The influence of crystallographic orientation on crack tip displacements of microstructurally small, kinked crack crossing the grain boundary

The paper presents an analysis of the effects of grain orientations on a short, kinked surface crack in a 316L stainless steel. The kinking of the crack is assumed to take place at the boundary between two neighbouring grains. The analysis is based on a plane-strain finite element crystal plasticity model. The model consists of 212 randomly shaped, sized and oriented grains, loaded monotonically in uniaxial tension to a maximum load of 0.96R_p0.2 (240 MPa). The influence that a random grain structure imposes on a Stage I crack is assessed by calculating the crack tip opening (CTOD) displacements for bicrystal as well as for polycrystal models, considering different crystallographic orientations. Since a Stage I crack is assumed, the crack is always placed in a slip plane. Results from a bicrystal case show that the maximal CTODs are directly related to the stiffness of the grain containing the crack extension. Anisotropic elasticity and crystal plasticity both contribute to this grain stiffness, resulting in maximal CTOD when Schmid factors are the highest on two slip planes. Such crystallographic orientation results in a soft elasto-plastic response. Anisotropic elasticity can additionally increase the softness of a grain at certain crystallographic orientations. Minimal anisotropic elasticity at the crystallographic orientations with the highest Schmid factors causes the CTOD to be maximized. Presuming that the crack will preferably follow the slip plane where the crack tip opening displacement is highest, we show that the crystallographic orientation can affect the CTOD values by a factor of up to 7.7. For a given grain orientation the maximum CTOD is attained when the crack extension deflection into a second grain is between −75.141° and 34°. For the polycrystal case we show that grains beyond the first two crack-containing grains change the CTOD by a factor of up to 3.3 and that the largest CTODs are obtained when placing the crack into a slip plane with crack extension that results in a crack extension being more perpendicular to the external load.     

Cyclic crystal plasticity analyses of stationary, microstructurally small surface cracks in ductile single phase polycrystals

ABSTRACT This paper explores the effects of microstructural heterogeneity on the cyclic crack tip opening and sliding displacements for stationary, microstructurally small transgranular surface cracks in a single phase metallic polycrystal using planar double slip crystal plasticity computations. Crack tip displacements are examined under plane strain conditions for stationary cracks of different lengths relative to grain size as a function of the applied nominal strain amplitude for tension-compression and cyclic shear. Nominal strain amplitudes range from well below to slightly above the nominal cyclic yield strength for each type of loading condition. Results indicate the complex nature of the crack tip sliding and opening displacements as functions of nominal strain amplitude and orientation of the nearest neighbour grains, the influence of the free surface in promoting the cyclic opening displacement even for cracks in the first surface grain, the rather restricted limits of applicability of linear elastic fracture mechanics, and very interesting crack tip plasticity effects which include crack tip displacement ratcheting or progressive accumulation, even for completely reversed, proportional applied loading. Results are compared for cases with and without crack face friction.     

Crack tip displacements of microstructurally small surface cracks in single phase ductile polycrystals

A planar double slip crystal plasticity model is applied to the evaluation of crack tip opening (CTOD) and sliding (CTSD) displacements for microstructurally small stationary cracks under monotonic loading for a material with nominal stress-strain behavior that is representative of a relatively high strength helicopter rotor hub material. Two-dimensional plane strain finite element calculations are presented for CTSD and CTOD of microstructurally small transgranular surface cracks in a polycrystal subjected to monotonic loading. The effects of crack length relative to grain size, orientation distribution of nearest neighbor grains, stress state and stress level are considered for nominal stress levels below the macroscopic yield strength. The CTOD and CTSD are computed for stationary crystallographic surface cracks with various realizations of crystallographic orientations of surrounding grains. It is found that (i) the opening displacement is dominant for remote tension even for crystallographic cracks oriented along the maximum shear plane in the first surface grain, (ii) there is a strong dependence of the CTOD on the proximity to grain boundaries, but lesser dependence of the CTSD, and (iii) that the elastic solutions for CTOD and CTSD are valid below about 30% of the 0.2% offset-defined yield strength.     

Crack tip displacements of microstructurally small cracks in 316L steel and their dependence on crystallographic orientations of grains

The paper presents an analysis of the effect of the grain orientations on a short Stage I surface crack in a 316L stainless steel. The analysis is based on a plane‐strain finite element crystal plasticity model. The model consists of 212 randomly shaped, sized and oriented grains that is loaded monotonically in uniaxial tension to a maximum load of 1.12R_p0.2 (280 MPa). The influence of random grain structure on a crack is assessed by calculating the crack tip opening (CTOD) and sliding displacements (CTSD) for single crystal and polycrystal models, considering also different crystallographic orientations. In the single crystal case the CTOD and CTSD may differ by more than one order of magnitude. Near the crack tip slip is activated on all the slip planes whereby only two are active in the rest of the model. The maximum CTOD is directly related to the largest Schmid factors. For the more complex polycrystal cases it is shown that certain crystallographic orientations result in a cluster of soft grains around the crack‐containing grain. In these cases the crack tip can become a part of the localized strain, resulting in a large CTOD value. This effect, resulting from the overall grain orientations and sizes, can have a greater impact on the CTOD than the local grain orientation. On the other hand, when a localized soft response is formed away from the crack, the localized strain does not affect the crack tip directly, resulting in a small CTOD value. The resulting difference in CTOD can be up to a factor of 4, depending upon the crystallographic set. Grains as far as 6xCracklength significantly influence the crack tip parameters. It was also found that among grains with favourable orientation the CTOD increased with the size of such a grain. Finally, a significant change in CTOD and CTSD was observed when extending the crack into the second grain and placing it in the primary or the conjugate slip plane.     

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.     

Oscillatory crack tip triangular elements for finite element analysis of interface cracks

In this paper a special crack tip element has been developed in which displacements and stresses have the same behaviour as those of bi‐material interface cracks with open tips. The element degenerates into a traditional triangular quarter point element in cases of homogeneous cracks. An isoparametric co‐ordinate system (ρ, t) is defined in this study, and numerical techniques using these co‐ordinates to evaluate Jacobian matrices, shape function derivatives, and element stiffness matrices are developed. Also, equations calculating the complex stress intensity factor using displacements are obtained in this study. Numerical results are in good agreement with known analytical solutions in two examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulating small crack growth behaviour using crystal plasticity theory and finite element analysis

Predictions of small crack growth under cyclic loading in aluminium alloy 7075 are performed using finite element analysis (FEA), and results are compared with published experimental data. A double‐slip crystal plasticity model is implemented within the analyses to enable the anisotropic nature of individual grains to be approximated. Small edge‐cracks in a single grain with a starting length of 6 μm are incrementally grown following a node‐release scheme. Crack‐tip opening displacements (CTOD) and crack opening stresses are calculated during the simulated crack growth, and da/dN against ΔK diagrams are computed. Interactions between the crack tip and a grain boundary are also considered. The computations are shown to accurately capture the magnitude and the variability normally observed in small crack fatigue data.     

Application of crack tip plasticity based fatigue model on high temperature alloy HAYNES® 282®

Confined crack tip plasticity model is employed to predict time independent fatigue crack growth rate (FCGR) behavior of HAYNES® 282® alloy at temperatures 1200F and 1400F. Crack growth tests were done in lab air, vacuum and steam environments at load ratios R = K_min/K_max ranging from 0.05 to 0.5. Calibrated model predicts average cyclic crack growth rate behavior of the material reasonably well. Predictions do not capture the accelerated fatigue crack growth rates observed in the data at low load levels. Such effects are believed to be caused by environmentally driven factors, which are not expected to be predicted by plasticity based models.     

Strip yield analysis of two collinear unequal cracks in an infinite sheet

Plastic zone sizes and crack tip opening displacements (CTODs) are obtained for two collinear cracks in an infinite sheet subjected to known remote stress. Analysis is conducted by assuming the crack accompanying plastic zones as a fictitious crack and formulating integral equations based upon traction free and no stress singularity conditions. In addition, critical remote stress, plastic zone sizes, and CTODs when the adjacent plastic zones touched are obtained by assuming the coalesced fictitious cracks as a single fictitious crack and formulating integral equations based upon no stress singularity and zero coalesced point displacement conditions. Extensive numerical results are presented.     

Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

Consideration of the mechanical behaviour of small fatigue cracks

The mechanical behaviour of small fatigue cracks is investigated for a low, medium and high strength material. At first an elastic consideration is performed which give a good impression how the stress fields change with crack size. In part 2 a full elastic-plastic analysis of short cracks is performed using a new numerical scheme to simulate the growth of shear bands emanating from the crack tip. The influence of material and loading paramters as well as of the crack size on the plastic crack tip opening displacement is discussed. It is also investigated how it is possible to get a conservative estimate of the crack tip deformation at small cracks.     

Effect of anisotropic plasticity on mixed mode interface crack growth

Crack growth along an interface between a solid with plastic anisotropy and an elastic substrate is modelled by representing the fracture process in terms of a traction-separation law specified on a crack plane. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. Conditions of small-scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity field is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and based on these results the dependence of the steady-state fracture toughness on the near-tip mode mixity is determined. Different initial orientations of the principal axes relative to the interface are considered and it is found that the steady-state fracture toughness is quite sensitive to this orientation of the anisotropy.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

Fatigue crack initiation life prediction for aluminium alloy 7075 using crystal plasticity finite element simulations

An experimentally-validated approach for predicting fatigue crack initiation life of polycrystalline metals is developed based on crystal plasticity finite element (CPFE) simulations. In this approach, the microstructure used in the simulations possesses statistically the same grain size and crystallographic orientations as those obtained from electron back-scatter diffraction experiments. A backstress model is incorporated into the CP constitutive model to describe the mechanical behaviour of aluminium alloy (AA) 7075 under cyclic loading. The key variables of the prediction model, the energy efficiency factor and plastic strain energy density, are calibrated using a fatigue test on a round-notched AA7075 specimen at room temperature. The proposed approach is then validated by using another fatigue test to predict 69.1–87.3% of the experimentally measured fatigue crack initiation life. The effects of the microstructure and texture on the energy efficiency factor and fatigue life prediction are quantitatively determined. It is shown that for a given range of energy efficiency factors a similar range of life prediction is obtained. Since the proposed approach considers the heterogeneity of the microstructure, it can well capture the grain scale deformation localisation and therefore improve the precision of fatigue life prediction.     

Effect of a variation in material properties on the crack tip opening displacement

This paper investigates the effect of a variation in material properties on the crack tip opening displacement, a parameter often used in the prediction of fatigue and fracture. This situation is typical when a component is subjected to a relatively slow temperature fluctuation or the material properties undergo direct changes, such as due to a phase transformation. An analytical strip‐yield model is developed using the small‐scale yielding assumption and theory of complex potentials. Four cases of crack tip plasticity behaviour are identified for the different combinations of parameters controlling the variation in material properties. Results of calculations over a wide range of material properties are presented and show a significant effect on the crack tip opening displacement. Finite element simulations are conducted to verify the analytical findings. The implications of the outcomes in relation to several practical situations are also discussed.     

A thermomechanical crystal plasticity constitutive model for ultrasonic consolidation

We present a micromechanics-based thermomechanical constitutive model to simulate the ultrasonic consolidation process. Model parameters are calibrated using an inverse modeling approach. A comparison of the simulated response and experimental results for uniaxial tests validate and verify the appropriateness of the proposed model. Moreover, simulation results of polycrystalline aluminum using the identified crystal plasticity based material parameters are compared qualitatively with the electron back scattering diffraction (EBSD) results reported in the literature. The validated constitutive model is then used to simulate the ultrasonic consolidation process at sub-micron scale where an effort is exerted to quantify the underlying micromechanisms involved during the ultrasonic consolidation process.     

Effect of constraint on void growth near a blunt crack tip

The growth of a single cylindrical hole ahead of a blunt crack tip was studied using large deformation finite element analysis in three-point bend specimens with different precrack depth. The effect of small second phase particles was taken into account by incorporating Gurson's constitutive equation. The effects of strain hardening and the initial distance from the hole to the crack tip were also investigated. The results show that the variation of crack tip opening displacement with load is not sensitive to constraint level. The effects of constraint on the growth of hole and ductile initiation toughness are diminished with decreasing initial distance from the hole to the blunt crack tip. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical modelling of plane strain plasticity induced crack closure effects for bimaterial interfacial cracks

The effects of plane strain plasticity induced crack closure on fatigue cracks located at the interface of dissimilar steel materials are presented using finite element modelling. Based on the study, it has been observed that bimaterial cracks produced unsymmetrical residual plastic strains and crack profiles in the crack wakes. It is seen that Young’s modulus and yield stress mismatch have profound effects on the development of unsymmetrical residual plastic strain and crack profiles, whereas the effect of Poisson’s ratio is insignificant. However, it has been found that for the material properties considered, low value of crack closure levels have been identified.     

Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet

The problem of two equal-length collinear cracks in an infinite sheet is treated using the weight function method. Exact weight functions for the inner and outer crack tips are derived based on the crack opening displacement solution for a reference load case. These weight functions are used to calculate stress intensity factors for different load cases, plastic zone sizes and crack tip opening displacements of the strip yield model. The approach is validated by the perfect agreement between the present strip yield model solutions and Collins and Cartwright’s analytical results based on the direct complex stress function formulation.     

Estimation of the effects of plasticity and resulting crack closure during small fatigue crack growth

On the basis of a plastic-strip model and the method of singular integral equations, a closed-form analytical solution of the problem of an elastic-plastic plate containing a rectilinear fatigue crack is considered. The solution is used for the prediction of fatigue growth of `mechanically-small' crack by accounting for reverse plastic yielding and plasticity-induced crack closure in the material. The main effects of these factors on the crack-growth rate are analyzed, and the predicted results are compared with experimental data on small fatigue-crack growth in a aluminum-lithium alloy 2091-T351 and Fe-3% Si alloy.