Discrete modelling of ductile crack growth by void growth to coalescence

Ductile crack growth is analyzed by discrete representation of the voids growing near a blunting crack-tip. Coalescence of the nearest void with the crack-tip is modeled, followed by the subsequent coalescence of other discretely represented voids with the newly formed crack-tip. Necking of the ligaments between the crack-tip and a void or between voids involves the development of very large strains, which are included in the model by using remeshing at several stages of the plastic deformation. The material is here described by standard isotropic hardening Mises theory. For a very small void volume fraction the crack-tip tends to interact with one void at a time, while larger void volume fractions lead to simultaneous interaction of multiple voids on the plane ahead of the crack-tip. In some cases a change from one of these mechanisms to the other is seen during growth through the many voids represented here. First uniformly spaced voids of equal size are considered, but also a few computations for a random distribution of the void spacings or of the void sizes are carried out.     

Overload effects in fatigue crack growth by crack-tip blunting

A basic mechanisms for fatigue crack growth in ductile metals is that depending on crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading. In the present paper, the effect of an overload in one of the cycles is studied based on this mechanism. In a standard numerical analysis accounting for finite strain, it is not possible to follow the blunting/re-sharpening process during many cycles, as severe mesh distortion at the crack-tip results from the huge geometry changes developing during the cyclic plastic straining. Here, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 700 full cycles by using remeshing at several stages of the plastic deformation. Crack growth results for purely cyclic loading are compared with predictions for cases where an overload is applied, and it is shown how crack growth slows down after the overload. Different load amplitudes, and an overload at different cycle numbers are considered.     

Springback investigation of anisotropic aluminum alloy sheet with a mixed hardening rule and Barlat yield criteria in sheet metal forming

A mixed hardening model has been implemented based on Lemaitre and Chaboche non-linear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. The Chaboche isotropic hardening theory is incorporated into the non-linear kinematic hardening model to introduce a surface of nonhardening in the plastic strain space. The bending and reverse bending case study has verified the effectiveness of the mixed hardening model by comparison with the proposed experiment results. Barlat’89 yielding criterion is adopted for it does not has any limitation while Hill’s non-quadratic yield criterion is for the case that the principal axes of anisotropy coincides with principal stress direction. The Backward–Euler return mapping algorithm was applied to calculate the stress and strain increment. The mixed hardening model is implemented using ABAQUS user subroutine (UMAT). The comparisons with linear kinematic hardening model and isotropic hardening model in NUMISHEET’93 benchmark show that the mixed hardening model coupled with Barlat’89 yield criteria can well reflect stress and strain distributions and give a more favorable springback angle prediction.     

Finite element analysis of elastic–plastic materials displaying mixed hardening

The problem of non-radial loading (in the stress space) of structures of elastic–plastic metals is discussed. A concept of combined isotropic and kinematic hardening is proposed. For a material displaying such mixed hardening, constitutive equations are derived and incorporated in a finite element program for the analysis of elastic–plastic 2D-structures. When determining the numerical solution of the non-linear equations, the program enables the investigator to select an optimal combination of step by step and iteration procedures. The computed results are compared with experimental results and with results from calculations based on the overlay model, Reference 25.     

Numerical simulation of fatigue crack growth behavior by crack-tip blunting

In ductile metals one of basic mechanisms for fatigue crack growth is that based on crack-tip blunting under the maximum load and re-sharpening of the crack-tip under minimum load. In this paper, simulations of fatigue crack growth by crack-tip blunting using ANSYS finite element code are presented. This investigation focuses solely on simulation of fatigue crack growth due to crack-tip plasticity only. As such, any material damage and its fracture is not considered. Due to high plastic deformation the present simulations utilize a remeshing technique which allows applying a number of load cycles without terminating the simulation due to the error caused by excessive mesh distortion. The simulations were conducted using a center cracked specimen under various loading conditions including different load ranges and load ratios R = −1, 0 and 0.333. It is shown that fatigue crack growth (FCG) slows down with number of cycles towards a steady state value. The simulated FCG data for constant amplitude loading follow the Paris power law relationship and also indicate a typical R-ratio dependence. It can be noted that for all load cases with load ratios R > 0 no crack closure in the vicinity of the crack-tip wake was observed.     

TiAl基合金层状组织中裂纹扩展区的塑性变形

本文通过观察ＴｉＡｌ合金层片组织四点弯曲试样侧面裂纹扩展与滑移迹线之间的关系，分析讨论了层状组织中塑性变形对裂纹扩展的影响。发现不论层片相对于裂纹处于何种取向，沿层片方向都有一定程度的塑性变形，这种变形对于平行、倾斜、垂直于片层扩展的主裂纹都可引起裂尖钝化，从而引发构成各种韧化机制     

A model on twinning-induced crack kinking out of a metal-ceramics interface

A continuum model is proposed to study the effects of deformation twinning on interface crack kinking in metal/ceramics layered materials. At the final stage of material failure, plastic work hardening exhausts and lattice rotation becomes main mechanism after competing with dislocation gliding. The crack-tip plasticity is established in terms of the second gradient of microrotation due to the coupling effect of the twins. The formed twinning structures not only shield the crack tip, but inhibit further dislocation emission by increasing the near-tip stress levels. A Dislocation-Free Zone (DFZ) can exist in the immediate vicinity of the tip. The model is based on the equivalence of the stresses derived from twin-based crack-tip plasticity, macroscopic plasticity and elasticity on the boundary. The two-parameter characterization of near-tip stress fields is used for the outer plastic zone to account for constraint effects. Crack kinking out of the interface follows the direction of the maximum flow stress from the crack-tip plasticity. The DFZ size and the crack-tip shielding ratio, as well as the kink angle, are obtained for various values of low hardening exponents and crack-tip constraints.     

Modeling the brittle–ductile transition in ferritic steels: dislocation simulations

We present a model for the brittle–ductile transition in ferritic steels based on two dimensional discrete dislocation simulations of crack-tip plasticity. The sum of elastic fields of the crack and the emitted dislocations defines an elasto–plastic crack field. Effects of crack-tip blunting of the macrocrack are included in the simulations. The plastic zone characteristics are found to be in agreement with continuum models, with the added advantage that the hardening behavior comes out naturally in our model. The present model is composed of a macrocrack with microcracks ahead of it in its crack-plane. These microcracks represent potential fracture sites at internal inhomogeneities, such as brittle precipitates. Dislocations that are emitted from the crack-tip account for plasticity. When the tensile stress along the crack plane attains a critical value σ _F over a distance fracture is assumed to take place. The brittle–ductile transition curve is obtained by determining the fracture toughness at various temperatures. Factors that contribute to the sharp upturn in fracture toughness with increasing temperature are found to be: the increase in dislocations mobility, and the decrease in tensile stress ahead of the macrocrack tip due to increase in blunting, and the slight increase in fracture stress of microcracks due to increase in plasticity at the microcrack. The model not only predicts the sharp increase in fracture toughness near the brittle–ductile transition temperature but also predicts the limiting temperature above which valid fracture toughness values cannot be estimated; which should correspond to the ductile regime. The obtained results are in reasonable agreement when compared with the existing experimental data.     

Bounding-surface plasticity for non-linear analysis of space structures

This paper presents a method for the non-linear analysis of space structures subjected to static and cyclic loading. A bounding-surface kinematic hardening plasticity model is used to simulate the hardening and hysteritic material behaviour. The model is used in conjunction with the lumped plasticity assumption coupled with the concept of a yield surface in force space. A hardening coefficient matrix which is a function of the plastic strain and the elastic stiffness matrix is introduced while the vectorial nature of the material memory parameters is maintained. This provides a smooth transition from the elastic to the plastic regime which simulates the hysteresis loops quite accurately. An updated Lagrangian formulation is used together with a predictor/corrector solution method. Several examples are presented to demonstrate the accuracy and efficiency of the method.     

Effect of underloads or overloads in fatigue crack growth by crack-tip blunting

Crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading is one of the basic mechanisms for fatigue crack growth in ductile metals. Here, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 700 full cycles by using re-meshing at several stages of the plastic deformation. A compressive underload in one of the cycles tends to increase the rate of cyclic crack growth, and this effect is studied in detail for a single underload, based on the blunting re-sharpening mechanism. Subsequently, the increased rate of crack growth due to periodically occurring underloads is analysed. A single overload has the opposite effect of giving a significant delay in the subsequent fatigue crack growth. An analysis is carried out to compare the effect of a small overload to that of a larger overload.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

Effects of plastic anisotropy on crack-tip behaviour

For a crack in a homogeneous material the effect of plastic anisotropy on crack-tip blunting and on the near-tip stress and strain fields is analyzed numerically. The full finite strain analyses are carried out for plane strain under small scale yielding conditions, with purely symmetric mode I loading remote from the crack-tip. In cases where the principal axes of the anisotropy are inclined to the plane of the crack it is found that the plastic zones as well as the stress and strain fields just around the blunted tip of the crack become non-symmetric. In these cases the peak strain on the blunted tip occurs off the center line of the crack, thus indicating that the crack may want to grow in a different direction. When the anisotropic axes are parallel to the crack symmetry is retained, but the plastic zones and the near-tip fields still differ from those predicted by standard isotropic plasticity.     

Constitutive modelling of fibre-reinforced composites with unidirectional plies using a plasticity-based approach

This paper presents the development of a constitutive model able to accurately represent the full non-linear mechanical response of polymer-matrix fibre-reinforced composites with unidirectional (UD) plies under quasi-static loading. This is achieved by utilising an elasto-plastic modelling framework. The model captures key features that are often neglected in constitutive modelling of UD composites, such as the effect of hydrostatic pressure on both the elastic and non-elastic material response, the effect of multiaxial loading and dependence of the yield stress on the applied pressure.The constitutive model includes a novel yield function which accurately represents the yielding of the matrix within a unidirectional fibre-reinforced composite by removing the dependence on the stress in the fibre direction. A non-associative flow rule is used to capture the pressure sensitivity of the material. The experimentally observed translation of subsequent yield surfaces is modelled using a non-linear kinematic hardening rule. Furthermore, evolution laws are proposed for the non-linear hardening that relate to the applied hydrostatic pressure.Multiaxial test data is used to show that the model is able to predict the non-linear response under complex loading combinations, given only the experimental response from two uniaxial tests.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

A perturbation solution for a crack in a power-law material under gross yielding

Abstract In order to develop an analytical method for quantifying the plastic-blunting behaviour of a short crack embedded in a notch plastic zone, the perturbation solution of He and Hutchinson is extended to include the effect of strain gradient. An edge-cracked plate subjected to a linearly varying remote strain is considered in this work to simulate the plastic deformation associated with a small crack at a notch root. The strain hardening of the material is assumed to obey a power-law. Comparison with finite-element (FE) computations demonstrates that this perturbation solution provides accurate values for the crack-tip opening displacement (CTOD) under gross-yielding conditions for a range of hardening parameters.     

The steadily propagating viscoplastic crack with elastic unloading

We analyze the crack-tip near field for a crack propagating under steady-state, quasi-static, antiplane strain conditions in a viscoplastic material. The constitutive model employed is a state variable model due to Robinson, which displays kinematic hardening characteristics, and which allows for the possibility of elastic unloading. Given a continuity assumption on the state variable components across the loading/unloading boundary, we find that the crack-tip near field is completely uninfluenced by conditions in the remote field. This conclusion is in agreement with earlier analyses in which the possibility of an elastic unloading zone was not admitted.