.40.81.21.62.0ahFig.2 A thick plate with a penny-shapedFig.3 SIF of penny-shaped crack lies on the centercrack.of infinite thick plate.2.4[0.402.2-一a/t=0.00.35 t一- -8/t=0.0--a/t=0.50.30- -a/t=0.30.25一-&/t=0.51.8......- =a/2h5 0.20一a/1=0.75 1.6-&n=fyD.Jπa1.4&=x-20]完0.10-0.051.00.000.0 0.2 0.4 0.6 0.8 1.0 1.21.41.61.82.00.0 0,2 0.40.6 0.81.01.21.41.6 1.8 2.0Fig. 4 SIFs of penny-shaped cracks embedded in thick plates with varied thickness.4↑↑4↑$$+↑↑42h12012b↓↓↓+。↓↓↓↓↓↓↓↓↓$rFig.5 Columns with penny shaped crack.interaction between the two cracks is little. When the radius of the column becomes larger, thepresent results become closer to those of Ref. [16] ( as shown by dashed lines) , which are abouttwo parallel penny-shaped cracks embedded in an infinite_ body. This demonstrates the effec-tiveness of the present method.中国煤化工4.3 Rectangular Plate with an Elliptical Crack Non.MHCNMHGThe last example is about a rectangular plate with an elliptical crack normal to the surfaces(Fig. 9). A distributed load is normally applied on crack faces. The surfaces and crack face aredivided into general 3D boundary elements as shown in Fig. 1(a). The present results and those.●46●ACTA MECHANICA SOLIDA SINICA20012.2a/c=2a/b0.5■Presentk= p.2pJ2.0. Ref. [13]0.4 '1.8| e/c=/bPresent0.3- -.- Ref. [13]Km= G.2IN(>-g1)心1.6a/c=0.0心小唇0.21.4Ref. [14]. Ref. [15]-- -- Ref. [13].2-0.1.00.00.10.20.3040.50.60.70.80σ0204°0808 T.oa/bFig.6 SIFs under tension.Fig. 7 SIFs under torsion.0.121.8。- 1-a/b=0.2- - -a/b=0.4- -- - a/6b=0.2一一a/b=0.40.10- -a- -a/b=0.6-▼一e/b=0.71.2一◆-a/b=0.80.081.1心0.06r 1.00.040.90.8a/b=0.6- - a/b=0.70.02◆- a/b=0.8........ a/b=0.0).00L0.7L0.50 1.00 1.50 2.00 2.50. 0.50 1.001.50 2.002.50Fig.8 SIFs of two coaxial penny- shaped cracks embedded in a long column under tcnsion.K=F.p~而0.8-一- Ref. [1.17]pointA2a|h风0.6-point B0.4-[ 2b|2t2w0.00.4b/tFig. 9 Rectangular plate with an lliptical crack.Fig. 10 SIFs of point A and B vs. b/t(a/w<0.2, h/w= 1,b/a=0.4).of Ref.[1] [17] are shown in Fig. 10. The results中国煤化工t with each other.This dernonstrates that the present method can be aplMHc N M H Gack problems effec-tively.Vol.14, No. 1Tao Weirming el al. : An Analysis of 3D Finite Cracked Bodies●47●V . CONCLUSIONAs there is no hypersingular integral, the present principal-typed traction BIEM for 3D fi-nite cracked bodies is fairly simple and convenient. It can be applied to axisymmetric crackproblems, crack intcraction problems as well as general 3D crack problems. From the exampleswe can also see that the accuracy is satisfactory, showing that the present method is effective.REFERENCES[1] A Handbook of Stress Intensity Factors, China Autronaution Institute, Bejing: Sxience Press, 1993.[2] Cruse,T. A., Two dimensional BIE，fracture mechanics analysis, Appl . Math. Modelling，VoL. 2,1978, 287.[3] Atkinson, C.，Stress analysis in fracture mechanics, In: Brebbia, C. A., Progress in Boundary ElementMethod, London: Pentech Press, Vol.2, 1983, 53- 100.[4]Yu,D. H. , The Mathematical Thoory of Natural BEM, Beijing: Science Press, 1993.[5 ] Sladek,J. and Sladek, V.，Dynamic stress intensity factorms studied by boundary integro differential equa~tions, Int. J. Numer. Methods Eng.. 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Y. and Ji, X., Analysis of 3D crack中国煤化Ili-analyical BEM,Commun. Nuner . Meth. Engng.，, Val. 13, 1997, 827 - 8YHCNMHG