Weibull stress solutions for 2-D cracks in elastic and elastic-plastic materials

Transcription

1 International Journal of Fracture 89: , Kluwer Academic Publishers. Printed in the Netherlands. Weibull stress solutions for 2-D cracks in elastic and elastic-plastic materials Y. LEI, N.P. O DOWD, E.P. BUSSO and G.A. WEBSTER Department of Mechanical Engineering, Imperial College, Exhibition Road, London, SW7 2BX, U.K. n.odowd@ic.ac.uk Received 7 August 1997; accepted in revised form 6 April 1998 Abstract. The Weibull stress is widely used as a measure of the probability of cleavage failure. In this work analytical and semi-analytical expressions for the Weibull stress are developed in terms of the remote loading parameters, J or K, and material properties. Results are presented for sharp cracks and notches in elastic and elastic-plastic materials under plane stress and plane strain conditions. The proposed relations enable Weibull stress estimates to be obtained without the need for costly finite element analyses and provide insight into the use of the Weibull stress as a parameter for the prediction of cleavage failure of cracked bodies. The expressions have been verified using finite element techniques and good agreement has been found throughout. The results of the analyses have been used to interpret the mesh size dependence of Weibull stress values obtained from finite element calculations. Key words: Weibull stress, cleavage fracture, fracture mechanics, finite element analysis. 1. Introduction The distribution function for the fracture strength of a solid subject to a homogenous stess field, can be expressed, on the basis of weakest link statistics, as a two-parameter Weibull distribution (1939) in terms of a modulus m 0 and a scaling parameter σ u0. The probability of failure, P f (σ ), of a solid under a homogenous tensile stress, σ, is then given by (Weibull, 1939), [ ( ) σ m0 ] P f (σ ) = 1 exp. (1) σ u0 Although Weibull statistics have been widely used to describe the scatter in strength and fracture toughness of brittle materials, (e.g. Freudenthal, 1968; Evans, 1978), the Weibull stress was first proposed as a micromechanics parameter by the Beremin group (1983). There, the driving force for cleavage fracture in a cracked nuclear pressure vessel steel was assumed to be controlled by the Weibull stress, σ w,definedas σ w = (σ1 i V ] 1/m )m i, (2) V o [ ne i=1 where V i is the volume of the ith material unit in the crack tip plastic zone experiencing a maximum principal stress σ i 1, n e is the number of material units in the plastic zone and m is the Weibull modulus associated with σ w. The reference volume V o is taken to be representative of corrected JEFF INTERPRINT frac4395 (frackap:engifam) v tex; 26/09/1996; 7:35; p.1

2 246 Y.Leietal. the material microstructure. If a finite element analysis is used to determine the Weibull stress, the unit volume V i in (2) represents the ith element volume in the FE model. The cumulative probability of failure, in terms of the modulus m and a scaling parameter, σ u associated with the Weibull stress, is given by P f = 1 exp [ ( σw σ u ) m ], (3) with the Weibull stress defined by (2). In (3) it is assumed that the failure probability is given by a two parameter Weibull distribution. In other recent work (e.g. Ruggieri and Dodds, 1996; Busso et al., 1997), a three parameter Weibull stress distribution has been examined which includes a threshold stress below which the probability of failure is negligible. Only the two parameter Weibull distribution is examined in this paper, though extension to include a threshold stress is, in principle, not difficult. The effect of incorporating a threshold stress will be discussed in Section 5. Since its introduction in (Beremin, 1988) the Weibull stress has been widely used to characterise the cleavage toughness of steels in the low temperature and brittle to ductile transition temperature regimes and to explain the scatter associated with fracture toughness data, e.g. (Ruggieri and Dodds, 1996; Sherry et al., 1995; Xia and Shih, 1996; Sanderson et al., 1996). In these works the Weibull stress was calculated using FE techniques. Although an approach toward developing an analytical solution is suggested in (Beremin, 1983), to our knowledge, no such closed form relations for the Weibull stress are currently available. The need for expensive numerical analyses to evaluate the Weibull stress has limited its use in practical situations and made systematical parametric investigations difficult. In this work, analytical and semi-analytical solutions for the Weibull stress are presented. These solutions allow direct calculation of the Weibull stress for cracked elastic and elastic plastic geometries under small scale yielding conditions. The paper is structured as follows: The Weibull stress definition used in the present work is outlined in Section 2. In Section 3, analytical Weibull stress solutions for a sharp crack and a notch are derived in terms of the remote loading parameters J and K and material properties. The Weibull stress solutions from Section 3 are compared with full field finite element calculations in Section 4. Here both small scale yielding conditions (where the remote field is given by the elastic K field) and a finite width geometry (a compact tension specimen) are examined. Section 5 contains a discussion of the results including the mesh size dependence of FE Weibull stress solutions and concluding remarks are presented in Section Definition of the Weibull Stress The original Weibull stress definition in (Beremin, 1983) (2), relies on the understanding that the reference volume V o should be large enough so that the statistical independence of neighbouring volumes V o may be assumed. In practise, however, as discussed in (Xia and Shih, 1996), the value of V o can be arbitrarily set as long as it is kept constant for the material under study. The effect of V o is simply to scale the magnitude of the Weibull stress and it may be seen from (2) that the ratio between the Weibull stresses for two different reference volumes, V o1 and V o2,isgivenby σ w (V o1 ) σ w (V o2 ) = ( Vo2 V o1 ) 1/m. (4) tex; 26/09/1996; 7:35; p.2

3 Weibull stress solutions 247 Y r θ (x,y) X Crack tip Figure 1. Coordinate system used for crack tip fields. In an FE calculation it could be argued, along the lines of (Beremin, 1983), that the element size should also be based on the requirement of statistical independence thus requiring that the smallest element be on the order of a few grain sizes. However, mathematically, this seems to be an unnecessary requirement as in the case of a nonsingular stress field, the computed Weibull stress value should be independent of the finite element mesh used. As will be seen below the Weibull stress does depend on the mesh size when the stress fields are singular, as is the case for a sharp crack. In this case the choice of a physical length scale for the mesh size may be appropriate. An alternative more physically meaningful approach is to assign a finite notch root radius to the crack tip. The latter approach is adopted here. For mathematical analysis it proves more convenient to replace the sum in (2) by an integration over the fracture process zone, as in (Ruggieri and Dodds, 1996). In transforming from a sum to an integral, the representative volume, V i, is shrunk to an infinitesimal size. The implicit assumption is that the microcrack distribution remains constant even for infinitesimally small elements. Equation 2 then becomes ( ) 1 1/m σ w = σ1 m d, (5) V o where is the volume of material inside the fracture process zone (FPZ), defined in (Ruggieri and Dodds, 1996) as the region where σ 1 > λσ o. Here λ is a parameter > 1 which scales the yield stress σ o.whenλ = 1 the FPZ is of similar size to the crack tip plastic zone. The dependence of the Weibull stress on the size of the FPZ, i.e., the magnitude of λ, will be discussed later in the text. For plane stress or plane strain conditions, (5) becomes, ( ) B 1/m σ w = σ1 m da, (7) V o A where B represents the out-of-plane thickness of the cracked body and A denotes the area of the FPZ normal to the crack front. We may rewrite (7) as ( ) 1 1/m σ w = σ1 m da, (8) L 2 A (6) tex; 26/09/1996; 7:35; p.3

4 248 Y.Leietal. Figure 2. Normalised Weibull stress vs. normalised K for a sharp crack in a linear elastic material. Figure 3. Coordinate system used for notch. with a length scaling parameter, L, defined as L = Vo B. (9) It will be seen in next section that L is the appropriate normalisation for the small scale yielding problem as it is the only additional length scale related to the Weibull stress under small scale yielding conditions. The Weibull stress for two-dimensional problems can be evaluated by integrating (8) when the stress field near the crack tip is known. In the next section, a range of Weibull stress solutions are presented. 3. Weibull stress solutions 3.1. ELASTIC MATERIAL Sharp crack solution The maximum principal stress in the vicinity of a crack tip in an elastic material can be obtained from the K field (Hertzberg, 1989) σ 1 = K (1 + sin 1 2πr 2 θ)cos 1 θ. (10) tex; 26/09/1996; 7:35; p.4

5 Weibull stress solutions 249 where K is the mode I stress intensity factor and r and θ are polar coordinates with the origin at the crack tip (see Figure 1). Substituting (10) into (8) and integrating over the FPZ, defined by (6), leads to the following relations, depending on the value of the Weibull modulus m, σ w λσ o = [ Ce (4 m) ( K 2 (λσ o ) 2 L ) 2 ] 1/m for m<4, σ w for m > 4. (11) where L is the length parameter defined in (9) and results naturally from the integration of (8). The constant C e is given by C e = 1 π [(1 + sin 1 π 2 2 θ)cos 1 2 θ]4 dθ. (12) 0 Integration of (12) yields C e = Equation (11) is plotted in Figure 2 for m = 3and λ =1to5. Equation (11) applies to both plane stress and plane strain states as the maximum principal stress fields for the two conditions are the same when the material is elastic. From (11), it may be seen that the Weibull stress for the elastic stress field remains finite as long as m<4in spite of the presence of the stress singularity at the crack tip. However, if m > 4, the Weibull stress tends to infinity. This is an important result since for most materials m is usually found to be larger than 4, e.g., a typical value for an enginering ceramic is m = 10 (Ashby and Jones, 1986). An infinite Weibull stress corresponds to a probability of cleavage failure of 1 (3), regardless of the applied load. Clearly this is a nonphysical result which restricts the applicability of (11). It should be pointed out that the use of a finite strain theory will not remove this singularity, e.g. refer to the finite strain singular crack tip elastic solutions in (Geubelle, 1995). The singularity in the Weibull stress may be removed by having a notch of finite root radius at the crack tip. Alternatively, the size of the near tip element may be taken to be some fixed finite material length. Both of these approaches are essentially equivalent as the near tip element will play the same role as a notch, (provided of course that singular elements are not used). In the next section the solution for the Weibull stress for a finite notch is presented Notch solution The solution for the stress field near a notch in an elastic material is given in (Tada et al., 1973). The maximum principal stress σ 1 near a notch in an elastic material can be expressed as σ 1 = K ( cos 1 2πr 2 θ + 1 ) (ρ ) 2 + sin 2 θ (13) 2 r where ρ is the radius of the notch (see Figure 3) and K is the magnitude of the surrounding K field. The extent of the fracture process zone (6) for this case can be determined by substituting (13) into (6) and solving for r, K 2 2πr(λσ o ) 2 ( cos( 1 2 θ) + 1 ) (ρ ) sin 2 θ 1 > 0. (14) 2 r tex; 26/09/1996; 7:35; p.5

6 250 Y.Leietal. Figure 4. Normalised Weibull stress versus L/ρ with a fixed value of normalised K, for a notch in an elastic material. For this situation, however, a closed form solution could not be obtained by combining (13) and (14) with (8). Instead (8) was integrated numerically. For the notch configuration, the Weibull stress can be expressed in terms of the following independent variables: notch radius, ρ, stress intensity, K, Weibull modulus, m, and length scale parameter, L. A plot in log-log coordinates of the normalised Weibull stresses, σ w /σ o, obtained from the numerical integration, versus the reciprocal of the normalized notch radius, L/ρ, for a given value of normalised K is shown in Figure 4. Here, it is found that curves for different values of m are straight lines, and that their slopes can be approximately represented by (m 4)/2m when m > 4. This suggests that the effect of ρ in the Weibull stress expression can be incorporated through a (L/ρ) (m 4)/2m term as follows σ w /σ o = ( ) L (m 4)/2m ˆF 1 (K,L,m). (15) ρ If the normalised Weibull stress is now plotted against the normalised K 2 for a given ρ in log-log coordinates, as shown in Figure 5, one again finds a linear dependency for a range of m values. Furthermore, for m > 4 the lines are parallel with a slope of 1/2. Hence, the functional dependency of σ w on both K and L can be expressed as σ w /σ o = Ĉ en (m) ( ) m 4 L 2m ρ K Lσo, for m > 4, (16) where Ĉ en (m) is a function to be determined. In Figure 6, normalised curves of σ w versus normalised K 2 are shown for a range of m values and for λ = 1 and 5. Note that when m > 4 the effect of λ on the Weibull stress is negligible. The function Ĉ en (m) has been determined numerically for m > 4 and is shown in Figure 7. For m > 15 the curve is well approximated by the relation Ĉ en = /m 0.02, shown by the dashed line in Figure 7. For values of tex; 26/09/1996; 7:35; p.6

7 Weibull stress solutions 251 Figure 5. Normalised Weibull stress versus normalised K with fixed value of ρ/l, for a notch in an elastic material. Figure 6. Effect of λ on the Weibull stress for a notch in an elastic material. m<4, it has been found that (11) (i.e. the sharp crack solution) gives a good estimate of the Weibull stress for the notch case. From (11), it can be seen that the Weibull stress depends on the size of the fracture process zone, that is through λ,form<4. However, the effect of λ is negligible when m > 4. This is due to the fact that, for large values of m, the major contribution to the Weibull stress comes from material points close to the notch root regardless of the value of λ tex; 26/09/1996; 7:35; p.7

8 252 Y.Leietal. Figure 7. Coefficient C en for a notch in a linear elastic material WEIBULL STRESS SOLUTIONS FOR AN ELASTIC-PLASTIC MATERIAL Sharp crack solution For a sharp crack in a power law hardening material, the near tip stress distribution is given by the HRR field (Rice and Rosengren, 1968; Hutchinson, 1968). Consider an elastic-plastic material whose stress-strain behaviour can be represented by a Ramberg Osgood material law, ɛ ɛ o = σ σ o + α ( ) σ n, (17) σ o where α is a material constant, n is the strain hardening exponent, and σ o and ɛ o a reference stress and strain, respectively. Here ɛ o = σ o /E with E the elastic modulus. The maximum principal stress in a polar coordinate system centered at the crack tip (see Figure 1) can be expressed as ( σ 1 /σ o = J αɛ o σ o I n r ) 1 n+1 f(θ,n), (18) where J represents the J integral and I n the HRR field constant related to the hardening exponent n (Rice and Rosengren, 1968; Hutchinson, 1968). The HRR principal stress function f(θ,n)can be obtained from the HRR field functions, σ rr, σ θ,θ and σ rθ via, f(θ,n)= 1 2 [( σ rr + σ θθ ) + ( σ rr σ θθ ) σ rθ 2 ]. (19) Integration of (8) then yields the following expression for the normalised Weibull stress, σ w λσ o = [ ˆ (n) 1 m 2(n+1) ( ) ] J 2 1/m for m<2(n + 1), αɛ o σ o Lλ n tex; 26/09/1996; 7:35; p.8

9 Weibull stress solutions 253 (a) (b) Figure 8. Details of the boundary layer mesh: (a) remote mesh, (b) mesh in the notch region. σ w for m > 2(n + 1), (20) where ˆ (n) is a function which depends only on the hardening exponent n, ˆ (n) = 1 [ ] 2(n+1) π f(n,θ) dθ. (21) In 2 0 σ o The function ˆ (n) can be evaluated numerically and depends strongly on n. For instance, ˆ (5) = and ˆ (10) = Note that (20) has the same form as (16) when n = 1. From (20), the Weibull stress for the elastic-plastic HRR field is finite as long as m< 2(n + 1) despite the stress singularity at the crack tip. However, for m > 2(n + 1) the Weibull stress is infinite. For many ductile engineering materials the latter inequality may hold, e.g. for low hardening steels, n 10, values quoted in the literature for m typically range from 10 to 30, (Beremin, 1983; Busso et al., 1997; Minami et al., 1992) Notch solution No suitable notch field solution analogous to (13) was found for a power law elastic-plastic material. Therefore a finite element analysis, using a boundary layer mesh, see e.g. (Moran et al., 1990), was carried out to assist in the formulation of a suitable closed form expression. In the boundary layer analysis, the elastic K-field is applied at the remote boundary of the FE mesh and the material deforms plastically in the vicinity of the notch. The Weibull stress is tex; 26/09/1996; 7:35; p.9

10 254 Y.Leietal. then obtained from (5) by integrating the FE stresses obtained within the FPZ. Further details of the FE analysis is provided in Section 4. Following a similar method to that used to obtain (16), the corresponding Weibull stress relation can be obtained, ( ) m 2(n+1) ( ) 1 σ w L m(n+1) J n+1 = Ĉ pn (m, n) σ o ρ αɛ o σ o L for m > 2(n + 1). (22) The function Ĉ pn in (22) depends on m and n and has been determined numerically from the FE analysis. A close approximation to Ĉ pn for the range of n and m values examined in this work is Ĉ pn = ( ) 1 2(n+1) ( ˆ (n) 1 + n + 1 ) 10(n + 1), for n > 3. (23) m 2 The effect of λ on the Weibull stress was found to be negligible for m > 2(n + 1), the trend is similar to that shown in Figure 6 for the notch in an elastic material. Hence λ has not been included in (22). For m < 2(n + 1), the sharp crack solution provides a good approximation to the Weibull stress around the notch. Again it may be seen that for n = 1, (22) reduces to the form of the elastic solution (16). Equation (22) is valid only if the notch radius is considerably smaller than the plastic zone size, i.e, ρ L J αɛ 0 σ 0 L. (24) The applicability of (22) over the full range of loading for a cracked specimen will be examined in the next section. 4. Finite element solutions for Weibull stress Small strain finite element analyses have been performed to verify the Weibull stress expressions derived in Section 3 under idealised K or J dominant conditions and to examine their applicability to a real crack geometry, in this case a compact tension specimen. The stress strain relationship for the elastic-plastic analyses in this section is described by (17). The finite element code ABAQUS (1996) has been employed and the Weibull stress is computed numerically from the FE results using (2). Four noded plane strain elements are employed throughout. The material properties for the SSY analyses are E/σ o = 500, ν = 0.3 and α = 1. Analyses have been carried out for a range of hardening exponents and these will be indicated where relevant FE ANALYSIS OF BOUNDARY LAYER MODEL Elastic analysis The elastic stress fields are obtained through a boundary layer or small scale yielding formulation (Moran et al., 1990). A sample FE mesh used in the analysis is shown in Figure 8(a). The K field is applied at the remote boundary which is typically times the smallest crack tex; 26/09/1996; 7:35; p.10

11 Weibull stress solutions 255 Figure 9. Comparison of normalised Weibull stress obtained numerically and analytically (11) for a sharp crack in a linear elastic material. tip element or notch root radius. Figure 8(b) shows a zoomed-in view of the mesh used for the notch analysis. The Weibull stress is then evaluated by inserting the FE stresses into (2). Figure 9 shows the results for a sharp crack in a linear elastic material for values of λ from 1 to 5. The predictions of (11) are plotted in the figure for comparison. It is clear that (11) agrees with the FE computations. The FE results for a notch in a linear elastic material are plotted in Figure 10 together with the predictions of (16) (solid lines in the figure). In this plot the notch radius varies but the ratio L/ρ remains fixed. In Figure 10, ρ o is a reference notch radius which is taken as 10 6 times the outer mesh radius where the boundary conditions are applied. Comparison between the predicted results and the FE solutions shows that good agreement is obtained throughout. Comparisons have also been made for different L/ρ ratios and similar agreement is achieved provided ρ is much smaller than the mesh outer radius (this ensures that the K field dominates in a sizeable region ahead of the notch) Elastic-plastic analysis The Weibull stress solution given by (20) is plotted in Figures 11(a) and (b) for m = 10 and m = 21, respectively. The hardening exponent of the material is taken as n = 10 in each case. (Note that the largest value of m which gives a finite Weibull stress for this value of n is m = 22.) The FE results for λ = 1 5 are also included for comparison. Good agreement is achieved in Figure 11(a) for large values of λ but agreement is poor for λ 6 2. This is due to the fact that (20) has been derived from the HRR fields. These fields, by definition, are dominant within a small region around the crack tip consistent with a large value of λ. Even under small scale yielding conditions it is well known that due to the effect of elasticity not accounted for in the HRR field derivation, the crack tip stress distributions do not agree precisely with the HRR field over distances greater than r/(j/σ o )>3, (e.g. see (O Dowd and tex; 26/09/1996; 7:35; p.11

12 256 Y.Leietal. Figure 10. Comparison of normalised σ w obtained numerically and analytically (16) for a notch in a linear elastic material. Shih, 1991)). When λ = 5, most of the material points around the crack tip which contribute towards the Weibull stress will be located entirely inside the valid HRR zone, i.e., the fracture process zone is considerably smaller than the zone of HRR validity. However for smaller values of λ, the contribution of elements outside the zone of HRR validity become significant. When taking λ = 5 only principal stresses greater than 5 times the yield stress contribute to the Weibull stress. Finite strain analyses of a blunting crack in an elastic plastic material show that for a low hardening material, n = 10, the maximum principal stress is about 3 times the yield stress (O Dowd and Shih, 1991). Thus, the applicability of (20) for low values of m may be questionable in practice. In Figure 11(b) results are presented for m = 21. In this case it is seen that good agreement between the analytical and numerical solution is obtained for all values of λ. When the m value approaches 2(n + 1), the largest contributions to the Weibull stress are from a small region close to the crack tip, i.e., in the HRR dominance region, regardless of the value of λ. An alternative to using the HRR field in (20) is to use the FE small scale yielding solution for a sharp crack to develop an expression for the Weibull stress in a manner analogous to that used for the plastic notch solution. The use of finite strain kinematics would also improve the applicability of the approach by including notch blunting in the analyses. Whether such approaches lead to a better prediction of the Weibull stress under small scale yielding conditions will be examined in future work. As discussed in Section 3.2.2, the form of the elastic plastic notch result (22) and the function Ĉ(n,m), while drawing from the analytical results for a linear elastic material, were determined from FE analyses. Figure 12(a) shows the FE results for a notch in an elasticplastic material for n = 10, L/ρ = and various values of m (the solid lines in the figure represent the predictions of (22)). FE results for different notch radii are plotted in tex; 26/09/1996; 7:35; p.12

13 Weibull stress solutions 257 (a) (b) Figure 11. Comparison of normalised σ w obtained numerically and analytically (20) for a sharp crack in elastic-plastic material, (a) m = 10, (b) m = tex; 26/09/1996; 7:35; p.13

14 258 Y.Leietal. (a) (b) Figure 12. Normalised σ w for a notch in an elastic-plastic material, (a) fixed notch radius, (b) a range of notch radii. Figure 12(b) where, as in the elastic analysis, ρ o takes a value of 10 6 times the mesh outer radius. As seen in Figure 12(b), if L/ρ and m are fixed the solutions corresponding to different notch radii collapse to a single curve as expected from (22) FE ANALYSES FOR A CT SPECIMEN We next compare the Weibull stress solutions with the results from an FE analysis of a CT specimen with a/w = 0.5 (illustrated schematically in Figure 13(a)). As in the previous section, small strain analyses were carried out and plane strain conditions examined. A typical FE mesh used in the analysis is shown in Figure 13(b), the mesh in the notch region being tex; 26/09/1996; 7:35; p.14

15 Weibull stress solutions 259 (a) P a W (b) P Figure 13. (a) Schematic of CT specimen (b) Finite element mesh. Figure 14. Comparison between normalised σ w obtained numerically and analytically (11) for a CT specimen with a sharp crack in a linear elastic material tex; 26/09/1996; 7:35; p.15

16 260 Y.Leietal. Figure 15. Comparison between normalised σ w obtained numerically and analytically (16) for a CT specimen with a notch in a linear elastic material. Figure 16. Estimation error of (16) for a CT specimen with a notch in a linear elastic material. similar to that shown in Figure 8(b). The material properties are taken as E/σ o = 420, ν = 0.3, n = 5andα = 1, which are typical values for a pressure vessel steel. (For the linear elastic analysis, n = 1.) Results are presented for the case, a/l = 25, ρ/l = The FE results for a sharp crack in a linear elastic material for m = 3 and different values of λ are plotted in Figure 14. The similarity with Figure 9 may be noted. The solution from (11) is represented by the solid line and the excellent agreement between the FE and analytical solutions is apparent tex; 26/09/1996; 7:35; p.16

17 Weibull stress solutions 261 Figure 15 shows the linear elastic FE results for a notch for various values of m. Solutions from (16) are plotted in solid lines in the figure. In Figure 16 the percentage differences between the FE and theoretical solution (16) are plotted against normalised load, K. Here σw TH FE is the Weibull stress estimated from (16) and σw the corresponding FE result. It may be seen that the estimate from (16) is within 5 percent of the numerical solution for all the cases examined. Figure 17 shows the elastic-plastic FE results of a sharp crack for n = 5andtwovalues of m, m = 6 (Figure 17(a)) and m = 11 (Figure 17(b)). Solutions from (20) are plotted in solid lines in the corresponding figures. As was seen in the small scale yielding analysis (Figure 11), for small values of m (Figure 17(a)), the agreement between the FE results and the estimation from (20) is dependent on the value of λ and for larger m values (Figure 17(b)), good agreement is obtained for all λ values. Figure 18 shows the elastic-plastic FE results of a notch for n = 5 and different values of m. Solutions from (22) are plotted in solid lines. The corresponding estimation errors are plotted in Figure 19 against the normalised load. Here P 0 is the limit load evaluated from the Electric Power Research Institute (EPRI) plane strain equation, (Kumar, 1981). It may be seen that the errors are largest at low loads. This is due to the fact that at low loads the specimen remains predominantly elastic, i.e., the plastic zone size is on the order of the notch root radius. Hence, at low loads, the Weibull stress is more closely approximated by the elastic solution, (16), rather than (22). In order to predict the Weibull stress accurately over the full range of loading from elastic to elastic-plastic conditions it would be necessary to develop an interpolation scheme along the lines of the EPRI scheme for J, (Kumar, 1981). At higher loads the effect of specimen geometry may also become significant, i.e., the near tip fields may no longer be well represented by the small scale yielding notch distributions from which (22) is derived. However, for the CT specimen this effect does not appear to be significant for the load levels examined (P 6 1.3P 0 ). 5. Discussion For a sharp crack, the elastic solution for the Weibull stress (11) can be derived from the elastic-plastic solution (20) by taking n = 1 since, for a linear elastic material, J/(αɛ o σ o L) is equivalent to K 2 /(σ 2 o L) and ˆ (n = 1) = C e /4. For the case of a notch, the elastic solution (16) can also be derived from the elastic-plastic solution (22) by taking n = 1 in the latter. However, it should be pointed out that the coefficient Ĉ pn (n, m) in (22) does not reduce to Ĉ en (m) in (16) when n = 1. Comparison between Ĉ en (m) and Ĉ pn (n = 1,m), (23), in Figure 20, reveals that (23) gives a good approximation for Ĉ en only for small m values. It has been found that the expression given for Ĉ pn (n, m) in (23) leads to a good approximation of the Weibull stress only for relatively low hardening, namely n > 3. Further work is required to determine the function Ĉ pn (n, m) which is valid for all values of n. As discussed earlier, under certain conditions the analytical solutions can predict an infinite value for the Weibull stress. In practise this will lead to mesh dependence of Weibull stress FE solutions for sharp cracks. Figure 21 shows two examples where the Weibull stress values were determined from an FE boundary layer analysis of the type discussed in Section 4. Figure 21 shows the normalised Weibull stress plotted against crack tip element size for a linear elastic material (Figure 21(a)) and an elastic-plastic material (Figure 21(b)) with hardening exponent n = 5. Here the element size is normalised by the mesh outer radius. From Figure 21, it may tex; 26/09/1996; 7:35; p.17

18 262 Y.Leietal. (a) (b) Figure 17. Comparison of normalised σ w obtained numerically and analytically (20) for a CT specimen with a sharp crack in an elastic-plastic material, (a) m = 6, (b) m = 11. be seen that for both the elastic and elastic-plastic case, the calculated Weibull stress values do not converge to a constant value when the crack tip element size is reduced to the order of 10 7 of the outer mesh radius for m = 10 and m = 30, respectively. However, for m = 3 in Figure 21(a) and m = 11 in Figure 21(b), the Weibull stress values are independent of the mesh size when the crack tip element size is less than 10 4 of the outer mesh radius. As discussed in Section 2 the mesh dependence may be eliminated by having a fixed mesh size tex; 26/09/1996; 7:35; p.18

19 Weibull stress solutions 263 Figure 18. Comparison of normalised σ w obtained numerically and analytically (22) for a CT specimen with a notch in an elastic-plastic material. Figure 19. Estimation error of (22) for a CT specimen with a notch in an elastic-plastic material tex; 26/09/1996; 7:35; p.19

20 264 Y.Leietal. Figure 20. Comparison of coefficient C en obtained from numerical analysis and (23) with n = 1. or a finite notch root radius, both of which have essentially the same effect, i.e., limit the maximum stress at the crack tip, and both introduce a physical length scale into the problem. The Weibull stress solutions for a linear elastic material (11) and (16) apply under both plane stress and plane strain conditions. For the elastic-plastic analysis, the results (20) and (22) are applicable only for plane strain conditions. Similar results could be obtained for plane stress from the plane stress HRR field and plane stress FE solutions. Furthermore constraint effects can be included into (20) and (22) by introducing a constraint factor, Q (O Dowd and Shih, 1994) into the stress fields in (18). The two fracture parameters, J and Q, will then be coupled into one failure criterion through the Weibull stress, σ w. The Weibull stress is used to determine the probability of cleavage failure. Rather than presenting the results in this paper in terms of the Weibull stress, one could equivalently write them as cleavage probability predictions. Substituting the relevant Weibull stress solutions into (3) yields { ( ) m ( ) } λσo C e K 4 P f (K) = 1 exp (4 m) Lλσo { = 1 exp σ u ( σo σ u ) m ( ) m 4 ( ) m } L 2 Ĉen K ρ Lσo m<4 m > 4 (25) for a linear elastic material. The notch solution has been used for m > 4 as the crack solution predicts P f = 1 in this case. Similarly for an elastic-plastic power law hardening material, { ( λσo P f (J ) = 1 exp σ u ) m ˆ ( 1 m 2(n+1) J αɛ o σ o Lλ n+1 ) 2 } m<2(n + 1) tex; 26/09/1996; 7:35; p.20

21 Weibull stress solutions 265 (a) (b) Figure 21. Mesh size effect on the Weibull stress for a sharp crack (a) linear elastic material, (b) elastic-plastic material. { = 1 exp ( σo σ u ) m ( ) m 2(n+1) ( ) m } L n+1 J n+1 Ĉpn m ρ αɛ o σ o L m > 2(n + 1). (26) Therefore, if m and σ u have been determined experimentally, (see e.g. (Sherry et al., 1995)) the probability of cleavage failure may be determined directly from (25) or (26). Of course the inherent limitations of the above equations needs to be considered before adopting such an approach tex; 26/09/1996; 7:35; p.21

22 266 Y.Leietal. By replacing J in the first of (26) with K, assuming small scale yielding conditions hold, and taking λ = 1, the following is obtained, ( ) ( ) 1 m ( σo K 2 ) 2 ln = C m 1 P f σ u σo 2L, for m<2(n + 1), (27) which is an alternative representation of the Weibull probability (Weibull, 1939). This equation is consistent with that presented in (Beremin, 1983). From the analysis presented in this paper, the constant C m may be seen to be C m = ˆ (1 ν 2 ) 2 1 m. (28) α 2(n+1) 2 The results presented in the paper are for small strain, two-dimensional stationary cracks. The effect of crack blunting has not been explicitly accounted for though it is expected that the notch solutions of Section 3, with ρ taken to be the current crack tip radius may be representative of the solutions which would be obtained from a finite strain analysis. Extension to the case of a three-dimensional crack, where typically J or K will vary along the crack front, is more difficult as full three-dimensional crack solutions are not currently available. However, plane strain and plane stress may be considered to provide upper and lower bounds, respectively, for the Weibull stress. Similarly, since it is known that small amounts of crack growth tend to reduce the magnitude of the Weibull stress (Ruggieri and Dodds, 1996; Busso et al., 1997), the stationary crack solution may be taken to be a conservative estimate of the actual Weibull stress for a growing crack. The methods used to determine the Weibull stress for the two parameter distribution, (2), could, in principle, be extended to the case of a three parameter distribution. The latter can incorporate, e.g., a threshold stress below which the contribution to the Weibull stress is negligible (e.g. (Ruggieri and Dodds, 1996)). The consequence of incorporating such a threshold stress on the cleavage failure probabilities for a ferritic weld steel has been investigated in (Busso et al., 1997). 6. Conclusions Analytical and semi-analytical Weibull stress expressions have been derived for sharp cracks and notches in terms of remote loading parameter, J or K, and material properties. The expressions have been verified numerically using the results from finite element analyses and good agreement has been obtained in all cases. The effect of notch acuity and choice of fracture process zone size on the magnitude of the Weibull stress as well as the limit of applicability of the proposed relations have been discussed. The expressions have been used to interpret the mesh size dependence of the Weibull stress values obtained from finite element calculations. For a sharp crack, the Weibull stress is finite as long as the Weibull modulus, m<2(n + 1), wheren is the hardening exponent of the material. Under these conditions, the Weibull stress calculated from the finite element analyses shows almost no dependence on the level of mesh refinement. For m > 2(n+1), the theoretical Weibull stress tends to infinity, with the values of Weibull stress obtained from finite element tex; 26/09/1996; 7:35; p.22

23 Weibull stress solutions 267 analysis being strongly mesh size dependent. For this case, the use of a finite notch at the crack tip is proposed. It has been shown that Beremin type cleavage failure probability expressions in terms of J or K and material properties can be obtained by the use of the above closed form Weibull stress solutions. Acknowledgements Support for this work by the IMC (Industry Management Committee of the United Kingdom) is gratefully acknowledged. The authors would also like to thank Dr. R. Moskovic from MEP and Dr. R. Ainsworth from NEL for their invaluable assistance with this work. The ABAQUS programme was provided under academic licence by Hibbitt, Karlsson, and Sorensen Inc., Providence, Rhode Island. References ABAQUS V.5.5 (1996). Hibbitt, Karlsson and Sorensen Inc., Providence, RI. Ashby, M.F. and Jones, D.R.H. (1986). In: Engineering Materials 2, Pergamon Press, UK. Beremin, F.M. (1983). A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metallurgy Transactions A 14, Busso, E.P., Lei, Y., O Dowd, N.P. and Webster, G.A. (1997). Submitted for publication. Evans, A.G. (1978). A general approach for the statistical analysis of multiaxial fracture. Journal of the American Ceramic Society 61, Freudenthal, A.M. (1968). Statistical approach to brittle fracture. Fracture: An Advanced Treatise, Vol. II. (Edited by H. Liebowitz), Academic Press, New York, Geubelle, P.H. (1995). Finite deformation effects in homogeneous and interfacial fracture. International Journal of Solids and Structures 32, Hertzberg, R.W. (1989). Deformation and Fracture Mechanics of Engineering Materials, 3rd edn., John Wiley and Sons. Hutchinson, J.W. (1968). Singular behavior at the end of a tensile crack in a hardening material. Journal of the Mechanics and Physics of Solids 16, Kumar, V., German, M.D. and Shih, C.F. (1981). An engineering approach for elastic-plastic fracture analysis. EPRI Topical Report NP-1931 Electric Power Research Institute, Palo Alto, California. Minami, F., Bruckner-Foit, A., Munz, D. and Trolldenier, B. (1992). Estimation Procedure for the Weibull parameters used in the local approach. International Journal of Fracture Moran, B., Ortiz, M. and Shih, C.F. (1990). Formulation of implicit finite elements methods for multiplicative finite deformation plasticity. International Journal for Numerical Methods in Engineering 29, O Dowd, N.P. and Shih, C.F. (1991). Family of crack-tip fields characterized by a triaxiality parameter: Part I Structure of fields. Journals of the Mechanics and Physics of Solids 39, O Dowd, N.P. and Shih, C.F. (1994). Two-parameter fracture mechanics: theory and applications. Fracture Mechanics: Twenty Fourth Volume ASTM STP 1207 (Edited by J.D. Landes et al.), American Society for Testing and Materials, Philadelphia, Rice, J.R., and Rosengren, G.F. (1968). Plane strain deformation near a crack tip in a power law hardening material. Journal of the Mechanics and Physics of Solids 16, Ruggieri, C. and Dodds, R.H. (1996). Transferability model for brittle-fracture including constraint and ductile tearing effects A probabilistic approach. International Journal of Fracture 79, Sanderson, D.J., Sherry, A.H., Beardsmore D.W., Howard, I.C. and Li Z.H. (1996). Draft Procedure for the Use of Local Approach to Predict the Cleavage and Ductile Fracture Behaviour of Steels. AEA Technology Report 0760 UK. Sherry, A.H., Sanderson, D.J., Lidbury, R.A. and Kikuchi, K. (1995). The application of local approach to assess the influence of in-plane constraint on cleavage fracture. American Society of Mechanical Engineers, Pressure Vessels and Piping Division PVP 304, tex; 26/09/1996; 7:35; p.23

24 268 Y.Leietal. Tada, H., Paris, P.C. and Irwin, G.R. (1973). The Stress Analysis of Cracks Handbook. Del Research, Hellertown, PA. Weibull, M. (1939). The phenomenon of rupture in solids. Ingeniors Vetenskapes Akademien, Handlingar 153, 55. Xia, L. and Shih, C.F. (1996). Ductile crack-growth III transition to cleavage fracture incorporating statistics. Journal of Mechanics and Physics of Solids 44, tex; 26/09/1996; 7:35; p.24