A centrally cracked thin circular disk, Part I: 3D Elastic–plastic finite element analysis

A full field solution, based on small deformation, three-dimensional elastic–plastic finite element analysis of the centrally cracked thin disk under mode I loading has been performed. The solution for the stresses under small-scale yielding and lo!cally fully plastic state has been compared with the HRR plane stress solution. At the outside of the 3D zone, within a distance of rσ_o/J=18, HRR dominance is maintained in the presence of a significant amount of compressive stress along the crack flanks. Ahead of this region, the HRR field overestimate the stresses. These results demonstrate a completely reversed state of stress in the near crack front compared to that in the plane strain case. The combined effect of geometry and finite thickness of the specimen on elastic–plastic crack tip stress field has been explored. To the best of our knowledge, such an attempt in the published literature has not been made yet. For the qualitative assessment of the results some of the field parameters have been compared to the available experimental results of K, gives a fair estimate of the crack opening stress near the crack front at a distance of order 10⁻² in. On the basis of this analysis, the Linear Elastic Fracture Mechanics approach has been adopted in analyzing the fatigue crack extension experiments performed in the disk (Part II).     

Effect of grain size on slow fatigue crack propagation and plastic deformation near crack tip

Fatigue crack growth and its threshold are investigated at a stress ratio of 0.5 for the three-point bend specimen made of Austenitic stainless steel. The effect of grain size on the crack tip plastic deformation is investigated. The results show that the threshold value Δk_th increases linearly with the square root of grain size d and the growth rate is slower for materials with larger grain size. The plastic zone size and ratio for different grain sizes are different at the threshold. The maximum stress intensity factor is k_max and σ_ys is the yield strength. At the same time, the characteristics of the plastic deformation development is discontinuous and anti-symmetric as the growth rate is increased from 2·10^—8 to 10⁻⁷ mm/cycle.A dimensionless relation of the form for collating fatigue crack starting growth data is proposed in which Δk_th represents the stress intensity factor range at the threshold. Based on experimental results, this relation attains the value of 0.6 for a fatigue crack to start growth in the Austenitic stainless steel investigated in this work. Metallurgical examinations were also carried out to show a transgranular shear mode of cyclic cleavage and plastic shear.     

Effect of stress ratio and specimen thickness on fatigue crack growth of CK45 steel

Presented are the effect of stress ratio and thickness on the fatigue crack growth rate of CK45 steel according to DIN 17200. Test results are obtained for constant amplitude load in tension with three stress ratios of R=0, 0.2 and 0.4 and three specimen thicknesses of B=6, 12 and 24 mm. Microgauge crack opening values were used to calculate ΔK_eff values from which the da/dN − ΔK_eff curves are obtained. Crack closure can be applied to explain the influence of mean stress and specimen thickness on the fatigue crack growth rate in the second regime of the two-parameter crack growth rate relation. An empirical model is chosen for calculating the normalized load ratio parameter U as a function of R, B and ΔK and, for correlating the test data.     

A reexamination of plasticity-induced crack closure in fatigue crack propagation

The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.     

Application of effective stress intensity factor crack closure model to evaluate train load sequence effects on crack growth rates

An effective stress intensity factor crack closure model is studied in relation to results of laboratory spectrum crack growth tests on compact tension specimens (CTS) fabricated from rail steel. Comparison of model predictions with test results for crack growth life is effected by means of an analysis of a center-cracked tension panel (CCT) subjected to an equivalent stress spectrum. The trends of the model predictions and test results agree as to the effect of changing cycle order in the spectrum, but the actual effect on crack growth life in the laboratory tests is found to be much stronger than the effect predicted by the crack closure model using the effective stress intensity factor.     

Effects of specimen thickness, hardening and crack closure for the plastic strip model

The triaxial stress constraint T_z and the effective yield stress distribution in the plastic zone for strain hardening materials are considered for the strip where the thickness effect is investigated by introducing a plastic constraint factor α. This factor depends on the specimen thickness, crack length, load level and hardening exponent. A simple expression of the plastic zone length and an expression involving α are obtained. Application of the strip model to crack closure shows that the specimen thickness has an influence and the results are compared with those found finite element.     

Finite deformation analysis of COD,J-integral and crack tip intense strain region in plane strain large-scale yielding

A finite element analysis was performed to simulate crack tip blunting and the development of the intense strain region in a small compact tension specimen (0.4 T CT) of SA533B-1 under plane strain large-scale yielding, with the condition of large-geometry change around the crack tip taken into consideration. The region where the equivalent plastic strain $\text{}\text{?}$g3_p is greater than 0.15 was defined as the intense strain region, which corresponded to the recrystallized-etched zone delineated experimentally around the blunting crack tip. The development of the intense strain region was discussed as a function of the J-integral and the crack opening displacement. A linear relationship was obtained between the plastic work W_p dissipated within the intense strain region and (J/σ_y)² or b², where b is the crack opening displacement, defined as the separation of the two points at which the boundary of the intense strain region surrounding the crack tip intersects with the free surfaces of the crack.     

Influence of debonding on the efficiency of crack patching

A simplified analysis of crack patching is presented in which the restraining effect of the external bonded reinforcement on the crack opening is simulated by distributed springs acting between the crack faces. It is suggested that elastic-perfectly plastic springs provide an adequate idealization for the case where debonding can occur. The analysis is thereby reduced to the solution of a one-dimensional integral equation involving only two material parameters: a spring constant k and a limit stress σ_L, both of which can be determined from the behaviour of a suitable overlap joint. The efficiency of the reinforcement is shown to be determined by the normalized crack length ka and the normalized stress σ/σ_L, so that a comprehensive characterization can be obtained from relatively little computation or experimental study.     

Influence of singularity dominated zone for propagating cracks in finite size specimens

The singularity dominated zones for straight as well as curved cracks propagating in finite size specimens were determined experimentally by using the optical method of dynamic photoelasticity using the near-field stress equations. Experimental data was carefully analyzed using improved numerical schemes to get the complete stress field around the propagating crack. This stress field was critically examined to evaluate the size of singularity dominated zones for cracks propagating in straight as well as curved paths. For this purpose, the exact solution was compared with the singular solution using stress components σ_x, σ_y, τ_xy and the maximum shear stress τ_max as a criterion respectively. For straight cracks where the stress field is symmetric about the crack path, the singularity dominated zones can be determined by using any one of the stresses. However, for a curved crack, the zones were unsymmetric. This study shows that σ_y, the crack opening stress, yields the best result for characterizing the singularity dominated zone around a running crack tip.     

CONSTRAINTS ON THE CRACK TIP PLASTIC FIELDS UNDER LARGE SCALE YIELD

Plane strain elastic-plastic finite element analyses are used to study the stress, strain fields around a straight crack in power hardening plastic material. Center crack panel (CCP), single edge crack panel (SECP) and double edge crack panel (DECP) tension specimens are analyzed with various crack lengths. Two local constraint parameters, i.e. in-plane stress ratioT _x and out-of-plane constraintT are analyzed, which are defined as tangential stress dividing normal (open) stress and out-of-plane stress dividing the sum of tangential stress and normal stress respectively. Numerical analyses indicate that the two local constraint parameters are nearly independent of the specimen geometry and the loading level in the plastic zone. Methods of estimating two constraint parameters and ways to define the boundary of the blunt zone are given. By using these parameters, the stress distributions in the plastic zone, especially in the blunt zone can be accurately predicted. Project supported by the National Foundation of Distinguished Young Scientists of China (No. 59625510).     

Fatigue life prediction of out-of-plane gusset welded joints using strain energy density factor approach

Fatigue growth behavior of out-of-plane gusset welded joints is studied using the strain energy density factor approach. Fatigue tests on two types of specimens with curvatures of ρ = 0 and ρ = 30 were performed in order to estimate fatigue strength under tension. Fatigue crack growth analysis is carried out to show the effects of initial crack shape, initial crack length and stress ratio. Fatigue crack growth parameters were obtained from crack growth curves assuming constant crack shapes. The results of analysis for the assumed crack shapes agreed well with the experimental data. Fatigue propagation life of the ρ = 30 specimen was larger than that of the ρ = 0 specimen.     

Influence of stress triaxiality on damage and crack tip opening displacement parameters for steels

The dependence of the void growth parameter on the local stress triaxiality and local effective plastic strain near the crack tip of ductile materials provides the motivation to seek for parameters that could rank the ductility of steels. Experimental data for AS 1405-180, AS 1204-350, HY-80 and C---Mn steels show that the crack tip opening displacement (CTOD) at initiation δ_c decreases with increasing crack tip stress triaxiality. This trend is confirmed by analysis. As the critical local effective plastic strain ε_ec also decreases with increasing local stress triaxiality, the ratio δ_c/ε_ec is found to remain nearly constant or independent of the local constraint, i.e., the stress triaxiality. These parameters are given for a class of steels in this paper. Their association with the resistance to ductile fracture remains to be investigated.     

Plasticity aspects of interfacial fracture for polymer/glass specimens

Experimental results suggest that the interfacial fracture resistance is minimal for approximate near tip Mode I accompanied by positive and negative near tip Mode II. Finite-strain FE analysis is made for an elastic–plastic medium bonded to an ideally elastic medium with an interface crack. Small-scale plasticity conditions are invoked and examined in relation to the elastic–plastic stress distribution along the bond line. Plasticity engenders a tendency to turn near tip biaxiality towards pure Mode I regardless of the mixed-mode loading. High levels of hydrostatic stress are attained. For different mode mixities of the applied load, the dependence of the elastic–plastic normal bond stress on load level is examined. It is found that under positive Mode II loading, the normal bond stress σ_yy tends to saturate as the load level rises. This does not occur for Mode I and negative Mode II loading. In addition, deformation patterns inside the plastic zone are examined for mixed-mode situations. A displacement criterion based on the normal bond crack opening suggests a dependence of the critical load level on the extent of mixed mode. Under positive mode II fracture, traces of the ductile material are found at the top of the elastic substrate. Some of these conclusions appear to be consistent with the fracture patterns observed for LD-polyethylene/glass interfacial mixed-mode fracture.     

Fully plastic center-cracked strip under anti-plane shear

The problem of a center-cracked strip subjected to uniform remote anti-plane shear stress is transformed to a problem in a hodograph plane which is solved exactly by Mellin transform and Wiener-Hopf technique. The material of the strip satisfies a pure power hardening stress strain relation and the results are valid for both deformation and flow theories of placticity. Numerical values are given for the crack opening displacement δ and Rice's path independent J integral for several values of the power hardening exponent n and crack width to strip width ratios. Approximate asymptotic formulas are presented for J and δ for large n.     

Characterization of crack tip stresses in plane-strain fracture specimens having weld center crack

Fracture toughness of metals depends strongly on the state of stress near the crack tip. The existing standards (like R-6, SINTAP) are being modified to account for the influence of stress triaxiality in the flaw assessment procedures. These modifications are based on the ability of so-called ‘constraint parameters’ to describe near tip stresses. Crack tip stresses in homogeneous fracture specimens are successfully described in terms of two parameters like J–Q or J–T. For fracture specimens having a weld center crack, strength mismatch ratio between base and weld material and weld width are the additional variables, along with the magnitude of applied loading, type of loading, and geometry of specimen that affect the crack tip stresses. In this work, a novel three-parameter scheme was proposed to estimate the crack tip opening stress accounting for the above-mentioned variables. The first and second parameters represent the crack tip opening stress in a homogeneous fracture specimen under small-scale yielding and are well known. The third parameter accounts for the effect of constraint developed due to weld strength mismatch. It comprises of weld strength mismatch ratio (M, i.e. ratio of yield strength of weld material to that of base material), and a plastic interaction factor (I_p) that scales the size of the plastic zone with the width of the weld material. The plastic interaction factor represents the degree of influence of weld strength mismatch on crack tip constraint for a given mismatch ratio. The proposed scheme was validated with detailed FE analysis using the Modified Boundary Layer formulation.     

Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics

Hancock and Cowling measured the critical crack tip opening displacements, δ_f, at fracture initiation in HY-80 steel specimens of six different configurations. δ_f varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δ_f measured by Hancock and Cowling correlates very well with the ratio ( ). δ_f is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).     

Kinetics of microcrack blunting ahead of macrocrack approaching shear wave speed

The fracturing of glass and tearing of rubber both involve the separation of material but their crack growth behavior can be quite different, particularly with reference to the distance of separation of the adjacent planes of material and the speed at which they separate. Relatively speaking, the former and the latter are recognized, respectively, to be fast and slow under normal conditions. Moreover, the crack tip radius of curvature in glass can be very sharp while that in the rubber can be very blunt. These changes in the geometric features of the crack or defect, however, have not been incorporated into the modeling of running cracks because the mathematical treatment makes use of the Galilean transformation where the crack opening distance or the change in the radius of curvature of the crack does not enter into the solution. Change in crack speed is accounted for only via the modulus of elasticity and mass density. For this simple reason, many of the dynamic features of the running crack have remained unexplained although speculations are not lacking. To begin with, the process of energy dissipation due to separation is affected by the microstructure of the material that distinguishes polycrystalline from amorphous form. Energy extracted from macroscopic reaches of a solid will travel to the atomic or smaller regions at different speeds at a given instance. It is not clear how many of the succeeding size scales should be included within a given time interval for an accurate prediction of the macroscopic dynamic crack characteristics. The minimum requirement would therefore necessitate the simultaneous treatment of two scales at the same time. This means that the analysis should capture the change in the macroscopic and microscopic features of a defect as it propagates. The discussion for a dual scale model has been invoked only very recently for a stationary crack. The objective of this work is to extend this effort to a crack running at constant speed beyond that of Rayleigh wave. Developed is a dual scale moving crack model containing microscopic damage ahead of a macroscopic crack with a gradual transition. This transitory region is referred to as the mesoscopic zone where the tractions prevail on the damaged portion of the material ahead of the original crack known as the restraining stresses, the magnitude of which depends on the geometry, material and loading. This damaged or restraining zone is not assumed arbitrarily nor assumed to be intrinsically a constant in the cohesive stress approach; it is determined for each step of crack advancement. For the range of micronotch bluntness with 0 < β < 30° and 0.2 σ_∞/σ₀ 0.5, there prevails a nearly constant restraining zone size as the crack approaches the shear wave speed. Note that β is the half micronotch angle and the applied stress ratio is σ_∞/σ₀ with σ₀ being the maximum of the restraining stress. For σ_∞/σ₀ equal to or less than 0.5, the macrocrack opening displacement COD is nearly constant and starts to decrease more quickly as the crack approaches the shear wave speed. For the present dual scale model where the normalized crack speed v/c_s increases with decreasing with the one-half microcrack tip angle β. There prevails a limit of crack tip bluntness that corresponds to β 36° and v/c_s 0.15. That is a crack cannot be maintained at a constant speed if the bluntness is increased beyond this limiting value. Such a feature is manifestation of the dependency of the restraining stress on crack velocity and the applied stress or the energy pumped into the system to maintain the crack at a constant velocity. More specifically, the transitory character from macro to micro is being determined as part of the unknown solution. Using the energy density function dW/dV as the indicator, plots are made in terms of the macrodistance ahead of the original crack while the microdefect bluntness can vary depending on the tip geometry. Such a generality has not been considered previously. The macro-dW/dV behavior with distance remains as the inverse r relation yielding a perfect hyperbola for the homogeneous material. This behavior is the same as the stationary crack. The micro-dW/dV relations are expressed in terms of a single undetermined parameter. Its evaluation is beyond the scope of this investigation although the qualitative behavior is expected to be similar to that for the stationary crack. To reiterate, what has been achieved as an objective is a model that accounts for the thickness of a running crack since the surface of separation representing damage at the macroscopic and microscopic scale is different. The transitory behavior from micro to macro is described by the state of affairs in the mesoscopic zone.     

随机载荷下疲劳裂纹的闭合和扩展估算模型

本文分析了裂尖前,后方塑性区对闭合性能的影响,提出从裂尖塑性钝化量和尾迹区残余塑性变形两个方面来确定裂纹面的残余变形,并讨论了压缩载荷对闭合应力的影响,由此建立了一个疲劳裂纹闭合模型,然后通过模型“有限记忆”性质的假定将它应用到随机加载情况。用此模型对铝合金2219——T851受飞机谱载荷的CCT平面应力试件进行了疲劳寿命估算,估算值与实验结果接近。     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

A phenomenological model of fatigue crack growth in perfectly plastic infinite plates under completely reversed uniaxial loading

The fatigue failure of a thin infinite center-cracked plate under completely reversed uniaxial loading is considered. A two-stage fatigue crack model including the incubation and crack propagation stages is constructed. The stress distribution in the vicinity of the crack tip is described using the concept of a conventional elastic crack. The crack-tip plastic zone is simulated by a Dugdale thin plastic zone, and the condition for the movement of the failure front is given by criteria of damage mechanics. It is shown that the fatigue crack growth rate in perfectly plastic materials with a plastic zone of constant length is a power-law function of the stress intensity factor range. This relationship is quadratic when the length of the plastic zone is not constant Published in Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 116–127, December 2005.