Transactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN

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1 Unitary weight functions for semi-infinite crack in the half-plane K.L. Molski Institute of Cranes and Heavy Machinery Engineering, Warsaw University of Technology, Narbutta 85, Warsaw, Poland ABSTRACT Weight function method, suggested by Bueckner [4] and Rice [5], is one of the most effective tools of stress intensity factor determination in cases of continuous load symmetrically distributed along both sides of the crack. To simplify integrating procedures and unify the weight function description - unitary weight function was proposed in [7], which assures high accuracy, very short computing time and easy application for computer use. In this paper the unitary weight functions as well as corresponding correction functions F for Mode I;, II and III are developed for a semi-infinite crack in the half-plane. In this case the "inverse geometry" method proposed by Bueckner [4] is used. Results obtained for Mode III are based on closed form solution. For Mode II and 1^ - with no rotation of crack sides permitted - basic pattern function is an approximate solution obtained by Hartranft and Sih [9], slightly corrected to agree with the known exact solution at the boundary of the body. All unitary weight functions obtained are represented in the unified form as fractional integrals. Accuracy of the transformation and limitations of the solutions are also discussed. INTRODUCTION Application of fracture mechanics to fatigue life estimation of structural elements usually needs determination of stress intensity factor K with high accuracy and at minimum cost. These basic contrary requirements cannot be satisfied at the same time. The problem which frequently appears in practice lies in finding a reasonable compromise between two extreme approaches: the first one - the use of methods of high accuracy with long computing time (analytical solutions, all numerical

2 418 Localized Damage techniques based on FEM and BEM etc.), and the other one - application of simplified formulae with limited validity, but with relatively short computing time. Besides, there are many difficulties in the use of particular solutions found in the literature, for example in [1,2,3], because they are related to a limited number of arbitrary loading conditions and are usually expressed in many different ways: graphs, tables, polynomial approximations etc., which often need additional effort for data preparation. WEIGHT FUNCTION Many existing methods of analysis of crack problems are based on continuous surface load symmetrically applied to both sides of the crack. One of them is the method of weight function proposed by Bueckner [4] and Rice [5], which enables one to relate any stress field - residual, local or caused by external load - to the crack path, by using the principle of superposition. The stress intensity factor is expressed then by Equation (1) as follows K=fo(x)m(x)dx */n (i) where "a" is a crack length, a(x) - surface stress and m(x) - weight function suitable for a particular geometry. One modification of this method can be found in [7], where the 'unitary weight function' has been introduced as well as its unified description adequate for use of computer data bases. In that case Equation (1) is just replaced by Equation (2), where all weight coefficients are considered as known values. 10 (2) f = l More details of this procedure can be found in [7] and in [6] for partially loaded crack surfaces.

3 SEMI-INFINITE CRACK Localized Damage 419 Let us consider a semi-infinite crack located in the half-plane and loaded continuously and symmetrically on both sides, Mode I, II and III, along a distance "a" from the crack tip, as shown in Figure 1. jrn 3OQQO000OQ Fig. 1. Semi-infinite crack of length "a" in the half-plane subjected to arbitrary continuous tension I and shear II and III on both sides. (No rotation of crack sides due to bending is permitted). where the a/b ratio may be interpreted now as the crack length to finite width parameter as it is usually applied for true finite width elements. The stress field surrounding the crack tip can be found by use of three stress intensity factors Kj, Kpj and KJJJ, which have to be determined. Regarding subsequent calculations of K values it is convenient to set the coordinate system out of the crack tip and join it to the body, or more precisely - to the nominal stress field of uncracked body, avoiding perpetual changing of surface load description while crack propagates. In this case no rotation of crack faces due to bending is allowed, so an influence of K% to the final solution is limited to simple tension - no bending considered. This implies a change of notation - Kj is replaced by K%. Anti-plane shear As shown in reference [4], the inverse geometry to the case of the semi-infinite crack in the half-plane is a simple edge crack in the half-plane and weight functions of the later are recognized here as pattern functions of the former. That implies changes in coordinate systems represented in Fig. 2.

4 420 Localized Damage Fig. 2. Complementary geometries and corresponding coordinate systems. In reference [2] the weight function for a single edge crack for Mode III is given in closed form, thus its unitary weight function w(x,l) is represented by Equation (3). (3) According to the inverse geometry method, normalized coordinate ^ = jcj/l is substituted now by a new parameter s = ^/fe" LIa), where ^ = *2/L, ^rid applied again to the Equation (3). Numerical integration of Eq. (3) is carried out to determine fractional values Q,- (alb) of the unitary weight function integrals vs. the alb ratio as well as the correction function F(a/b). Numerical results are shown in Figure 3 and Figure 5. Tension and longitudinal shear In these cases, which are essentially identical if no rotation is allowed, approximate solution of Hartranft and Sih [9] is used as a unitary pattern function, given by Equation (4): m 1+/W (4) wherefit) = (I-**) [ f t* f ] and t= x/l.

5 Localized Damage 421 Subsequent procedures of coordinate transformations and numerical integration are identical with the ones from the previous case. Besides, the f(t) function have to be slightly corrected, namely the value is now to satisfy the known exact solution at the boundary. Numerical results are shown in Figure 4 and Figure , a A Fig. 3. Fractional values of the unitary weight function integrals vs. the a/b ratio for Mode III.

6 422 Localized Damage o.i Vb Fig. 4. Fractional values of the unitary weight function integrals vs. the a/b ratio for Mode It and II. 1.0 T 1 1 T I 0.4 I L a/b Fig. 5. Correction functions for Mode It, II and III.

7 DISCUSSION OF NUMERICAL RESULTS Localized Damage 423 From Fig. 3 and Fig. 4 one may conclude that unitary weight functions for Mode II and Mode III coincide in both extremes of the a/b ratio and are slightly different in the remaining range. Maximum difference does not exceed 12%. Further analysis will be necessary to confirm this behaviour for other geometries. The correction functions F%, FJJ and Fm shown in Figure 5 are similar in shape to those for infinite strip with a central crack although the initial points are different. Accuracy of the transformation is analysed by comparing the results for Mode III (Fig. 3) to the ones obtained by Stallybrass [8], for a semi-infinite crack in the half-plane. Maximum differences does not exceed 0.3% for biquadratic polynomial description of Q,- (alb). CONCLUSIONS In the presented case of the semi-infinite crack in the half-plane the method of inverse geometry associated by complementary fields appeared to be a very effective tool for evaluation of the unitary weight function as well as correction function F(a/b). Accuracy of numerical transformation to the new geometry by using biquadratic polynomials for description of fractional integral values of the unitary weight function does not exceed 0.3%, if the pattern function to be transformed is a closed form solution - as for Mode III in this case. If not - additional errors may be expected. It is now difficult to say, whether minor differences between unitary weight functions for Mode III and 1^ or II, shown in Fig. 3 and Fig. 4, are due to inexact description of the later pattern function or they really have to be slightly different. In other words, it should be determined, if for the same geometry and different loading Modes I, II and III, corresponding weight functions are just proportional to their own correction functions F or not. This problem is very important for many geometries and will be investigated in the nearest future. The method of inverse geometry in many cases does not take into consideration other loading effects - as for example bending - which may occur in transformed body, making the whole analysis useless. Thus additional constraints should be imposed on the body, as it has been done here, or include these effects in the solution by using a different method.

8 424 Localized Damage ACKNOWLEDGMENTS The investigation described in this paper is a part of the research project No. 7 S sponsored by Polish State Committee for Scientific Research. REFERENCES 1. Sih, G.C., Handbook Stress Intensity Factors, Lehigh University, Bethlehem, Pennsylvania, Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis ofcracks Handbook, Del Research Corp., Hellertown Pa., Murakami, Y, (Ed). Stress Intensity Factors Handbook, Pergamon Press, Bueckner H.F., 'Field Singularities and Related Integral Representations', Chapter 5, Methods of Analysis and Solutions ofcrack Problems, (Ed). Sih, G.C., Vol.1, pp , Noordhoff International Publishing, Leyden, The Netherlands, Rice, J.R.,'Some Remarks on Elastic Crack-tip Stress Fields', Solids and Structures, Vol.8, pp , Molski K.L., 'Factor de Intensidad de Esfuerzos para Grietas Cargadas Superficialmente',Mem0ri0 del X Congreso de la Academia Nacional de Ingenieria, pp , Mexico Molski K.L., 'Computer Aided Assessment of Stress Intensities for Cracks', Computational Methods in Fracture Mechanics, Vol 2, Computational Mechanics Publications, pp , Stallybrass M.P., 'A Semi-Infinite Crack Perpendicular to the Surface of an Elastic Half-Plane', Int. Journal ofengineering Sciences, Vol. 9. p. 133, Hartranft, R.J. and Sih, G.C.,'Alternating Method Applied to Edge and Surface Crack Problems', Methods ofanalysis and Solutions of Crack Problems, (Ed). Sih, G.C., Vol.1, pp , Noordhoff International Publishing Leyden, The Netherlands, 1973