Limit loads and approximate J estimates for axial through-wall cracked pipe bends

This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bending. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bending is proposed.     

Fatigue strength assessment of laser stake‐welded web‐core steel sandwich panels

This paper investigates the fatigue strength assessment of web‐core steel sandwich panels. The production of these structures is made possible by laser stake welding. The investigation in this study considered two series of panels, one being an empty steel structure and the other filled with in situ polyurethane foam in order to increase the panel stiffness. Both series were tested under cyclic bending loading condition (R = 0) until one of the panel joints failed completely. A 3D panel bending response was analysed using finite element method. The J‐integral values at the panel joints were obtained by means of plane strain finite element analysis and by using displacements from 3D panel response. The influence of the weld geometry on the J‐integral value was investigated. It was found that the J‐integral value is similar in the cases of the average and critical geometry. The contact between the joint plates is possible in some cases, but its influence proved to be insignificant for the fatigue strength assessment. The study further shows that by using the average geometry, the J‐integral approach was able to identify the critical panel joints and present the fatigue strength results from both panel series in a narrow scatterband. The fatigue strength at two million cycles obtained for the panels within this study was in agreement with the laser stake welds and other steel joint types from previous studies. However, the slope of the panels fatigue resistance curve was found to be shallower than in the case of joints.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Bending modified J–Q theory and crack-tip constraint quantification

It is well known that the J–Q theory can characterize the crack-tip fields and quantify constraint levels for various geometry and loading configurations in elastic–plastic materials, but it fails at bending-dominant large deformation. This drawback seriously restricts its applications to fracture constraint analysis. A modification of J–Q theory is developed as a three-term solution with an additional term to address the global bending stress to offset this restriction. The nonlinear bending stress is approximately linearized in the region of interest under large-scale yielding (LSY), with the linearization factor determined using a two-point matching method at each loading for a specific cracked geometry in bending. To validate the proposed solution, detailed elastic–plastic finite element analysis (FEA) is conducted under plane strain conditions for three conventional bending specimens with different crack lengths for X80 pipeline steel. These include single edge notched bend (SENB), single edge notched tension (SENT) and compact tension (CT) specimens from small-scale yielding (SSY) to LSY. Results show that the bending modified J–Q solution can well match FEA results of crack-tip stress fields for all bending specimens at all deformation levels from SSY to LSY, with the modified Q being a load- and distance-independent constraint parameter under LSY. Therefore, the modified parameter Q can be effectively used to quantify crack-tip constraint for bending geometries. Its application to fracture constraint analysis is demonstrated by determining constraint corrected J–R curves.     

A comparative study of dynamic, ductile fracture initiation in two specimen configurations

In this work a single edge notched plate (SEN(T)) subjected to a tensile stress pulse is analysed, using a 2D plane strain dynamic finite element procedure. The interaction of the notch with a pre-nucleated hole ahead of it is examined. The background material is modelled by the Gurson constitutive law and ductile failure by microvoid coalescence in the ligament connecting the notch and the hole is simulated. Both rate independent and rate dependent material behaviour is considered. The notch tip region is subjected to a range of loading rates J by varying the peak value and the rise time of the applied stress pulse. The results obtained from these simulations are compared with a three point bend (TPB) specimen subjected to impact loading analysed in an earlier work [3]. The variation of J at fracture initiation, J _c, with average loading rate J is obtained from the finite element simulations. It is found that the functional relationship between J _c and J is fairly independent of the specimen geometry and is only dependent on material behaviour.     

EXPERIMENTAL J EVALUATION FROM A LOAD-DISPLACEMENT CURVE FOR HOMOGENEOUS AND OVERMATCHED SENB OR CCT SPECIMENS

Abstract This paper addresses the problem of the determination of J-intergral from experimentally measured quantities for homogeneous and overmatched single edge notched bending specimens (SENB) or through thickness centre cracked panels loaded in tension (CCT). Commonly, the experimental J-integral is calculated from the area under the load versus load-line displacement curve. Nevertheless, in the case of gross-section yielding, which currently occurs for short cracked specimens or overmatching cases, this methodology mis-estimates the real J-integral. A new proposal, based on analytical considerations and finite element calculations, is made to estimate the J-integral from the area under the load versus CMOD curves.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

Advances in J‐integral estimation of circumferentially surface cracked pipes

This paper describes enhanced J‐integral estimation schemes for pipes with circumferential semi‐elliptical cracks subjected to tensile loading, global bending and internal pressure. These schemes are given in two different forms to cover the wide ranges of geometries and material parameters; the modified GE/EPRI method and the modified reference stress method. In the former method, new plastic influence functions for fully plastic J‐integral estimation are developed based on extensive three‐dimensional finite element calculations. In the latter method, new optimized reference loads are suggested and utilized to predict the J values. To verify the feasibility of these two schemes, J‐integral values obtained from further detailed FE analyses are compared to those from the proposed schemes. Because the estimated J‐integrals agree fairly well with the detailed FE analysis results, the new solutions can be applied for accurate structural integrity assessment of different size pipes with a circumferential surface crack.     

J-integral estimation for single edge notch specimens in tension

A new J-integral estimation scheme for single edge notch specimens in tension (SENT) is presented. The scheme consists of equations obtained by analysing finite element results of SENT specimens under plane strain conditions, and separate equations have been derived for two forms of idealized material stress-strain curves-power law and bilinear representations. Non-dimensional J values are presented as functions of non-dimensional far field strains which eliminate the dependency of J on the aspect ratios of specimens. It was found that both the material law and the crack depth to specimen depth ratio had significant influences on the predicted J values and by presenting equations for different parts of the J versus strain curves, the scheme is able to take these influences into account.     

A simple class of finite elements for plate and shell problems. II: An element for thin shells,with only translational degrees of freedom

This paper is the second of a pair which discuss the development of a class of overlapping hinged bending finite elements, which are suitable for the analysis of thin-shell, plate and beam structures. These elements rely on a simple physically appealing analogy, in which overlapping hinged facets are used to represent bending effects. They are based on quadratic overlapping assumed displacement functions, which results in constant stress/strain representation. Only translational nodal degrees of freedom are necessary, which is a significant advantage over most other currently available beam, plate and shell finite elements which employ translational, rotational and higher-order nodal variables. In paper I the hinged bending element concept has been introduced, and the hinged beam bending (HBB) and hinged plate bending (HPB) elements formulated. In the present paper these concepts are extended to develop a hinged shell bending (HSB) element. The HSB element can be readily combined with the constant strain triangular (CST) plane stress finite element for the modelling of thin-shell structures; and the combined HSB-CST element is tested against a number of 'standard' thin-shell problems. The present paper, like paper I, is conducted entirely in the context of small-displacement elastic behaviour.     

Approximate J estimates for axial part-through surface-cracked pipes

This paper provides net‐section limit pressures and a reference stress based J estimation method for pipes with constant depth, internal axial surface cracks under internal pressure. Based on systematic small strain finite element (FE) limit analyses using elastic perfectly plastic materials, net‐section limit pressures are firstly determined, and based on FE results, a closed‐form limit pressure solution is proposed. Furthermore, based on the proposed limit pressure solution, a method to estimate elastic–plastic J is proposed based on the reference stress approach. When the reference stress is defined by the proposed (global) limit pressure, estimated J values based on the reference stress approach are overall slightly lower than FE results, implying that the method is non‐conservative. By re‐defining the reference using optimised reference loads, resulting J estimates agree well with FE results.     

Mixed-mode fracture load prediction in lead-free solder joints

Double cantilever beam (DCB) fracture specimens were made by joining copper bars with both continuous and discrete SAC305 solder layers of different lengths under standard surface mount (SMT) processing conditions. The specimens were then fractured under mode-I and various mixed-mode loading conditions. The loads corresponding to crack initiation in the continuous joints were used to calculate the critical strain energy release rate, J_ci, at the various mode ratios using elastic–plastic finite element analysis (FEA). It was found that the J_ci from the continuous joint DCBs provided a lower bound strength prediction for discrete 2 mm and 5 mm long joints at the various mode ratios. Additionally, these J_ci values calculated from FEA using the measured fracture loads agreed reasonably with J_ci estimated from measured crack opening displacements at crack initiation in both the continuous and discrete joints. Therefore, the critical strain energy release rate as a function of the mode ratio of loading is a promising fracture criterion that can be used to predict the strength of solder joints of arbitrary geometry subject to combined tensile and shear loads.     

Fatigue strength assessment of laser stake‐welded T‐joints subjected to reversed bending

The paper investigates the fatigue strength of laser stake‐welded T‐joints subjected to reversed bending. The fatigue tests are carried out with the load ratio, R ≈ ?0.8. The experimental data is firstly analysed using the nominal stress approach and then by the J‐integral as the local fatigue strength parameter in the finite element (FE) assessment. The nominal stress approach demonstrated that the fatigue strength of the investigated T‐joints is lower than encountered for any other steel joint under reversed tensile loading. The results also showed that the fatigue strength of this joint under the load ratio R ≈ ?0.8 increases with respect to R = 0 bending by 22.6% in the case of the nominal stress approach and 13% in the case of the J‐integral approach. However, the slopes of the fatigue resistance curves for different load ratios appear very similar, suggesting that the load ratio has an insignificant influence to the slope. In contrast to the similar slopes, the scatter indexes were different. The nominal stress approach shows that the scatter index is 3.4 times larger for R ≈ ?0.8 than R = 0 bending. The J‐integral approach showed that the scatter index for R ≈ ?0.8 is only 67% larger than in the R = 0 case because the weld geometry is modelled in the FE analysis.     

A finite element analysis of dynamic fracture initiation by ductile failure mechanisms in a 4340 steel

In some recent dropweight impact experiments [5] with pre-notched bend specimens of 4340 steel, it was observed that considerable crack tunneling occurred in the interior of the specimen prior to gross fracture initiation on the free surfaces. The final failure of the side ligaments happened because of shear lip formation. The tunneled region is characterized by a flat, fibrous fracture surface. In this paper, the experiments of [5] (corresponding to 5 m/s impact speed) are analyzed using a plane strain, dynamic finite element procedure. The Gurson constitutive model that accounts for the ductile failure mechanisms of micro-void nucleation, growth and coalescence is employed. The time at which incipient failure was observed near the notch tip in this computation, and the value of the dynamic J-integral, _J d, at this time, compare reasonably well with experiments. This investigation shows that J-controlled stress and deformation fields are established near the notch tip whenever J _d , increases with time. Also, it is found that the evolution of micro-mechanical quantities near the notch root can be correlated with the time variation of J _d .The strain rate and the adiabatic temperature rise experienced at the notch root are examined. Finally, spatial variations of stresses and deformations are analyzed in detail.     

Characterization of crack-tip field and constraint for bending specimens under large-scale yielding

Elastic-plastic crack-tip fields and constraint levels in bending specimens under large-scale yielding (LSY) are examined. The J–A ₂ three-term solution is modified by introducing an additional term caused by the global bending. Three different methods, i.e. two-point matching, constant A ₂ and elastic stress estimation method, are proposed to determine the fourth term. It is shown that the elastic stress estimation method is the simplest, yet effective, in that the fourth term can be derived from the strength theory of materials and the concept of plastic hinge, and effectively quantifies the contribution of the global bending moment on the crack-tip field. Consequently, the modified J–A ₂ solution, with the inclusion of the correction for global bending, does not introduce any new parameter. The two parameters remain as the loading (J and M) and the constraint level (A ₂). To validate the present solution, detailed finite element analyses (FEA) were conducted for a Three Point Bend (TPB) specimen with a/W=0.59 in A285 steel, and Single Edge Notched Bend (SENB) specimen subjected to pure bending with a/W=0.5 in A533B steel at different deformation levels ranging from small-scale yielding (SSY) to LSY. Results show that the modified J–A ₂ solution matches fairly well with the FEA results for both TPB and SENB specimens at all deformation levels considered. In addition, the fourth stress term is (a) proportional to the global bending moment and inversely proportional to the ligament length; (b) negligibly small under SSY; and (c) significantly large under LSY or fully plastic deformation. Accordingly, the present model effectively characterizes the crack-tip constraint for bending dominated specimens with or without the large influence from the global bending stress on the crack-tip field.     

Crack growth on elastic-plastic bimaterial interfaces

Finite element calculation based on finite strain theory is carried out to simulate the crack growth on bimaterial interfaces under the assumption of small scale yielding and plane strain condition. The modified Gurson's constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. The crack growths in homogeneous material and in bimaterials are compared. It is found from the calculation that the critical macroscopic fracture toughness for crack growth J _IC is much lower in bimaterials than in homogeneous material. For bimaterial cases, the J _IC of a crack between two elastic-plastic materials which have identical elastic properties with different yield strength is lower than that of a crack between an elastic-plastic material and a rigid substrate. It seems that the difference in yield strength between the dissimilar materials has more significant influence on the void nucleation and crack growth than the difference in hardening exponent.     

A simple class of finite elements for plate and shell problems. I: Elements for beams and thin flat plates

This is the first paper of a pair which together discuss the development of a class of overlapping hinged bending finite elements which are suitable for the analysis of thin-shell, plate and beam structures. These elements rely on a simple physical analogy, involving overlapping hinged facets. They are based on quadratic overlapping assumed displacement functions. Only translational nodal degrees of freedom are necessary, which is a significant simplification over most other currently available beam, plate and shell finite elements which employ translational, rotational and higher-order nodal variables. In this paper the hinged bending element concept is introduced, and the hinged beam bending (HBB) and hinged plate bending (HPB) elements are formulated. In paper II these concepts are extended to develop a hinged shell bending (HSB) element. The HSB element can be readily combined with the constant strain triangular (CST) plane stress finite element for the modelling of thin-shell structures.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

A numerical study on the crack tip constraint of pipelines subject to extreme plastic bending

This study investigates the fracture response and crack tip constraint of thick wall pipelines subject to large plastic bending. Such a circumstance frequently occurs during the installation of offshore pipelines (such as the reeling method), and accidental overloading, both inducing inelastic bending. The near-tip stress and strain fields are obtained through the fully nonlinear 3D finite element models constructed to examine the response of a practical range of cracked pipeline geometries and material properties. It is observed that throughout the loading history (up to the large scale yielding of the pipeline), by incorporation of the J–Q two parameter fracture theory, the near crack tip fields do indeed resemble those obtained from a K–T modified boundary layer formulation. This analogy provides sufficient proof for the applicability of the similitude concept inherent and fundamental to any fracture assessment procedure. All the pipelines considered in this study, which had realistic crack sizes, exhibited low constraint behavior (i.e. −1.4 < Q < −0.4). Additionally, Q was observed to decrease as a linear function of the global bending strain. Based on this correlation, simplified design equations are presented by which the constraint of such pipelines could be effectively estimated. The equations would be suitable for incorporation in the constraint-matched integrity assessment procedures that would in turn overcome the overt conservatism produced by the use of single parameter fracture mechanics approaches. Suitability of the low constraint laboratory specimens for fracture toughness measurements is also confirmed.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.