The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

Stress-intensity factors for circular hollow section V-joints with a rack-plate chord

This study investigates the fatigue crack‐driving force, measured by the linear‐elastic stress‐intensity factors (SIFs), for a surface crack at the root of the welds in a thick‐walled, circular hollow section (CHS) V‐shape joint, typically installed in modern offshore jack‐up platforms. The primary (chord) member of the V‐joint consists of two half CHSs welded to both sides of a thick rack plate, while the secondary (brace) member adopts thick‐walled CHSs. The surface‐breaking crack considered in this study locates at the interface between the rack plate and the weld metal joining the half CHS, and represents an initial flaw introduced by lack of penetration in the welding procedure. The finite‐element model incorporates a very detailed, local crack‐front mesh in a global continuous mesh through a mesh‐tying procedure, which ensures displacement continuity between the independent master surface and the dependent slave surface. A simple plate model verifies the mesh‐tying procedure in computing the linear‐elastic stress‐intensity factors for two remote loading conditions. The computation of the stress‐intensity factors employs a linear‐elastic interaction integral approach. The comparison of the computed SIF values with a previous experimental measurement for a CHS T‐joint verifies the accuracy and feasibility of the interaction integral approach in computing SIF values for surface cracks in welded tubular connections. Subsequent numerical analysis on the gapped V‐joints examines the mixed‐mode SIF values for different loading conditions and includes an array of practical joint geometric parameters and crack sizes. The nondimensional mode I stress‐intensity factors generally increases with the following variations in the joint geometric parameters: an increase in the chord radius to the wall thickness ratio (γ=d₀/2t₀) , an increase in the brace diameter to the chord diameter ratio (β=d₁/d₀) , a decrease in the crack depth ratio (a/t) or an increase in the crack length c. The current study identifies a practical group of V‐joints that requires detailed treatment in the fatigue assessment procedure. These V‐joints adopt a large β ratio and demonstrate high mode‐mixity angles [ψ= tan^?1(K_II/K_I)] with correspondingly high mode I and mode II stress‐intensity factors.     

Characterization of the computational accuracy in surface crack problems

The paper focuses on the comparison of different methods for calculating stress intensity factors (K_I) in surface crack problems based on results of numerical analyses of elastic crack-tip fields. The computational accuracy is quantified by means of the so-called ‘averaged error estimation technique’ which is extended to the evaluation of local errors in the determination of stress intensity factors at characteristic points of the crack front. Numerical data involved in the present study are obtained from boundary-element calculations. Three values of the stress intensity factor, i.e. those defined from nodal tractions, displacements and energy-release rate, are provided. The highest error level is found for the displacement-based data, while the energy-release calculations yield the best accuracy. A considerable increase in the error value is noticed near the intersection of the crack front with a body surface where the conventional assumption on the square-root stress singularity is, in general, not applied. It is shown that the accuracy of stress intensity factor analysis can be improved by eliminating uncertainties associated with the local stress state along the crack front. © 1998 John Wiley Sons, Ltd.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

This paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

On the I–II mixed mode fracture of granite using four‐point bend specimen

Four‐point bend experiments on black granite are conducted. The fracture behaviours of granite under pure mode I, pure mode II and I–II mixed mode are investigated, and the corresponding stress intensity factors K_I , K_II and the non‐singular term T‐stress are obtained through numerical–experimental method. The results are compared with the theoretical predictions of generalized maximum tangential stress criterion and other conventional criteria. It shows that generalized maximum tangential stress criterion fits the experimental results better for considering the effect of T‐stress. Contrasting with other loading configurations, the values of T‐stress for asymmetric four‐point bend specimens are much smaller, especially for pure mode II specimens, which provide an asymmetric deformation field where the T‐stress is approaching zero.     

Weight functions and stress intensity factors for quarter-elliptical corner cracks in fastener holes

Approximate weight functions for a quarter‐elliptical crack in a fastener hole were derived from a general weight function form and two reference stress intensity factors. Closed‐form expressions were obtained for the coefficients of the weight functions. The derived weight functions were validated against numerical data by comparison of stress intensity factors calculated for several nonlinear stress fields. Good agreements were achieved. These derived weight functions are valid for the geometric range of 0.5 ≤a/c≤ 1.5 and 0 ≤a/t≤ 0.8 and R/t= 0.5; and are given in forms suitable for computer numerical integration. The weight functions appear to be particularly suitable for fatigue crack growth prediction of corner cracks in fastener holes and fracture analysis of such cracks in complex stress fields.     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

An improvement of the EDI method in linear elastic fracture mechanics by means of an a posteriori error estimator in G

In this paper, an error estimator that quantifies the effect of the finite element discretization error on the computation of the stress intensity factor in linear elastic fracture mechanics is presented. In order to obtain the proposed estimator, a shape design sensitivity analysis (SDSA) is applied to the fracture mechanics problem. Following this approach, one of the most efficient post‐processing techniques for computing the strain energy release rate G, the well‐known EDI method, may be interpreted as a continuum method of the SDSA. The proposed error estimator is based on the recovery of the gradient fields and its reliability has been checked by means of numerical problems, yielding very good estimations of the true error. The new estimator remarkably improves the results given by a previous error estimator, which is based on a discrete analytical approach of SDSA. As a consequence, the combination of the new error estimator and the result given by the EDI method provides a much more accurate estimation of G. Copyright © 2003 John Wiley & Sons, Ltd.     

On the time compression (test acceleration) of broadband random vibration tests

Many broadband random vibration tests are time compressed. This is done by increasing test intensity according to the Basquin model of cyclic fatigue. Conventionally, the test level is accelerated from the root mean acceleration and an assumed power constant (k = 2) is applied. Using conventional analysis the potential error in test severity can be very large if k is incorrect. The Miner–Palmgren hypothesis of accumulated fatigue is used to re‐assess the potential error in test severity accounting for the non‐stationarity found in road distribution. This shows a substantially reduced sensitivity to the value of k depending on the distribution of actual vibration intensities around the time‐compressed test intensity. Using an example of a leaf‐sprung vehicle, the conventional level of time compression is shown to have low sensitivity to errors in k, whereas for an example of an air‐ride vehicle a lower level of time compression is needed to reduce error sensitivity. Copyright © 2010 John Wiley & Sons, Ltd.     

T‐stress corrected notch stress intensity factors with application to welded lap joints

The definition, content and application of the notch stress intensity factors (NSIFs) characterizing the stress field at rounded slit tips (keyholes) is discussed. The same is done in respect of the T‐stress transferred from the corresponding pointed slit tips. A T‐stress based correction of the NSIF K_1,ρ is found to be necessary. The applicability of the T‐stress term supplemented by higher‐order terms in Williams’ solution to the slit tip stresses in tensile‐shear loaded lap joints is discussed in more detail. The role of the T‐stress in constituting the near‐field stresses of rounded slit tips is shown to cause a difference between internal and external slit tip notches. The notch stress equations for lap joints proposed by Radaj based on structural stress and by Lazzarin based on a finite element model of the rounded notch are reconsidered and amended based on the derivations above.     

Calculation of the crack tip parameters in the holed‐cracked flattened Brazilian disk (HCFBD) specimens under wide range of mixed mode I/II loading

In this paper, the crack tip parameters including the stress intensity factors (K_I and K_II), T‐stress and the third terms of the stress field (A₃ and B₃) are determined comprehensively for a disk‐type sample named holed‐cracked flattened Brazilian disk (HCFBD) under various combinations of mode I and mode II loading. The HCFBD specimen is a circular disk containing a central hole in which the initial cracks are created radially from the hole circumference. Moreover, the ends of HCFBD are flattened for the sake of convenient loading. Performing enormous finite element analyses and calculating the stress intensity factors K_I and K_II, the states of pure mode II are determined for different configurations of HCFBD. Furthermore, the sign and magnitude of parameter A₃ which plays an important role to justify the geometry and size effects on the fracture toughness of quasi‐brittle materials are also determined for HCFBD with different geometrical ratios.     

Adaptive finite element analysis of mixed-mode fracture problems containing multiple crack-tips with an automatic mesh generator

A fully automatic advancing front type mesh generator to take care of crack problems has been presented. It is coupled with the Zienkiewicz and Zhu error estimator and the refinement methodology depends on the concept of strain energy concentration for completely automatic adaptive analysis of mixed-mode crack problems. For the first time energy based path independent M ₁-integral has been used to extract mixed-mode stress intensity factors in randomly changing quadratic triangular meshes. To fulfill the objective of automatic adaptive procedures, an approach has been suggested and validated for generation of integration paths automatically without user intervention. Stress intensity factors have been obtained within engineering accuracy.     

Adaptive finite element computation of dielectric and mechanical intensity factors in piezoelectrics with impermeable cracks

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.     

Fatigue crack growth in fibre metal laminates with multiple open holes

The fatigue crack growth behaviour of hybrid S2‐glass reinforced aluminium laminates (Glare) with multiple open holes was investigated experimentally and analytically. It was observed that the presence of multiple‐site fatigue damage would increase crack growth rates in the metal layers as two propagating cracks converged. An analytical crack growth model was established for predicting crack growth rates based on empirical Paris equation. The effective stress intensity factor at crack tips is a function of mode I far‐field stress intensity factor, crack opening stress intensity factor and effective non‐dimensional stress intensity factor that incorporated the crack‐bridging effect in fibre metal laminates. The predicted results under different applied stress can capture the trend of averaged crack growth rates in experiments, although deviation exists in the predictions.     

Fatigue of shot peened 7075-T7351 SENB specimen – A 3-D analysis

As‐received or shot peened 7075‐T7351 single‐edged notch bend (SENB) specimens, 8.1‐mm thick, were fatigued at a constant maximum load and at stress ratios of R= 0.1 and 0.8 to predetermined numbers of fatigue cycles or to failure. The SENB specimens were then fractured by overload and the tunnelling crack profiles were recorded. The crack‐growth rate, da/dN, after crack initiation at the notch was determined by crack‐profile measurement and fractography at various fatigue cycles. The shot peened surface topography and roughness was also evaluated by three‐dimensional (3‐D) laser scanning microscopy. Residual stresses in the as‐received specimens and those generated by shot peening at Almen scales of 0.004A, 0.008A, 0.012A and 0.016A, were measured by an X‐ray diffraction stress analyser with an X‐ray target, CrK, every 0.1 mm to a depth of 1 mm. The 3‐D stress intensity factor of the curved crack front was determined by the superposition of the 3‐D finite element solutions of the stress intensity factor of the loaded SENB specimen without the residual stress and the stress intensity factor of the unloaded SENB specimen with a prescribed residual stress distribution. da/dN versus the resultant stress intensity factor amplitude, ΔK_I, plots showed that while the residual stress locally retarded the crack‐growth rate it had no effect on the overall crack‐propagation rate.     

Adaptive finite element analysis of mixed‐mode crack problems with automatic mesh generator

Fully automatic advancing front type mesh generator to take care of crack and fracture problems has been presented. It is coupled with Zienkiewicz and Zhu error estimator, the refinement methodology depends on the concept of strain energy concentration for adaptive analysis of mixed‐mode crack problems. No investigation is reported in this direction so far. It has been found that the above combination proved to be very powerful for adaptive finite element analysis of mixed‐mode crack problems in two‐dimensional isotropic solids. Very accurate stress intensity factors have been obtained for a target error of 10 per cent with a minimum number of steps. Copyright © 2000 John Wiley & Sons, Ltd.     

Limited weld residual stress measurements in fatigue crack propagation: Part II. FEM-based fatigue crack propagation with complete residual stress fields

Using a limited set of residual stress measurements acquired by neutron diffraction and an equilibrium‐based, weighted least square algorithm to reconstruct the complete residual stress tensor field from the measured residual stress data, the effect of weld residual stress on fatigue crack propagation is investigated for 2024‐T351 aluminium alloy plate joined by friction stir welding. Through incorporation of the least squares, complete equilibrated residual stress field into a finite element model of the Friction Stir Weld (FSW) region, progressive crack growth along a direction perpendicular to the welding line is simulated as part of the analysis. Both the residual stress redistribution and the stress intensity factor due to the residual stress field, K_res, are calculated during the crack extension process. Results show that (a) incorporation of the complete, self‐equilibrated residual stress field into a finite element (FE) model of the specimen provides a robust, hybrid approach for assessing the importance of residual stress on fatigue crack propagation, (b) the calculated stress‐intensity factor due to the residual stress field, K_res, has the same trend as measured experimentally by the ‘cut‐compliance method’ and (c) the da/dN results are readily explained with reference to the effect of the residual stress field on the applied stress intensity factor.