Unified triaxiality at the crack tip subjected to plane strain condition under mixed‐mode (I + II) fracture

The new model of stress triaxiality, subjected to plane strain condition under mixed‐mode (I + II) loading, at the yield loci of the crack tip, has been formulated using unified strength theory. It evaluates critical values of triaxiality for various convex and non‐convex failure criteria, unlike the existing model. It shows the effects of Poisson's ratio and intermediate principal stress for materials whose strength in tension and compression is either equal or unequal. Further, on this basis, the crack initiation angles are predicted for various crack inclinations and compared with those obtained from other fracture criteria. The plastic zone shapes supplement the results. Critical yield stress factor, a significant parameter at the crack tip got lowered as the difference among the three principal stresses reduced to a minimum. The crack initiation angles obtained from the model showed good agreement with those obtained from G‐, S‐, and T‐criterion.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

Elastic and plastic stress analysis of an interface crack between two dissimilar layers

In the present paper, the mixed-mode Dugdale model is applied to investigate the plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers. In the analysis, both normal and shear stresses are assumed to exist in the plastic zones and satisfy the Von Mises yield criterion. The plastic zone sizes can be determined on condition that the stress intensity factors caused by the stresses in the plastic zones and applied loading are zero. Then, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, the plane stress condition is considered. The plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers under a uniform load are examined. The effects of Dundurs’ parameters and the thickness of materials on the plastic zone size and the crack tip opening displacement are investigated in detail. Numerical results show that in the case of small thickness, the values of the normalized plastic zone size and the normalized crack tip opening displacement decrease with increasing Dundurs’ parameters, α and β, while, in the case of infinite thickness, the value of the normalized plastic zone size is independent of α, and the value is symmetric about the axis on which β = 0.     

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.     

Representation of crack-tip plasticity in pressure sensitive geomaterials

This paper presents a model based on dislocation theory for representation of crack tip plasticity in a pressure sensitive material. The basic equations are derived from a simple model of the crack tip plasticity which represents the plasticity by superdislocations placed at the effective centres of the complete slip process distributed around the crack tip. The positions and strengths of the superdislocations are determined by imposing the following conditions:(i) the total stress-intensity factor at the crack tip is zero(ii) the total stress minus the self stress of the superdislocation acting at the superdislocation position is just equal to the friction stress due to the Mohr-Coulomb yield criterion and(iii) the total crack opening displacement produced by the model is maximized.Condition (iii) enables the angle of the slip band on which the superdislocation lies to be determined. The results are presented in a dimensionless form allowing the discussion of particular cases and the recognition of the dominant parameters. A series of parametric studies are carried out to demonstrate that the model can capture the essentials of the crack tip plasticity.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

Elastoplastic modelling of subsurface crack growth in rail/wheel contact problems

Propagation of small subsurface cracks subjected to shear under repeated rolling contact load is studied. An analytical crack model (Dugdale) with plastic strips at the two crack tips is employed. Compressive stresses promoting crack closure and friction between crack faces are considered. The triaxial stress state is used in the yield criterion. A damage criterion is suggested based on experimental LCF data. In a numerical study, critical crack lengths are found below which propagation of an existing crack should be effectively suppressed.     

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.     

Dynamic growth of anti-plane shear cracks in ideally plastic crystals

A near-tip asymptotic analysis is given for the stress and deformation field near the tip of crack propagating dynamically under anti-plane shear in an ideally plastic single crystal. A paricular class of orientations of the crack relative to the crystals is considered so that the yield is so simple diamond shape (relative to directions along perpendicular to the crack) in the plane of the anti-plane shear stresses. The near-tip solution is shown to consists of sectors which carry constant stresses, at yield levels, corresponding to adjacent vertices on the diamond-shaped yield locus, and which are joined along an elastic-plastic shock discontinuity. All plastic flow in the near-tip region occurs in the shock. Plastic strains and particle velocity are finite at the crack tip. The plastic strain is proportional to the elastic strain at onset of yielding and is inversely proportional to the elastic Mach number associated with the speed of crack growth.     

A numerical study of the deviation of the crack ending at the interface in a multilayer

The study of the deviation of a crack reaching the interface separating two materials in a multilayer in a 4-point deflection is considered. This study is based on the use of special isoparametric finite elements.¹ (The elements surrounding the crack tip have nodes placed at a distance α from the crack tip, and in this way we impose a singularity of the stresses in the neighbourhood of the crack.) The Erdogan² criterion that estimates the deviation angle shows that this angle is a function of the initial angle of the crack, and the properties of each component of the multilayer.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

On the effect of the residual stresses on the crack opening displacement in a cracked sheet

The residual stress and displacement fields caused by localized plastic flow near a mode I crack tip in a sheet under plane stress conditions are investigated. The present study founds on the classical Dugdale scheme of the plastic flow localization. The residual stress field is considered to be induced by reversed plastic flow near the crack tip caused by an unloading. As it is found the residual stresses around the crack compress the crack tip, while the residual tensile stresses in a distant from the crack tip zone occur. It is shown the maximum residual tensile stresses can reach the significant value of the one third of the yield limit. The length of the compressed plastic zone and the residual displacement distributions are obtained. The exact formula for the residual crack opening displacement to estimate the crack closure is found. Then the next loading of the cracked plate is considered. It is shown that the second loading causes the origin of two plastic zones localized near the crack tip and at the point, where the maximum residual tensile stresses are concentrated. Again, according to the Dugdale scheme of the plastic localization, both the plastic flow zones are modelled as narrow stripes on the line extending the crack. To determine three non-dimensional parameters, characterizing the position of the segment-like plastic flow zones, a non-linear system of equations is obtained and analyzed. The exact formula for the crack opening displacement after a loading–unloading cycle is obtained. An asymptotic analysis (as the linear dimension of the distant plastic flow zone compared with the actual crack length is small) is given. It shows that the effect of the distant plastic flow zone appears as some complementary crack closure.     

The effects of constraint on crack tip fields and fracture toughness

The detailed stresses, deformations and porosities in the vicinity of a blunting crack in different fracture specimens, Single Edge Notch (SEN), Three Point Bending (TPB) specimens, and Small Scale Yielding (SSY) model were studied by the larger deformation finite element method. The presence and subsequent growth of smaller scale voids were taken into account by using a modified Gurson's model to describe the constitutive behavior of the material. The dependences of the stresses, deformations and porosities on specimen configurations were associated with the crack tip constraint. The porosity in the tip region, along with the void coalescence criterion, were used to predict the macroscopic fracture toughness as a function of the constraint, and a comparison with experimental data was performed in this paper.     

The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

Development of secondary plastic strips near tensile cracks in the plate

We suggest a method for the solution of the elastoplastic problem of biaxial tension-compression of a thin cracked plate under the assumption that plastic strains at each tip of the crack are localized along three slip lines. The plastic zone is simulated by three inclined slip strips at the tips of the crack. It is assumed that the material of the plate is absolutely elastoplastic and the yield criterion is satisfied along the slip lines. The strips lying on the continuation of the crack are simulated by segments of discontinuity of normal stresses, while the other strips are simulated by segments of discontinuity of tangential stresses Thus, the elastoplastic problem under consideration is reduced to the boundary-value problem of the linear theory of elasticity for a body weakened by branched cuts with unknown lengths and orientations of side branches. These parameters are determined in the process of numerical solution of the problem by the method of singular integral equations. Numerical results are presented for the semiinfinite plane with edge crack and under the conditions of uniform tension at infinity. We present numerical values of some parameters of nonlinear fracture mechanics, such as crack tip opening displacement and lengths and angles of inclination of plastic strips, for various combinations of the components of external load.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 2, pp. 7–13, March – April, 1995.     

Effect of thermomechanical loads on the propagation of crack near the interface brittle/ductile

The purpose of this paper is to understand the combined effect of thermal and mechanical loading on the initiation and behaviour of sub-interface crack in the ceramic. In this study a 2D finite element model has been used to simulated mixed mode crack propagation near the bimaterial interface. The assembly ceramometalic is subjected simultaneously to thermomechanical stress field. The extent of a plastic zone deformation in the vicinity of the crack-tip has a significant influence on the rate of its propagation. The crack growth at the joint specimen under four-point bending (4PB) loading and the influence of residual stresses was also evaluated by the maximum tensile stress criterion. The J-integral at the crack tip is generally expressed by the thermomechanical local stresses. The results obtained show the effect of the temperature gradient ΔT, the size of the crack and the applied stresses on the J-integral.     

Asymptotic analysis of growing crack stress/deformation fields in porous ductile metals and implications for stable crack growth

Asymptotic stress and deformation fields near a quasi-statically growing plane strain tensile crack tip in porous elastic-ideally plastic material, characterized by the Gurson-Tvergaard yield condition and associated flow rule, are derived for small uniform porosity levels throughout the range 0 to 4.54 percent. The solution configuration resembles that for crack growth in fully dense, elastically compressible, elastic-ideally plastic Huber-Mises material for this porosity range, except that the angular extents and border locations of near-tip solution sectors vary with porosity level, as do the stress and deformation fields within sectors. Increasing porosity is found to result in a dramatic reduction in maximum hydrostatic stress level, greater than that for a stationary crack; it also causes a significant angular redistribution of stresses, particularly for a range of angles ahead of the crack and adjacent to the crack flank. The near-tip deformation fields derived are employed to generalize a previously-developed, successful ductile crack growth criterion. Our model predicts that for materials having the same initial slopes of their crack growth resistance curves, but different levels of uniform porosity, higher porosity results in a substantially greater propensity for stable crack growth.     

A MODEL TO PREDICT THE FATIGUE LIFE OF FIBRE-REINFORCED TITANIUM MATRIX COMPOSITES UNDER CONSTANT AMPLITUDE LOADING

A model based on micro-mechanical concepts has been developed for predicting fatigue crack growth in titanium alloy matrix composites. In terms of the model, the crack system is composed of three zones: the crack, the plastic zone and the fibre. Crack tip plasticity is constrained by the fibres and remains so until certain conditions are met. The condition for crack propagation is that fibre constraint is overcome when the stress at the location of the fibre ahead of the crack tip attains a critical level required for debonding. Crack tip plasticity then increases and the crack is able to propagate round the fibre. The debonding stress is calculated using the shear lag model from values of interfacial shear strength and embedded fibre length published in the literature. If the fibres in the crack wake remain unbroken, friction stresses on the crack flanks are generated, as a result of the matrix sliding along the fibres. The friction stresses (known as the bridging effect) shield the crack tip from the remote stress, reducing the crack growth relative to that of the matrix alone. The bridging stress is calculated by adding together the friction stresses, at each fibre row bridging the crack, which are assumed to be a function of crack opening displacement and sliding distance at each row. The friction stresses at each fibre row will increase as the crack propagates further until a critical level for fibre failure is reached. Fibre failure is modelled through Weibull statistics and published experimental results. Fibre failure will reduce the bridging effect and increase the crack propagation rate. Calculated fatigue lives and crack propagation rates are compared with experimental results for three different materials (32% SCS6/Ti-15-3, 32% and 38% SCS6/Ti-6-4) subjected to mode I fatigue loading. The good agreement shown by these comparisons demonstrates the applicability of the model to predict the fatigue damage in Ti-based MMCs.     

A Griffith crack shielded by a dislocation pile-up

The dislocation free zone at the tip of a mode III shear crack is analyzed. A pile-up of screw dislocations parallel to the crack front, in anti-plane shear, in the stress field of a crack has been solved using a continuous distribution of dislocations. The crack tip remains sharp and is assumed to satisfy Griffith's fracture criteria using the local crack tip stress intensity factor. The dislocation pile-up shield the sharp crack tip from the applied stress intensity factor by simple addition of each dislocation's negative contribution to the applied stress intensity value. The analysis differs substantially from the well known BCS theory in that the local crack tip fracture criteria enters into the dislocation distributions found.