Pressure with friction of a perfectly rigid die upon an elastic half space with cracks

We study the interaction of a rigid die with a base of any shape and the surface of an elastic half space containing cracks in the presence of friction in the contact zone. The solution of the plane contact problem of the theory of elasticity is obtained by the method of singular integral equations. The detailed analysis of the problem is performed for the case where the base of the die is parabolic and a crack is rectilinear and appears on the surface of half space. We also investigate the effects of the friction coefficient, crack length, its orientation, and location on stress intensity factors KI and KII at the crack tip and the distribution of contact stresses under the die.     

Fracture mechanics assessment of stress concentrations in incomplete fretting contacts

Life assessment of fretting fatigue has been studied for decades. Crack-analogy methods have been proposed for analyzing fretting fatigue of flat contact pairs. In the present work we re-consider the stress field near fretting contact pairs and study the feasibility of using known fracture parameters to assess incomplete fretting contact problems. Both analytical and FEM analysis reveal that the stress field near the discontinuous round corner of a friction pad, in which the round surface has been idealized without contacting the workpiece, is the same as that of crack tip. The stress field is described by the known stress intensity factors, K_I and K_II. For sticking contact these two fracture parameters are independent, whereas for the slipping contact K_II is linearly correlated with K_I. Therefore, the stress field around the slipping contact can be characterized only by one fracture parameter, together with friction coefficient. For the continuous contact pairs with finite round contact surface, the local stress concentrations near the contact edge are finite and can be characterized by K_I and K_II, either, in analogy to the blunting crack tip due to finite strains. Detailed computations confirm that using the fracture parameters to characterize the fretting contact failure is affected by both loading condition and friction pad geometry. The dominance zone around the pad corner decreases more significantly with vertical press load than the horizontal friction load. In the bi-material contact friction pair the stress field can be described by K_I and K_II in the same form.     

Fretting fatigue limit as a short crack problem at the edge of contact

This paper proposes a local stress concept to evaluate the fretting fatigue limit for contact edge cracks. A unique S–N curve based on the local stress could be obtained for a contact edge crack irrespective of mechanical factors such as contact pressure, relative slip, contact length, specimen size and loading type. The analytical background for the local stress concept was studied using FEM analysis. It was shown that the local stress uniquely determined the ΔK change due to crack growth as well as the stress distribution near the contact edge. The condition that determined the fretting fatigue limit was predicted by combining the ΔK change due to crack growth and the ΔK_th for a short crack. The formation of a non‐propagating crack at the fatigue limit was predicted by the model and it was experimentally confirmed by a long‐life fretting fatigue test.     

Finite element analysis of a subsurface crack

The problem of a subsurface crack parallel to the surface of a half space was studied by the finite element method. Without using the interface or gap elements over the crack faces, the crack faces would penetrate into each other for the traction-free boundary condition under shear loading, which is physically impossible. Using the gap elements, this problem was avoided, and a contact zone was observed near one crack tip. The size of the contact zone decreases but the maximum contact pressure at the closed crack tip increases as the crack approaches the surface. For tensile and shear loadings, both K _I (mode I stress intensity factor) and K _II (mode II stress intensity factor) increase as the crack approaches the surface. For shear loading there is no K _I at the closed tip and the K _I and K _II at the open tip are comparable as the crack approaches the surface.     

Numerical solution for multiple crack problem in an infinite plate under compression

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K ₁) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K ₂) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.     

Calculation of KII in crack face contacts using X-FEM. Application to fretting fatigue

In this work, the modeling of LEFM problems that imply crack face closure and contact using the extended finite element method (X-FEM) is presented aiming at its application to fretting fatigue problems. An assessment of the accuracy in the calculation of K_II is performed for two different techniques to model crack face contacts in X-FEM: one is based on the use of additional elements to establish the contact and the other on a segment-to-segment (or mortar) approach. It is concluded that only the segment-to-segment approach can lead to optimal convergence rates of the error in K_II. The crack face contact modeling has also been applied to a fretting fatigue problem, where the estimation of K_II under crack closure conditions plays an important role in the stage I of fatigue crack propagation. The effect of the crack face friction coefficient has been studied and its influence on the range of K_II has been ascertained during loading and unloading cycles.     

Fatigue crack propagation simulation in an aircraft engine fan blade attachment

This paper discusses the computation of three-dimensional fatigue crack growth rates in a typical military aircraft engine fan blade attachment under centrifugal and aerodynamic loads. The three-dimensional crack growth simulations utilize FRANC3D, a state-of-the-art crack propagation software developed at Cornell University, which uses boundary elements and linear elastic fracture mechanics. With an existing three-dimensional finite element contact stress analysis with a prescribed coefficient of friction (COF) along the contact surface, the displacements and stress intensity factors are calculated on the crack leading edge to yield crack propagation trajectories and growth rates. Due to complex geometry of the fan blade attachment and loading conditions, all three-fracture modes are considered and the associated stress intensity factors (SIF) are calculated using the Crack Opening Displacement (COD) approach. Crack propagation trajectories under mixed-mode conditions are obtained using the planar and maximum tangential stress crack-extension criteria. The fatigue crack in the blade attachment is subjected to an over speed mission cycle that includes high cycle frequencies (i.e., spectrum load) and the crack growth rate is predicted utilizing the Forman–Newman–de Koning (FNK) model. Scanning Electron Microscope (SEM) images of a cracked component from an engine ASMET (Accelerated Simulated Mission Endurance Test) are used to evaluate and compare the simulation results. The calculated SIF's from the simulations indicate a strong Mode-I (K_I) and Mode-III (K_III) interaction at the edge of contact (EOC). However, on the free surface it is primarily a crack opening (K_I) condition only. The crack growth rates are determined using the planar extension criterion which correlates better with the test data than the maximum tangential stress extension criteria.     

Multi-crack analysis of hydraulically pumped cone fracture in brittle solids under cyclic spherical contact

The evolution of surface damage in bilayers due to cyclic spherical indentation in the presence of incompressible lubricant is studied using an all-transparent glass/polycarbonate system as a model for more practical applications such as dental crowns and rolling contact fatigue. In situ observations and post-mortem material sectioning reveal that inner cone cracks evolve sequentially from the contact edge inward by slow growth in a process controlled by stress shielding from preceding cracks. The embryonic cracks are then accelerated by the action of fluid pressure into the flexural tensile stress at the lower part of the coating, where crossover fracture leading to delamination between the coating and substrate may ensue. A consistent FEM brittle fracture analysis incorporating multiple cracks, rate-dependent toughness and liquid pressure is used to follow the damage evolution in the coating. Crack trajectories are determined incrementally under the dual constraint K _I = K _II = 0, which maximize the tension at the crack tip upon the application of fluid pressure. The latter, evaluated at each increment with the aid of a fluid entrapment model, helps drive the leading crack past the compression zone beneath the contact via a hydraulic pump like action. In the early stages of fracture, the liquid pressure is reasonably well approximated by the Hertzian radial surface stress at the crack mouth. Fluid trapped in secondary cracks accentuate the compression beneath the contact. This helps squeeze more liquid into the tip of the leading crack in a zipping like action, which further enhance the crack driving force in the far field. The analytic predictions generally collaborate well with the tests.     

Frictional contact induced crack initiation in incompressible substrate

Surface crack initiation in an incompressible substrate induced through frictional fretting contact is analyzed using an energy-based fracture mechanics model. A closed-form energy release rate for surface crack initiation at the contact boundary has been derived with the crack growth angle determined by the mixed mode singular stress field at the contact edge. The driving forces in the form of J_i-integral, the critical energy release rate and the critical load for crack initiation from the crack free surface have been formulated. The relations between the friction coefficient and crack initiation angle, critical load have been specified.     

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.     

Mixed mode crack tip parameters for different wheel positions relative to a vertical crack at the rail foot

Finite element method is used to analyze a rail with a vertical bottom up crack at its foot, under the axle load and surface traction of a wheel. The possibility of crack formation at the foot of the rail in the neighborhood of a welding connection is discussed. A brief review on the importance of T‐stress in brittle fracture is presented. Seven cases with different locations of the crack relative to rail's sleeper contact region are considered. Numerous positions of the wheel are considered, and in each case, 3 crack parameters K_I, K_II, and T‐stress are calculated. Then, the biaxiality ratio and the mixity parameter for each loading and crack condition are calculated. It is shown that the location of crack and wheel can create mixed mode loading in the cracked rail and that the magnitude of crack tip parameters are strongly dependent on these geometric variables. In particular, the magnitudes of T‐stress and biaxiality ratio are significant in some cases. The effect of friction between the crack faces in the presence of compressive mode I loading on the mode II stress intensity factor is studied. Under mixed mode loading, due to the axle load and surface traction, the most critical condition is the formation of vertical cracks near the sleeper contact region.     

An insight into mode II fracture toughness testing using SCB specimen

The effect of friction forces between the test specimen and its bottom supports on the mode II fracture toughness values obtained using the semicircular bend (SCB) specimen is investigated. First, a number of experiments were conducted on SCB specimen in order to determine the mode II fracture toughness of polymethyl methacrylate (PMMA) according to the conventional approaches available in the literature. Three different types of supports that have been frequently employed by researchers in recent years were used to evaluate the effect of support type on the fracture loads. It was found that the friction forces between the supports and the SCB specimen have a significant effect on the value of mode II fracture toughness measured using the SCB samples. Then, the specimen was simulated using finite element method for more detailed investigation on the near crack tip stress field evolution when friction forces increase between the supports and the SCB specimen. The finite element results confirmed that the type of support affects not only the stress intensity factors K_I and K_II but also the T‐stress. The experimental and numerical results showed that the use of the crack tip parameters available in literature for frictionless contact between the supports and the SCB specimen can result in significant errors when the mode II experiments are performed by using the fixed or roller‐in‐grove types of supports.     

THE EFFECTS OF MICROSTRUCTURE AND FRACTURE SURFACE ROUGHNESS ON NEAR THRESHOLD FATIGUE CRACK PROPAGATION CHARACTERISTICS OF A TWO-PHASE CAST STAINLESS STEEL

In this study, the effects of stress ratio, microstructure and fracture surface roughness on the fatigue properties of a two-phase cast stainless steel were investigated. This behaviour was examined by means of the fracture mechanics approach and fractography. The fatigue crack growth rate decreased with decreasing stress ratio. The stress ratio markedly influenced the fatigue crack growth rate as ΔK approached the ΔK_th value. The roughness of the fracture surface was greater in the as-cast material than in the heat-treated material. Analysis of the crack growth data using ΔK_eff showed that the effect of R ratio could be explained but that the effect of microstructure on crack growth rate could not.     

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.     

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.     

Interactions of a sub-surface crack with a center of dilatation in an elastic half-plane

This paper examines the interaction between a crack parallel to the free surface of an elastic half-plane and an internal center of dilatation. The problem is decomposed into two auxiliary problems. When the center of dilatation approaches the crack tip, two kinds of singularity are analytically obtained. If the overburden stress and the friction on crack surface are neglected, both modes I and II stress intensity factors (K_I and K_II) are induced at the crack tips. The maximum of K_I and K_II occurs when the center of dilatation is located in front of the crack tips. The tensile cracking is likely to be prohibited by the overburden stress, while shear cracking remains possible even including the effects of both overburden and friction on the crack surface.     

Effect of crack shape,inclination angle,and friction coefficient in crack surface contact problems

In rolling/sliding contact fatigue, it is known that the crack propagates at a characteristic angle =15–30 deg to the surface. To analyze the mechanism, however, the body force method has been widely used assuming 3D crack models for =45–90. In this study, therefore, the unknown body force densities are newly approximated by using fundamental density functions and polynomials. Then, a semi-elliptical crack model is analyzed for =15–90 under compressive residual stresses and Hertzian contact loads. The stress intensity factors K _II, K _III are calculated with varying the crack shape b/a, inclination crack angle , and crack face friction coefficient . The calculations show that the present method is useful for the analysis for =15–30 deg with high accuracy. It is seen that the K _II-values when b/a0 are larger than the ones when b/a=1 by 0–24% for both under compressive residual stress and Hertzian contact load. Regarding the maximum K _II values under Hertzian contact load, the results of =15 deg are smaller than the ones of =45 deg by 23–34%. Regarding the amplitude of (K _{II max}–K _{II min}), the results of =15 deg are smaller than the ones of =45 deg by 4–24%. With increasing the value of friction coefficient for crack faces the value of K _II decreases significantly. When the crack is short and the inclination angle is small, the value of friction coefficient f for Hertzian contact load largely affect the K _II value.     

Fretting fatigue crack analysis in Ti–6Al–4V

A study was conducted to verify the efficacy of a fracture mechanics methodology to model the crack growth behavior of fretting fatigue-nucleated cracks obtained under test conditions similar to those found in turbine engine blade attachments. Experiments were performed to produce cracked samples, and fretting fatigue crack propagation lives were calculated for each sample. Cracks were generated at 10⁶ cycles (10%-of-life) under applied stress conditions previously identified as the fretting fatigue limit conditions for a 10⁷ cycle fatigue life. Resulting cracks, ranging in size from 30 to 1200 μm, were identified and measured using scanning electron microscopy. Uniaxial fatigue limit stresses were determined experimentally for the fretting fatigue-cracked samples, using a step loading technique, for R=0.5 at 300 Hz. Fracture surfaces were inspected to characterize the fretting fatigue crack front indicated by heat tinting. The shape and size of the crack front were then used in calculating ΔKth values for each crack. The resulting uniaxial fatigue limit and ΔKth values compared favorably with the baseline fatigue strength (660 MPa) for this material and the ΔKth value (2.9 MPa√m) for naturally initiated cracks tested at R=0.5 on a Kitagawa diagram.Crack propagation lives were calculated using stress results of FEM analysis of the contact conditions and a weight function method for determination of ΔK. Resulting lives were compared with the nine million-cycle propagation life that would have been expected in the experiments, if the contact conditions had not been removed. Scatter in the experimental results for fatigue limit stresses and fatigue lives had to be considered as part of an explanation why the fatigue life calculations were unable to match the experiments that were modeled. Analytical life prediction results for the case where propagation life is observed to be very short experimentally were most accurate when using a coefficient of friction, μ=1.0, rather than for the calculations using μ=0.3     

Shear mode threshold for a small fatigue crack in a bearing steel

The fatigue behaviour of small, semi‐elliptical surface cracks in a bearing steel was investigated under cyclic shear‐mode loading in ambient air. Fully reversed torsion was combined with a static axial compressive stress to obtain a stable shear‐mode crack growth in the longitudinal direction of cylindrical specimens. Non‐propagating cracks less than 1 mm in size were obtained (i) by decreasing the stress amplitude in tests using notched specimens and (ii) by using smooth specimens in constant stress amplitude tests. The threshold stress intensity factor ranges, ΔK_IIth and ΔK_IIIth, were estimated from the shape and dimensions of non‐propagating cracks. Wear on the crack faces was inferred by debris and also by changes in microstructure in the wake of crack tip. These effects resulted in a significant increase in the threshold value. The threshold value decreased with a decrease in crack size. No significant difference was observed between the values of ΔK_IIth and ΔK_IIIth.     

Calculation of mode III stress intensity factor by BEM for cracked axisymmetric bodies

This paper is concerned with the numerical calculation of Mode III stress intensity factor by BEM for cracked axisymmetric bodies, under torsion. Mode III stress intensity factors K _III are obtained using the asymptotic displacement field in the vicinity of the crack border. The asymptotic field is derived by integration along the boundary of the meridian of the cylinder. For traction free cracks no discretization of the crack surface was found necessary. Numerical results proving the efficiency of the proposed method are presented and compared with results given in the literature and with those obtained by FEM.