Experimentally determined key curves for fracture specimens

Crack extension during fracture toughness tests of ferritic structural steels cannot be determined from measurements of unloading compliance or electric potential change when the specimen is dynamically tested. Measurements of crack extension in fracture toughness tests are also very difficult when the test temperature is high or the test environment is aggressive. To circumvent this limitation, researchers for years have been developing key curve and normalization function methods to estimate crack extension in standard elastic-plastic fracture toughness test geometries. In the key curve method (Ernst et al., 1979; Joyce et al., 1980) a load-displacement curve is measured for a so-called `source' specimen that is sub size or has a blunt notch so that the crack will not initiate during elastic-plastic loading. The load and displacement are then converted to normalized stress-strain units to obtain a key curve that can be used to predict crack extension in geometrically similar `target' specimens of same material loaded at similar loading rates and tested under similar environmental conditions. More recently Landes and coworkers (Herrera and Landes, 1990; Landes et al., 1991) proposed the normalization data reduction technique – Annex A15 of ASTM 1820 specification – that presents an alternative to the standard E1820 unloading compliance procedure. Although the normalization method works well in many cases, it has serious drawbacks: the load, displacement and crack length at the end of the test must be measured; the prescribed functional form that is fitted to the initial and final data may not be accurate for all materials; and the iterative method of inferring crack length from the combination of the data and the normalization function is complex. The compliance ratio (CR) method developed in this paper determines key curves for predicting crack extension as follows. First, a statically loaded source specimen with the unloading compliance procedure specified in ASTM 1820. Second, the so-called CR load-displacement curve is calculated for the source specimen, which is the load-displacement record that would have been obtained if the crack had not extended. Third, non-dimensionalizing the CR load by the maximum load and the displacement by the elastic displacement at the maximum load, P ^* _i/P _max and v _i/v _{el max} from the source specimen yields the adjusted key curve. Analysis of extensive data shows that the key curve is independent of notch type, initial crack length and temperature. But it is dependent on specimen size and steel type. Assuming that the key curves of the source and target specimens are one and the same, the compliance of the target specimens are calculated with a reverse application of the compliance ratio method, and the crack length is obtained using the equations in ASTM E1820. The CR Method is found to be much simpler than the normalization method described in the Annex A15 of ASTM 1820. With the compliance ratio method, Joyce et al. (2001) successfully predicted crack extension in dynamically loaded specimens using a key curve of a statically loaded specimen.     

The Crack Extension Force of a Curved Crack Derived from the Principle of Virtual Work

A general method to determine the crack extension force F related to a finite segment of a curved crack is presented. A path independent integral expression that holds in curvilinear coordinates is derived from the principle of virtual work. An appropriate virtual displacement field allows variation of the position of a crack tip. F is related to the path independent integral expression through variation of a total energy expression. To illustrate the applicability of the method F is derived for the conical crack in axisymmetric loading. For a plane crack with a straight front F is identical to the J-integral, Rice (1968).     

Decomposition of Eshelby’s energy momentum tensor and application to path and domain independent integrals for the crack extension force of a plane circular crack in Mode III loading

Vanishing divergence of Eshelby’s (energy momentum) tensor allows formulation of path or domain independent integral expressions of the crack extension force. In this work, a decomposition scheme of this tensor is presented, which results in zero divergence decomposed parts that allow formulation of expressions yielding the Mode I, II and III crack tip parameters J and K, with particular emphasis on Mode III, at present. By using the Mode III decomposed part of Eshelby’s tensor and the virtual crack extension method, a path and a domain independent integral, both new, for the crack extension force of a plane circular crack in axi-symmetric Mode III loading, are derived as examples of application.     

Characterization of stable crack extension in aluminium sheet material using the crack tip opening angle determined optically and by the δ5 clip gauge technique

Tests standards aimed at deriving fracture toughness data and crack resistance curves under low constraint condition have recently been finished by ASTM and ISO. These standards cover various experimental methods for determining critical crack tip opening angles, CTOA, for characterising stable crack extension in sheet material. In this paper, some key items of these standard methods are validated, namely the experimental determination of the crack tip opening angle by optical observation and using the δ₅ clip gauge method. When applying such standard methods to material characterization it is of particular interest to know how CTOA-data derived by different methods compare with each other. This paper compares CTOA-data as derived by the optical method with those derived by using the δ₅ clip gauge method. In order to study possible specimen size and geometry effects the methods have been applied to a wide range of specimen geometries. The results demonstrate that CTOA-data derived by the optical method are well suited to provide a specimen size and geometry independent characterization of stable crack extension where the thus obtained CTOA-data are constant over a large amount of stable crack extension. In contrast to this result, CTOA-data obtained from the δ₅ clip gauge method revealed a complex pattern of size and geometry effects, and only in case of compact specimens with a selected size the two CTOA-methods provide nearly identical CTOA-data over a large amount of crack extension.     

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.     

A method of determining the critical value of the J-integral under conditions of stable crack growth

A method is given for determining cracking resistance based on the strain diagram for stable crack growth; the specimen is loaded to give a set rate of crack extension or with given speed for the clamps. The critical value of the Jintegral has been determined by calculating the work of strain for a hypothetical specimen composed of a nonlinearly elastic material whose cracking resistance is equal to that of the actual one. Two forms of implementation are considered: simplified and numerical. The results are compared with those from standard methods and show that the proposed method is promising as it can be based on tests on a single specimen with a simple technique.Translated from Problemy Prochnosti, No. 3, pp. 18–25, March, 1992.     

3-D elastic-plastic investigation of fracture parameters in side-grooved compact specimen

This paper presents an analytical comparison of a standard and a side-grooved compact tension specimen. Both specimens were analyzed using 3-13 finite element techniques in the elastic regime as well as under large-scale yielding conditions. The standard specimen reveals large variations of both the crack opening displacement and the energy release rate along the crack front clearly indicating that the crack will tend to propagate in a tunneling mode. The side-grooved specimen, on the other hand, provides a much more uniform variation of both parameters thereby promoting both flat fracture and a uniform crack growth. The results clearly indicate that for ductile materials loaded well into the plastic range, a much more uniform fracture process can be obtained with the side-grooved specimen. We also show that the Merkle-Corten method of estimating theJ-integral from an experimentally obtained load versus load-line displacement record is in good agreement with the average energy release rate calculated by the virtual crack extension method.     

The influence of specimen's geometry on the crack extension angle

The objective of the present paper is to study the dependence of the angle at which a crack would propagate under mixed mode loading conditions on specimen's geometry. The centre-and edge-cracked specimens under tension and bending are examined. For each type of specimen the crack extension angle is determined for various values of the crack inclination angle and the ratio of the crack length to specimen's width. The minimum strain energy theory developed by Sih is used. Numerical results establishing the influence of specimen's geometry on the crack extension angle are given and useful conclusions are disclosed.     

J-resistance curve and ductile tearing of a mild steel

The onset of ductile tearing at room temperature of mild steel BS 15 was studied using side grooved compact tension specimens. The approach to this problem was divided conveniently into two basic parts: first, identification and evaluation of the toughness at initiation of crack extension, and second, assessment and characterization of the subsequent crack growth behaviour. The critical value of the J integral for crack initiation, J_c was obtained using two different techniques: the multispecimen method and a single specimen compliance test. It was found that the latter could be used with much larger unloading than originally proposed. This has the advantage of greater accuracy in the determination of the compliance, and consequently in the evaluation of crack extension. In the second part of the work, resistance curves were obtained applying two different approaches. The resistance curves obtained following the more exact method were used for the determination of the tearing modulus T of the material, and the values thus derived, compared with a selection of other steels.     

The measurement of J-R curves and J-integral values at crack initiations for metallic materials with three curve method of single specimen

The measurement of J-resistence curves and their proof values at crack initiations are carried out with the method of three curves from a single specimen by using 12CrlMoV, 1.25Cr0. 5MoSi, 16MnR steels and two weld metals. The results of the single specimen are in agreement with those of the standard multi-specimen method. It shows from the experiments that the connected analyses of the three curves from a single specimen are in accordance with the physical fracture processes of precracked three point bending specimens. The ductile crack extension of the materials tested follows the power law relation.     

Finite element calculation of energy release rate for 2-D rubbery material problems with non-conservative crack surface tractions

The energy release rate G for 2-D rubbery material problems with non-conservative crack surface tractions is calculated by a modified version of virtual crack extension method with finite element solutions. The formulation is demonstrated to be ‘patch-independent’ and therefore a complicated finite element model around the crack tip is not required.     

Computations of energy release rate under monotonic and cyclic loading conditions

In case of an elastic–plastic fracture mechanics analysis, the determination of the energy release rate distribution is a crucial point. In the present paper, three numerical techniques: the virtual crack closure technique (VCCT), J-integral and energy derivative technique (EDT), are used to compute the energy release rate in a middle-crack tension specimen with the combined isotropic/kinematic hardening model. The results obtained by these methods are compared with each other under monotonic and cyclic loading conditions. Finally, it comes out that the difference of the VCCT method to the J-integral is rather insensitive to load increasing, especially when the traction >40% of yield stress, however, the deviation of the VCCT and J-integral results are within 10%, suggesting that one may use the VCCT for plastic cracked specimen analysis. The computations show that the EDT provides the same values for the monotonic as the J-integral if the plastic deformations are not large, but for high plastic loading the EDT overestimates the fracture energy. For cyclic loading case, VCCT method offers closer results as the elastic analytical results, also suggesting that the whole plastic dissipated energy in the loading process should be integrated. While EDT method gives the smaller results than the J-integral because of the energy dissipated in the unloading phase is considered in the loading process.     

CTOD-R curve construction from surface displacement measurements

The construction of a fracture resistance δ–R (or J–R) curve requires the appropriate measurement of crack-tip opening displacement (CTOD) as a function of crack extension. This can be made by different procedures following ASTM E1820, BS7448 or other standards and procedures (e.g., GTP-02, ESIS-P2, etc.) for the measurement of fracture toughness. However, all of these procedures require standard specimens, displacement gauges, and calibration curves to get intrinsic material properties. This paper deals with some analysis and aspects related to the measurement of fracture toughness by observing the surface of the specimen. Tests were performed using three-dimensional surface displacement measurements to determine the fracture parameters and the crack extension values. These tests can be conducted without using a crack mouth opening displacement-CMOD or load-line displacement gauge, because CMOD can be calculated by using the displacement of the surface points. The presented method offers a significant advantage for fracture toughness testing in cases where a clip gauge is not easy to use, for example, on structural components. Simple analysis of stereo-metrical surface displacements gives a load vs. crack opening displacement curve. Results show that the initiation of stable crack propagation can be easy estimated as the point of the curve’s deviation. It is possible to determine the deviation point if the crack opening displacement measurements are close to crack tip in the plastic zone area. The resistance curve, CTOD-R, is developed by the local measurement of crack opening displacement (COD) in rigid body area of specimen. COD values are used for the recalculation with the CMOD parameter as a remote crack opening displacement, according to the ASTM standard.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

The observation of compliance change in the transition from hydrogen to air environment during the fatigue test

A compliance change was observed during fatigue testing of ASTM A710 HSLA steel using constant “K” CDCB specimen. The compliance decreased from 1.296 × 10⁻⁵ mm/N to 1.235× 10⁻⁵ mm/N when the environment was changed from hydrogen to air under the fatigue test conditions of f = 0.2 Hz, R = 0.1 and Δ K = 10 MPa✓m. The compliance change was observed in all fatigue testing while changing the environment from hydrogen to air. This compliance change can be explained numerically using the differential method for the design factors of the CDCB specimen. It was found from the calculation that the compliance change corresponded to a 6.3% change in Young’s modulus. It is proposed that the increased compliance resulted from the decreased Young’s modulus, the reduced Young’s modulus resulted from the increased lattice dilation which in turn resulted from a significantly increased hydrogen concentration at the crack tip region. The increased hydrogen concentration at the crack tip resulted from stress-induced hydrogen diffusion at the crack tip region.This work was conducted at Illinois institute of Technology (IIT).     

Limit load solution and loading behavior of C(T) fracture specimen

Deformations far from the crack tip and plastic collapse at limit load may control the fracture of specimens fabricated from highly ductile materials. To investigate the behavior of test specimen under plastic fracture, this paper derives solutions for the limit load of the C(T) specimen based on slip-line analysis. The modified Green's solution gives the most accurate results. Analysis of test data for many types of metal materials shows that, after some initial crack extension, the specimens reach the limit load. Yet, previous investigators analyzed the resistance to crack extension in these specimens with J-R curves. The large plastic deformations and unloading of the material in the wake of crack extension violate the basic assumptions of J-integral, thus invalidating the J-R analysis. It is, therefore, necessary to perform limit load analysis in the investigation of ductile crack extension.     

Electrical potential technique for monitoring subcritical crack growth at elevated temperatures

The electrical potential technique has been successfully used to monitor crack extension under fatigue as well as sustained loading at elevated temperatures in the presence of gross creep deformation. Calibration curves for actual crack extension vs change in electrical potential were determined for two specimen geometries, namely the compact type (CT) and the center crack tension (CCT) type, for an ASTM, grade A470 class 8 steel at 538°C (1000°F) and for a type 304 stainless steel at 594°C (1100°F).

A normalizing factor for expressing crack extension has been derived for the CT specimen. This factor accounts for changes in calibration due to small differences in initial crack length and, it also makes the calibration curve independent of the test temperature and material. Hence, the calibration curves presented herein are applicable to other materials and temperatures provided the specimen geometry and size is the same and the current input and potential leads are also located at the same position.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth

Predicting mixed-mode fatigue crack propagation is an important and troublesome issue in structure assessment for decades. In the present paper an extended finite element method (XFEM) combined with a new cyclic cohesive zone model (CCZM) is introduced for simulating fatigue crack propagation under mixed-mode loading conditions, which has been implemented in the commercial general purpose software ABAQUS. The algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without re-meshing. The cyclic cohesive zone model is based on the known S–N curves and Goodman diagram for metallic materials and validated by uniaxial tension results. Furthermore, the sensitivity of the model parameter is investigated for mixed-mode fatigue. The virtual crack closure technique has been extended to the cohesive zone model and proposed to calculate the energy release rate for the generalized Paris’ law. Finally, the crack propagation rate and direction under mixed-mode fatigue loading conditions are studied.