THE DUGDALE S TYPE CRACK MODEL FOR ELLIPTIC CRACK AT MULTIAXIAL LOADING

Transcription

1 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY HE DUGDALE S YPE CRACK MODEL FOR ELLIPIC CRACK A MULIAXIAL LOADING Gregor GALAENKO SP imoshenko Institute of Mechanics, National Academ of Sciences of Ukraine, Department of Fracture Mechanics of Materials, Kiv, Ukraine ABSRAC he 3-D task for elliptic crack ( x 2 / a 2 2 / b 2 1) + of mode I with plastic one in plane = is considered at uni-, bi- and triaxial loading in infinit Material is suggested b ideall elastoplastic with ielding condition in general form he plastic one is modeled b elliptic ring with unknown sies AB, along of the axis Ox and O he stresses in the plastic one are determined from ielding condition and condition of plane strain in local coordinate sstem connected with external contour of plastic one he following cases of loading are considered: 1) uniaxial loading in infinit = p; 2) biaxial loading ( ) = p, = q = g ; 3) triaxial loading = p, = q, = g he maximum x 51 x value of stresses in plastic one is analed in dependence from sign and values of stresses q, g at von Mises ielding condition In the first and third cases at q = g the sies A, B of the plastic one are obtained in analtical form he limit cases of small-scale ielding and full plastic state are considered too Kewords: elliptic crack, plastic one, multiaxial loading, plastic constraint factor 1INRODUCION he Dugdale crack model [1] was suggested to thin plates ( plane stress) onl Direct application of this model to plane strain or 3-D problem is not correct due to triaxial stress state at the crack front Guo [2] and Neimit [3] used a semi-analtical method to rationalie effects of out-of-plane and in-plane constraints for through- thickness cracked bodies Analsis in these works is focused on small- scale ielding (SSY) conditions in which the crack-tip plastic one is full enclosed b K or K dominated elastic field Galatenko proposed the Dugdale-tpe crack model to plane strain [4] and to circular diskshaped crack [5] without limitations on the plastic one sies Dependence of the plastic constraint factor (PCF) m on external loads level and loads acting parallel to the crack plane was obtained for reska and von Mises ielding conditions At SSY conditions the PCF ( m m k ) is limited b imposed restrictions on the -stresses he report presented deals with extension of the proposed crack model to elliptic crack he results are compared with those for circular disk-shaped crack

2 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY 2 MODEL OF HE ELLIPIC CRACK Let us consider 3-D volume with elliptic crack S { S : x 2 / a 2 2 / b 2 1} + located in plane =, under acting of the triaxial stresses at infinit = p, = q, = g he stresses p are x tensional and q, g can be as tensional as compression he material is assumed to be elasticperfectl plastic with ielding condition F ( 1, 2, 3) = Y, where 1, 2, 3 are principal stresses, Y is a ield stress he plastic one S at the crack front is modeled b elliptic ring in crack plane with unknown sies A, B of the external contour In the plastic one the stresses,,, τ τ = τ = satisf to ielding condition After subtraction of the applied ( ) x x x stress state = p, = q, = g the auxiliar task can be obtained for plastic one S x = x S = p = q = g = (2), (, ) :, x x,, τx τx; In the local orthogonal coordinate sstem nt (Fig1), located at the external plastic one contour the expression (2) have the form = x S = p = = = (3), (, ) :, n n n, t t t, τnt τnt τnt Here,, τ are stresses in plane = without crack which are connected with n t nt, b relations x 52

3 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY τ n t nt x x + = + x x + = = x sin 2 θ, cos 2 θ; cos 2 θ; 2 where θ is angle between normal n and axis Ox It is well known [6] that asmptotic stress field in the coordinate sstem nt corresponds to plane strain = n, t = ν( + n) = 2 ν, τnt = at =, r (4) he stresses are limited and continuous on the boundar of elastic and plastic ones hen from (3),(4) we have p = ; = 2 ν( p); τ τ = (5) n n t t nt nt he ielding condition and relations (5) substitute the full equations sstem to determine the stresses in plastic one In particular, for von Mises ielding condition the obtained sstem have the form p = n n ; t t = 2 ν( p); τnt = τnt ; (6) ( n) + ( n t ) + ( t ) + 6τnt = 2 Y After calculating of stresses in the plastic one the boundar task of elasticit theor for crack S + S can be formulated ( x, ) S: = p;, (, ) :, = x S = p ( ) x, S + S: u = he further solution of problem (7) is similar to the Leonov-Panasuk-Dugdale problem [7] he stresses or PCF m = / characteries constraint effects on the crack front he solution (7) gives the following formula for Y 1 q+ g 3 2 q+ g q g 2 ( ) = (1 4 ν) p+ ( q g)cos 2θ + 4Y 3( p cos 2 θ) 3 q g sin 2 θ 2(1 (8) In according to (8) the stresses depend from angle θ he extremum points can be obtained from the necessar condition 53 d / dθ = which gives the following equation 2 q+ g q g 2 ( ) ( ) q+ g 3 q g sin 2θ 4Y 3( p cos 2 θ) 3 q g sin 2 θ p+ ( q g) cos 2θ = (9) Equation (9) has a solution at three cases: 1) q= g, p his case corresponds to axismmetric loading when independent of θ and has the form 1 = (1 4 ν) p+ q+ 4Y 3( p q) 2(1 In particular, at q= g =, p we have uniaxial loading = p at which (7)

4 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY 1 = (1 4 ν) p+ 4Y 3 p (11) 2(1 Results at these schemes of loading are analed in paper [8] 2) Condition 2 q+ g q g 2 ( ) q+ g 3 4 3( p cos 2 θ) 3 q g sin 2 θ p+ ( q g) cos 2θ =, (12) after transformations, can be written as he last equation corresponds to full ductile fracture because the external loads satisf to ielding condition 3) sin 2θ = he critical points θ = πk/ 2 ( k =,1,2,3) can be obtained which coincide with ends of the axes of elliptic crack We consider the third case at bi- and triaxial nonaxismmetric ( q g) loads on infinit 3 RESULS AND DISCUSSION 31 Biaxial loading 1 At = p, = q, = the distribution x are determined b formulas 1 q 3 2 q q 2 = (1 4 ν) p+ qcos2θ+ 4 3( p cos2 θ) 3q sin 2 θ 2(1 2 ν) ( ) ( ) ( ) q g + g p + p q = 2 Y (13) (13) Fig2 shows the dependence of PCF m on angle θ at different stresses q / Y and ν=, 3 he curves 1,2,3,4,5 correspond to values q / =,5;,1;;,1;,5 For given load the maximum values equal 54

5 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY 1 max = ( θ = ) = (1 4 ν) p q+ 4Y 3( p q), 2(1 q< ; 1 max = ( θ = π / 2) = (1 4 ν) p+ 2q+ 4Y 3 p, 2(1 q > In particular, at q= ν p we have from the second expression of (14) p 4 3p max = ( θ = π /2) = ( ν ) he result (15) corresponds to plane strain case [4] for central crack 2b 2 At = p = g we have the following distribution (14) (15) 1 g 3 2 g g 2 = (1 4 ν) p+ + gcos 2θ + 4 3( p + cos 2 θ) 3g sin 2 θ 2(1 (16) Dependence m on angle θ is shown on Fig3 Maximum m takes place on the ends of larger axis of ellipse at g > and on the ends of smaller axis at g < When g = ν p, the ends of larger axis of elliptic crack have the same constraint that at plane strain 31 riaxial nonaxismmetric loading In general case of triaxial loading the distribution is determined b formulas (8) riaxial loading on infinit can be presented as superposition of axismetric (q=g) state and uniaxial tension or compression along of one axis of elliptic crack he first state doesn t influence on the position of critical points Consequentl, their position is determined b the second state Fig4 shows m θ dependence at values of stresses acted along of elliptic axes : 1-55

6 8 Uluslar Aras K lma Konferans Bildiriler Kitab 7 9 Kas m 27 Prooceedings of 8th International Fracture Conference 7 9 November 27 Istanbul/URKEY q=, 5, g =,1 ; 2 - q=,1, g =,5 ; 3 - q=,5, g =,1 ; 4 - q=,5, g =,1 ; 5 - q= g = In the first and third schemes of loads maximum m takes place at θ = π /2 When g > q then maximum m is realied at θ = 4 CONCLUSIONS Analing the results, we can formulate a rule for determination of the maximum constraint points of plastic strains at the elliptic crack front under multiaxial loading: the maximum constraint (maximum ) occurs at those points on the crack peripher where the external tangential tensile stress are maximum It means that in dependence on stresses qg,, acted along of elliptic crack axes, the fracture beginning can be on the ends of larger axis in spite of minimum of stress intensit factor in these points REFERENCES 1 Dugdale, DS Yielding of steel sheets containing slits, J Mech and Phs of Solids,V8, N2,196 2 Guo,W hree-dimensional analses of plastic constraint for through-thickness cracked bodies, EngFractMech, 62, Neimit, A, Dugdale model modification due to the geometr induced plastic constraints, EngFractMech, 67, 2 4 Galatenko, GV, o elastoplastic crack model at plane strain, Prickladna Mechanika, V28,N9, Galatenko, GV, Circular disk-shaped crack at triaxial loading, Prickladna Mechanika, V28,N8, Cherepanov, GP, Mechanics of brittle fracture [in Russian], Nauka Publishing, Moscow, Panasuk, VV, he limiting state of brittle bodies with cracks [ in Russian], Naukova Dumka Publishing, Galatenko, GV, A ring-shaped plastic one at an elliptic crack under triaxial axismmetric loading, Prickladnaa Mechanika, V4,N2, 24 56