Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

Approximate J estimates for tension-loaded plates with semi-elliptical surface cracks

This paper provides a simplified engineering J estimation method for semi-elliptical surface cracked plates in tension, based on the reference stress approach. Note that the essential element of the reference stress approach is the plastic limit load in the definition of the reference stress. However, for surface cracks, the definition of the limit load is ambiguous (“local” or “global” limit load), and thus the most relevant limit load (and thus reference stress) for the J estimation should be determined. In the present work, such limit load solution is found by comparing reference stress based J results with those from extensive 3-D finite element (FE) analyses. Based on the present FE results, the global limit load solution proposed by Goodall for surface cracked plates in combined bending and tension was modified, in the case of tension loading only, to account for a weak dependence on w/c and was defined as the reference normalizing load. Validation of the proposed equation against FE J results based on actual experimental tensile data of a 304 stainless steel shows excellent agreements not only for the J values at the deepest point but also for those at an arbitrary point along the crack front, including at the surface point. Thus the present results provide a good engineering tool for elastic-plastic fracture analyses of surface cracked plates in tension.     

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/t values of 0.2, 0.4, 0.6 or 0.8 and a/c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2?a/c?1 and 0?a/t?0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.     

A new constraint based fracture criterion for surface cracks

A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic-plastic finite element analyses were used to determine the variations of a constraint parameter (α_h) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jα_h is a maximum. The criterion is verified with test results from surface cracked specimens.     

Elastic-plastic J and COD estimation schemes for throughwall circumferentially cracked elbow under in-plane closing moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic-plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic-plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, no such schemes are available for throughwall circumferentially cracked elbow (or pipe bend), an important component for LBB analysis. In this paper, simple J and COD estimation schemes are proposed for throughwall circumferentially cracked elbow subjected to closing bending moment. The ovalisation of elbow cross-section has a significant bearing on its fracture behavior. Therefore, unlike conventional deformation theory plasticity analysis, incremental flow theory is adopted considering both material and geometric non-linearities in the development of the proposed estimation schemes. Although it violates Ilyushin’s theorem, it has been shown that the resulting estimation schemes is still reasonably accurate for engineering purposes. Finally, experimental/numerical validation has been provided by comparing the J-integral and COD between numerical/test data and predictions of the proposed estimation schemes.     

Residual stresses and fracture mechanics analysis of a crack in welds of high strength steels

This study presented the characteristics of residual stresses in welds of high strength steels (POSTEN60, POSTEN80) whose tensile strengths were 600 MPa and 800 MPa, respectively. Three-dimensional thermal elastic-plastic analyses were conducted to investigate the characteristics of welding residual stresses in welds of high strength steels through the thermal and mechanical properties at high temperatures obtained from the elevated temperature tensile tests. A finite element analysis method which can calculate the J-integral for a crack in a residual stress field was developed to evaluate the J-integral for a centre crack when mechanical stresses were applied in conjunction with residual stresses.The results show that the volumetric changes associated with the austenite to martensite phase transformation during rapid cooling after welding of high strength steels significantly influence on the development of residual stresses in the weld fusion zone and heat-affected zone. For a centre crack in welds of high strength steels where only residual stresses are present, increased tensile strength of the steel, increased the J-integral values. The values of the J-integral for the case when mechanical stresses are applied in conjunction with residual stresses are larger than those for the case when only residual stresses are present.     

An alternative formulation of the Ritchie-Knott-Rice local fracture criterion

In the paper an alternative formulation of the RKR local fracture criterion is proposed. It is based on the features of the stress distribution in front of a blunted crack in an elastic-plastic material. The stress distribution is computed using the finite strain option in the finite element method. It is postulated that the opening stress in front of the crack should be greater than the critical one, σ_c, over the distance l ? l_c, where l_c is considered as a material parameter. The hypothesis is applied to estimate the influence of the in-plane constraint on fracture toughness. New formulas to compute the critical value of the J-integral are derived both for the small scale yielding and large plastic deformations in front of the crack. The results obtained are compared with the Sumpter and Forbes experimental results and with the O’Dowd analytical formula concerning the J_c = J_c(J_IC,Q) relation.     

Reference stress based elastic-plastic fracture analysis for circumferential through-wall cracked pipes under combined tension and bending

This paper proposes an engineering method to estimate the J-integral and the crack opening displacement (COD) for circumferential through-wall cracked pipes under combined tension and bending. The proposed method to estimate the COD is validated against three published pipe test data, generated from a monotonically increasing bending load with a constant internal pressure, which shows excellent agreements. Further validation is performed against three-dimensional, elastic-plastic finite element results using actual tensile data of a TP316 stainless steel at the temperature of 288°C. The FE results of the J-integral and the COD, resulting from six cases of proportional and non-proportional combined tension and bending, compare very well with those estimated from the proposed method.Excellent agreements of the proposed method with experimental data and the detailed FE results firstly provide sufficient confidence in the use of the proposed method to the LBB analysis of through-wall cracked pipes under combined tension and bending. More importantly, these validations suggest that the proposed method can be used to any combination of the bending-to-tension ratio, not only for proportional loading but also for non-proportional loading. Finally the proposed method is simple to use, which gives significant merits in practice, and thus is easy to be generalised to more complex situations.     

Three-dimensional stress state around quarter-elliptical corner cracks in elastic plates subjected to uniform tension loading

Detailed full-field three-dimensional (3D) finite element analyses have been conducted to study the out-of-plane stress constraint factor T_z around a quarter-elliptical corner crack embedded in an isotropic elastic plate subjected to uniform tension loading. The distributions of T_z are studied in the forward section (0° ? θ ? 90°) of the corner cracks with aspect ratios a/c of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. In the normal plane of the crack front line, T_z drops radially from Poisson’s ratio at the crack tip to zero beyond certain radial distances. Strong 3D zones (T_z > 0) exist within a radial distance r/a of about 4.6-0.7 for a/c = 0.2-1.0 along the crack front, despite the stress-free boundary conditions far away. At the same radial distance along the crack front in the 3D zones, T_z increases from zero on one free surface to a peak value in the interior, and then decreases to zero on another free surface. The distributions of T_z near the corner points are also discussed. Empirical formulae describing the 3D distributions of T_z are obtained by fitting the numerical results, which prevail with a sufficient accuracy in the valid range of 0.2 ? a/c ? 1.0 and 0° ? θ ? 90° except very near the free surfaces where T_z is extremely low. Combined with the K-T solution, the transition of approximate plane-stress state near the surfaces to plane-strain state in the interior can be characterized more accurately.     

High-rate J-testing of toughened polyamide 6/6: Applicability of the load separation criterion and the normalization method

This paper examines the applicability of the load separation criterion and the normalization method in determining J_R curve of a toughened polyamide 6/6 at high loading rates (1 m/s). The analysis of procedure problems associated to this high experimental rate is performed. The results obtained using the normalization method are then compared with those measured via multi-specimen testing procedures proposed by ESIS (Technical Committee 4). The results show that, unlike low loading rate tests, the presence of the oscillations in the load vs displacement traces, due to the inertial effects produced during the impact, complicate considerably the elaboration of the data, with particular reference to the identification of the separable blunting region. The comparison of J_Ic values obtained according to the different procedures examined indicates that the values of J_Ic = J_0.2 (taken at 0.2 mm crack growth) are in good agreement, whereas consistent differences among the values of J_Ic = J-blunting (taken at the blunting line) are observed.     

Effect of biaxial loads on elastic-plastic <Emphasis Type="Italic">J</Emphasis> and crack tip constraint for cracked plates: finite element study

Based on detailed two-dimensional (2-D) and three-dimensional (3-D) finite element (FE) analyses, this paper attempts to quantify in-plane and out-of-plane constraint effects on elastic-plastic J and crack tip stresses for a plate with a through-thickness crack and semi-elliptical surface crack under positive biaxial loading. For the plate with a through-thickness crack, plate thickness and relative crack length are systematically varied, whereas for the plate with a semi-elliptical surface crack, the relative crack depth and aspect ratio of the semi-elliptical crack are systematically varied. It is found that the reference stress based approach for uniaxial loading can be applied to estimate J under biaxial loading, provided that the limit load specific to biaxial loading is used, implying that quantification of the biaxiality effect on the limit load is important. Investigation on the effect of biaxiality on the limit load suggests that for relatively thin plates with small cracks, in particular with semi-elliptical surface cracks, the effect of biaxiality on the limit load can be neglected for positive biaxial loading, and thus elastic-plastic J for a biaxially loaded plate could be estimated, assuming that such plate is subject to uniaxial load. Regarding the effect of biaxiality on crack tip stress triaxiality, it is found that such effect is more pronounced for a thicker plate. For plates with semi-elliptical surface cracks, the crack aspect ratio is found to be more important than the relative crack depth, and the effect of biaxiality on crack tip stress triaxiality is found to be more pronounced near the surface points along the crack front.     

Mode I fatigue delamination growth in GFRP woven laminates at low temperatures

The cryogenic fatigue delamination behavior of glass fiber reinforced polymer woven laminates under Mode I loading has been investigated experimentally and numerically. Fatigue delamination tests were conducted using double cantilever beam specimens at room temperature, liquid nitrogen temperature (77 K) and liquid helium temperature (4 K). Fracture surface examination using scanning electron microscopy revealed delamination mechanisms under fatigue loading. A finite element analysis was also employed to calculate the J-integral range and damage distributions. The effects of temperature and loading condition on the fatigue delamination growth rates were discussed.     

Methods for calculating stress intensity factors in anisotropic materials: Part II—Arbitrary geometry

The problem of a crack in a general anisotropic material under conditions of linear elastic fracture mechanics (LEFM) is examined. In Part I, three methods were presented for calculating stress intensity factors for various anisotropic materials in which z = 0 is a symmetry plane and the crack front is along the z-axis. These included displacement extrapolation, the M-integral and the separated J-integrals.In this study, general material anisotropy is considered in which the material and crack coordinates may be at arbitrary angles. A three-dimensional treatment is required for this situation in which there may be two or three modes present. A three-dimensional M-integral is extended to obtain stress intensity factors. It is applied to several test problems, in which excellent results are obtained. Results are obtained for a Brazilian disk specimen made of isotropic and cubic material. Two examples for the latter are examined with material coordinates rotated with respect to the crack axes.     

J integral solution for semi-elliptical surface crack in high density poly-ethylene pipe under bending

The goal of this work is to analyse the severity of semi-elliptical crack defects and to study the degree of damage in the poly-ethylene pipe in bending during the crack propagation. The semi-elliptical cracks are considered in this work located in different position in the wall of the pipe. The three finite element method based on the computation of the J integral was used to analyse the fracture behaviour of these structures. The effect of the position, shape and size of the crack on the J integral values was highlighted. The effects of strain rate and the temperature on the J integral values were also examined. The obtained results show that the strain rates have a strong influence on the J integral values especially for circumferential crack at higher bending moment. However, the energy for circumferential crack is more important compared to axial crack. The effect of the depth of the crack becomes important when the ratio (a/t) reaches a critical value of 0.6 (a/t = 0.6), especially when the ratio a/c is weak (semi-elliptical crack, a/c = 0.2) where the J integral values becomes independently of the crack depth, this conclusion is opposite to the above for the poly-ethylene pipe subjected to internal pressure. We recall finally, that the temperature effect on circumferential cracks behaviour is more important compared to the axial cracks at critical crack size (a/c = 0.2 and a/t = 0.6). It is also shown that in the wall of pipe, the internal cracks are more dangerous than the external cracks.     

Crack resistance curves of GLARE laminates by elastic compliance

The objective of this work was to study the applicability of the elastic compliance technique for crack resistance curves evaluation of commercial GLARE laminates using small SE(B) and C(T) specimens. The experimental evaluation of R-curves of 25.0 mm wide SE(B) specimens of unidirectional GLARE 1 3/2 0.3 and 50.0 mm wide C(T) specimens of bidirectional GLARE 3 5/4 0.3 was performed. Fracture toughness was measured through a recently proposed experimental methodology based on standardized specimens and elastic-plastic methodologies (J-integral and CTOD δ₅), whereas crack growth was measured optically and estimated by elastic compliance. According to the results the elastic compliance technique seemed to be applicable to GLARE laminates, accurately predicting stable crack growth during the tests.     

T-Stress solutions for a semi-elliptical axial surface crack in a cylinder subjected to mode-I non-uniform stress distributions

This paper presents the T-stress solutions (T₁₁ and T₃₃) for semi-elliptical axial surface cracks in a cylinder subjected to mode-I non-uniform stress on the crack surface. Two cylindrical geometries with inner radius (R_i) to wall thickness (t) ratios R_i/t = 5 and 10 were considered. The T-stresses were applied along the crack front for normalized crack depth values a/t of 0.2, 0.4 and 0.5 and aspect ratios a/c of 0.2, 0.4, 0.6 and 1.0. Three stress distribution; uniform, linear and parabolic were applied to the crack face. In addition to these solutions, concrete formulation of the superposition principle is given for the T₃₃-stress, which is known as an elastic parameter that describes the out-of-plane crack tip constraint effect. Then, the validity of the formulation was shown through application of our T-stress solutions to the problem of an axial semi-elliptical surface crack in a cylinder subjected to internal pressure, and checking that the principle of superposition holds for the problem.     

Determination of K Id at or below NDTT using instrumented drop-weight testing

Two kinds of surface cracked specimens are numerically analyzed by the three dimensional elastic-plastic FEM. Near tip regions are divided into fine elements, and the stress and displacement fields at the crack tip are compared with HRR solutions.At first, surface cracked specimens subjected to bending are analyzed by changing the aspect ratio and depth/thickness ratio. The effect of the loading condition, crack shape and the crack depth on the stress and displacement fields are discussed. Then the pipe with surface crack subjected to bending is analyzed and the availability of the J-integral concept to the LBB analysis is discussed.In every specimen, it is shown that in the regions very near to the crack tip, the displacement field is similar to HRR solutions of plane strain. In the outer regions, however, the stress and displacement fields depend strongly on the shape, thickness, and loading conditions.     

Calculation of the energy J-integral for bodies with notches and cracks

The approximate solutions for calculation of the energy J-integral of a body both with a notch and with a crack under elastic-plastic loading have been obtained. The crack is considered as the limit case of a sharp notch. The method is based on stress concentration analysis near a notch/crack tip and the modified Neuber's approach. The HRR-model and the method based on an equation of equilibrium were also employed to calculate the J-integral. The influence of the strain hardening exponent on the J-integral is discussed. New aspects of the two-parameter J ^* _c-fracture criterion for a body with a short crack are studied. A theoretical investigation of the effect of the applied critical stress (or the crack length) on the strain fields ahead of the crack tip has been carried out.     

Some new practical equations for rapid calculation of J-integral in plates weakened by U-notches under bending

The paper deals with calculations of the J-integral for a plate weakened by U-notches under Mode I bending loading in the case of a material obeying a linear elastic law. The main aim of the study is to suggest some new equations suitable for calculations of the J-integral under bending loading. Although some closed-form solutions exist in the literature based on elliptic integrals, some expressions, simple to be implemented, are useful as a rapid engineering tool.