Load-displacement Record Research Articles

Crack propagation rate as a function of crack tip characteristics

The dependence of crack growth rate on various crack tip parameters was studied. Experiments were performed on thin sheets of 6063-T6 Al-alloy having a central notch, to find crack tip opening displacement, total strain range, plastic strain range, crack opening stress and crack growth rate. Crack tip opening displacement and crack opening stress were measured, using a surface measurement technique, with small crack opening displacement gauges. The theoretical predictions of crack tip opening displacement compare fairly well with the experimental values. It is found that crack propagation rate vs total strain range-plastic strain range gives a straight-line fit on a log-log graph and, for positive stress ratios, the fatigue crack growth rates are found to be independent of R. Experimental results show that the crack opening stress is not affected by the position of the gauge when it is mounted behind and near the crack tip. The effect of mechanical properties and loading on crack growth were also studied. The specimens were fatigue cracked to a predetermined length and some specimens were annealed and again loaded cyclically. The application of cyclic loads to annealed specimens caused significant increase in crack propagation rates in comparison with the specimens having no heat-treatment. The load-displacement record was found to stabilize in about 10 cycles; the crack then extended slowly as a fatigue crack. Crack propagation rates for different values of R for annealed and work-hardened material were plotted against a crack tip parameter, ΔK∗, based on notional crack lengths. Since the results of d a d N vs ΔK∗ for both states of material (as-received and annealed) seem to lie on the same straight line on a log-log graph, the study provides a hope that the results for a material tested in any state (annealed or work-hardened) for positive values of R (0·0≤R≤0·3) will lie on this line, thus eliminating fatigue tests on the same material under different work-hardening conditions for different values of R. Models for d a d N have been developed using various crack tip parameters.

Single specimen test method for determining fracture energy ( [formula omitted]) of highly deformable materials

A single specimen fracture-test method for determining the fracture energy of highly deformable materials is presented. This test method is based on the locus method, which determines the critical J- integral value ( J c ) by partitioning the essential energy for crack propagation from the total absorbed energy. The locus method uses the locus of crack initiation points on the load-displacement record of specimens which differ only in initial crack size. With a proper specimen shape and loading condition, the locus line is predictable. When the locus line is predictable J c can be evaluated from a single specimen. Experimental results show that the fracture energy determined by the single specimen method agrees well with those by the tearing energy approach and multiple specimen locus method.

The [formula omitted] concept for elastic-plastic material behavior

THE CONSIDERATION of conservation laws within the theory of elasticity lead to the observation by Eshelby[l] that the generalized (material) force on a defect in an elastic solid could be expressed as a surface integral of the energy momentum tensor. Cherepanov[2] and independently Rice[3] introduced path-independent integrals for the evaluation of two-dimensional crack problems. Rice’s work especially gave a significant impulse towards the application of the J integral in cases beyond the limits of linear elastic fracture mechanics (LEFM) based on the K concept. The starting point of Rice’s considerations was the assumption of elastic (linear or non-linear) material behavior. This implies that the stresses can be derived from a potential, the strain energy density, which is a unique function of stresses or strains. On that basis it was not only shown that J is path independent but also that J is equivalent to the energy release rate. This latter interpretation established the correlation with the K concept of LEFM via the energy release rate G and, therefore, the potential of J as a fracture parameter became evident. Begley and Landes[4,5] used J as a fracture criterion parameter and proposed a multispecimen procedure to measure the material fracture toughness J,,. To avoid principal difficulties arising from a multi-specimen technique, Rice et aZ.[6] proposed estimation techniques for J from a single load-displacement record. By plotting J from their original results vs crack length change, Au, Begley and Landes[l derived the J integral resistance curve as a characterization of the material resistance against ductile tearing after initiation. Clarke et a1.[8] simplified this approach and proposed the partial unloading technique where in the course of a single-specimen experiment, the amount of crack extension is determined from the change of the elastic compliance during repeated small unloadings. The application of J-resistance curves to the evaluation of crack stability under ductile conditions was established by Paris et u1.[9] and Hutchinson and Paris[lO] who introduced a tearing instability theory. The acceptance of the J-integral concept was promoted by the fact that J was easily determined in the course of numerical analyses, especially by the finite element method. In two-dimensional cases J may be computed directly from the contour integral since all quantities entering J are available. More efficient and easily extended to three-dimensional situations is the stiffness derivative method introduced by Parks[l 1,121. DeLorenzi[l3] presented a node shifting and releasing technique to model crack extension for plane problems. In a plastic fracture handbook Kumar et al.[l4] compiled a series of approximate solutions for J and J(Aa) based on finite element results. A major concern regarding the application of J to materials described by an incremental or flow theory of plasticity (as is the case for most of the structural materials like e.g. steels or aluminium alloys) came from the fact that the theoretical basis of J was within a deformation theory of plasticity. For proportional loading up to initiation it could be argued that the stressstrain curves are identical for nonlinear elastic and incrementally plastic material laws and therefore the near-tip stress and strain field is the same. The J integral evaluated on the basis of deformation theory, was considered a valid fracture parameter, provided that J uniquely defines the stress and strain field in the vicinity of the crack. For plane situations and for certain approximations to the stress-strain curve, asymptotic solutions for the stress and strain field in the vicinity of a crack, the so called HRR-field have been derived by Hutchinson[lS] and Rice and Rosengren[l6]. In the case of crack extension this assumption is not valid, but the possible error was considered tolerable if the relative amount of crack extension stayed within certain limits and if it was ensured that the

Formula omitted] estimation analysis for circumferential throughwall cracked pipes

J- integral estimation solution is derived for pipes containing a circumferential throughwall crack. Bending moment and axial tension loadings are considered. These solutions are useful for calculating J from single load-displacement record obtained as part of pipe fracture testing, and are applicable for a wide range of flaw length to pipe circumference ratios. Results for J at initiation of crack growth generated using the solution developed in this paper agree well with J results from finite elements analyses.

Ductile fracture mechanics methodology for complex cracks in nuclear piping

Limit load and J- integral estimation solutions are developed for circumferentially complex-cracked pipes in bending. The limit load solution is developed using thick-walled cylinder analysis which included the effects of flaw depth accurately. J- integral estimation solutions are developed that are suitable for a wide range of loading from linear elastic, elastic-plastic to net-section yielding of the flawed section. Mode I stress intensity factor solution is developed from experimental compliance data. Two types of J solutions are developed. First, J solutions for determining the J- resistance curve from single load-displacement record are presented. Next, elastic-plastic J solution in the format of EPRI J estimation scheme is presented. The latter solution was used to predict the load carrying capacity of complex-cracked pipes made of Type-304 stainless steel, Inconel 600, and A106 GrB materials. Predictions were compared against pipe tests to demonstrate the accuracy of the limit load and J estimation solutions.

A method for determining [formula omitted] of mild steel by means of one single specimen

A new method for determining J Ic of steel from the load-displacement (P-Δ) record of one specimen is proposed. Four kinds of mild steel were tested to verify the method. Test results showed that J Ic values obtained using the proposed method were close to the J Ic values obtained by the standard multiple specimen method.

A new method for measuring [formula omitted] curve of mild steel

The J R curve is getting more and more use in evaluating a material's resistance to crack growth, but the standard method for measuring it needs several specimens. In this paper, a new method is proposed to determine the J R curve of mild steel from the load-displacement record of a cracked TPB specimen and a blunt notched TPB specimen. The J R curve obtained using the proposed method is in accordance with the J R curve obtained by the standard multiple specimen method.

J-Integral Analysis for Notched Round Bar in Tension

J-integral and crack-associated displacement formulas are derived for a notched round bar in tension. The formulas are applicable for a wide range of flaw length to bar radius ratio. The formulas are particularly useful for calculating J from single load-displacement record obtained as part of fracture toughness test.

A method to evaluate critical J-integral value: Locus method

A method which evaluates the critical J-integral value (J c) by utilizing the locus of crack initiation points on load-displacement records is presented. This method partitions the fracture energy along the locus line based on an energy rate interpretation. The shape of the locus line depends on both the material and the configuration of the specimen. Since the fracture energy can be partitioned even along a nonlinear locus line, the method is also suitable for highly deformable materials, which usually exhibit a nonlinear locus line. In order to validate this point, a thermoplastic rubber was used for experimental study. The experimental results show that J c can be evaluated by this method without restricting the ratio of the initial crack length to the specimen width (a/w) or the specimen length provided that the locus line can be located on the load-displacement record.

The equivalent energy method for KIC determination

The equivalent energy concept of using areas under the load-displacement record of a fracture test to any point on the linear portion of the curve and to the point of maximum load has been found to result in fracture toughness values that overestimated those determined according to the ASTM standard. The discrepancy was minimised if the 5% secant load point was used instead.

A direct measurement of shear fracture energy in rocks

Shear fracture energy has been measured for a granite and a limestone using a torsional method. We monitor failure with periodic stiffness determinations and measure strain energy released by integrating the load‐displacement record. Values of strain energy release rate (G3C) obtained for zero normal stress lie in the range 10²–10³ Jm−2. These fall between the values of G1C from tensile experiments and of shear fracture energy measured under high normal stress in triaxial tests.

Experimental analysis of crack propagation in granite

An experimental approach is used to demonstrate the concept of an effective crack length with double-edge-notched specimens of Charcoal and Rockville granite. Crack propagation in rock is characterized by microcracking around the crack tip and interlocking along a portion of the crack; this region is called the fracture process zone, and together with the fraction free length defines the effective crack length.Twenty closed-loop, strain-controlled fracture tests were conducted on Charcoal and Rockville granite. Crack growthwas monitored with a travelling behaviour of both granites, it is suggested that the process zone is larger for the larger grain-sized rock (Rockville). Conventional linear elastic fracture mechanics techniques are used to calculate the apparent fracture toughness of Charcoal granite at various crack lengths. In addition, a J-integral expression is derived for the double-edge-notched geometry in terms of the area under the load-displacement record. The shortcomings of both analyses are discussed. An attempt is made to explain the inadequacies by including the process zone in the calculations of the fracture toughness.

The J-integral and tearing modulus for a sen specimen under bending and tension

An expression for determining J from the load displacement record of SEN specimen is given. The result is compared with other analytical expressions and with FE calculations. A correction for crack growth similar to the one given in ASTM standard E813 is introduced. The applied tearing modulus is given both “exactly” and for an ideally plastic material. Finally J is evaluated in terms of failure curves in a failure-assessment diagram to show that there is no systematical difference between different bending-tension combinations.

Fracture toughness of turbine rotor materials and validity criterion forK IC

In this paper, the plane strain fracture toughness,K IC, of 2.8Ni-Cr-Mo-V steels was evaluated at the lowest service temperature (around room temperature) by using a large-size specimen, 345 mm thick and 920 mm in diameter, which was prepared from a low pressure turbine rotor shaft. In parallel with this, the validity criterion forK IC was investigated by using various sizes of compact tension specimens. It was proposed as a validity criterion forK IC thatP max=PQ which means the approximate linearity of load-displacement record.

A Plastic Fracture Mechanics Prediction of Fracture Instability in a Circumferentially Cracked Pipe in Bending—Part I: J-Integral Analysis

A method of evaluating the J-integral for a circumferentially cracked pipe in bending is proposed. The method allows a J-resistance curve to be evaluated directly from the load-displacement record obtained in a pipe fracture experiment. This method also permits an analysis for fracture instability in a circumferential crack growth using a J-resistance curve and the tearing modulus parameter. The influence of the system compliance on fracture instability is discussed in conjunction with the latter application. The results suggest that a compliant piping system containing a crack can exhibit ductile fracture instability after some stable crack growth. The importance of using a J-resistance curve that is consistent with the type of constraint for a given application is emphasized.

A Study on J-Integral as a Fracture Criterion (2nd Report)

In the previous paper, an improved method to evaluate J-value from single load-displacement record has been proposed.In this paper, the effects of material properties on J-value for 3-point bend specimens with different notch ratios are investigated by elasto-plastic FEM calculations using 8-noded isoparametric quadrilaterals. It is concluded that yield stress (σY) and strain hardening rate (H') have less effect on the relationship between non-dimensional J-value (EJ/aσ2Y) (E : Young's modulus, a : notch length) and stress ratio (σN/σY) (σN : nominal skin stress in the notched section), or non-dimensional deflection (Eu/dσY) (u : deflection at load point, d : depth of specimen).Several practical methods to evaluate J-value from single load-displacement record, that is, Rice's method, the more convenient method using total displacement and the authors' method proposed in the previous paper are compared one another, and it is shown that the authors' method gives the best approximation out of the three methods compared, but it still gives overestimation especially for shallow notched specimen in highly load range.On the basis of numerical results with varied conditions, a more simple and accurate J evaluation method for 3-point bend specimen is newly proposed and is expressed by Eq. (17) and Eq. (18). Using this proposed method, J-value can be estimated from crack opening at edge or applied load which is easily measured in fracture testings.

Determination of nonlinear energy toughness values for cyclic loading applications

For several years the nonlinear energy method proposed by Liebowitz and Eftis has been examined as a failure criterion for static testing of center-cracked and compact tension specimens. Since the method appears to be valid under conditions of crack-tip plasticity, subcritical crack growth and load relaxation, tests have been conducted to ascertain the merit of this method as a failure criterion under cyclic loading conditions. The nonlinear energy toughness for cyclic loading, G ̃ fc , is obtained from an envelope of the cyclic load-displacement record, which naturally imposes some restrictions on the loading program. The cyclic toughness parameter, G ̃ fc , has been evaluated for thin, center-cracked sheets of 2024-T3 and 7075-T6 aluminum alloys. The specimen dimensions were held constant and the load parameters were varied so that a significant variation of the cyclic life was obtained. Both alloys exhibited a significant reduction of G ̃ fc with increasing cyclic life in a manner similar to the classical S-N diagram. For example, the ratio of cyclic to static toughness, G ̃ fc/G c , was found to be about 0.8 when failure occurred after approximately 150 cycles. There appeared to be a tendency for the curve to level off at this point, which suggests that these curves may represent compressed S-N curves. It is felt that this method may serve the design process by allowing the establishment of a fracture toughness parameter capable of including the effects of the entire loading history of a structure into the fracture toughness requirements.

A Study on J-Integral as a Fracture Criterion

The J-integral has been proposed as an engineering fracture criterion for elastic-plastic behavior in metals.The J-integral, as is defined, has several advantages over some other currently used fracture parameters. But the experimental evidence so far obtained to show its practical usefulness have been limited.Using an ordinary ship steel, the authors investigated some practical aspects of the J-integral as a fracture criterion. The value of J was evaluated from several different methods including numerical (finite element analysis) as well as experimental (load vs. load point displacement record) procedure. The specimen used are center-notched tension, compact tension and bend specimens with varied dimensions. The critical values of J were evaluated from brittle fracture tests using these specimens.The method to evaluate J-value from single load-displacement record, originally proposed by Rice et al., has been modified by the authors to yield better results. Relation between J and COD and their equivalence are discussed. It is concluded that J concept is almost equivalent to the COD concept and thus JC would be one of the useful measures of fracture toughness, especially in case of fracture under large scale yielding condition.

Fracture toughness characterization of several aluminum alloys in semi-brittle fracture

A method has recently been developed for determining a nonlinear fracture toughness parameter defined by the relation G ̃ c = C ̃ G c where G c is the critical elastic strain energy rate as defined by Irwin. The C ̃ term is a function of the nonlinearity of the load-displacement test record and has been evaluated using the three parameter Ramberg-Osgood approach, although other curve fitting techniques could be applied as well. The method is quite straightforward and is applicable to plane stress, plane strain and mixed mode testing although only plane stress conditions are considered in this paper. For the case of a linear load-displacement record C ̃ → 1 and G ̃ c reduces to the linear elastic result. The toughness parameter G ̃ c has been evaluated for a number of high strength aluminum alloys and compared with published G c values for these materials. The tests were conducted on center-cracked sheets of 2014-T6, 2024-T81, 7075-T6 and 7475-T61 aluminum alloys under conditions of varying specimen geometry and displacement gage length. It was found that the values of G ̃ c obtained from displacement readings with a gage length of 2 in. generally agreed with published values of G c = K c 2 E . The G ̃ c values were found to vary inversely with gage length and a/w ratios. The variation in values for G ̃ c is of the same order of magnitude as the scatter in published values for G c . However, G ̃ c appears to be less sensitive than G c to changes in a/ w.

On nonlinear effects in fracture mechanics

Linear elastic treatment of fracture is considered applicable for net section stress up to about 0.8 the uniaxial tensile yield stress. Crack front plastic yield is still small enough to be viewed and treated as a small perturbation to the local crack front elastic stress field. Assuming these same circumstances and adopting the same point of view, an approach is presented for incorporating the nonlinear effects of small scale crack front plastic yield and slow crack extension in determination of the energy release rate and fracture toughness. Deviation from linearity of the load-displacement record in a fracture toughness test offers a quantifiable measure of these effects and is used to calculate the energy release rate G ̃ , which turns out to have the form G ̃ = (1 + P ̃ )G , where G is Irwin's elastic energy release rate. The factor P ̃ is directly determined in each test case, and when added to unity defines a nonlinear correction C ̃ to G, which has a lower bound value of one. Fracture toughness values for one-eight inch thick 7075-T6 center cracked aluminum sheet based on this approach are compared with uncorrected values and with values obtained by the Irwin y ∗ method of plasticity correction.