STRESS ANALYSIS FOR AN INFINITE STRIP WEAKNED BY PERIODIC CRACKS

Applied Mathematics and MechanicsPublished by Shanghai University,(English Edition, Vol 25, No 11, Nov 2004)Shanghai, China◎Editoriai Comittee of Appl . Math. Mech. , ISSN 0253-4827Article ID: 0253-4827(2004)11-1298-06STRESS ANALYSIS FOR AN INFINITE STRIPWEAKNED BY PERIODIC CRACKS *CHEN Yi-zhou (陈宜周)( Division of Engineering Mechanics, Jiangsu UniversityZhenjang, Jiangsu 212013, P.R. China)(Communicated by WANG Yin-bang)Abstract: Stress analysis for an infinite strip weakened by periodic cracks is studied. Thecracks were assumed in a horizontal position, and the strip was applied by tension “p”iny- direction. The boundary value problem can be reduced into a complex mixed one. It isfound that the EEVM ( eigenfunction expansion variational method) is efcient to solve theproblem. The stress intensity factor at the crack tip and the T-stress were evaluated. Fromthe deformation response under tension the cracked strip can be equivalent t0 an orthotropicstrip without cracks. The elastic properties in the equivalent orthotropic strip were alsoinvestigated. Finally, numerical examples and results were given .Key words: eigenfunction expansion variational method; periodic crack; stress intensityfactor; T- stressChinese Library Classification: 0346Document code: A2000 Mathematics Subject Classification: 74R10; 74S30; 74K20IntroductionMaterials such as ceramics and rocks contain many cracks. In this situation, multiple crackproblems in plane elasticity were investigated by many investigators- 1 ~35. Doubly periodic crackproblem is a particular one in this field. The problem was studied by severa! investigators-4-61 .The previously obtained results are limited to evaluating the stress intensity factor. Also， aconsiderable discrepancy has been found between the referenced sources. In the cracked stripcase, investigator paid attention to a strip weakened by several cracks7]. Recently, the stiffnessfor solids containing inclusions and microcracks was studied81. Also， the effective elasticproperties of solids with many cracks were investigated:91.In this paper, a strip weakened by periodic cracks is studied. The cracks are assumed in ahorizontal position, and the tension p is applied in y- direction (Fig.1). In this case, one can cuta rectangular cell from the cracked strip. From the symmetric condition of loading and geometry,中国煤化工* Received date: 2003-08-20; Revised date: 20Foundation item: the National Natural Science F:YHCN MHG3)Biography: CHEN Yi-zhou (1935 ~ )，Professor, Doctor (Tel: + 86-511-8780780; E-mail:yizhou922 @ yahoo. com. cn;chens @ ujs. edu.cn)1298Stress Analysis for an Infinite Strip W eakned1299v=V=V%Cng:=0σx=Ony=0,σg=σnv=0212不2b. v=T=-V/σs=0(a) An infinite strip with(b) A cracked cellthe periodic cracksFig.1it is found that the boundary value problem is invariably1.reduced into a complex mixed one .0.9-1,0On the other hand, investigators have found that0.8-1v/b=0.42.0the EEVM ( eigenfunction function variation method) is 0.70.%an efficient approach to solve the problem. For 1 0.60.8example, the crack problems in the mixed boundary0.4|value condition case and in the dissimilar material casehave been solved by using EEVM[10.11]. Generally, the .0.3EEVM is better than usual boundary collocation method0.0 0.10.20.3 0.40.50.6 0.70.8.in fracture analysis, simply because it does not dependon the boundary collocation scheme as well as it hasa Fig.2 Normalized .clastic constantclear physical explanation10,111. On the other hand, ifC(h/a,a/b) (C1 = E2/ Eo,EEVM is used one can obtain the T- stress from thesee Eq.(24))solution. This goal is not easy to reach by using theusual singular integral equation method in which we can only get the stress intensity factor at thecrack tip. Recently, the effective modulus of the cracked medium becomes an important topic insolid mechanicst91. In this paper, the displacement at the boundary of the cracked solids can beeasily obtained after solving the boundary value problem. Therefore， the average strains arobtainable. From the constitutive equation, the effective moduli can be derived. In the mentionedcase , the cracked strip can be equivalent to an orthotropic strip without cracks . Finally , numericalresults are given to demonstrate the influence of the crack geometry to the stress intensity factorand the effective elastic moduli .1 Analysis for a Long Strip Weakened by P中国煤化工The following analysis depends on the comple:TYHCNMH Ghod in planeelasticity(12]. In the method， the stresses (σ,σ,,oxy)， the displacements (u,0), and theresultant force function ( X, Y) are expressed in terms of two complex potentials 中(z) ,w(z) such1300CHEN Yi-zhouthatσx + σ, = 4Reψ' (z),σ,-io,y = *'(z)+ (z-z)(z) + w'(z)，(1)f=- Y+iX = $(z) +(z-z)中(z) + w(z),(2)2G(u + iv) = c中(z) - (z- z)中(z) - w(z),(3)where G is the shear modulus of elasticity, rt = (3- )/(1 + n) for the plane stress problem,v is the Poisson' s ratio .In the following analysis, the strip with periodic cracks is shown in Fig.1, and the remotetension isσ, = p. The relevant elatic constants are denote by vo, Go,Eo(Eo = 2Go(1 + vo)),respectively. r0 = 0.3 is used in this paper. In analysis, it is convenient to cut a rectangularcracked cell from the strip (Fig. 1). Clearly, the boundary condition for the cracked cell will bev=i=士0b，σxy=0(- b≤x≤b，y=土h),(4a)σx=0,σxy=0(x =士b,-h≤y≤h).(4b)In Eq. (4a), 06 is an undetermined value, which will be determined by|.o,(x,h)dx = bp.(5)From Eq. (4a) we see that the boundary condition shown by Eq.(4a) is a complex mixed one，since both the displacement and traction boundary value conditions are involved in Eq. (4a).To solve the problem, we introduce the following complex potentialo :φ(z) =jx,φ(k)(z)， w(z) =ZxXxwo(a)(z),(6)where[φ(k)(z) = w(*)(z) =√z2- a2z2k-2 (1≤k≤M)，(7)(6(2)(z) =-w(b)(z) = 2(-M)-I1(M+1≤k≤2M).Note that， the introduced complex potential satisfies the traction free condition along thecrackl10]. In Eqs.(6) and (7), 2M is the number of truncated terms in the eigenfunctionexpansion form.In the simply mixed boundary casel10], the goverming equation for evaluating X:(k = 1,2,.. ,2M) takes the form2AmX% = Bm (m = 1,2,. ,2M),(8) .Amk=Akm=ofnμ{")ds-. omnpu()ds(k,m = 1,2,..,2M)， {9)Bm = | p;u?m)ds -|。o"nji;ds(m = 1,2,... ,2M),(10)where σ{k) , u{t) denote the stresses and displacements in the h- th expansion term, Pi is thetraction applied on the portion Cp of boundary, u; is the中国煤化工) the portion Cuof boundary .HCNMHGBelow, the EEVM ( eigenfunction expansion variational method) is used to solve thementioned boundary value problerm[10] . In the present case (Fig. 1), the boundary“AC" belongsStress Analysis for an Infinite Strip Weakned1301to a complicated mixed boundary. In this case, for evaluating the integral along the boundary“AC”in Eq.(9)， one simply uses the“C,” type integral for the assumed boundary tractioncondition(σxy = 0in Eq.(4a)), and use the “C。”type integral for the assumed displacementcondition(v = u =土Ub in (4a)). In this case, we finally haveAmk =Akm =i [σ{)um) - o;mbp(4)]dx +J ACf [o$2)u(m) + ofh)u(m)]dy(k,m = 1,2,. ,2M),(11)Bm =-oσ;")idx(m = 1,2,. ,2M).(12)In Eqs.(11) and (12), for example, AC means that the integration is performed along the lineAC in Fig.1, σm) represents the σy stress component of the m- th expansion form, and i is thegiven displacement along the line AC which was shown in the condition (4a). Physically, wesolve the problem under the condition (4b) and the condition D =元=土1,oxy = 0 (- b≤x≤b,y=土h), and Ub is obtained from Eq .(5).After solving Eq. (8)，the stress intensity factor and the T-stress can be obtainedfrom'[10,13]K, = 2(2r)!/2 lim√z- a$(z)or K, = 2(ra)/2Xxra2k-2,(13)T = 4XM+I.(14)In the case of using M = 8, the calculated results for the stress intensity factor and the T stresscan be expressed as follows:K = A.(h/b,a/b)p(ra)|/2,(15)T =- B(h/b,a/b)1- (a/b)P.(16)The calculated A,(h/b,a/b) and B:(h/b,a/b) values are listed in Tables 1 and 2 ,respectively.Clearly, from the displacement response in y- direction the cracked strip can be modeled byan orthotropic strip without cracks. It is known that the constitutive equation in the orthotropiccase takes the formll4]121Ex =Oy，Ey =-σx +Oy，Yxy =σxy.(17)EE2G12Table 1 Normalized stress intensity factor AI (h/b,a/b ) in the cracked strip problem (seeFig.1 and Eq.(15))\ 4960.10.20.40.50.60.70.80.9660.9000.8540.8480.8860.9861.1701.5270.9930.9780.9701.0381.1451.3381. 6881 .001.1.0071.0241.061中国煤化工1.7621.01.0051.0191.0481.095YHCNMHG1.51 .0061 .0251.0591.1111.190 I .3081.4831.8022.01.0061.024 .1.0581. 1061. 1801.2891.4591.7571302CHEN Yi-zhouTable 2 Normalized T-stress B (h/b,a/b ) in the cracked strip problem (see Fig.1 andEq.(16 ))B9%0.0.20.30.40.50.60.70.8h力0.8460.6360.4350.2780. 1990.0900.044 .0.0180.8830.7450.6020.4680.352 .0.2550.1760.1120.8970.7880.6800.5750.4770.3850.2980.2121.00.902).8090.7190.6330.548 .0.4640.3750.2751.50.906.). 8230.7460.6710.5980.518.0.4250.3242.00.9060.822 .0.6680.5920.5100.4160.309In Eq.(17) there is a relation as follows:(E1v21)/(E2v12) = 1.(18)To obtain the effective elastic constants, we introduce a particular trivial solutionVoσx = 1，O,=0，Ex=REy =-Eo'(19)After substituting Eq.(19) into Eq.(17), we have a solutionE=Eo，Y12=vo.(20)Secondly, from the numerical solution mentioned above, we have other particular solutionσx =0，Oy,av=p，Ex,av=万，. Ey,av=元，(21)where the subscript “av”means that the relevant quantity is understood in the sense of average onthe some portion of the boundary, and uav is defined byuav=动。u(b,y)dy.(22)Similarly, substituting Eq.(21) into Eq.(17) yieldshuavE2 =(23)bvbClearly, the obtained results for E:,V12,E2，v12 shown by Eqs.(20) and (23) may notsatisfy the relation (18) exactly. In fact, in the range for h/b and a/b used in the followingnumerical examples, the ratios ( E|221)/( E2v12) are varying with the ranges 0.9987 to 1 .001 3.That is to say the proposed assumption coincides the physical situation very well.The calculated E2 values are expressed asEz = C;(h/b,a/b) Eo.(24)The relevant dimensionless values Ci(h/a,a/b) are plotted in Fig.2. From Fig.2 we see thatC:(h/b, a/b) values are always less than unity within the ranges of study.2 ConclusionsAll the quantities that were interested in such as中国煤化工at the crack tip,the T- stress and the effective moduli of elasticity forYHCN M H Gbtained from thesuggested EEVM. Thus, the EEVM provides an effective way to solve the periodic crack problemfor an infinite strip.Stress Analysis for an Infinite Strip Weakned1303It is found that the infinite strip with periodic cracks is equivalent to an orthotropic platewithout cracks. This will be useful for evaluating the effect of damage in continuum medium. Inaddition，from the calculated results we see if the relative crack length is larger， the stressintensity factor and the Young’s modulus of elasticity are more affected . For comparison, we citetwocases, 1) In the case ofh/b = 1.0and a/b = 0.1, thereisK; = 1.005 p(πa)2,E2 =0.985 E;2) In case ofh/b = 1.0anda/b = 0.6, it becomes K = 1.288 p(πa)I/2,Ez =0.583 Eo. Therefore, the elastic constant E2 is reduced considerably when a/b is changing froma/b=0.1toa/b=0.6,inthecaseofh/b=1.0.References:1 ] Savnuk M P. Two dimensional Problems of Elasticity for Body with Cracks [ M]. Kiev: NaukaDumka, 1981.[ 2] CHEN Yi-zhou. A survey of new integral equations in plane elasticity crack problem[J]. EngngFract Mech , 1995 ,51(5) :387- 394.[ 3] CHEN Yi-zhou,Lee K Y. An infinite plate weakened by periodic cracks[J]. J Appl Mech ,2002 ,69(3):552 - 555.[ 4 ] Isida M, Usijima N, Kishine N. Rectangular plate , strips and wide plates containing intermal cracksunder various boundary conditions[J]. Trans Japan Soc Mech Engrs , 1981 ,47:27 - 35.[ 5 ] Delameter W R, Herrmann G, Barnett D M. Weakening of elastic solid by a rectangular array ofcracks[J]. J Appl Mech , 1975 ,42(1):74- 80.[ 6] Parton V Z,Perlin P I. Integral Equations in Elasticity[ M]. Moscow: Mir, 1982 .[ 7 ] BenthemJ P, Koiter W T. Asymptotic approximations to crack problems[A]. In:GC Sih Ed. Me-chanics of Fracture[C]. 1973,1:131- 178.[ 8 ] Huang Y,Hu K X, Chandra A. Stiffness evaluation for solids containing dilute distributions of in-clusions and microcracks[J]. J Appl Mech ,1995 ,62(1):71- 77.[9] Kachanov M. Elastic solids with many cracks and related problems[ A]. In:J W Hutchinson,T WuEds . Advances in Applied Mechanics[ C].1993 ,30:259 - 445.[10] CHEN Yi-zhou. An investigation of the stress intensity factor for a finite intermally cracked plate byusing variational method[]. Engng Fract Mech , 1983 , 17(5) :387 - 394.[11] Wang W C,Chen J T. Stress analysis of finite interfacially cracked bimaterial plates by using varia-tional method[J]. Comput Methods Appl Mech Engrg , 1989,73:153- 171.[12] Muskhelishvili N I. Some Basic Problems in the Theory of Elasticity [ M] . Gronigen: Noordhoff,[13] CHEN Yi- zhou. Closed form solution of T-stress in plane elasticity crack problems[J]. Internat JSolids and Structures ,2000 ,37(11):1629- 1637.[14] Lekhnitsky S G. Theory of Elasticity of Anisotropic Elastic Body[ M] . San Francisco; Holden-Day ,1963.中国煤化工MYHCNMHG