Maximum Average Tangential Stress Criterion for Prediction of the Crack Path

The concept of the average stress has been employed to propose the maximum average tangential stress (MATS) criterion for predicting the direction of fracture angle. This criterion states that a crack grows when the maximum average tangential stress in the fracture process zone ahead of the crack tip reaches its critical value and the crack growth direction coincides with the direction of the maximum average tangential stress along a constant radius around the crack tip. The tangential stress is described by the singular and non-singular (T-stress) terms in the Williams series solution. The predicted directions of fracture angle are consistent with the experimental data for the mixed mode I/II crack growth behavior of Guiting limestone.     

Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading

The mixed mode I/II fracture initiation angle and the crack growth trajectory of a soft rock (Guiting limestone) were investigated experimentally and theoretically for two different shaped test specimens with various sizes. It was observed that for similar mode mixities in the two specimens, the fracture paths grew in two different trajectories. It is shown that the observed crack path and the fracture initiation angle can be predicted theoretically by using a generalized form of the maximum tangential stress criterion. The main difference in the fracture initiation angles was found to be related to the magnitude and sign of the T-stress.     

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.     

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.     

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.     

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.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T ₁₁, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T ₃₃, acting parallel to the crack front, decreases with the crack front curvature.     

The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading

The purpose of this paper is to revisit the maximum tensile stress (MTS) criterion to predict brittle fracture for mixed mode conditions. Earlier experimental results for brittle fracture of polymethylmethacrylate (PMMA) using angled cracked plates are also re-examined. The role of the T -stress in brittle fracture for linear elastic materials is emphasized. The generalized MTS criterion is described in terms of mode I and II stress intensity factors, K _I and K _II and the T- stress (the stress parallel to the crack), and a fracture process zone, r _c . The generalized MTS criterion is then compared with the earlier experimental results for PMMA subjected to mixed mode conditions. It is shown that brittle fracture can be controlled by a combination of singular stresses (characterized by K ) or non-singular stress ( T -stress). The T -stress is also shown to have an influence on brittle fracture when the singular stress field is a result of mode II loading.     

A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation

This paper addresses mixed-mode crack growth in two-dimensional functionally graded solids under thermomechanical loads, and investigates the effect of mechanical and thermal loads as well as the T-stress on their crack growth behavior. A novel residual strain-based formulation in the interaction integral method is developed and used for the accurate evaluation of mixed-mode stress intensity factors and/or the T-stress. Simulation of mixed-mode crack propagation in functionally graded materials including solid oxide fuel cells under thermomechanical loads is performed by means of the finite element method and the generalized interaction integrals in conjunction with a remeshing algorithm. An iterative procedure is used for crack growth simulation including the calculation of mixed-mode stress intensity factors and/or the T-stress by means of the generalized interaction integral method, determination of crack growth direction and crack initiation condition based on selected fracture criteria, and local automatic remeshing along the crack path. The present approach employs a user-defined crack increment at the beginning of the simulation. Crack trajectories and fracture parameters obtained by the present simulation for thermomechanical loads are assessed for some numerical examples in comparison with those for mechanical loads.     

On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials

The centrally cracked Brazilian disc specimen has been used frequently in the past for investigating mixed mode I-II fracture toughness in rock materials. However, a review of the available test results reveals that the conventional fracture criteria like the maximum tangential stress criterion always underestimate the mixed mode I-II fracture toughness data obtained from the Brazilian disc specimen. In this paper, a generalized maximum tangential stress criterion which takes into account the effects of the three fracture parameters K_I, K_II and T-stress is used for predicting the mixed mode fracture toughness data available in the literature for several types of rock materials tested with the Brazilian disc specimen. It is shown that the generalized maximum tangential stress criterion provides significantly improved predictions for the experimental results.     

Constraint effects during stable transient crack growth

The effect of plastic-flow constraint on the field of a crack, during stable propagation within the area, covered by the plastic zone at initiation of growth, has been examined. Plane strain, mode I and contained yielding have been considered. The material is homogeneous with elastic-plastic behavior, described by Hook's law, J ₂-flow theory and isotropic hardening. The numerical investigation has been performed within the framework of a boundary layer formulation, whereby the remote loading is fully specified by the first two terms in Williams' expansion, characterized by K and T. It is shown, that a self-similar state is reached, after growth of the order of the fracture process zone size at initiation. The characteristic length of the self-similar field is the fracture process zone size. Under contained yielding, the self-similar field depends on an appropriately normalized T/K ratio, which is a measure of the deviation of the stress field from small scale yielding distribution at distances of the order of the damage process zone size. According to the analysis, the effect of constraint on near-tip triaxiality, during transient growth is moderate. Also the effect of constraint on crack growth resistance is weak at initiation and increases with crack propagation.     

Effects of T-stresses on fracture initiation for a closed crack in compression with frictional crack faces

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to K _II and non-singular T-stresses T _x and T _y parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive T _y increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive T _x decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2T _x  = T _y  ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.     

A criterion for brittle fracture in U-notched components under mixed mode loading

A failure criterion is proposed for brittle fracture in U-notched components under mixed-mode static loading. The criterion, called UMTS, is developed based on the maximum tangential stress criterion and also a criterion proposed in the past for mode I failure of rounded V-shaped notches [Gomez FJ, Elices M. A fracture criterion for blunted V-notched samples. Int J Fracture 2004;127:239-64]. Using the UMTS criterion, a set of fracture curves are derived in terms of the notch stress intensity factors. These curves can be used to predict the mixed mode fracture toughness and the crack initiation angle at the notch tip. An expression is also obtained from this criterion for predicting fracture toughness of U-notched components in pure mode II loading. It is shown that there is a good agreement between the results of UMTS criterion and the experimental data obtained by other authors from three-point bend specimens.     

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.     

Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads

In this paper, the brittle fracture initiation characteristics under general combination of the opening mode (Mode I), sliding mode (Mode II) and tearing mode (Mode III) were investigated both theoretically and experimentally.

First, the perfectly brittle fracture tests were conducted on specimens of PMMA (Polymethylmethacrylate) for all possible combinations of the fracture modes including respective pure modes. The experimental fracture strengths were compared with those predicted by the fracture criteria which are represented in terms of: (1) maximum tangential stress, [σ_gq]_max, extended to general combined modes, (2) maximum energy release rate at the propagation of a small kinked crack, [G^k(γ)]_max, and (3) newly derived maximum energy release rate at the initiation of a small kinked crack, [G(γ)]_max. It was found that the [G^k(γ)]_max or [G(γ)]_max criterion was very effective to predict both the direction of initial crack propagation and the fracture strength. These energy release rates are expressed in closed forms, and the interaction curves of the brittle fracture strength under arbitrary combinations of Modes I, II and III were derived.

Next, for fracture accompanied by plastic deformation, tests were carried out on specimens of mild steel (SM 41) imposing bi-axial tensile loads at various low temperatures. Then, brittle fracture with plastic deformation occurs under a combination of Modes I and II. In the case of brittle fracture with small scale yielding, the [G(γ)]_max criterion predicts well the direction of initial crack propagation but estimates only lower fracture strength than the experimental one. In the cases of brittle fracture with large scale yielding and under general yielding, it was found from the fracture tests that the direction of initial crack propagation was nearly normal to the resultant vector of the crack opening displacements in the opening and sliding modes at the notch tip. To this type of fracture, the modified COD criterion predicts well the direction of initial crack propagation, but lower fracture strength.

When brittle fracture occurs under the influence of plastic deformation, in such cases as the last three mentioned above, the actual fracture strength is higher than what the most reliable criterion predicts and it increases as deformation in Mode II becomes larger.     

The influence of the T-stress on the directional stability of cracks

The sign of the T-stress has been widely used for deciding whether directional stability prevails for straight cracks subjected to mode I loading under small scale yielding. However, there is little evidence for the reliability of such a criterion. On the contrary, it is shown that a local criterion is not applicable and that directional stability generally depends on body and load geometry as well as on material parameters, whereas the sign of the T-stress is totally irrelevant in most cases.     

Cracked Brazilian disc specimen subjected to mode II deformation

The centrally cracked Brazilian disc specimen has been used by many researchers to study mode I and mode II brittle fracture in different materials. However, the experimental results obtained in the past from this specimen indicate that the fracture toughness ratio (K_IIc/K_Ic) is always significantly higher than the theoretical predictions. It is shown in this paper that the increase in the ratio K_IIc/K_Ic can be predicted if a modified maximum tangential stress (MTS) criterion is used. The modified criterion takes into account the effect of T-stress in addition to the conventional singular stresses. The fracture toughness ratio K_IIc/K_Ic is calculated for two brittle materials using the modified criterion and is compared with the relevant published experimental results obtained from fracture tests on the cracked Brazilian disc specimen. A very good agreement is shown to exist between the theoretical predictions and the experimental results.     

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.     

The effective T-stress estimation and crack paths emanating from U-notches

The concept of the T-stress as a local constraint factor has been extended to U-notch tip stress distribution as the effective T-stress. The effective T-stress has been estimated as the average value of the T-stress in the region corresponding to the effective (characteristic) distance ahead of the notch tip. The T-stress is evaluated by finite element method using the experimental load for crack initiation and computing the difference between principal stresses along ligament. A large range of critical effective T-stress values is investigated for different specimen configurations and notch aspect ratios. Crack stabilisation and crack bifurcation for fracture emanating from notches according to the critical effective T-stress is discussed. A model involving the influence of the critical effective T-stress on void growth for ductile failure in the vicinity of the notch tip has been proposed.