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.     

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.     

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.     

Fatigue crack opening stress based on the strip-yield model

The modified strip-yield model based on the Dugdale model and two-dimensional approximate weight function method were utilized to evaluate the effect of in-plane constraint, transverse stress, on the fatigue crack closure. The plastic zone sizes and the crack opening stresses considering transverse stress were calculated for four specimens: single edge-notched tension (SENT) specimen, single edge-notched bend (SENB) specimen, center-cracked tension (CCT) specimen, double edge-notched tension (DENT) specimen under uniaxial loading. And the crack opening behavior of the center-cracked specimen under biaxial loading was also evaluated. Normalized crack opening stresses σ_op/σ_max for four specimens were successfully described by the normalized plastic zone parameter Δω^′_rev/ω^′ considering transverse stress, where Δω^′_rev and ω^′ are the size of the reversed plastic zone at the moment of first crack tip closure and the size of the forward plastic zone for maximum stress, respectively. The normalized plastic zone parameter with transverse stress also was satisfactorily correlated with the behavior of crack closure for CCT specimen under biaxial loading.     

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.     

An analysis of ductile rupture modes at a crack tip

An elastic-Viscoplastic model of a ductile, porous solid is used to study the influence of the nucleation and growth of micro-voids in the material near the tip of a crack. Conditions of small scale yielding are assumed, and the numerical analyses of the stress and strain fields are based on finite strain theory, so that crack tip blunting is fully accounted for. An array of large inclusions or inclusion colonies, with a relatively low strength, results in large voids near the crack tip at a rather early stage, whereas small second phase particles in the matrix material between the inclusions require large strains before cavities nucleate. Various distributions of the large inclusions, and various critical strains for nucleation of the small scale voids between the inclusions, are considered. Localization of plastic flow plays an important role in determining the failure path between the crack tip and the nearest larger void, and the path is strongly sensitive to the distribution of the large inclusions. Values of the J-integral and the crack opening displacement at fracture initiation are estimated, together with values of the tearing modulus during crack growth, and these values are related to experimental results.     

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.     

Effects of the strain-hardening exponent on two-parameter characterizations of surface-cracks under large-scale yielding

The influence of strain hardening exponent on two-parameter J-Q near tip opening stress field characterization with modified boundary layer formulation and the corresponding validity limits are explored in detail. Finite element simulations of surface cracked plates under uniaxial tension are implemented for loads exceeding net-section yield. The results from this study provide numerical methodology for limit analysis and demonstrate the strong material dependencies of fracture parameterization under large scale yielding. Sufficient strain hardening is shown to be necessary to maintain J-Q predicted fields when plastic flow progresses through the remaining ligament. Lower strain hardening amplifies constraint loss due to stress redistribution in the plastic zone and increases the ratio of tip deformation to J. The onset of plastic collapse is marked by shape change and/or rapid relaxation of tip fields compared to those predicted by MBL solutions and thus defining the limits of J-Q dominance. A radially independent Q-parameter cannot be evaluated for the low strain hardening material at larger deformations within a range where both cleavage and ductile fracture mechanisms are present. The geometric deformation limit of near tip stress field characterization is shown to be directly proportional to the level of stress the material is capable of carrying within the plastic zone. Accounting for the strain hardening of a material provides a more adjusted and less conservative limit methodology compared to those generalized by the yield strength alone. Results from this study are of relevance to establishing testing standards for surface cracked tensile geometries.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

Plastic strip model of cracked weld joint with residual stresses

The effect of residual stresses on the fracture behavior of a cracked weld joint is studied by making use of the continuous dislocation formulation. Considered are the plastic zone length of the strip model zone and the opening displacement of a crack that is normal to both weld line and base metal boundary; they depend on the character of the yield stresses for the base metal (BM), weld material (WM), and heat affected zone (HAZ). The crack driving force is found to increase with the tensile residual stress while crack initiation and growth are suppressed if the residual stress is compressive. Moreover, the plastic zone and crack opening displacement are found to decrease linearly with the HAZ yield strength as the HAZ width is increased for HAZ yield strength greater than that of BM.     

Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics

Hancock and Cowling measured the critical crack tip opening displacements, δ_f, at fracture initiation in HY-80 steel specimens of six different configurations. δ_f varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δ_f measured by Hancock and Cowling correlates very well with the ratio ( ). δ_f is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).     

Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens

In this paper, the effect of constraint induced by the crack depth on creep crack-tip stress field in compact tension (CT) specimens is examined by finite element analysis, and the effect of creep deformation and damage on the Hutchinson–Rice–Rosengren (HRR) singularity stress field are discussed. The results show the constraint induced by crack depth causes the difference in crack-tip opening stress distributions between the specimens with different crack depth at the same C*. The maximum opening stress appears at a distance from crack tips, and the stress singularity near the crack tips does not exist due to the crack-tip blunting caused by the large creep deformation in the vicinity of the crack tips. The actual stress calculated by the finite element method (FEM) in front of crack tip is significantly lower than that predicted by the HRR field. Based on the reference stress field in the deep crack CT specimen with high constraint, a new constraint parameter R is defined and the constraint effect in the shallow crack specimen is examined at different distances ahead of the crack tip from transient to steady-state creep conditions. During the early stages of creep constraint increases with time, and then approaches a steady state value as time increases. With increasing the distance from crack tips and applied load, the negative R increases and the constraint decreases.     

Fracture of bicrystal metal/ceramic interfaces: A study via the mechanism-based strain gradient crystal plasticity theory

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plasticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative study of stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen. Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive modelling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reached ahead of the crack tip are found to be two times larger than the stresses obtained from conventional crystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrinsic material length to the size of plastic zone ratio (l/R₀). It is found that the larger the (l/R₀) ratio, the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. An insight into the role of cohesive strength and work of adhesion in macroscopic fracture is also presented which can be used by experimentalists to design better bimaterials by varying cohesive strength and work of adhesion.     

Effect of cracked weld joint and yield strength dissimilarity on crack tip stress triaxiality

This paper is concerned with an elasto-plastic analysis of a weld joint containing a central crack in the weld material (WM) whose yield strength may differ from that of the base material (BM). Stress triaxiality along the path of expected crack extension is found to be influenced not only by the applied tensile load level and crack length relative to the specimen width but also by the degree of BM/WM mismatch in the yield strength.Three different cases are analyzed by application of a two-dimensional finite element analysis. They are referred to as under-match, even-match and over-match which correspond, respectively, to the WM yield strength being less than, equal to and greater than that of the BM. In general, the stress triaxiality along the crack front tends to increase for an under-matched weld and decrease for an over-matched weld using the even-matched case as a reference. As the crack length is reduced for a given specimen width, the stress triaxiality decreases accordingly and the BM/WM material dissimilarity becomes more obvious. Displayed graphically are also the crack front plastic zone size that increases with the applied tensile load level and suffers a discontinuity across the weld line.     

A reexamination of plasticity-induced crack closure in fatigue crack propagation

The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.     

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.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Influence of stress triaxiality on damage and crack tip opening displacement parameters for steels

The dependence of the void growth parameter on the local stress triaxiality and local effective plastic strain near the crack tip of ductile materials provides the motivation to seek for parameters that could rank the ductility of steels. Experimental data for AS 1405-180, AS 1204-350, HY-80 and C---Mn steels show that the crack tip opening displacement (CTOD) at initiation δ_c decreases with increasing crack tip stress triaxiality. This trend is confirmed by analysis. As the critical local effective plastic strain ε_ec also decreases with increasing local stress triaxiality, the ratio δ_c/ε_ec is found to remain nearly constant or independent of the local constraint, i.e., the stress triaxiality. These parameters are given for a class of steels in this paper. Their association with the resistance to ductile fracture remains to be investigated.