Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending

Accurate yield surfaces of plane strain single-edge-cracked specimens having shallow as well as deep cracks are developed using finite element limit analyses and monotonic interpolation functions. Fully plastic shallow crack configurations are classified based on certain aspects of the yield surfaces. Relationships between incremental plastic crack tip and crack mouth opening displacements and incremental load point displacement/rotation are obtained for a wide range of relative crack depths and loading ratios. Fully plastic crack-tip fields for a sufficiently deep crack in a single-edge cracked specimen are examined to provide the stress triaxiality and the angular orientation of flow line at the crack tip in terms of the remotely applied tension-to-bending ratio. Evidence for fully plastic crack-tip stress fields consisting of an incomplete Prandtl fan and a crack plane constant state region is discussed.     

Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure

The understanding of fatigue crack closure has been proved to be a challenging and controversial topic among the fatigue community over the last three decades. The effect of the specimen (or component) thickness has been shown to have a significant effect on closure behaviour and this seems to be related to the relative size of the plastic zone. Real cracks are inherently three-dimensional; plane stress-like behaviour is found close to the region where the crack front intersects the free surface, whereas most of the crack front will experience something close to plane strain. The aim of the present work is to investigate the influence of specimen thickness on closure behaviour (both close to and remote from the surface) and on fatigue crack propagation. The paper will present results from a simple experimental program, which consists of fatigue testing CT specimens with different thicknesses. Fatigue crack propagation is measured optically. Crack closure is assessed using traditional compliance techniques (clip gauge and back face strain gauge) and Digital Image Correlation methods. Experimental results are compared with two and three-dimensional simulations of plasticity-induced fatigue crack closure. The implications of thickness effects for predicting the propagation of three-dimensional fatigue cracks are discussed.     

MODELLING THE INFLUENCE OF STRAIN HARDENING AND PLASTIC CONSTRAINT ON CRACK CLOSURE OF ARBITRARILY THICK CCT-SPECIMENS

The triaxial stress constraint and the effective yield stress distribution in the plastic zone for strain hardening materials are studied, and then a modified strip-yield model is proposed to investigate the thickness effect of CCT specimens. Consequently, a plastic constraint factor α is defined and analysed in detail. The results show that the factor α can comprehensively account for the influence of thickness, crack length, loading level and hardening exponent. A simple expression for the plastic zone length and a fitting expression involving α are obtained. Application of the modified strip model to Newman’s crack closure model, and comparison with FEM results, show that the model can account for the influence of thickness on crack closure.     

Separable functions in load separation for the ηpl and ηpl CMOD plastic factors estimation

The objective of this paper is to develop the load separation method for evaluating the _pl and _pl ^CMOD plastic factors used in the J estimation approach based on load versus displacement records. Appropriate forms for the geometry and deformation functions have been suggested from the EPRI Handbook solutions to produce the separable form for the load. The obtained functions are applied to evaluate the _pl and _pl ^CMOD plastic factors for center cracked tension specimen. The present load separation method gave results which are somewhat different from the estimated values of _pl given in the literature. For shallow cracks, the _pl and ^CMOD _pl plastic factors show considerable variation with crack size and the strain hardening exponent. For a deeply cracked CCT specimen, the ^CMOD _pl factor tends to the _pl factor and equals approximately unity. Abbreviations: CCT – center cracked specimen; CMOD – Crack Mouth Opening Displacement; EPRI – Electric Power Research Institute; FEM – Finite Element Method; LLD – Load Line Displacement.     

Evaluation of elastic fracture mechanics parameters for bend specimens

Existing solutions for stress intensity factor and mouth opening displacement of three-point bend specimens are shown to overestimate these quantities for shallow cracks by up to ±4.5 percent, because they do not account for the disturbance of the bending stress distribution by the concentrated force at the loading point. This error is far larger than the accuracy claimed by these solutions (0.2 to 0.5 percent).New expressions are therefore developed for stress intensity, crack mouth opening displacement and crack mouth open angle of single edge notched bend specimens loaded in three-point bending. As a reference, and to show the accuracy of the solutions, also the pure bending situation is treated. Rigorously derived asymptotic solutions are used for the shallow and deep crack limits, in order to prescribe both the proper limit values and gradients to the interpolation functions, of which the intermediate values are derived from refined finite element analyses.The crack mouth opening angle solutions are primarily intended to facilitate crack mouth opening measurement at other locations then the specimen surface, i.e. at an offset from the specimen surface as is the case when removable knife edges are applied. No solutions of the crack mouth opening angle of three-point bend specimens were available until now. For use with unloading compliance crack length measurement, also an inverse crack mouth opening relation is developed. This also includes crack mouth opening measurement at an offset from the specimen surface, which is lacking in presently available expressions of this kind.     

Finite element analysis of crack mouth opening displacement compliance in crack length evaluation for clamped single edge tension specimens

In the unloading compliance method developed for clamped single edge tension (SE(T)) specimens, six crack mouth opening displacement (CMOD)‐based compliance equations (i.e. a/W = f(BCE′)) were proposed for the crack length evaluation without clearly clarifying the corresponding predictive accuracies. In addition, the effective elastic modulus (E_e) that reflects the actual state of stress should also be introduced in the crack length evaluation for SE(T) specimens, because the actual state of stress in the remaining ligament of the test specimen is neither plane stress (E) nor plane strain (E′). In this study, two‐dimensional (2D) plane strain and three‐dimensional (3D) finite element analyses (FEAs) are carried out to investigate predictive accuracies of the six compliance equations. In both 2D and 3D FEA, specimens with a wide range of crack lengths and geometric configurations are included. For a given specimen, the value of E_e that presents the equivalent stress state in the remaining ligament is calculated on the basis of 3D FEA data. A set of formulae for the clamped SE(T) specimen is proposed that allows to evaluate E_e from the corresponding CMOD compliance. This approach is verified using numerical data. The observations of the numerical verification suggest that the use of E_e instead of E or E′ in CMOD‐based compliance equations markedly improves the accuracy of the predicted crack length for clamped SE(T) specimens.     

Study of slant fracture in ductile materials

Slant fracture is widely observed during crack growth in thin sheet specimens made of ductile materials, providing a good case for investigating three-dimensional criteria for mixed-mode ductile fracture. To gain an understanding of slant fracture events and to provide insight for establishing a slant fracture criterion, stable tearing fracture experiments on combined tension-torsion (nominal mixed-mode I/III) specimens and nominal Mode I Arcan specimens made of Al 2024-T3 are analyzed using the finite element method under three-dimensional conditions. Two types of finite element models are considered for the study of slant fracture: (a) combined tension-torsion specimens containing stationary, flat and slant cracks subject to loads corresponding to the onset of crack growth, and (b) stable tearing crack growth with slanting in a nominal Mode I Arcan specimen. Analysis results reveal that there exists a strong correlation between certain features of the crack-front effective plastic strain field and the orientation of the slant fracture surface. In particular, it is observed that (a) at the onset of crack growth in the combined tension-torsion experiments, the angular position of the maximum effective plastic strain around the crack front serves as a good indicator for the slant fracture surface orientation during subsequent crack growth; and (b) during stable tearing crack growth in the Mode I Arcan specimen, which experiences a flat-to-slant fracture surface transition, the crack growth path on each section plane through the thickness of the specimen coincides with the angular position of the maximum effective plastic strain around the crack front.     

Finite element methodology for elastic-plastic fracture problems in three dimensions

The governing finite element system for elastic-plastic analysis of fracture specimens in three dimensions is formulated. The formulation accounts for mixed material hardening, finite strains, finite rotations and plastic incompressibility. The implementation of these aspects into a computational formula is presented and alternative formulations are compared. Small strain theory is recovered as a special case of the present formulation. Analysis is performed on a finite thickness centre-cracked specimen. The grid characteristics required for converged solutions are discussed. The effects of material hardening model and specimen thickness are studied. The local yield state is examined as a gauge of the local deformation processes. The implications for the fracture behaviour of the specimen are discussed. Local surface displacements are compared to experimentally measured yield surfaces. The formulation is shown to predict extremely accurate local deformation in the neighbourhood of the crack front. Contrary to the few three-dimensional fracture studies carried out to date, this analysis concentrates on the local deformation behaviour which ultimately controls fracture. Accurate resolution of this behaviour is essential before meaningful fracture criteria in three dimensions can be developed.     

SHORT CRACK BEHAVIOUR IN NODULAR CAST IRON

Fatigue crack growth rates have been determined on standard specimens containing long cracks (∼5–10mm) and on specimens containing two-dimensional short cracks (∼0.10–0.50mm). Large differences have been observed indicating that at a given stress intensity factor short cracks propagate much faster than long cracks. Mouth opening displacement measurements for both specimen geometries have shown that the crack closure effect is largely responsible for the observed effect. These results are used to rationalize the behaviour of short cracks initiated from natural sites which were either graphite nodules or microshrinkage pores. The three-dimensional aspect of these natural small cracks is analysed and discussed in detail.     

Further developments in <Emphasis Type="Italic">J</Emphasis> evaluation procedure for growing cracks based on LLD and CMOD data

Laboratory testing of fracture specimens to measure resistance curves (J − Δa) have focused primarily on the unloading compliance method using a single specimen. Current estimation procedures (which form the basis of ASTM E1820 standard) employ load line displacement (LLD) records to measure fracture toughness resistance data incorporating a crack growth correction for J. An alternative method which potentially simplifies the test procedure involves the use of crack mouth opening displacement (CMOD) to determine both crack growth and J. However, while the J-correction for crack growth effects adopted by ASTM standard holds true for resistance curves measured using load line displacement (LLD) data, it becomes unsuitable for J-resistance measurements based upon the specimen response defined in terms of load-crack mouth opening displacement (CMOD). Consequently, direct application of the evaluation procedure for J derived from LLD records in laboratory measurements of resistance curves using CMOD data becomes questionable. This study provides further developments of the evaluation procedure for J in cracked bodies that experience ductile crack growth based upon the eta-method and CMOD data. The introduction of a constant relationship between the plastic components of LLD (Δ _p ) and CMOD (V _p ) drives the development of a convenient crack growth correction for J with increased loading when using laboratory measurements of P-CMOD data. The methodology broadens the applicability of current standards adopting the unloading compliance technique in laboratory measurements of fracture toughness resistance data (J resistance curves). The developed J evaluation formulation for growing cracks based on CMOD data provides a viable and simpler test technique to measure crack growth resistance data for ductile materials.     

Experimental and analytical study of SIMCON tension members

The strength and ductility of slurry infiltrated mat concrete (SIMCON) tension members were investigated both experimentally and analytically to construct a mechanical model for simulating tensile force–displacement relationships. In addition to standard strength testing, special tests were conducted on tension specimens with preset cracks to determine the interaction between steel fibers and the cement matrix near an opening crack. These tests were conducted on two sets of preset-crack specimens: (i) with symmetrically inclined fibers and (ii) with aligned fibers having variable debonded lengths on each side of the crack. Using measured bridging forces of inclined fibers, an efficiency factor of plane random fibers, compared to aligned fibers, was determined to be approximately 0.58. It was found that the ductility of SIMCON mainly stems from plastic deformation of steel fibers rather than fiber pull-out. SIMCON tensile response was characterized by elastic, nonlinear hardening and softening regimes. The hardening response was notch insensitive without multiple crack formation. In the elastic regime, only minute stiffness reduction was observed. The nonlinear hardening regime was characterized by internal damage growth without visible crack formation and ended with the appearance of a co-linear set of partial cracks. The softening regime was described by a localized failure of fibers with variable failure strains at the co-linear cracks. Based upon the experimental observation that a co-linear set of partial cracks form at the ultimate composite stress, upper and lower bounds of the SIMCON stress–strain relation in the hardening regimes were obtained.     

Numerical simulations of specimen size and mismatch effects in ductile crack growth - Part I: Tearing resistance and crack growth paths

The Gurson-Tvergaard-Needleman (GTN) model has been used for detailed numerical simulations of the effects of specimen size and yield stress mismatch on ductile crack growth behaviour in two different finite specimen geometries. For deep cracked bending specimens the crack growth resistance, expressed through the far-field J, increases as the specimen size is reduced, most strongly seen in case of low hardening. An opposite effect can be seen to some extent for shallow cracked specimens loaded in tension for low and intermediate hardening. For the yield stress mismatch cases low hardening and bend loading are found to promote crack growth deviation away from the initial crack plane.     

NUCLEATION AND SHORT CRACK GROWTH IN FATIGUED POLYCRYSTALLINE COPPER

Surface evolution in polycrystalline copper specimens with a shallow notch has been studied in interrupted constant strain amplitude cyclic loading. The inhomogeneous strain distribution close to stress amplitude saturation leads to the formation of extrusions and intrusions along persistent slip bands within the grain and also in suitably oriented grain boundaries. Numerous primary cracks within a grain or at a grain boundary are nucleated. Some cracks can grow further either by linking with existing cracks or by nucleation of new elementary cracks ahead of the crack tip. Crack growth rates of individual cracks fluctuate considerably but for each strain amplitude, which results in a saturated plastic strain amplitude, a crack growth rate of an equivalent crack can be established. This crack growth rate was found to depend strongly on the plastic strain amplitude in agreement with the Manson-Coffin law.     

Plastic limit load and its application to the fracture toughness testing for heterogeneous single edge notch tension specimens

The current investigation pursues the confirmation of the applicability of the limit load solutions in determination of the η factors necessary for fracture toughness testing protocols. The procedure begins with the correct calculation of limit load values in welded single edge notch tension (SE(T)) fracture specimens containing centreline cracks. Hence, the η factor is inferred through the principle of potential energy. Additionally, such results are compared with those obtained from finite element analyses, including strain hardening effects available in the literature. SE(T) specimens subject to pin‐loading display that the η factors are insensitive to the configurational effects and hardening properties. On the other hand, in clamped SE(T) specimens, such effects become meaningful, making its usage in fracture toughness experiments questionable. This work provides an alternative methodology to compute fully plastic proportionality coefficients (η) based on limit load solutions for heterogeneous cracked SE(T) specimens. These analyses also consider the limitations and potentialities of such an approach in experimental measurements of ductile crack growth.     

Experimental J estimation equations for single-edge-cracked bars in four-point bend: homogeneous and bi-material specimens

The present paper provides the plastic η factor solutions for J testing of single-edge-cracked specimens in four-point bend (SE(PB)), based on limit (slip line field) and detailed finite element limit analyses. Both homogeneous specimens and bi-material specimens with interface cracks are considered. Moreover, two different η factor solutions are developed, one for J (or C^* in creep crack growth testing) estimation based on the load-load line displacement (LLD) records, η_p^VLL, and the other J estimation based on the load-crack mouth opening displacement (CMOD) records, η_p^CMOD.It is found that the use of load-LLD records can be recommended only for testing deeply cracked SE(PB) specimens. Moreover, depending on how the LLD is measured, a different value of η_p^VLL should be used. On the other hand, the use of load-CMOD records is recommended for all possible crack lengths. Moreover, the proposed η_p^CMOD solution can be used not only for a homogeneous specimen but also for any bi-material specimen with an interface crack.     

Short crack growth and fatigue life in austenitic-ferritic duplex stainless steel

The kinetics of short crack growth has been studied in austenitic‐ferritic 2205 duplex stainless steel. Smooth cylindrical specimens and specimens with shallow notch were subjected to constant plastic strain amplitude loading. The crack growth was studied in notched specimens. The notch area has been mechanically and electrolytically polished to facilitate the observation of crack initiation and growth. The initiated cracks were observed in an SEM (scanning electron microscope). The crack growth was studied using long distance QUESTAR optical microscope equipped with high‐resolution camera. In constant plastic strain amplitude loading the microcracks were initiated and their growth kinetics has been studied. The characteristic features of the crack growth at different plastic strain amplitudes were recorded. Two approaches to analyse the crack growth rates were adopted. The comparison of the prediction of the fatigue life using the plastic‐strain‐dependent crack growth rate was compared with Manson–Coffin law and the relation between parameters of this law and parameters of the short crack growth law were established.     

Determination of dynamic rotation factor for DWTT specimens by strain analysis and hardness measurement

As to accurately ascertain the value of dynamic rotation factor r^*, strain analysis‐based measurement was performed on the bent low‐blow specimens by means of two approaches. (i) Surface strain analysis on the ligament indicates a critical transition zone from the necking to swelling, where the total strain reaches the minimum as undeformed material. (ii) Hardness measurement on the thickness‐reduction specimen also reflects a typical V‐shape distribution of interior plasticity along the ligament as observed from the surface, suggesting the location of true rotation centre relative to the neutral strain axis where the hardness is nearly invariable without any work hardening. As a result, the obtained value of r^* maintains nearly 0.45 for (B × 4B) size drop‐weight tear test‐type specimens after their crack initiation and propagation, well consistent with the values calculated from the slip‐line field theory.     

Fracture prediction of thin plates under localized impulsive loading. Part II: discing and petalling

The onset of fracture and subsequent propagation of radial cracks of thin clamped circular plates under localized impulsive loading were predicted analytically and numerically for discing and petalling stages with increasing intensity of applied impulse and various radii of loaded area. The equivalent plastic strain times the average stress triaxiality was introduced as a ductile fracture criterion in the numerical simulation. The strain hardening law and critical damage/fracture function was calibrated from tensile test on round specimen and a parallel numerical simulation. Based on the critical damage value, and calculated distributions and histories of stress and strain, the initiation site and extent of fracture were predicted for a range of loading radii and intensity of applied impulse. It was clearly demonstrated that the crack length and final deformed shapes of plates are strongly influenced by the spatial distribution and intensity of impulsive loading. A comparative study on the propagation of radial cracks was also presented. Finally, the numerically obtained crack length was shown to agree well with the closed form solution derived earlier by one of the present authors.     

Recrystallization-etch approach to study the plastic energy absorbtion at crack initiation and extension of pressure-vessel steel A533B-1

To evaluate the elastic-plastic fracture toughness parameter of nuclear pressure-vessel steel A533B-1, a newly developed technique (the recrystallization-etch technique) for plastic strain measurement was applied to different sizes of compact tension specimens with a crack length/specimen width of 0.6–0.5 that were tested to generate resistance curves for stable crack extensions. By means of the recrystallization-etch technique, the plastic energy dissipation or work done within an intense strain region at the crack tip during crack initiation and extension was measured experimentally. Furthermore, the thickness effects on this crack tip energy dissipation rate were examined in comparison with other fracture-parameter J integrals. Thickness effects on critical energy dissipation and energy dissipation rate during crack extension were obtained and the energy dissipation rate dW _p/da in the mid-section shows a constant value irrespective of specimen geometry and size, which can be used as a fracture parameter or crack resistance property.