Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Analysis of deformation-induced crack tip toughening in ductile single crystals by nano-indentation

This paper reports on the experimental examination of the deformation characteristics near a crack tip in a cyclically work-hardened copper single crystal using a 2D surface scans with nano-indentation. The experimental methodology enables the characterization of the primary deformation field near a crack tip via the modulation of the imposed secondary deformation field by a nano-indentation. In a heavily deformed 4-point bend specimen, the measurements showed an existence of an asymptotic field around the crack tip at a distance of R ⩾ 2.5J/σ₀. The measurements also showed the qualitative details of toughness evolution within the high-gradient deformation field around the crack tip. The nature of dislocation distribution (i.e. statistically distributed vs. distributions necessitated by geometry) around the crack tip is quantified. The measurements indicate the dominance of the geometrically necessary dislocation within the finite deformation zone ahead of the tip up to a distance of R ≈ 3J/σ₀. Thereafter, it is confined in radial rays coinciding with the sector boundaries around the crack tip. These measurements elucidate the origin of the inhomogeneous hardening and the size dependent macroscopic response close to crack tip.     

Acoustic-emission crack monitoring in fracture-toughness tests for AISI 4340 and SA533B steels

The feasibility of using acoustic-emission techniques for the characterization of fracture resistance in AISI 4340 and SA533B steels is examined. The critical value of J-integral (J _IC ) is measured with a single small specimen (compact tension) loaded in the elastic-plastic range at room temperature. Initiation is detected during loading by acoustic emission. A new AE procedure for crack-growth monitoring in fracture-toughness specimens has been proposed. The possibility of discriminating AE signals from noncritical sources, such as void nucleation during crack-tip plastic deformation and signals from a growing crack, is discussed.     

Fully plastic center-cracked strip under anti-plane shear

The problem of a center-cracked strip subjected to uniform remote anti-plane shear stress is transformed to a problem in a hodograph plane which is solved exactly by Mellin transform and Wiener-Hopf technique. The material of the strip satisfies a pure power hardening stress strain relation and the results are valid for both deformation and flow theories of placticity. Numerical values are given for the crack opening displacement δ and Rice's path independent J integral for several values of the power hardening exponent n and crack width to strip width ratios. Approximate asymptotic formulas are presented for J and δ for large n.     

Effect of specimen size and crack depth on 3D crack-front constraint for SENB specimens

Three-dimensional (3D) elastic–plastic finite element analyses (FEA) are performed to study constraint effect on the crack-front stress fields for single-edge notched bend (SENB) specimens. Both rectangular and square cross-section of the specimens with a deep crack of a/W=0.5 are considered to investigate the effect of specimen size. A square-cross-section specimen with a shallow crack of a/W=0.15 is also considered to examine the effect of crack depth. Stresses from FEA at the crack front on different planes of the specimen are compared with those determined by the J–A₂ three-term solution. Results show that in-plane stress fields can be characterized by the three-term solution throughout the thickness even in the region near the free surface. Cleavage fracture toughness data is compared to predict the effects of specimen size and crack depth on fracture behavior. It is found that the distributions of crack opening stress are nearly the same for the SENB specimens at the critical J which is consistent with the RKR model. Furthermore our results indicate that there is a distinct relationship between the crack-front constraint and the cleavage fracture toughness. By introducing the failure curves, the minimum fracture toughness and scatter band can be well captured using the J–A₂ approach.     

Finite deformation analysis of COD,J-integral and crack tip intense strain region in plane strain large-scale yielding

A finite element analysis was performed to simulate crack tip blunting and the development of the intense strain region in a small compact tension specimen (0.4 T CT) of SA533B-1 under plane strain large-scale yielding, with the condition of large-geometry change around the crack tip taken into consideration. The region where the equivalent plastic strain $\text{}\text{?}$g3_p is greater than 0.15 was defined as the intense strain region, which corresponded to the recrystallized-etched zone delineated experimentally around the blunting crack tip. The development of the intense strain region was discussed as a function of the J-integral and the crack opening displacement. A linear relationship was obtained between the plastic work W_p dissipated within the intense strain region and (J/σ_y)² or b², where b is the crack opening displacement, defined as the separation of the two points at which the boundary of the intense strain region surrounding the crack tip intersects with the free surfaces of the crack.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

In-Situ SEM J-Testing and Crack-Tip Micromechanics of Zircaloy-4 by Three-Point Bending

A three-point bend fixture has been designed, fabricated, and utilized to demonstrate the feasibility of performing in-situ J-testing at ambient and elevated temperatures inside a scanning electron microscope (SEM). Using the three-point bend test technique, in-situ SEM J-testing has been performed to measure the crack mouth opening displacement and crack extension as a function of the applied load in order to generate J-R curves for Zircaloy-4 at 25°C and 316°C. Once the J-R curve is determined, an equivalent K_J-resistance (K_J-R) curve is computed on the basis of a relationship between the J-integral (J) and the stress intensity factor (K). The J-R and K_J-R curves of Zircaloy-4 exhibit a rising R-curve behavior, while the elastic K-R curve underestimates the fracture resistance of Zircaloy-4 once substantial crack extension has occurred. For the specimen dimensions considered, the J-R curves generated by in-situ SEM J-tests are not sensitive to the specimen geometry and measure the actual fracture resistance of the material. Furthermore, the onset of crack extension is dictated by the emission of one or more slipbands from the crack tip, and a change in the crack-tip displacement field, followed by void formation along the slipband, and linkage of the voids with the main crack.     

Effect of residual surface stress on the fracture of nanoscale materials

The problem of a mode I crack in nanomaterials under a remote mechanical load is investigated. The effect of the residual surface stress on the crack surface is considered and the solutions to the crack opening displacement (COD) and the stress intensity factor (K_I) are obtained. The results show that the surface effect on the crack deformation and crack tip field are prominent at nanoscale. Moreover, COD and K_I are influenced by the residual surface stress not only on the surface near the crack tip region but also on the entire crack surface.     

On the fracture behavior of inhomogeneous materials––A case study for elastically inhomogeneous bimaterials

This paper presents a case study, examining the influence of a sharp bimaterial interface on the effective crack driving force in a fracture mechanics specimen. The inhomogeneity of the elastic modulus in linear elastic and non-hardening and hardening elastic–plastic bimaterials is considered. The interface is perpendicular to the crack plane. The material properties and the distance between the crack tip and the interface are systematically varied. The effect of the material inhomogeneity is captured in form of a quantity called “material inhomogeneity term”, C_inh. This term can be evaluated either by a simple post-processing procedure, following a conventional finite element stress analysis, or by computing the J-integral along a contour around the interface, J_int. The effective crack driving force, J_tip, can be determined as the sum of C_inh and the nominally applied far-field crack driving force, J_far. The results show that C_inh can be accurately determined by both methods even in cases where J_tip-values are inaccurate. When a crack approaches a stiff/compliant interface, C_inh is positive and J_tip becomes larger than J_far. A compliant/stiff transition leads to a negative C_inh, and J_tip becomes smaller than J_far. The material inhomogeneity term, C_inh, can have the same order of magnitude as J_far. Based on the numerical results, the dependencies of C_inh on the material parameters and the geometry are derived. Simple expressions are obtained to estimate C_inh.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

The surface-forming energy release rate based fracture criterion for elastic–plastic crack propagation

The J-integral based criterion is widely used in elastic–plastic fracture mechanics. However, it is not rigorously applicable when plastic unloading appears during crack propagation. One difficulty is that the energy density with plastic unloading in the J-integral cannot be defined unambiguously. In this paper, we alternatively start from the analysis on the power balance, and propose a surface-forming energy release rate (ERR), which represents the energy available for separating the crack surfaces during the crack propagation and excludes the loading-mode-dependent plastic dissipation. Therefore the surface-forming ERR based fracture criterion has wider applicability, including elastic–plastic crack propagation problems. Several formulae are derived for calculating the surface-forming ERR. From the most concise formula, it is interesting to note that the surface-forming ERR can be computed using only the stress and deformation of the current moment, and the definition of the energy density or work density is avoided. When an infinitesimal contour is chosen, the expression can be further simplified. For any fracture behaviors, the surface-forming ERR is proven to be path-independent, and the path-independence of its constituent term, so-called J_s-integral, is also investigated. The physical meanings and applicability of the proposed surface-forming ERR, traditional ERR, J_s-integral and J-integral are compared and discussed. Besides, we give an interpretation of Rice paradox by comparing the cohesive fracture model and the surface-forming ERR based fracture criterion.     

Enhancement of fracture toughness of steel with defects by warm-prestressing

The technique of warm-prestressing to improve the resistance of structural steel with defects against low temperature fracture has received considerable attention. It is found that warm-prestressing can improve the fracture toughness and change the COD or δ_c, especially the crack tip plastic opening δ_p.The experimental results obtained from three-point bending tests of 42Mn2 steel specimens at −60°C and −20°C are analyzed. Experiments are also made on the bursting of pressure vessels manufactured from #20 steel. The results indicate that warm-prestressing at room temperature increased the bursting pressure at −40°C for d/t = 0.2 to 0.4, where d is the depth of surface crack and t the vessel thickness.     

The equivalence of K Ic and J Ic fracture-toughness measurements in Ni−Cr−Mo steels

Several series of fracture-toughness measurements were made in 4340 type steels, using both the standardK _Ic test method and theJ _Ic test method described by Landes and Begley.K _Ic results andJ _Ic results converted toK _Ic units are nearly identical for a given steel over a range of specimen size. The fracture toughness of steels produced by vacuum-degassing, remelt, and airmelt processes are compared over a range of yield stress. SimplifiedJ _Ic test procedures are suggested for use with Ni?Cr?Mo steels within certain ranges of yield stress and specimen geometry.     

Study on stress wave propagation in fractured rocks with fractal joint surfaces

This paper presents the experimental and theoretical investigation of property of stress wave propagation in jointed rocks by means of SHPB technique and fractal geometry method. Our aim focuses on the influence of the rough joint surface configuration on stress wave propagation. The comparison of behavior of reflection and transmission waves, deformation and energy dissipation of a rough joint surface characterized by its fractal feature with that of a smooth plane joint has been carried out. It has shown that the rough joint surface distinctly affects the stress wave propagation and energy dissipation in the jointed rocks. The rougher the joint surface was, the more permanent deformation occurred and the more attenuation stress wave took place as well. A nonlinear relationship between the normalized energy dissipation ratio W_J/W_I of the jointed rock and the joint roughness in terms of the fractal dimension has been formulated. It seems that the ratio W_J/W_I, presenting how much energy has been dissipated in the joint, nonlinearly increased with the increment of the fractal dimension D of the jointed surface. The ratio W_J/W_I of a roughly jointed rock, however, tends to be the same as that of a smoothly jointed rock if the fractal dimension is less than a critical value D_c = 2.20. The energy dissipation ratio at the critical point D_c seem to be a constant, not dependent of rock type but fractal joint configuration.     

Transient crack growth under creep conditions

Transient crack growth in an elastic/power-law creeping material is investigated under antiplane shear loading and small-scale-yielding conditions. At time t = 0 the solid is suddenly loaded far from the crack by tractions that correspond to the elastic crack-tip stress distribution. At that time the crack begins to propagate at a constant velocity. The stress fields evolve in a complex manner as the crack propagates due to the competing effects of stress relaxation due to constrained creep and stress elevation due to the instantaneous elastic material response to crack growth. From detailed finite element calculations it is shown that these fields can be approximated by a simple matching of three asymptotic singular crack-tip solutions. A characteristic stress, distance and time are defined for this problem which provide a normalization that accounts for any crack velocity, loading and all material properties for a given creep exponent n. Results are presented for crack-tip stresses, strains, crack opening displacements and creep zones.     

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.