A Peierls criterion for the onset of deformation twinning at a crack tip

A criterion for the onset of deformation twinning (DT) is derived within the Peierls framework for dislocation emission from a crack tip due to Rice (J. Mech. Phys. Solids 40(2) (1992) 239). The critical stress intensity factor (SIF) is obtained for nucleation of a two-layer microtwin, which is taken to be a precursor to DT. The nucleation of the microtwin is controlled by the unstable twinning energyγ_ut, a new material parameter identified in the analysis. γ_ut plays the same role for DT as γ_us, the unstable stacking energy introduced by Rice, plays for dislocation emission. The competition between dislocation emission and DT at the crack tip is quantified by the twinning tendencyT defined as the ratio of the critical SIFs for dislocation nucleation and microtwin formation. DT is predicted when T>1 and dislocation emission when T<1. For the case where the external loading is proportional to a single load parameter, T is proportional to . The predictions of the criterion are compared with atomistic simulations for aluminum of Hai and Tadmor (Acta Mater. 51 (2003) 117) for a number of different crack configurations and loading modes. The criterion is found to be qualitatively exact for all cases, predicting the correct deformation mode and activated slip system. Quantitatively, the accuracy of the predicted nucleation loads varies from 5% to 56%. The sources of error are known and may be reduced by appropriate extensions to the model.     

Heat generation at the tip of a moving crack

The high energy concentration at the tip of a moving crack causes irreversible deformations and produces heat as a consequence. The resulting temperatures were calculated by consideration of the crack tip as a moving heat-source of rectangular shape. In brittle materials with very small plastic zones and high crack velocities, these temperatures are predicted to be higher than 1000 K. For the experimental verification of these calculations, a very sensitive radiation thermometer was developed. It registers the intensity of the radiation at four wavelengths. By comparison of these intensities with that of black body radiation, the temperature was determined as 3200 K for glass and 4700 K for quartz.     

Stress distribution at the tip of a growing crack

High-speed motion-picture photography and an optical method of stress analysis have been used to study the distribution of elastic stress fields at the tip of a crack growing at a fast rate. The existence of several specific properties of the field characteristic of fast crack propagation rates has been established, and the results obtained are used to explain the branching of cracks.The authors convey their thanks to G. I. Barenblatt for sponsoring this work.The authors convey their thanks to G. I. Barenblatt for sponsoring this work.     

Singular stress at the ends of a brittle crack for zero value of the J integral

Novosibirsk State Technical University, 630092 Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 6, pp. 108–112, November–December, 1995     

Reverse shear on inclined planes at the tip of a sharp crack

Plane strain plastic yielding at a crack tip has been represented by edge dislocations with Burgers vectors parallel to symmetrical planes inclined at 70° and 45° to the plane of the crack. The plastic displacement and the stresses near the crack tip were calculated by a numerical method and the effect of a reduction in applied stress was determined. Removal of the whole or a part of the initial load produces reverse shear in regions of the slip band nearest the crack tip. The amount of reverse shear depends only on the reduction in the load and not on its initial value. The reverse shear is associated with the presence of negative dislocations and the stresses near the crack tip may become compressive even though the applied (remote) stress is still tensile. The degree and extent of compression depends on the reduction in applied stress and on its original value. It is argued that the residual compressive stresses produced under fluctuating loads may produce crack closure and crack arrest. The effect of residual plasticity in a slip band left behind a growing crack has been estimated. It is shown that after an overload the excess residual plasticity opposing crack opening rises to a maximum value when the crack tip has advanced some distance from the point where the overload was applied.     

A note on the stress singularity at a non-uniformly moving crack tip

The stress-singularity at a crack tip moving arbitrarily in an elastic plate under plane strain conditions is investigated. By formulating the wave-equations in a polar coordinate system attached to the crack-tip, it is found by an asymptotic analysis that the angle-dependence of the singularity is only dependent on the instantaneous cracktip velocity. This result is used to derive a relation between the dynamic stress-intensity factor and the energyrelease rate.     

Energy domain integral applied to solve center and double-edge crack problems in three-dimensions

A three-dimensional Boundary Element Method (BEM) implementation of the energy domain integral for the numerical computation of the crack energy release rate is presented in this paper. The domain expression of the energy release rate is naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using the appropriate boundary integral equations. The pointwise crack energy release rate is evaluated along the three-dimensional crack front over a cylindrical domains that surround a segment of the crack front. The accuracy of the implementation is demonstrated by solving several problems, which include geometries containing straight as well as curved crack fronts.     

Energy condition at the end of a nonstationary crack

The plane problem of the theory of elasticity is considered. It is assumed that in the neighborhood of the tip of an arbitrarily moving crack the stresses have a singularity of order r^–1/2. On this assumption a general expression is obtained for the distribution of the stresstensor components in the given neighborhood. This distribution is determined by the two parameters N and P. In the case of stresses symmetrical about the line of the crack (P=0) the angular distribution does not depend on the intensity coefficient N and is determined only by the velocity of the crack at the given instant and the transverse and longitudinal wave velocities. On the same assumptions it is shown that the energy condition obtained by Craggs for the particular case of steady-state motion is a necessary condition for the arbitrarily moving crack. Irwin [1] and Cherepanov [2] have studied these questions in the quasi-static approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 175–178, May–June, 1969.     

On the temperature rise at the tip of a fast running crack2

The high energy concentration at the tip of a running crack leads to irreversible deformations, and a great amount of the deformation energy is set free as heat. Assuming that this moving heat source is of circular shape, the temperature distribution around the crack tip has been calculated. The temperatures are dependent on the radius of the heat source and the crack velocity. Some examples for the material glass are given. The very high temperatures computed lead to the supposition that the observed light emission during fast fracture is of thermal origin.     

Strength reduction of metals upon hydrogen chemisorption at the tip of a crack

When hydrogen interacts with the freshly formed surface, the interatomic bonds at the tip of a crack in the single crystal of a metal are known to break. The decrease in the brittle strength of the single crystals of metals in the presence of hydrogen in the crack compared with the strength of the same metal in the absence of hydrogen is estimated quantitatively using comparative criteria of brittle strength. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 173–178, May–June, 1998.     

Deformation and stability loss of a part of the atomic chain at the tip of a crack

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 160–172, May–June, 1993.     

Calculation of the evolution of plastic yielding at the tip of a crack and related phenomena

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 154–160, May–June, 1993.     

Yielding on inclined planes at the tip of a crack loaded in uniform tension

Plane-strain yielding from a crack in an infinite elastic body is represented here by a distribution of edge dislocations on two planes inclined at angles ±ga to the crack plane, and the equilibrium condition is solved numerically. Approximate analytical expressions are obtained for the plastic-zone length, the crack opening displacement, and the J-integral, as functions of the applied stress and α. A comparison with a co-planar model of the plastic zone gives very similar results for α ≈ 65°. It is shown that fracture criteria based either on a critical crack opening displacement (COD) or on a critical value of J are always different, and the use of the former may lead to critical defect-sizes which are twice as large as those given by the latter. Furthermore, COD appears not to be a well-defined material property. The critical J criterion gives a fracture stress which is α-dependent : this may be responsible for deviations towards results of linear elastic fracture mechanics when post-yield fracture mechanics is used to analyse extensive yielding. The changes in the stress field of the crack due to the existence of the plastic zone are also discussed.     

Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case

This paper describes an asymptotic analysis of the strain and stress fields at the tip of a crack in a sheet of incompressible hyperelastic material. The investigations are carried out within the framework of finite elastostatics and for the class of Generalized Neo-Hookean materials. Both the symmetric (mode I) and non-symmetric (mixed-mode) cases are considered. It is shown that the latter situation corresponds locally to a rigid body rotation of the symmetric fields. The effect of the hardening parameter on crack tip blunting is investigated analytically and numerically.     

A brief history of the crack tip stress intensity factor and its application

The primary objective of this work is to discuss the origins, background and development of the elastic crack tip stress intensity factor, K, as they occurred. The further development of the three modes and the compilations of related formulas in the literature are discussed. The origins of applications to static crack growth stability, and sub-critical growth due to fatigue and environmental effects are included. Significant events such as the formation of the ASTM committee on Fracture Mechanics, the adoption of Damage Tolerance Analysis by the aircraft industry using Fracture Mechanics as a basis, and the further extension of the methods to large-scale plasticity conditions are presented. Finally a discussion of early predictions of crack paths is discussed.     

Singular behavior at the tip of a crack in an elastic-plastic material with plastic orthotropy

The two-dimensional stress field at the tip of a crack in a plastically orthotropic material is analyzed by the total deformation theory of plasticity in conjunction with the J-integral. A model of a plastically orthotropic material is constructed by the use of the theory proposed by R. Hill (1950) and the uniaxial stress-strain relation suggested by W. Ramberg and W.R. Osgood (1943). It is found that the stresses in the vicinity of the crack tip have a singularity of the same order as that in the case of isotropic materials, but their amplitudes are greatly influenced by the plastic orthotropy. Numerical work is carried out for two typical metals, and the effect of the plastic orthotropy is examined for the stress field, the crack opening displacement, the strain energy density, and the shape of the elastic-plastic boundary.     

Finite strains at the tip of a crack in a sheet of hyperelastic material: II. Special bimaterial cases

An asymptotic analysis of the strain and stress near-tip fields for a crack in a sheet of Generalized Neo-Hookean materials is presented in this second in a series of three papers. The analysis is based on the nonlinear plane stress theory of elasticity and concerns two special cases of the interface crack problem: in the first situation both components have the same hardening behavior; next, we investigate the particular case of a sheet of Generalized Neo-Hookean material bonded to a rigid substrate. The transition between the two special cases is studied in detail. The analytical results are also compared with a full-field finite element solution.     

Finite strains at the tip of a crack in a sheet of hyperelastic material: III. General bimaterial case

In this last in a series of three papers, we summarize an asymptotic analysis of the near-tip stress and deformation fields for an interface crack between two sheets of Generalized Neo-Hookean materials. This investigation, which is consistent with the nonlinear elastostatic theory of plane stress, allows for an arbitrary choice, on both sides of the three parameters characterizing this class of hyperelastic materials. The first three terms of the approximation series are obtained, showing the existence of a non-oscillatory and contact-free solution to the interface crack problem. The analytical results are compared with a full-field solution obtained numerically using the finite element method.