Size effects in tearing instability: An analysis based on energy dissipation rate

It is standard practice to use a G or J resistance curve (R-curve) to describe stable tearing. One important question is whether a J_R curve generated in a fully yielded specimen is the same as the G_R curve in a structure under small scale yielding. This paper argues that both the J_R and G_R curves are best viewed as derivatives of the energy dissipation rate D. Energy dissipation rate is not a material property, but various simple rules can be proposed to describe its likely variation with crack extension and with geometry. It is concluded that near size invariance of J_R curves is approached when D is high. As the tearing resistance reduces, cracks in structurally relevant configurations will begin to extend in contained yield. Differences between J_R and G_R curves may then cause problems. In general, the G_R curve has a shallower slope as instability approaches. If the whole J_R curve (from the small specimen) is very low, it is unlikely that a material would be accepted in a structure; but, if a material has a moderately high initiation toughness, J_0.2, combined with a low dJ_R/da (say less than 25 MPa) it becomes dangerous to rely on J_R curve theory to predict crack stability in the structure. An alternative approach using energy dissipation rate is explained.     

The energy dissipation rate approach to tearing instability

Stable and unstable tearing in metals is currently analysed by J integral theory, or by the G_R curve approach. This paper explains an alternative analysis route based on energy dissipation rate, D. It is shown that the implication of increasing toughness with crack growth in G_R and J_R curves is misleading. Even in small scale yielding (SSY), it is possible to have stable tearing under increasing G or J whilst at the same time D is constant (or even reducing) with crack growth. New terms: C for crack driving force, D^* for geometry normalised D, D_ssy for D in SSY, and crack stability index are explained. A D based fracture analysis diagram is introduced. Comparisons are made between energy dissipation rate, J integral, and G_R curve instability prediction methods. It is shown that, in most instances, these different approaches are compatible; but that the use of J_R curves derived in fully yielded test pieces to predict failure in SSY has the potential to lead to an unconservative instability prediction. The practical advantage of the energy dissipation rate approach is that it can be applied to all product thicknesses at any extent of crack growth. The major advantage compared to the G_R approach is that toughness measurements can be made on much smaller specimens.     

Energy rates and crack stability in small scale yielding

Crack stability in small scale yielding is traditionally analysed using the R-curve approach with toughness indexed by either of the linear elastic fracture mechanics parameters K or G. In ductile materials stable tearing commences well before crack instability and progresses under increasing G_R. This is often assumed to mean that toughness is increasing with crack growth. It is shown in this paper that a rising G_R curve is generated even when a crack propagates with constant toughness (constant energy dissipation rate). The paper demonstrates that this apparent anomaly occurs because G does not represent the energy input rate for a crack advancing under increasing load in an elastic-plastic material. The constant energy dissipation rate model is consistent with a size independent G_R curve; also crack instability predictions are identical with both theories. The G_R curve approach has practical advantages, but use of energy dissipation rate offers better physical insight and greater versatility when analysing tough materials.     

J–R curves corrected by load-independent constraint parameter in ductile crack growth

The constraint effect on J–resistance curves of ductile crack growth is considered under the condition of two-parameter J–Q^* controlled crack growth, where Q^* is a modified parameter of Q in the J–Q theory. Both J and Q^* are used to characterize the J–R curves with J as the loading level and Q^* as a constraint parameter. It is shown that Q^* is independent of applied loading under large-scale yielding or fully plastic deformation, and so Q^* is a proper constraint parameter during crack growth. An approach to correct constraint effects on the J–R curve is developed, and a procedure of transferring the J–R curves determined from standard ASTM procedure to nonstandard specimens or real cracked structures is outlined.The test data of fracture toughness, J_IC, and tearing modulus, T_R, by Joyce and Link (Engng. Fract. Mech. 57(4) (1997) 431) for a single-edge notched bend specimen with various depth cracks are employed to demonstrate the efficiency of the present approach. The variation of J_IC and T_R with the constraint parameter Q^* is obtained, and then a constraint-corrected J–R curve is constructed for the test material of HY80 steel. Comparisons show that the predicted J–R curves can match well with the experimental data for both deep and shallow cracked specimens over a reasonably large amount of crack extension.Finally, the present approach is applied to predict the J–R curves of ductile crack growth for five conventional fracture specimens. The results show that the effect of specimen geometry on the J–R curves is generally much larger than the effect of specimen sizes, and larger specimens tend to have lower crack growth resistance curves.     

Parametrizing ductile tearing resistance by four parameters

The energy dissipation rate, R, is considered as a measure of resistance to crack extension in elasto-plastic fracture mechanics. It can be re-evaluated from J_R test records of bend and tensile specimens. Three types of R(Δa)-curves are identified. If crack initiation occurs close to or at maximum load, the energy dissipation rate is decreasing with crack extension and approaches a stationary value. This type of R(Δa)-curves can be described by an exponential function with three parameters, namely the initial value, the final stationary value, and a transition length. The cumulative J_R-curves for different specimen geometries can be derived by integration. The three parameters of the R(Δa)-fit together with an integration constant, the initiation value, J_i, characterize ductile fracture resistance both quantitatively and physically interpretable. Constraint effects on R-curves can be quantified in terms of these parameters. A procedure for transferring J_R-curves from one geometry to another is proposed.     

Establishing methodology to predict fracture behaviour of piping components by numerically predicting specimen fracture data using tensile specimen test

Using large deformation FEM analyses in SA333Gr.6 carbon steel material, the present study demonstrated the assessment of SZWc value that leads to J_SZWc and finally compares with the respective experimental results. It also includes numerical prediction of specimen J-R curve using Gurson-Tvergaard-Needleman parameters obtain from tensile specimen tests. Using numerically predicted results, the crack initiation and instability stages in circumferentially through-wall cracked elbows is finally predicted and compares with experimental results. The present study gives evidence that the non-linear FEM analysis supported with proper tensile test data can be helpful in assessing the safety of bend pipes with through-wall crack.     

A load–deformation formulation with fracture representation based on the J–R curve for tubular joints

This study presents a new fracture formulation to describe the ductile tearing and unstable fracture failure for circular hollow section (CHS) joints under monotonically increasing brace tension. The initiation of the ductile tearing occurs when the crack driving force in an assumed initial shallow crack reaches the material fracture toughness determined from a standard fracture toughness test. The joint behavior prior to the ductile crack initiation follows a previously proposed nonlinear formulation based on the latest strength equations recommended by the International Institute of Welding. The load–deformation characteristics beyond the crack initiation assume that the energy release rate and the amount of crack extension adhere to the experimentally measured J–R curve, prior to the unstable fracture failure. Unstable fracture, which leads to the total loss of the joint capacity, occurs when the crack driving force reaches the maximum fracture resistance determined from the J–R curve test. The proposed load–deformation representation for tubular joints, when implemented in the large-scale K-frame pushover analysis with a material fracture toughness test, predicts successfully the global frame response governed by the joint fracture failure, as observed in the frame test.     

A concept for scaling J R-curves by plastic constraint factors

Ductile tearing resistance can be characterized in a physically meaningful way by the energy dissipation rate, R. The respective R(a)-curves can be re-evaluated from experimental J _R-curves, and the latter can in reverse be obtained from the former by integration. For fully yielded specimens the geometry dependence of R(a)-curves can be scaled by plastic limit-load factors. This normalization opens a possibility for transferring a J _R-curve from one specimen geometry to another.     

Crack-opening angle and dissipation-rate analysis of R-curves for side-grooved pieces of HY130 steel in bending

The behaviour of side-grooved deep-notch three-point bend test pieces of 20 mm thick HY130 steel has been studied for large amounts of crack growth in three different widths. Growth occurs at limit load and the conventional R-curves follow the pattern that wider pieces give lower R-curves. Analysis of this behaviour is made in terms of the crack-tip opening angle, (CTOA) and the energy dissipation rate, dW _dis/Bda, or D, from which a particular R-curve, J _dis, can be formed. After an initial transient regime of about 2 mm growth, a steady-state region develops in terms of both CTOA and D. The steady state CTOA reduces with increase of initial width. The energy rate, D, is split into areal and volumetric components, γ and ρ, and, with neglect of the elastic components, ρ is related to the steady-state CTOA. The cumulative dissipation defined by J _dis is compared to several conventional R-curves. It is concluded that the interpretation of steady-state crack growth in deep-notch three-point bend pieces can be expressed in terms of either CTOA or D, but that transference of data even from one size of a side-grooved piece to another, let alone to another configuration, cannot yet be made except on a lower bound basis.     

The geometry dependence of the JR curve for circumferential growth of through-wall cracks in cylindrical pipes subject to bending loads. II: Effect on the instability criterion

Proceeding from the experimentally vindicated basis that circumferential growth of a through-wall crack in a circular cylindrical Type 304 stainless steel pipe subject to bending loads is associated with a constant crack tip opening angle, an earlier paper has predicted the shape of the J_R curve in terms of geometrical and material parameters. The predictions compare favourably with experimental results for 4 in. (～ 10 cm.) diameter pipes containing circumferential through-wall cracks with different sizes. In this paper, the implications of the geometry dependence of the J_R curve are considered in relation to the criterion for the instability of circumferential crack growth. Limitations of the tearing modulus approach are accordingly highlighted, particularly if it is used for a material with a low crack growth resistance.     

Determination of dynamic crack resistance of ductile cast iron using the compliance ratio key curve method

The compliance ratio method is an analytical approach for instantaneous crack length determination in dynamic single-specimen J-R curve testing of ferritic ductile cast iron (DCI). Comparison testing at room temperature and −40 °C was applied to PCVN and SE(B)15 specimens to examine their performance and suitability for the dynamic key curve method for DCI. An experimental reference database of dynamic crack resistance curves was set up by low-blow multiple-specimen tests and used to validate the results of the CR method. The influence of test temperature, microstructure, loading rate and specimen geometry on fracture behavior of the tested DCI was investigated in great detail and these parameters were linked to fracture mechanical properties. The results obtained show that the CR method is suited to establish valid dynamic crack resistance curves for both types of specimen. Nevertheless, SE(B)15 specimens are preferred for dynamic J-R curve determination of DCI based on their advantages such as higher accuracy.     

The jump-like crack growth model, the estimation of fracture energy and JR curve

In this paper the jump-like crack growth model for monotonic loading is applied to re-examine both the onset of crack growth and process of stable crack growth. In the former case the fracture energy associated with a new surface creation is estimated and the in-plane constraint influence on this quantity is examined using the J-A₂ approach. In the later case the formula to compute the J-resistance curve is re-examined and compared with the one known from the standards. In the analysis the plane strain model of a structural element made of elastic-plastic material is assumed.     

R-curve behaviour of a structural steel

The ductile tearing behaviour-of BS4360 50D structural steel has been studied using two bending specimen geometries: square (W = B) cross section three point bend specimens with different initial fatigue crack length, ranging from a/W = 0.20 to about 0.75, and rectangular cross section (W = 2B) specimens with initial fatigue crack length a/W = 0.3.With the first set of specimens it was intended to investigate the possible dependence of the tearing behaviour on initial crack length. It was concluded that specimens with shorter values of initial crack length present higher resistance to ductile tearing. Two different methods of calculating the J-resistance curve, one based on the approximate equation J = 2A/(W?a)B and a more elaborate calculation procedure introduced by Garwood were used. The curves determined following the more accurate procedure were used to determine the value of the tearing modulus T, which is related to the occurrence of tearing instabilities in structures. It was found that T decreases slightly with increasing value of initial crack length.The tests using the wider specimen geometry were intended to examine the tearing behaviour over a larger amount of crack growth allowed by the wider geometry used. It was found that the resistance curves, using the COD concept, present a maximum after which the resistance begins to drop. This maximum value of the COD-resistance curve is higher than the conventional δ_max measured at max load. These tests were performed using the compliance technique of evaluating the crack length. Tests performed with 10% unloading presented slightly lower values of resistance than tests performed with complete unloading.It was found that the relationship J = mσ_flowδδ presents increasing values of m with crack growth. This is attributed to an increasing local yield stress, caused by the through thickness deformation.     

High rate toughness of ductile polymers

Impact toughness of two highly ductile polymers: acrylonitrile-butadiene-styrene (ABS) terpolymer and polypropylene block copolymer (PPBC) - was evaluated using the essential work of fracture (EWF) - and a J-R resistance single specimen curve - S_pb techniques. The EWF has proved to be capable of determining toughness from the total fracture energy of several samples differing in initial ligament length and the linear regression of the data. On the other hand, the S_pb method, which is based on the load separation principle, is able of constructing J-R curves by inferring instantaneous crack growth length from the sole comparison between one sharp and one blunt-notched load-displacement traces. Results show that both methodologies can be used under impact conditions when evaluating ABS polymers. However, ABS impact fracture toughness value yielded by the EWF method, w_Ie, was larger than the J_0.2 value obtained from the S_pb method. This difference was imputed to the more progressive development of the necking zone in front of the crack tip under plane strain conditions. On the contrary, for very ductile fracture behavior like that demonstrated by PPBC in which J-controlled conditions were not achieved and hence J-R curves could not be built the EWF appeared as a valuable alternative to characterize impact toughness.     

A study of stable crack growth using experimental methods,finite elements and fractography

Systematic analysis of the in-plane constraint influence on J-resistance curves is presented. J_R curves were also recorded and analyzed beyond the limits of crack extension inside which the stress field can be assumed to be dominated by J-integral. Three steels and four types of specimen: SEN(B), SEN(T), CCT and DENT have been tested. Along with the J_R curves the fracture mechanisms have been analyzed with the help of scanning microscopy. The numerical, finite element analysis has been adopted to compute the Q-stresses, as a measure of the in-plane constraint prior to the onset of crack growth. The analysis of the stress field in front of the crack has been performed to check whether the state of stress prior to the crack growth can predetermine the way the crack will grow. It turns out that characteristic features in the J_R curves runs can be predicted qualitatively from the Q(a/W) and Q(J) curves. However, there is a good correlation between Q-stress and voids diameters on fractured surfaces. Several patterns in J_R curves runs have been observed for tested specimens; e.g. no influence of specimen thickness on J_R curves runs was observed for side-grooved specimens. Strong influence of specimen thickness on J_R curve shape was observed for non-side-grooved specimens. J_R curve run higher for thinner specimens unless they are dominated by plane stress. For bent specimens J_R curves run higher for shorter cracks but they run lower for specimens in tension.     

Engineering assessment of ductile tearing in uniaxial and biaxial tension

To decide between clearly different approaches for engineering assessment of plane stress tearing, we performed uniaxial and biaxial tensile tests on middle-cracked specimens at various in-plane constraint states. Attention was focused on determination of the crack tip opening spacing δ_n, crack tip opening angle ψ_c, crack mouth opening angle α_s, energy dissipation rate R, and specific work of fracture A_s. Our experimental results demonstrate that the values of δ_n, ψ_c, α_s, R, and A_s for a high-strength low-hardening aluminium alloy all depend on the specimen geometry, its size, and the load biaxiality ratio. However, assessment of ductile tearing by interconnected characteristic quantities α_s and A_s is more preferable over the use of δ_n, ψ_c, and R values for a number of basically important reasons discussed in this paper.     

The crack growth resistance of thin steel sheets under eccentric tension

The stable crack growth in thin steel sheets is the topic of this paper. The crack opening was observed using a videoextensometry system, allowing the crack extension determination. J_R-curve and δ_R-curve were established from obtained data. The ductile tearing properties of different thin sheets of steel were determined, including the impact of the specimen orientation, from test performed on compact tension specimens loaded under two conditions. The effect of the material, the rolling direction, and loading rate on the crack growth resistance of thin steel sheets was analyzed. In addition to the crack growth resistance, J-integral values for crack initiation were also estimated. The relation between J _i and J_0.2 was assessed using the basic mathematical and statistical methods. This relation was described by a linear regression model.     

Fatigue crack growth law of API X80 pipeline steel under various stress ratios based on J‐integral

Load‐controlled three‐point bending fatigue tests were conducted on API X80 pipeline steel to investigate the effects of stress ratio and specimen orientation on the fatigue crack growth behaviour. Because of the high strength and toughness of X80 steel, crack growth rate was measured and plotted versus ΔJ with stress ratio. The fatigue crack length is longer in the transverse direction, whereas the fatigue crack growth rates are nearly the same in different orientations. Finally, a new fatigue crack growth model was proposed. The effective J‐integral range was modified by ΔJ_p in order to correlate crack closure effect due to large‐scale yield of crack tip. The model was proved to fit well for fatigue crack growth rate of API X80 at various stress ratios of R > 0.     

CHARACTERIZATION OF DUCTILE TEARING RESISTANCE BY ENERGY DISSIPATION RATE

The J integral, which is widely used in elastic-plastic fracture mechanics, is not the true driving force any more if the crack is propagating. This leads to some inconsistencies when ductile tearing resistance is characterized in terms of J, especially for large crack extensions. Instead, Turner has proposed the energy dissipation rate as a physically more meaningful quantity. His concept is discussed and more evidence is given that will provide a better understanding of ductile tearing. It is shown how this quantity can be determined by measuring the heat production ahead of a fast running crack, or calculated in a finite element analysis, or re-evaluated from J-R test records of bend and tensile specimens. The energy dissipation rate is decreasing with crack extension in gross plasticity and approaches a stationary state. From these relations, the shapes of the cumulative J-R curves can be derived for different specimen geometries.     

Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.