High strain-rate crack growth in rate-dependent plastic solids

At high crack velocities in metallic materials nearly all plastic strain accumulates at very high strain-rates, typically in the range 10³ s^?1 to 10⁵ s^?1. At these rates, dislocation motion is limited by dynamic lattice effects and the plastic strain-rate increases approximately linearly with stress. The problem for a crack growing at high velocity is posed for steady-state, small scale yielding in elastic/rate-dependent plastic solids. A general expression is derived for the near-tip stress intensity factor in terms of the remote intensity factor, or equivalently for the near-tip energy release-rate in terms of the overall release-rate. An approximate calculation of the plastic strain-rates provides this relation in analytical form. Imposition of the condition that the near-tip energy release-rate be maintained at a critical value provides a propagation equation for the growing crack. A single, nondimensional combination of material constants emerges as the controlling parameter. Implications for dynamic crack propagation are discussed.     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

A stress analytical solution for Mode III crack within modified gradient elasticity

The strain gradient exists near a crack tip may significantly influence the near-tip stress field. In this paper, the strain gradient and the internal length scales are introduced into the basic equations of mode III crack by the modified gradient elasticity (MGE). By using a complex function approach, the analytical solution of stress fields for mode III crack problem is derived within MGE. When the internal length scales vanish, the stress fields can be simplified to the stress fields of classical linear elastic fracture mechanics. The results show that the singularity of the shear stress is made up of two parts, r^−1/2 part and r^−3/2 part, and the sign of the stress σ_yz changes. With the increase of l_x, the peak value of σ_yz decrease and its location moves farther from the fracture vertex. The influence of strain gradient for mode III crack problem cannot be ignored.     

A numerical study of crack tip constraint in ductile single crystals

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.     

Crack-tip fields for fast fracture of an elastic-plastic material

An asymptotic analysis of the near-tip fields is given for transient crack propagation in an elastic-plastic material. The material is characterized by J₂ flow theory together with a bilinear effective stress-strain curve. Both plane stress and plane strain conditions have been considered. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variation of the near-tip stresses, all as functions of the crack-tip speed and the ratio of the slopes of the two portions of the bilinear stress-strain relation. It was found that the results are much more sensitive to the elastic-plastic constitutive relation than to the crack speed. This result is important for numerical analyses of dynamic crack propagation problems.     

Plane stress dynamic fields near a propagating crack-tip in a power-law material

Based on the plastic-dynamic equations, the asymptotic behaviour of the near-tip fields for a plane stress tensile crack propagating in a power-law material has been studied in this paper. It is shown that the stress and strain singularities are, respectively, of the order and , whereA is a constant which is related to the size of plastic region,r is the distance to the crack tip,n is the power-law exponent. Projects sponsored by the National Science Foundation.     

Crack-tip fields in steady crack-growth with linear strain-hardening

Singular stress and strain fields are found at the tip of a crack growing steadily and quasi-statically into an elastic-plastic strain-hardening material. The material is characterized byJ₂ flow theory together with a bilinear effective stress-strain curve. The cases of anti-plane shear, plane stress and plane strain are each considered. Numerical results are given for the order of the singularity, details of the stress and strain-rate fields, and the near-tip regions of plastic loading and elastic unloading.     

Development of a multiaxial fatigue theory by considering constraint effects on small mixed-mode cracks

The effects of the transverse strain (the normal strain in the crack-line direction) on the near-tip fields of small shallow surface cracks (Case A cracks) in power-law hardening materials are investigated by finite element analyses. The small Case A cracks are under plane stress, general yielding, and mixed mode I and II conditions. Constant effective stress contours representing the intense straining zones near the tip, deformed crack-tip profiles and near-tip mode mixity factors are presented for different transverse strains in the crack-line direction. Based on the concept of characterization of fatigue crack growth by the cyclic J-integral, the effects of the transverse strain on J are investigated. The results suggest that the fatigue life prediction based on multiaxial fatigue theories and the critical plane approach should include the constraint effects due to the transverse strain. Consequently, the concept of constant fatigue life contour on the Γ-plane in multiaxial fatigue theories is generalized to the constant fatigue life surface in the Γ-space where the shear strain and the two normal strains are the three axes. Finally, a damage parameter as a function of the shear strain and the two normal strains is proposed for evaluation of fatigue damage under multiaxial loading conditions.     

Finite-deformation analysis of the crack-tip fields under cyclic loading

The plane-strain crack subjected to mode I cyclic loading under small scale yielding was analysed. The influence of the load range, load ratio and overload on the near-tip deformation-, stress- and strain-fields was studied. Although the near-tip zones of appreciable cyclic plastic flow for all loading regimes matched closely one another, when scaled with (ΔK/σ_Y)², the activities of plastic flow within them manifested dependence on K_max and K_min, as well as on overload. Cyclic trajectories of the crack-tip opening displacement (CTOD) converged to stable self-similar loops of the sizes proportional to ΔK² and positions in CTOD-K plane dependent on the maximum K along the whole loading route, including an overload. Computed near-tip deformation evidenced plastic crack advance, this way visualising of the Laird–Smith concept of fatigue cracking. This crack growth by blunting-resharpening accelerated with rising ΔK and was halted by an overload. Crack closure upon unloading had no place. The affinities were revealed between computed near-tip stress–strain variables and the experimental trends of the fatigue crack growth rate, such as its dependence on K_max and K_min (or ΔK and K_max), and retardation by overload. Thus, the effects of loading parameters on fatigue cracking, hitherto associated with crack closure, are attributable to the stress–strain fields in front of it as the direct drives of the key fatigue constituents – damage accumulation and bond breaking.     

Calculation of near-tip non-singular stresses for pressurized cracks

When the crack surfaces are traction-free, there is only one constant term T in the near-tip stress field, which contributes uniformly to the stress component acting in the direction parallel to the crack flank. As to pressurized cracks, the non-singular part of the asymptotic stresses appears to be more complicated and is no longer characterized only by the constant T. In this work, an effective numerical approach is developed for calculation of the non-singular parts of the asymptotic near-tip stresses under the action of nonuniform crack surface pressures. With this approach, the near-tip non-singular stress field can be accurately evaluated by direct use of regular numerical methods such as finite elements.     

Stable crack growth along a brittle\ductile interface—II. Small scale yielding solutions and interfacial toughness predictions

The problem of a crack growing steadily and quasi-statically along a brittle\ductile interface under plane strain, mixed mode, and small scale yielding conditions is considered. The ductile material is assumed to be characterized by the J₂-flow theory of plasticity with linear strain hardening, while the brittle material is assumed to be linear elastic. A displacement-based finite element method, exploiting the convective nature of the problem, is utilized to solve the relevant boundary value problem. In Part I of this work, the corresponding asymptotic problem was solved. This paper addresses the full-field problem in order to validate the asymptotic solutions, and to explore the physical implications of the results. The numerical full-field results are found to be in good agreement with the analytical asymptotic solutions. In particular, the full-field results strongly suggest that the stress fields in the vicinity of the crack tip are variable-separable of the power singular type; and also that the mode mix of the near-tip stress fields is, to a large extent, independent of the applied elastic mode mix. The amplitude (the plastic stress intensity factor) and the regions of validity of the asymptotic fields are estimated from the full-field results, and are observed to be strongly dependent on the applied mode mix. The remote elastic loading fields appear to influence the near-tip fields, primarily, through the plastic stress intensity factor. The present work also explores the suggestion made by Bose and Ponte Castaneda, 1992 that the solutions to the small scale yielding problem may be used in the context of a standard crack growth criterion, requiring that continued growth take place with a fixed near-tip crack opening profile, to obtain theoretical predictions for the dependence of interfacial toughness on the applied mode mix. Based on the numerical results, predictions for mixed mode toughness of the brittle\ductile interface are reported. The results, which are in qualitative agreement with available experimental data and also with some recent theoretical results, predict a strong dependence of interfacial toughness on mode mix. This suggests that ductility provides the main operating mechanism for explaining the dependence of interfacial toughness on the mode mix of the applied loading fields, during steady crack growth.     

Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime.Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.     

High-magnification moiré interferometer for crack tip analysis of steels

A high-magnification moiré interferometer, particularly suitable for near-tip field analysis in cracked materials, is described. It has a submillimeter field of view, a high-resolution image sensor (1.4 million pixels), X-Y-Z translation stage and an optical fiber light delivery system. These features enable the microscope head to observe the crack tip while the specimen is loaded in a standard tensile test machine. Automated fringe pattern analysis, using temporal phase shifting and spatial phase unwrapping, enables thex ory displacement component to be measured and the corresponding in-plane strain component computed. The displacement placement accuracy is better than 40 nm, and the effective strain gage dimension is ∼ 25 μm. Furthermore, the interferometer has a built-in white light microscope that allows the observation of the specimen granular microstructure in exact registration with the displacement field. The interferometer has hence been employed to investigate the near-tip fields of a precracked stainless steel specimen under load. The influence of the grain boundaries on the measured displacement fields was relatively minor. The near-tip strain field shows a significant asymmetrical behavior despite pure mode lloading conditions.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

幂硬化介质中平面应力动态裂纹的尖端弹塑性场

本文采用塑性动力学方程,对幂硬化介质中平面应力动态裂纹尖端场进行了渐近分析,其结果表明:在裂纹尖端附近,应力具有的奇异性,应变具有的奇异性,其中A是一个与塑性区尺寸有关的常数因子,r是离开裂纹尖端的距离,n为硬化指数,文中给出了尖端场的控制参量D,它依赖于马赫数;并且给出了各物理量的角函数。     

Dynamic steady crack growth in elastic-plastic solids—propagation of strong discontinuities

The near-tip field of a mode I crack growing steadily under plane strain conditions is studied. A key issue is whether strong discontinuities can propagate under dynamic conditions. Theories which impose rather restrictive assumptions on the structure of an admissible deformation path through a dynamically propagating discontinuity have been proposed recently. Asymptotic solutions for dynamic crack growth, based on such theories, do not contain any discontinuities. In the present work a broader family of deformation paths is considered and we show that a discontinuity can propagate dynamically without violating any of the mechanical constitutive relations of the material. The proposed theory for the propagation of strong discontinuities is corroborated by very detailed finite element calculations. The latter shows a plane of strong discontinuity emanating from the crack tip (with its normal pointing in the direction of crack advance) and moving with the tip. Elastic unloading ahead of and/or behind the plane of discontinuity and behind the crack tip have also been observed.The numerical investigation is performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field, characterized by K₁, and T. It is shown that the family of near-tip fields, associated with a given crack speed, can be arranged into a one-parameter field based on a characteristic length, L_g, which scales with the smallest dimension of the plastic zone. This extends a previous result for quasi-static crack growth.     

Finite element analysis for steady-state hydride-induced fracture in metals by composite model

Delayed hydride cracking, which is observed in hydride-forming metals, due to the precipitation of hydrides near the crack tip, is investigated under conditions of constant temperature and crack velocity, plane strain and small-scale hydride-precipitation. The coupling of the operating physical processes of hydrogen-diffusion, hydride precipitation and material deformation is taken into account. The material is assumed to be an elastic composite made of hydrides and solid solution, with properties depending locally on the volume fraction of the hydrides. In the present analysis, the composite elastic properties have been derived by a generalized self consistent model for particulate composites. With respect to hydride-precipitation, two cases have been considered: (i) precipitation in a homogeneous medium with elastic properties, equal to the effective properties of the composite and (ii) precipitation in an inhomogeneous medium, where the expanding hydride has different elastic properties than those of the surrounding solid solution. The differences between the near-tip field distributions, produced by the two precipitation models, are relatively small. The effect of the hydrogen concentration far from the crack tip, on the near-tip field is also studied. It is shown that for small crack growth velocities, near the threshold stress intensity factor, the remote hydrogen concentration weakly affects the normalized stress distribution in the hydride-precipitation zone, which is controlled by the thermodynamically required hydrostatic stress, under hydrogen chemical equilibrium. However, for values of the applied stress intensity factor and the crack tip velocity, away from the threshold stress intensity factor and crack arrest, the effect of remote hydrogen concentration on the normalized near-tip stress field is strong. Reduction of the remote hydrogen concentration generally leads to reduction of the hydride-precipitation zone and increase of the near-tip stresses. Also reduction of the remote hydrogen concentration leads to distributions closer to those under hydrogen chemical equilibrium.     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

Growth of crack caused by temperature gradients with change in surface insulation

This work is concerned with thermoelastic stress and failure analysis of a centrally cracked panel subjected to temperature gradients while the insulation on the crack surface is varied. The corresponding temperature and thermoelastic stress fields are obtained by application of the finite element method. According to the strain energy density criterion, the crack grows incrementally when the maximum of the minimum strain energy density function reaches a critical value for a given material. Crack growth resistance curves involving plots of the strain energy density factor S versus the half crack length a are developed for crack surfaces with varying degree of heat resistance. The resulting curves are straight lines satisfying the condition dS/da = const. and useful for determining combined influence of thermal loading and structural geometry that lead to global instability.     

Near tip field for pseudo Mode II crack growth: Power law hardening material

The asymptotic behavior of stress and strain near the tip of a Mode II crack growing in power law hardening material is analyzed by assuming that the crack grows straight ahead even though tests show otherwise. The results show that the stress and strain possess the singularities of (ln r)^2/(n−1) and (ln r)^2n/(n−1) respectively. The distance from the crack tip is r, and n is the hardening exponent, i.e. σⁿ. The amplitudes of the stress and strain near the crack tip are determined by the asymptotic analysis.