Cyclic crack growth in elastic plastic solids: a description in terms of dislocation theory

Whenever the plastic deformation is in the order of some Burgers vectors, it appears to be reasonable to describe crack tip plasticity by means of “mathematical” dislocations. A discrete dislocation model is presented for the simulation of mode I fatigue crack propagation. In order to take into account the crack face contact behind the crack tip a procedure was developed which enables the computation of the dislocation motion even when crack closure occurs.     

The flow theory of mechanism-based strain gradient plasticity

The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials [J. Mater. Res. 9 (1994) 1734; Scripta Metall. Mater. 31 (1994) 1037; Interface Sci. 3 (1996) 169].     

Potential theory approach to a dislocation problem

The BCS dislocation model of a crack under anti-plane shear is interpreted in terms of classical plasticity and a yield criterion of BCS type. The model leads to complete and consistent elastic-plastic and limit load solutions which, however, exhibit unbounded stresses at the crack tip. Using a potential theoretic formulation a generalization of the BCS yield criterion is dealt with and a slight modification of the yield criterion results in more physically acceptable bounded stresses. However, it is necessary that the yield strength vanish at the crack tip. It is felt that the BCS model is not valid in obtaining stress and strain fields in a real material. The model is extremely useful for measuring crack tip displacements.     

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.     

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.     

The distributed dislocation technique for calculating plasticity-induced crack closure in plates of finite thickness

An analytical method for calculating plasticity-induced fatigue crack closure in plates of finite thickness is presented. The developed method utilizes the distributed dislocation technique (DDT) and Gauss-Chebyshev quadrature. Crack tip plasticity is incorporated by adopting a Dugdale type strip yield model. The finite plate thickness effects are taken into account by using a recently obtained three-dimensional solution for an edge dislocation in an infinite plate. Numerical results for the ratio of the size of the crack tip plasticity zones are presented for the cases of uniform thickness wake and linearly increasing wake for a range of plate thickness to crack length ratios and applied load ratios. The results show a very good agreement with previous analytical solutions in the limiting cases of very thick and very thin plates. Further results for the opening stress to maximum stress ratio are also provided and are compared with known three-dimensional finite element (FE) solutions. A good agreement is observed. The developed method is shown to be an effective and very powerful tool in modeling the crack closure phenomenon.     

Simulation of simplified zigzag crack paths emerging during fatigue crack growth

When subjected to fatigue loading, microstructurally short cracks form zigzag shapes in single shear mechanisms due to nucleation, glide and annihilation of dislocations. Such crack progression is geometrically complicated and computationally expensive to model. This paper investigates the possibilities of efficiently modelling such growth behaviour by geometrical simplifications of the zigzag path. The crack path and the emerging plasticity was modelled by two different dislocation formulations and it was found that, irrespective of dislocation modelling technique, a valid geometrical approximation was to disregard the surface roughness except for the zigzag section closest to the crack tip.     

Shielding and amplification of a penny-shape crack due to the presence of dislocations

The interaction between a penny-shape crack and a dislocation in crystalline materials is investigated within the framework of dislocation dynamics. The long-range and singular stress field resulting from the crack is determined by modeling the crack as continuous distribution of dislocation loops. This distribution is determined by satisfying the traction boundary condition at the crack face, resulting into a singular integral equation of the first kind that is solved numerically. This crack model is integrated with the dislocation dynamics simulation technique to yield the stress field of the combine system of crack and different types of dislocations situated at different positions in a three dimensional space. The integrated system is then used to investigate the dislocation behavior and its influence on the crack opening displacement and the characteristic of the stress field near the crack tip. It is shown that, depending on the relative position of the dislocation and its character, the dislocation may result in reduction in the stress amplitude at the crack tip and in some cases in closure of the crack tip. These analyses yield shielding and amplification zones near the crack providing an insight of the dislocation influence on the crack. The full dislocation dynamic analysis reveals the nature of the crack dislocation interaction and the manner in which the dislocation morphology changes as it is attracted to the crack surfaces, as well as the changes it causes to the crack profile.     

Crack tip fields at a ductile single crystal-rigid material interface

Small-scale yielding around a stationary crack along a ductile single crystal–rigid material interface is analyzed. Plane strain conditions are assumed to prevail and geometry changes are neglected. The analyses are carried out using both continuum slip and discrete dislocation plasticity theory for model fcc and bcc crystal geometries having either two or three slip systems. Numerical and analytical asymptotic solutions are presented for continuum slip plasticity theory. Solutions exhibiting both slip bands and kink bands are obtained. The addition of a third slip system to ductile single crystals having two slip systems is found to have a significant effect on the interface crack-tip fields. The results illustrate the role that each of the formulations considered can play in elucidating crack tip fields in single crystals.     

Fracture toughness of layered structures: Embrittlement due to confinement of plasticity

The fracture toughness of a layered composite material is analyzed employing a combined two dimensional dislocation dynamics (DD)-cohesive zone (CZ) model. The fracture mechanism of an elastic-plastic (ductile) material sandwiched within purely elastic layers approaches ideally brittle behaviour with decreasing layer thickness. We investigate the influence of different constitutive parameters concerning dislocation plasticity as well as the effect of cohesive strength of the ductile material on the scaling of fracture toughness with layer thickness. For a constant layer thickness, the results of the numerical model are consistent with the expectation that fracture toughness decreases with increasing yield strength, but increases with the cohesive strength of the material. The scaling behaviour of the fracture toughness with layer thickness depends on these material parameters, but also on the dislocation microstructure in the vicinity of the crack tip. Strain localization due to easy dislocation generation right at the crack tip improves toughness in thin layers and leads to a jump-like increase of fracture toughness with layer thickness. However, the fracture toughness for films that are thick enough to exhibit bulk behaviour proves to be higher when the distribution of dislocations is more homogeneous, because in this case the crack grows in a stable fashion over some distance.     

Distribution of dislocations at a mode I crack tip and their shielding effect

Distribution of dislocations at a finite mode I crack tip is formulated. Closed form solutions for the dislocation distribution function, the dislocation-free zone (DFZ), the local stress intensity factor and the crack tip stress field are obtained. The dislocation distribution has similar features to a mode III crack model. Under a given applied stress, there may exist different configurations of plastic zone and DFZ. Crack tip shielding by dislocations depends on both applied stresses and the configuration.     

Application of the distributed dislocation technique for calculating cyclic crack tip plasticity effects

This paper describes a method for modelling cyclic crack tip plasticity effects based on the distributed dislocation technique (DDT). A strip‐yield model is utilised to allow for the determination of the crack opening displacement, size of the plastic zones and in the case of a fatigue crack, the wake of plasticity. The DDT can be easily implemented for a wide range of cracked geometries with reliable control over the accuracy and convergence. Thickness effects can also be incorporated through a recently obtained solution for an edge dislocation in an infinite plate of finite thickness. Results for finite length cracks that have had limited growth, such that there is no plastic wake, are presented for a range of applied loads and R‐ratios. Further results are provided for a steady‐state fatigue crack in a plate of finite thickness. The present results are compared with analytical solutions and they show an excellent agreement.     

The brittle-to-ductile transition and dislocation activity at crack tips

Dislocation activity in the vicinity of a crack tip and the brittle-to-ductile transition (BDT) are analysed using discrete dislocation dynamics simulations. The comparison of these simulations with fracture experiments on tungsten single crystals helps to identify the decisive mechanisms for the BDT of this material. Dislocation nucleation and the availability of active sources are shown to be limiting plasticity at low temperatures and partly in the semi-brittle regime. At elevated temperatures, fracture toughness, crack tip plasticity and the BDT itself can all be viewed as thermally activated processes, which can all be scaled by the same activation energy. It is concluded that they must be controlled by dislocation mobility. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

A threshold fatigue crack closure model: Part I – model development

A fatigue crack closure model is developed that includes the effects of, and interactions between, the three closure mechanisms most likely to occur at threshold; plasticity, roughness, and oxide. This model, herein referred to as the CROP model (for Closure, Roughness, Oxide, and Plasticity), also includes the effects of out‐of‐plane cracking and multi‐axial loading. These features make the CROP closure model uniquely suited for, but not limited to, threshold applications. Rough cracks are idealized here as two‐dimensional sawtooths, whose geometry induces mixed‐mode crack‐tip stresses. Continuum mechanics and crack‐tip dislocation concepts are combined to relate crack face displacements to crack‐tip loads. Geometric criteria are used to determine closure loads from crack‐face displacements. Finite element results, used to verify model predictions, provide critical information about the locations where crack closure occurs. The CROP model is verified with experimental data in part II of this paper.     

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.     

Elastic interaction between screw dislocations and the internal crack near a free surface

The elastic interaction between screw dislocation and the internal crack near a free surface has been investigated. The stress intensity factor at the crack tip, crack extension force, the image force on the dislocation are affected by the free surface. The number and nature of dislocations, m, inside the crack also play an important role in fracture. In order to understand the plastic zone, the zero-force points of dislocation along the x-axis are involved. The dislocation emitted from the right-hand crack tip is enhanced by positive m and reduced by negative m. On the other hand, if the internal crack is closer to the free surface, a dislocation generated from the right-hand crack tip is easier for negative m and more difficult for positive m. However, the role of m on the dislocation emission for the left-hand crack tip is opposite to that for the right-hand crack tip. Finally, three special cases can be obtained from our results. (1) The interaction between a dislocation and a surface crack; (2) the interaction between a dislocation and an internal crack; (3) the interaction between two dislocations.     

Damage mechanisms of fatigue fracture in a metastable beta Ti-15-3 alloy

The mechanism of crack tip deformation in metastable beta Ti-15-3 alloy under fatigue loading has been examined. In spite of the small thickness of the test specimens (1 mm), the plastic zone revealed plane strain conditions which was transformed to a plane stress zone when its size became 0.25 of the crack length. Slip processes whose density increased with crack length were the dominant microscopic feature of crack tip plasticity. Microcracks emanating from the main crack appeared as a result of extensive slip damage. Transmission electron microscopy (TEM) and X-ray evidence indicate the absence of twinning or phase transformation and that dislocation processes constitute the microstructural origin of crack propagation resistance in the alloy. Energy calculations show that the specific energy of slip, 20 MJ m⁻³, exceeds that of microcracking by three orders of magnitude.     

Deformation and thermodynamic response for a crack dislocation model of brittle fracture

Concepts from dislocation and microcrack models are used to define idealized attributes describing the tip region of a crack. The attributes characterize both geometry and deformation; and this identifies a material defect species called a crack dislocation. With the attributes as variables, a scalar density function for each crack dislocation species can be defined.The crack dislocation is different from the common edge or screw species because it creates a new surface area as well as a displacement discontinuity as it propagates through a crystalline lattice. To model the discontinuities, a relative deformation functional is developed which depends on the crack dislocation density function. Since the crack dislocations are the only material defects assumed to contribute deformation discontinuities, the model is for brittle fracture only.The thermodynamic response uses primarily the methodology established by Gibbs. The existence of an internal energy functional is assumed. The methodology results in a definition for a thermodynamic potential for crack dislocation kinetics, a generalization of the Griffith crack propagation concept, and a local measure for the surface strain energy density changes on the crack dislocation line. The equilibrium thermodynamic potential for crack dislocation kinetics introduces a thermodynamic concept to demarcate crack dislocation density transitions from stationary to non-stationary; hence, it provides an incipient fracture criterion that has a thermodynamic basis. For nonequilibrium thermodynamics, the Onsager formalism is used to model the rates and fluxes of the thermodynamic functions.     

Interaction of a dislocation with a surface crack in nonlocal elasticity

The interaction of a screw dislocation with a surface crack is analyzed in nonlocal elasticity under applied shear loading. The stress field is determined near the crack and the dislocation. The classical singularities at the crack tip and the dislocation core are eliminated. A maximum of the shear stress near the dislocation and a minimum between the crack and the dislocation are found, which reveals the interaction of the dislocation with the crack. Two particular cases are obtained: one is no dislocation and the other is no crack in the medium. Numerical calculations show the results from nonlocal elasticity are more rational physically than those from the classical one.