Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along [10[`1]][10\overline 1] direction. Solutions corresponding to different stress intensity rates [(K)\dot]\dot{{K}}, T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with [(K)\dot]\dot{{K}} which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.     

Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions

The fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories were compared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predicted the fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The values of the mixed-mode stress intensity factors K_I, and K_II as well as the corresponding constraint level characterized by T-stress were calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configuration the effect of T-stress on the crack path is not significant. On the other hand the effect of constraint on the fatigue crack propagation rate is more pronounced.     

Stress intensity factor K and the elastic T-stress for corner cracks

The stress intensity factor K and the elastic T-stress for corner cracks have been determined using domain integral and interaction integral techniques. Both quarter-circular and tunnelled corner cracks have been considered. The results show that the stress intensity factor K maintains a minimum value at the mid-plane where the T-stress reaches its maximum, though negative, value in all cases. For quarter-circular corner cracks, the K solution agrees very well with Pickard's (1986) solution. Rapid loss of crack-front constraint near the free surfaces seems to be more evident as the crack grows deeper, although variation of the T-stress at the mid-plane remains small. Both K and T solutions are very sensitive to the crack front shape and crack tunnelling can substantially modify the K and T solutions. Values of the stress intensity factor K are raised along the crack front due to crack tunnelling, particularly for deep cracks. On the other hand, the difference in the T-stress near the free surfaces and at the mid-plane increases significantly with the increase of crack tunnelling. These results seem to be able to explain the well-observed experimental phenomena, such as the discrepancies of fatigue crack growth rate between CN (corner notch) and CT (compact tension) test pieces, and crack tunnelling in CN specimens under predominantly sustained load.     

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.     

Cracked asphalt pavement under traffic loading – A 3D finite element analysis

An asphalt pavement containing a transverse top-down crack is investigated under traffic loading using 3D finite element analysis. The stress intensity factors (SIFs) and the T-stress are calculated for different distances between the crack and the vehicle wheels. It is found that all the three Modes (I, II and III) are present in the crack deformation. The signs and magnitudes of K_I, K_II, K_III and T are significantly dependent on the location of the vehicle wheels with respect to the crack plane. The magnitude of T-stress is considerable, if compared to the stress intensity factors, when one of the wheels is very close to the crack plane.     

A fracture mechanics model for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems

A computational model is presented for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems. This model assumes the appearance of an initial microcrack on the contact surface due to the mechanical or thermal treatment of the material, and as a consequence of an on-going process in early the stage of exploitation. The discretised model of the contacting mechanical elements is subjected to normal loading (Hertzian contact pressure), tangential loading (friction between contacting surfaces) and internal pressure to the crack surfaces. Crack propagation is predicted as follows: (1) using modified maximum tangential stress criterion, which takes into account the influence of stress intensity factors K_I and K_II, T-stress, stress on the crack’s surface caused by lubricant pressure inside the crack, and the critical distance ahead of the crack tip and (2) the classical maximum tangential stress criterion, which only takes into account the influence of the stress intensity factors K_I and K_II. The stress intensity factor based on these two criteria is then used in a short crack growth theory to determine the fatigue life of an initial crack to extent up to micro-pit. The developed model is applied to a real spur gear pair.     

Time-derivative equations for fatigue crack growth in metals

Predicting fatigue crack growth in metals remains a difficult task because available models are based on cycle-derivative equations, such as the Paris law, while service loads are often far from being cyclic. The main objective of this paper is therefore to propose a set of time-derivative equations for fatigue crack growth. The model is based on the thermodynamics of dissipative processes. For this purpose, three global state variables are introduced in order to characterize the state of the crackthe crack length a, the plastic blunting at crack tip and the intensity of crack opening C. Thermodynamics counterparts are introduced for each variable. Special attention is paid to the elastic energy stored inside the crack tip plastic zone, because, in practice, residual stresses at crack tip are known to considerably influence fatigue crack growth. The stored energy is included in the energy balance equation, and this leads to the appearance of a kinematics hardening term in the yield criterion for the cracked structure. No dissipation is associated with crack opening, but to crack growth and to crack tip blunting. Finally, the model consists in two laws: a crack propagation law, which is a relationship between d dt and da/dt and which observes the inequality stemmed from the second principle, and an elastic-plastic constitutive behaviour for the cracked structure, which provides d dt versus applied-load. The model was implemented and tested. It reproduces successfully the main features of fatigue crack growth as reported in the literature, such as the Paris law, the stress-ratio effect and the overload retardation effect.     

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.     

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.     

Crack sharpness effects in fracture testing of polymers

Tests on five polymers are described in which the fracture toughness,K _b, was determined in three-point bending using cracks with a range of tip radii. The variation ofK _b with tip radius is modelled using a two criterion elastic model, a stress and a length, and using these it is possible to estimate the sharp crack values and the effects of blunting arising from the plastic zone. A suggestion for a possible standard test is given.     

Application of a conservation integral to an interface crack interacting with singularities

A mutual integral, which has the conservation property is applied to the problem of an interfacial crack. The stress intensity factors K ₁, K ₂, K ₃ and T-stress for the problem in an infinite medium are easily obtained by using the mutual integral without solving the boundary value problem. In doing so the auxiliary solutions are required and they are taken from the known asymptotic solutions. This method is amenable to numerical evaluation of the stress intensity factors and T-stress if the interfacial crack in a finite medium is considered.     

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.     

On fracture of kinked interface cracks – The role of T-stress

This paper presents a modified maximum tangential stress criterion (MMTS) for prediction of the fracture initiation conditions in kinked bi-material cracks. The criterion takes into account the effect of T-stress as well as the stress intensity factors (K_I and K_II) to predict the mixed mode fracture toughness of interface cracked specimens. First the fracture criterion is developed and the effect of sign and magnitude of T-stress on mixed mode fracture toughness is studied analytically. Then, the suggested criterion is evaluated using the experimental data reported for some epoxy/Aluminum Brazil-nut-sandwich specimens in the literature. The MMTS criterion is also compared with the conventional maximum tangential stress (MTS) criterion and hence, significantly improved estimates were achieved for mixed mode fracture toughness of the tested specimens.     

Crack growth prediction and non-linear analysis for an elasto-plastic solid

In this paper, a variable radius for the plastic zone is introduced and a maximum principal stress criterion is proposed for the prediction of crack initiation and growth. It is assumed that the direction of crack initiation coincides with the direction of the maximum principal stress. The von Mises yield criterion is applied to define the plastic zone, instead of assuming a plastic zone with a constant distance r from the crack tip. An improvement is made to this fracture criterion, and the criterion is extended to study the crack growth characteristics of mixed mode cracks. Based on the concept of frictional stress intensity factor, k_f, the rate of fatigue crack propagation, db/dN, is postulated to be a function of the effective stress intensity factor range, Δk_eff. Subsequently, this concept is applied to predict crack growth due to fatigue loads. The proposed crack growth model is discussed by comparing the experimental results with those obtained using the maximum principal stress criterion.     

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.     

Estimations of the T-stress for small cracks at notches

The T-stress is increasingly being recognized as an important additional stress field characterizing parameter in the analyses of cracked bodies. Using T-stress as the constraint parameter, the framework of failure assessments including the constraint effect has been established; and the effect of T-stress on fatigue crack propagation rate has been investigated by several researchers. In this paper, a simple method for determining the T-stress for small notch-emanating cracks is presented. First, the background on the T-stress calculation using the superposition principle and the similarities between the elastic notch-tip stress fields described by two parameters: the stress concentration factor (K_t) and the notch-tip radius (ρ), are summarized. Then, the method of estimating T-stress for both short and long cracks at the notches is presented. The method is used to predict T-stress solutions for cracks emanating from an internal hole in a wide plate, and cracks emanating from an U-shaped edge notch in a finite thickness plate. The results are compared to the T-stress results in the literature, and the T-stresses solutions obtained from finite element analysis. Excellent agreements have been achieved for small cracks. The method presented here can be used for a variety of notch crack geometries and loading conditions.     

Shakedown of a cracked body consisting of kinematic hardening material

In this paper, the shakedown behaviour of a cracked body is studied. The main idea is to consider the crack as a notch. Then no singular stresses appear at crack tip. Due to the local character of the problem, Melan's shakedown theorem is used. By solving shakedown as an optimization problem, the limited stress intensity factor (SIF) for shakedown K_sh is obtained. It is found that the shakedown limit SIF of a cracked body is proportional to the initial yield stress σ_y of the material times the square root of the effective crack tip radius π, i.e. K_sh ∝ σ_y√ϱ. Comparison of shakedown limit SIFs with fatigue thresholds for certain materials, so far as can be found in literature, shows that these two quantities agree well with each other. This agreement indicates that shakedown of the cracked body is one of the reasons for arrest of the crack under cyclic loads. Shakedown investigation is then a new method for predicting the fatigue threshold of a cracked body. Thus, a transition from shakedown to cyclic fracture mechanics has been achieved.     

Study of the fatigue failure of an anti-return valve of a high pressure machine

This paper presents a study of the fatigue failure of an anti-return valve, designed to work in the high pressure system (500 MPa) of a high pressure processing machine. To do this, the crack propagation has been simulated by means of the linear-elastic fracture mechanics approach under mixed-mode loading conditions. From an initial crack, which size is related with the microstructure and superficial finish, the crack growth has been simulated using the stress intensity factors K_I and K_II of the cracked valve axisymmetric geometry. The crack propagation path has been obtained step by step, applying the criterion of the maximum circumferential stress at the crack tip. The experimental and simulated crack propagation paths have been compared and, as a consequence of the reliable results obtained, the fatigue life of the valve has been calculated using the Paris law of the material with an effective stress intensity factor K_eff. The good agreement with experimental fatigue life allows to perform new improved designs using the methodology presented.     

A brief note on elastic T-stress for centred crack in anisotropic plate

The stress intensity factors (SIFs) and the T-stress for a planar crack with anisotropic materials are evaluated by the fractal finite element method (FFEM). The FFEM combines an exterior finite element model and a localized inner model near the crack tip. The mesh geometry of the latter is self-similar in radial layers around the tip. A higher order displacement series derived from Laurent series and Goursat functions is used to condense the large numbers of nodal displacements at the inner model near the crack tip into a small set of unknown coefficients. In this study, the variations of the SIFs and the T-stress with material properties and orientations of a crack are presented. The separation of the analytical displacement series into four fundamental cases has shown to be necessary in order to cover all the material variations and the orientations of a crack in the plate with general rectilinear anisotropic materials.     

Numerical modelling of crack path in the lubricated rolling–sliding contact problems

A two-dimensional computational model for simulation of contact fatigue of lubricated rolling–sliding contact problems is presented. In the model it is assumed that the initial crack of length 0.02 mm is initiated at the surface due to previous mechanical or heat treatment of the material as well as a consequence of running process in an early stage of exploitation. The discretised model with the initial crack is then subjected to the normal contact pressure and tangential loading due to friction between the contacting surfaces. The model also considers the moving contact of the contacting surfaces and fluid trapped in the crack. The crack propagation path is predicted with the MTS and modified MTS criterion, which takes into account the influence of the stress intensity factor K_I and K_II, T-stress, stress on the crack surface caused by lubricant pressure inside the crack and critical distance ahead the crack tip. The numerical results correspond well with available experimental data.