Plastic zone near the crack tip in a material with strain anisotropy caused by loading along spatial rectilinear trajectories

The equation of the boundary of the plastic zone near the tip of a mode I crack is deduced for the case of a plate made of a material with strain anisotropy. It is assumed that the anisotropy is caused by hardening in the process of plastic deformation performed prior to the appearance of the crack under loading along arbitrary rectilinear trajectories in the space of the stress tensor. An analysis of this equation demonstrates that the main factors affecting the shape and size of the plastic zone are the level of plastic strains accumulated in the process of preloading, their sign, and the orientation of the crack relative to the axes of anisotropy.     

Correlation of a plastic zone length near the tip of a mode I crack in an orthotropic material with anisotropy parameters

We derived equations that relate the length of a plastic zone near a mode I crack tip in a plate made of an orthotropic material with yield strength levels in the direction of the anisotropy axes. The case of crack orientation along one of the anisotropy axes is examined, with the latter being determined by the strain hardening of a material at the stage preceding the crack nucleation. The growth of yield strength along the axes lying in the plane of the plate is shown to result in smaller sizes of the plastic zone. An increase in yield strength in the direction of the normal to the above plane leads to an increase in its length. Ukrainian Regional Research and Design Institute of Civil Engineering, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 32–37, July–August, 1999.     

A model of fatigue crack propagation in metals

A model of the formation and evolution of a local plastic deformation zone at the crack tip is proposed based on the analysis of the main physical processes taking place in a metallic material under the action of cyclic loads. An equation of fatigue crack growth rate curves, which explicitly accounts for the loading frequency, was derived. The equation applies to the whole range of crack lengths from short cracks to macroscopic ones. __________ Translated from Problemy Prochnosti, No. 1, pp. 35–43, January–February, 2009.     

‘Pressurized’ Mode III Crack in an Elastic Perfectly Plastic Solid

This paper presents the solution for the elastic plastic boundary of a fully ‘pressurized’ mode III crack in an elastic perfectly plastic solid. By fully ‘pressurized’ (in analogy to an internally pressurized mode I crack) is meant that the crack faces are subjected to an externally applied (shear) stress equal to the (shear) yield stress. The dislocations in stress space method is used to find the solution. At the crack plane a plastic zone for the fully pressurized crack extends out a distance 0.273 a from a crack tip, where a is the half width of the crack. (The comparable distance found from a Bilby–Cottrell–Swinden crack is 0.414 a.) The plastic zones originating at the two crack tips merge, in a point, at the center of the crack. This revised version was published online in August 2006 with corrections to the Cover Date.     

Propagation of cracks in structural alloys at a temperature of 4.2 K

We study distinctive features of the process of crack propagation in structural materials under conditions of discontinuous yield and under the action of pulses of electric current at a temperature of 4.2 K. The kinetics of fracture processes in steel under the indicated conditions is studied by the method of stereofractographic analysis. We investigate the applicability of approaches to the evaluation of the characteristics of the crack-growth resistance of materials that take into account the effect of discontinuous yield and pulses of electric current on the process of formation of a plastic zone at a crack tip. Translated from Problemy Prochnosti, No. 6, pp. 36–40, November–December, 1997.     

Influence of deformation anisotropy on the size of the plastic zone at the tip of an opening mode crack

The form and dimensions of the plastic zone at the tip of an opening mode crack in a plate made of a material with deformation anisotropy were investigated within the limits of the elastic solution. The anisotropy was caused by strengthening during plastic deformation until formation of cracks by loading in a straight trajectory located in the plane of the plate. It was shown that in the case of anisotropy caused by loading in a trajectory which is oriented on a normal to the crack edges the size of the plastic zone decreases and its boundaries are rotated in the direction opposite to the crack growth. Loading in a trajectory in the direction of crack growth leads to broadening of the plastic zone in the transverse direction.Translated from Problemy Prochnosti, No. 1, pp. 73–76, January, 1990.     

Effects of notch radius and anisotropy on the crack tip plastic zone

Factors which influence the shape and size of the plastic zone in the immediate vicinity of a crack tip in isotropic materials at small loads are investigated. The plastic zone dimensions for the opening mode (Mode I) have been calculated over a range of values for the crack tip radius. An increase in tip radius results in an increase in the plastic zone dimension. In anisotropic materials, the orientation of crack slit and the anisotropic yield constants are other factors that affect the plastic zone size and shape. In this paper, typical curves for the shape and size of plastic zone are given to illustrate the influence of normal or shear anisotropic yield constants. For sheet metals the effects of anisotropy on the plastic zone dimensions can be evaluated in terms of R values. Suggested values of constant b for isotropic materials are given if the “radius” approximation is employed for small applied stresses.     

Elastic and plastic fracture analysis of a crack perpendicular to an interface between dissimilar materials

Elastic and plastic fracture analysis of a Mode I crack perpendicular to an interface between dissimilar materials is carried out. Continuously distributed dislocations are used to simulate the crack. The simulation will cause singular integral equations with Cauchy kernel. By solving the singular integral equations numerically, the effects of crack depth (distance from the interface to the crack middle point) and Dundurs’ parameters on the Mode I stress intensity factor are investigated systematically. Then, based on the Dugdale model, the plastic zone size, and the crack tip opening displacement of the crack under uniform loadings are investigated. The effects of uniform loadings, crack depth, and Dundurs’ parameters on the plastic zone size and the crack tip opening displacement are examined. Numerical results show that when the crack is embedded in a stiffer material, the values of both the normalized plastic zone size and the normalized crack tip opening displacement are larger than 1. On the contrary, if the crack is embedded in a softer material, the values of both the normalized plastic zone size and the crack tip opening displacement are less than 1.     

Generalized griffith problem of shear with regard for the roughness of the crack surfaces

We study the generalized Griffith problem of longitudinal and transverse shear with regard for the roughness of the crack surfaces. The crack lips are modeled in the form of teeth. We study the process of contact of the teeth and its influence on the stress-strain state at the crack tip. The prefracture zone is modeled by cuts whose lips are in the complex stressed state and, hence, suffer the action of average shear and tensile stresses satisfying the Huber-Mises relations. As a result, we obtain expressions for the length of the plastic zone and crack-tip displacements in the cases of both transverse and longitudinal shear. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36, No. 2, pp. 49–54, March-April, 2000.     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Towards a new model of crack tip stress fields

This work introduces a novel mathematical model of the stresses around the tip of a fatigue crack, which considers the effects of plasticity through an analysis of their shielding effects on the applied elastic field. The ability of the model to characterize plasticity-induced effects of cyclic loading on the elastic stress fields is assessed and demonstrated using full-field photoelasticity. The focus is on determining the form of the shielding stress components (induced by compatibility requirements at the elastic–plastic interface along the crack flank and via the crack tip plastic zone) and how they influence the crack tip elastic stress fields during a load cycle. The model is successfully applied to the analysis of a fatigue crack growing in a polycarbonate CT specimen.     

Three-stage model of an elastoplastic cracked body

We propose a three-stage model of an elastoplastic cracked body, which takes into account the evolution of stresses in the prefracture zone according to the complete tensile stress-strain diagram of the investigated material. The zone of loosening corresponding to the last (descending) part of the tensile stressstrain diagram lies in the immediate vicinity of the crack tip. The zone of plastic deformation of the material is located after the zone of loosening. These zones are simulated by separate rectilinear stress-strain diagrams taking into account the effect of linear hardening of the material. Within the framework of the proposed model, we formulate a deformational fracture criterion and determine the characteristics of crack resistance of D16T steel. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 1, pp. 59–64, January–February, 1998.     

Fractographic and numerical study of hydrogen–plasticity interactions near a crack tip

This paper offers a fractographic and numerical study of hydrogen–plasticity interactions in the vicinity of a crack tip in a high-strength pearlitic steel subjected to previous cyclic (fatigue) precracking and posterior hydrogen-assisted cracking (HAC) under rising (monotonic) loading conditions. Experiments demonstrate that heavier cyclic preloading improves the HAC behaviour of the steel. Fractographic analysis shows that the microdamage produced by hydrogen is detectable through a specific microscopic topography: tearing topography surface or TTS. A high resolution numerical modelling is performed to reveal the elastoplastic stress–strain field in the vicinity of the crack tip subjected to cyclic preloading and subsequent monotonic loading up to the fracture instant in the HAC tests, and the calculated plastic zone extent is compared with the hydrogen-assisted microdamage region (TTS). Results demonstrate that the TTS depth has no relation with the active plastic zone dimension, i.e., with the size of the only region in which there is dislocation movement, so hydrogen transport cannot be attributed to dislocation dragging, but rather to random-walk lattice diffusion. It is, however, stress-assisted diffusion in which the hydrostatic stress field plays a relevant role. The beneficial effect of crack-tip plastic straining on HAC behaviour might be produced by the delay of hydrogen entry caused by residual compressive stresses and by the enhanced trapping of hydrogen as a consequence of the increase of dislocation density after cyclic plastic straining.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.     

Damage modelling of adhesively bonded joints

A cohesive zone model (CZM) has been used in conjunction with both elastic and elasto– plastic continuum behaviour to predict the response of a mixed mode flexure and three different lap shear joints, all manufactured with the same adhesive. It was found that, for a specific dissipated CZM energy (Γ₀) there was a range of CZM tripping tractions (σ_u) that gave a fairly constant failure load. A value of σ_u below this range gave rise to global damage throughout the bonded region before any crack propagation initiated. A value above this range gave rise to a discontinuous process zone, which resulted in failure loads that were strongly dependent on σ_u. A discontinuous process zone gives rise to mesh dependent results. The CZM parameters used in the predictions were determined from the experimental fracture mechanics specimen test data. When damage initiated, a deviation from the linear load–displacement curve was observed. The value for σ _uwas determined by identifying the magnitude that gave rise to the experimentally observed deviation. The CZM energy (Γ ₀) was then obtained by correlating the simulated load-crack length response with corresponding experimental data. The R-curve behaviour seen with increasing crack length was successfully simulated when adhesive plasticity was included in the constitutive model of the adhesive layer. This was also seen to enhance the prediction of the lap shear specimens. Excellent correlation was found between the experimental and predicted joint strengths.     

Microscale characterization of granular deformation near a crack tip

This paper presents a study of microscale plastic deformation at the crack tip and the effect of microstructure feature on the local deformation of aluminum specimen during fracture test. Three-point bending test of aluminum specimen was conducted inside a scanning electron microscopy (SEM) imaging system. The crack tip deformation was measured in situ utilizing SEM imaging capabilities and the digital image correlation (DIC) full-field deformation measurement technique. The microstructure feature at the crack tip was examined to understand its effect on the local deformation fields. Microscale pattern that was suitable for the DIC technique was generated on the specimen surface using sputter coating through a copper mesh before the fracture test. A series of SEM images of the specimen surface were acquired using in situ backscattered electronic imaging (BEI) mode during the test. The DIC technique was then applied to these SEM images to calculate the full-field deformation around the crack tip. The grain orientation map at the same location was obtained from electron backscattered diffraction (EBSD), which was superimposed on a DIC strain map to study the relationship between the microstructure feature and the evolution of plastic deformation at the crack tip. This approach enables to track the initiation and evolution of plastic deformation in grains adjacent to the crack tip. Furthermore, bifurcation of the crack due to intragranular and intergranular crack growth was observed. There was also localization of strain along a grain boundary ahead of and parallel to the crack after the maximum load was reached, which was a characteristic of Dugdale–Barenblatt strip-yield zone. Thus, it appears that there is a mixture of effects in the fracture process zone at the crack tip where the weaker aspects of the grain boundary controls the growth of the crack and the more ductile aspects of the grains themselves dissipate the energy and the corresponding strain level available for these processes through plastic work.     

Two methods of investigation of the influence of hydrogen on the propagation rate of a crack and behavior of fracture of high-strength steels

We present two methods for the investigation of the influence of hydrogen on the propagation rate of a crack and behavior of fracture of high-strength steels. The method for investigation of the influence of electrolytic hydrogenation on a subcritical growth of a crack in high-strength steels is based on the use of simple beam specimens of a certain geometry and on the application of lateral loading in such a way that the stress intensity factor can be constant at the tip of a preliminary induced crack. The method is of great importance for the performance of comparative experiments in evaluating the influence of active media and structural anisotropy of specimens made of high-strength steels with limited sizes on their corrosion crack resistance. Typical examples of the application of the method to investigation of the role of electrolytic hydrogenation in subcritical propagation of cracks and their branching in highstrength steels are given. The method for investigation of heat release under strain and fracture of hydrogenated specimens involves the use of microcalorimetric devices, which allow one to study the influence of hydrogenation on peculiarities of the kinetics of elastic and plastic strains of high-strength steels. We illustrate the efficiency of the method proposed by plotting the “load-elongation” curves and corresponding (in time) characteristics of heat release power in the process of strain and fracture of specimens made of a high-strength steel. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 4, pp. 113–120, July–August, 1998     

A low-cycle fatigue approach to fatigue crack propagation

The damage accumulation hypothesis is used to derive a fatigue crack growth rate equation. The fatigue life of a volume element inside the plastic zone is evaluated by using low-cycle fatigue concepts. Crack growth rate is expressed as a function of cyclic material parameters and plastic zone characteristics. For a given material, crack growth increment, is predicted to be a fraction of the plastic zone size which can be expressed in terms of fracture mechanics parameters,K andJ. Hence, the proposed growth rate equation has a predictive capacity and is not limited to linear elastic conditions.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

Effect of martensite transformation on fracture behavior of shape memory alloy NiTi in a notched specimen

In this paper, the finite element calculation of the stress–strain distribution in front of a notch tip were carried out for two materials. One is a shape memory alloy NiTi with the stress-induced martensite transformation, and another is a fully transformed martensite NiTi without the transformation. Based on the results obtained, and combining a model of the fracture process zone, effect of martensite transformation on the fracture behavior of the shape memory alloy NiTi in a notched specimen of plane stress state is comparably analyzed. The results show that the martensite transformation increases the load to produce plastic deformation in the transformed martensite at the notch tip and decreases the maximum normal stress and plastic strain near the notch tip, and tends to suspend the crack nucleation and propagation in the fully transformed martensite in front of the notch tip, and thus increases the fracture load and improves the toughness. A quantitative analysis based on the model of the fracture process zone shows that the martensite transformation in the SMA NiTi causes about 47% increase in the apparent fracture toughness.