Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.     

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.     

J dominance in mixed mode loading

Elastic-plastic plane-strain crack problems subject to combined mode I and mode II loadings have been analysed with modified boundary layer formulations using the first two terms, K and T of the asymptotic elastic field. Corresponding full field calculations have been performed on geometries in which the mode I component arises largely from bending or tension and in which the T stress varies from tensile to compressive. The conditions for J dominance have been considered in terms of the effect of the T stress on the asymptotic field. As in related work on the pure mode I problem, positive T stresses are shown to favour J dominance, while compressive T stresses cause the stresses to fall from the HRR field.     

A Photoelastic Study of T‐stress in Centrally Cracked Brazilian Disc Specimen Under Mode II Loading

Abstract: In the traditional formulation of the stress field near a crack tip, the presence of the T‐stress is generally considered only under mode I or mixed mode I and II conditions. In this paper its presence in almost pure Mode II is experimentally investigated by mean of photoelasticity and its effects on the isochromatic fringe patterns are discussed. The test specimens are Brazilian discs containing sharp central cracks. After crack generation, all residual stresses are removed with thermal treatment of the specimens. Then, a compressive load is applied in a specific angle to induce mode II deformation. The observed isochromatic fringes show very good consistency with theoretical predictions. Experimental results indicate that this specimen has a negative T‐stress in mode II condition. The results calculated for K_II and T from photoelastic experiments agree well with numerical results available from finite element method.     

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.     

T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

The non-singular T-stress provides a first-order estimate of geometry and loading mode, e.g. tension vs. bending, effects on elastic–plastic, crack-front fields under mode I conditions. The T-stress has a pronounced effect on measured crack growth resistance curves for ductile metals – trends most computational models confirm using a two-dimensional setting. This work examines T-stress effects on three-dimensional (3D), elastic–plastic fields surrounding a steadily advancing crack for a moderately hardening material in the framework of a 3D, small-scale yielding boundary-layer model. A flat, straight crack front advances at a constant quasi-static rate under near invariant local and global mode I loading. The boundary-layer model has thickness B that defines the only geometric length-scale. The material flow properties and (local) toughness combine to limit the in-plane plastic-zone size during steady growth to at most a few multiples of the thickness (conditions obtainable, for example, in large, thin aluminum components). The computational model requires no crack growth criterion; rather, the crack front extends steadily at constant values of the plane-stress displacements imposed on the remote boundary for the specified far-field stress intensity factor and T-stress. The specific numerical results presented demonstrate similarity scaling of the 3D near-front stresses in terms of two non-dimensional loading parameters. The analyses reveal a strong effect of T-stress on key stress and strain quantities for low loading levels and less effect for higher loading levels, where much of the plastic zone experiences plane-stress conditions. To understand the combined effects of T-stress on stresses and plastic strain levels, normalized values from a simple void-growth model, computed over the crack plane for low loading, clearly reveal the tendency for crack-front tunneling, shear-lip formation near the outside surfaces, and a minimum steady-state fracture toughness for T = 0 loading.     

Studies of elastic and elastic–plastic J integral for mixed mode cracked plate under biaxial loading

Analyses of I–II mixed mode central cracked plate by finite element method are performed in this paper, and some different phenomena are found. First for I–II mixed mode crack, the distribution of J integral along crack tip thickness depends on biaxiality factors because of the existence of vertex (corner) singularity, which is unlike that for mode I or mode II crack. Then J integrals at middle layer keep constant for any cracked plates with different inclined angles β when the biaxiality ratio is equal to 1 or ?1, which implies that the inclined angle or the extent of I–II mixed mode has no effect on the J integral for positive or negative equal axial loading conditions. And the decreasing trend of J integral with the inclined angle β for biaxiality ratio λ being between?1 and 1 is just opposite with that for biaxiality ratio λ being larger than 1 and smaller than ?1. Finally, proposed h₁ (a/W, n, λ, β) of cracked plate with different inclined angles under different biaxial loading are calculated.     

Threshold and growth mechanism of fatigue cracks under mode II and III loadings

ABSTRACT The fatigue crack growth behaviour of 0.47% carbon steel was studied under mode II and III loadings. Mode II fatigue crack growth tests were carried out using specially designed double cantilever (DC) type specimens in order to measure the mode II threshold stress intensity factor range, ΔK_IIth. The relationship ΔK_IIth > ΔK_Ith caused crack branching from mode II to I after a crack reached the mode II threshold. Torsion fatigue tests on circumferentially cracked specimens were carried out to study the mechanisms of both mode III crack growth and of the formation of the factory‐roof crack surface morphology. A change in microstructure occurred at a crack tip during crack growth in both mode II and mode III shear cracks. It is presumed that the crack growth mechanisms in mode II and in mode III are essentially the same. Detailed fractographic investigation showed that factory‐roofs were formed by crack branching into mode I. Crack branching started from small semi‐elliptical cracks nucleated by shear at the tip of the original circumferential crack.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of microstructure on mixed-mode, high-cycle fatigue crack-growth thresholds in Ti-6Al-4V alloy

Effect of microstructure on mixed‐mode (mode I + II), high‐cycle fatigue thresholds in a Ti‐6Al‐4V alloy is reported over a range of crack sizes from tens of micrometers to in excess of several millimeters. Specifically, two microstructural conditions were examined—a fine‐grained equiaxed bimodal structure (grain size ～20 µm) and a coarser lamellar structure (colony size ～500 µm). Studies were conducted over a range of mode‐mixities, from pure mode I (ΔK_II/ΔK_I = 0) to nearly pure mode II (ΔK_II/ΔK_I ～ 7.1), at load ratios (minimum load/maximum load) between 0.1 and 0.8, with thresholds characterized in terms of the strain‐energy release rate (ΔG) incorporating both tensile and shear‐loading components. In the presence of through‐thickness cracks—large (> 4 mm) compared to microstructural dimensions—significant effects of mode‐mixity and load ratio were observed for both microstructures, with the lamellar alloy generally displaying the better resistance. However, these effects were substantially reduced if allowance was made for crack‐tip shielding. Additionally, when thresholds were measured in the presence of cracks comparable to microstructural dimensions, specifically short (～200 µm) through‐thickness cracks and microstructurally small (< 50 µm) surface cracks, where the influence of crack‐tip shielding would be minimal, such effects were similarly markedly reduced. Moreover, small‐crack ΔG_TH thresholds were some 50–90 times smaller than corresponding large crack values. Such effects are discussed in terms of the dominant role of mode I behaviour and the effects of microstructure (in relation to crack size) in promoting crack‐tip shielding that arises from significant changes in the crack path in the two 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.     

Calculation of mixed mode stress intensity factors for an elliptical subsurface crack under arbitrary normal loading

Normal loading causes mixed fracture modes in an elliptical subsurface crack because of the nonsymmetrical geometry with respect to the crack face. In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks in an elastic semi‐infinite space under normal loading are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis, under uniform normal loading are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are used to calculate MMSIFs for any point of the crack front under linear and nonlinear two‐dimensional (2D) loadings. So, in order to evaluate the fatigue crack growth phenomenon under complicated 2D stress distributions, MMWFs can be easily used. The comparison between the MMSIFs obtained from the MMWFs and finite element analysis indicates high accuracy.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.     

Influence of mode mixity and loading conditions on the fatigue crack growth behaviour of an epoxy adhesive

Adhesive joints usually experience mixed mode and mostly cyclic stresses conditions during their service life. The aim of the current research is to investigate the fatigue behaviour of a structural epoxy adhesive. Pure modes I and II and mixed mode tests were carried out to study the fracture and fatigue crack growth (FCG) behaviour of the adhesive. Compliance‐based beam method was considered for experimental fracture energy measurement. The effects of load level and load ratio on the mode I FCG behaviour and Paris law parameters were also investigated. Result showed that the effect of load level on fatigue crack propagation is more pronounced for lower R ratios. It was found that when the crack faces are closer during the unloading process, the difference between the R² and G_min/G_max is higher. Some possibilities are the crack closure phenomenon, difficulty in measuring the G_min , and the employed data reduction approach.     

Fracture Toughness and Creep Crack Growth in Polypropylene Under Combined Modes I and II Loading

Fracture toughness and creep crack growth characteristics under combined mode I and II loadings were studied using the compact tension shear (CTS) specimens of polypropylene. The K _I - K _II envelope for crack initiation was obtained under various combined mode loadings. The creep crack growth rates da/dt under combined mode I and mode II loadings can be correlated with a single effective stress intensity factor K _Ieff based on the combined mode fracture envelope.     

Fatigue crack growth of a railway wheel

Typically, fatigue crack propagation in railway wheels is initiated at some subsurface defect and occurs under mixed mode (I–II) conditions. For a Spanish AVE train wheel, fatigue crack growth characterization of the steel in mode I, mixed mode I–II, and evaluation of crack path starting from an assumed flaw are presented and discussed.Mode I fatigue crack growth rate measurement were performed in compact tension C(T) specimens according to the ASTM E647 standard. Three different load ratios were used, and fatigue crack growth thresholds were determined according to two different procedures. Load shedding and constant maximum stress intensity factor with increasing load ratio R were used for evaluation of fatigue crack growth threshold.To model a crack growth scenario in a railway wheel, mixed mode I–II fatigue crack growth tests were performed using CTS specimens. Fatigue crack growth rates and propagation direction of a crack subjected to mixed mode loading were measured. A finite element analysis was performed in order to obtain the K_I and K_II values for the tested loading angles. The crack propagation direction for the tested mixed mode loading conditions was experimentally measured and numerically calculated, and the obtained results were then compared in order to validate the used numerical techniques.The modelled crack growth, up to final fracture in the wheel, is consistent with the expectation for the type of initial damage considered.     

A new testing method to obtain mode II fatigue crack growth characteristics of hard materials

Flaking type failure in rolling‐contact processes is usually attributed to fatigue‐induced subsurface shearing stress caused by the contact loading. Assuming such crack growth is due to mode II loading and that mode I growth is suppressed due to the compressive stress field arising from the contact stress, we developed a new testing apparatus for mode II fatigue crack growth. Although the apparatus is, as a former apparatus was, based on the principle that the static K_I mode and the compressive stress parallel to the pre‐crack are superimposed on the mode II loading system, we employ direct loading in the new apparatus. Instead of the simple four‐point‐shear‐loading system used in the former apparatus, a new device for the application of a compressive stress parallel to the pre‐crack has been developed. Due to these alterations, mode II cyclic loading tests for hard steels have become possible for arbitrary stress ratios, including fully reversed loading (R=?1); which is the case of rolling‐contact fatigue. The test results obtained using the newly developed apparatus on specimens made from bearing steel SUJ2 and also a 0.75% carbon steel, are shown.     

J-Dominance in Mixed Mode Ductile Fracture Specimens

The asymptotic mixed mode crack tip fields in elastic-plastic solids are scaled by the J-integral and parameterized by a near-tip mixity parameter, M _{_p} . In this paper, the validity and range of dominance of these fields are investigated. To this end, small strain elastic-plastic finite element analyses of mixed mode fracture are first performed using a modified boundary layer formulation. Here, a two term expansion of the elastic crack tip field involving the stress intensity factor |K| the elastic mixity parameter M _{_e} as well as the T-stress is prescribed as remote boundary conditions. The analyses are conducted for different values of M _{_e} and the T-stress. Next, several commonly used mixed mode fracture specimens such as Compact Tension Shear (CTS), Four Point Bend (4PB), and modified Compact Tension specimen are considered. Here, the complete range of loading from contained yielding to large scale yielding is analyzed. Further, different crack to width ratios and strain hardening exponents are considered. The results obtained establish that the mixed mode asymptotic fields dominate over physically relevant length scales in the above geometries, except for predominantly mode I loading and under large scale yielding conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Creep Crack Growth in Polyethylene Under Combined Mode I and Mode II Loading

Creep crack growth characteristics under various combined mode I and mode II loadings were studied using the compact tension shear (CTS) specimens of polyethylene. Creep crack growth rates da/dtunder combined mode I and mode II loading can be correlated with a single effective stress intensity factor K _Ieffderived from the combined — mode fracture toughness envelope. The steady state or constant crack growth rates which occupy the significant part of creep failure life increase with the increasing initial effective stress intensity factor.     

Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth

During a service loading fatigue cracks can be subjected to a mixed mode loading if, due to the alteration of the loading direction, the basic crack modes (Modes I, II and III) are combined. An alteration of the loading direction, e.g. can occur either occasionally paired with an overload (mixed mode overload) or permanently in terms of a mixed mode block loading as a combination of normal and shear stresses.Within the scope of this paper, experimental investigations on both mixed mode overloads, which are interspersed into a Mode I baseline level loading, and mixed mode block loadings are presented. The experimental investigations show that the retardation effect decreases with an increasing amount of Mode II of the overload. Due to the block loading, the fatigue crack growth rate is retarded as well, and the crack is also deflected. The kinking angle depends on the fraction of shear stresses. Furthermore, a detailed elastic–plastic finite element analysis of the fatigue crack growth after mixed mode overloads is presented in order to understand the mechanism of the load interaction effects. By such numerical simulations, it can be shown that, due to mixed mode overloads, plastic deformations occur, which on the one hand reduce the near-tip closure and on the other hand cause a far-field closure. Also the stress distribution before and after the crack tip changes. A mixed mode overload causes lower closure and the crack tip deformations become asymmetrical, which is a reason for the smaller retardation effect of a mixed mode overload.