EPRI Handbook Research Articles

New CMOD based η-factor equations considering R-O strain hardening for circumferential through-wall cracked pipe under bending moment

The fracture study of circumferential through-wall cracked (CTWC) pipe under bending load is of interest for integrity assessment of conventional piping system and especially important for Leak-Before-Break (LBB) analysis of primary heat transport piping system of nuclear power plants (NPP). The fracture study in the elastic-plastic regime needs to obtain material J-resistance curve where J-integral is usually calculated from test data using the η-factor methodology. The evaluation of J-R curve for CTWC pipe is also important for study of transferability of material fracture resistance data from small specimen to full scale components. Further, η-factor to evaluate J-integral may be based on either load-lone-displacement (LLD) or crack mouth opening displacement (CMOD). The conventionalη-factor equations for CTWC pipe are LLD based and derived based on the assumption that the material is elastic perfectly plastic. However, use of LLD based η-factor needs to subtract no-crack displacement which poses additional difficulty and real materials always show some form of strain hardening. Therefore, an effort has been made in this paper to obtain the CMOD based η-factor for CTWC pipe considering material strain hardening as per the Ramberg-Osgood (R-O) relation. In the present study, plastic η-factor (ηpl) has been evaluated by elastic-plastic finite element analyses (FEA) over a wide range of crack angles (2θ = 30°–150°) and R-O strain hardening exponent (n = 2–7) for three R/t ratios (5, 10 and 20). To capture the ovalization of pipe cross section during deformation, geometric non-linearity is also considered in FEA. Based on these FEA, new ηpl equations are proposed by 2-D surface curve fitting for different R/t ratios. For validation of FE model, a separate set of geometrically linear FEA was also carried out for comparison of FE J-integral with EPRI Handbook solutions. Further, semi-analytical η-factor derived from EPRI relations of J-integral and CMOD were also compared with the present (geometric-linear) finite element results. In both the cases, the matching was very good.

Plastic J-integral calculations using the load separation method for the double edge notch tension specimen

The load separation method was used to determine the plastic work factor ( η pl) for the double edge notch tension (DENT) geometry for power-law hardening materials. The resulting expression for η pl compares favorably to the results of Rice, Paris, and Merkle. The results were also compared to η pl calculated from the plastic J results in EPRI Handbook NP-1931 and substantial differences were noted. A dependence on gage length was identified in the load separation results which indicate a gage length to half-width ratio of at least 3 is required to ensure that all of the plastic displacement is fully sensed and that η pl is independent of gage length. Additionally, a coupling term between crack size and material hardening exponent was identified in the geometric load separation function G.

Separable functions in load separation for the ηpland ηplCMODplastic factors estimation

The objective of this paper is to develop the load separation method for evaluating the ηpl and ηplCMOD 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 ηpl and ηplCMOD plastic factors for center cracked tension specimen. The present load separation method gave results which are somewhat different from the estimated values of ηpl given in the literature. For shallow cracks, the ηpl and ηCMODpl plastic factors show considerable variation with crack size and the strain hardening exponent. For a deeply cracked CCT specimen, the ηCMODpl factor tends to the ηpl factor and equals approximately unity.

Plastic J-integral calculations using the load separation method for the center cracked tension specimen

The load separation method was used to determine the plastic work factor ( η pl) for the center crack tension geometry for power law hardening materials with Ramberg–Osgood hardening exponents n ranging from 2 to 20 and crack sizes a/ W ranging from 0.4 to 0.8. The resulting expression for η pl compares favorably to the analytical work of Rice, Paris, and Merkle and to Sharobeam and Landes. The results were also compared to η pl calculated from the plastic J results in EPRI Handbook NP-1931. Unlike the EPRI results, η pl from the load separation method are not a function of crack size.

J-integral function for a centre-cracked tensile specimen

The equation which relates J to the area measured between the load v crack mouth opening displacement (CMOD) and the secant offset line in a centre-cracked tensile (CCT) specimen is investigated in this paper. The limits of the equation are evaluated for a range of power law hardening material behaviour using data given in the EPRI Handbook. Comparison with finite element results confirms that the equation gives good agreement with J contour integral values for crack length to specimen width ratios ( 0·25 ≤ a w ≤ 0·75 ) of practical interest and a wide range of material behaviour.

J Estimation Method for a Semi-Elliptical Surface Flaw in a Cylinder

Abstract With the emergence of readily available simple J solutions for various types of structures through J estimation techniques, J-integral-based elastic-plastic fracture mechanics has become a common tool for analyzing ductile materials. This paper presents elastic-plastic solutions for semi-elliptical surface flaws of four different crack depths. Solutions are developed from a series of finite element analyses using the ABAQUS computer program with a deformation plasticity material constitutive law. Although the solutions are directly applicable to a single set of Ramberg-Osgood material parameters, they may be extended to include various amounts of strain hardening by a ratioing technique utilizing calibration functions from the EPRI handbook for continuous flaws. This paper addresses cylindrical structures loaded by internal pressure, and excludes any consideration of thermal loadings. Elasticplastic J solutions, determined for pressures up to 5000 psi (12700 Pa), or twice the design pressure, depart from plastic-zone-corrected linear elastic predictions at approximately 3000 psi (7620 Pa) pressure. The h1 plastic calibration functions derived in this paper are limited to the point of greatest crack depth in 6:1 aspect ratio semi-elliptical inside surface flaws in cylindrical pressure vessels.

Inconsistencies between CTOD andJCalculations

AbstractCrack-tip opening displacement (CTOD) determined using the plastic hinge model included in the ASTM E 1290 (Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement) is inconsistent with CTOD determined using the ASTM E 813 (Test Method for JIc. Measure of Fracture Toughness) combined with an equation relating J to CTOD. The plastic hinge model is also inconsistent with the EPRI handbook estimation scheme. The plastic hinge model is highly inaccurate for hardening materials. To explore these inconsistencies, the EPRI scheme was used to generate theoretical load-displacement curves of a compact tension (CT) specimen for a wide range of materials and crack sizes. An alternate equation, derived for a power law hardening material, is shown to be consistent with both the ASTM E 813 and the EPRI handbook estimation scheme. The alternate equation accounts for nonlinearities in the relationship between CTOD and plastic load-line displacements. Differences between the alternate equation and the EPRI scheme are discussed. Experimental results demonstrate the inaccuracy of the plastic hinge model and the accuracy of the power law model.

Dugdale model for strain hardening materials

A Dugdale model for hardening materials under plane stress conditions is proposed. Crack tip opening displacements (CTODs) of different geometries, compact, center crack tension, double edge crack tension and three point bend specimens, have been calculated by means of the weight function method. The results have been compared with the J- integral from EPRI's handbook [Y. Kumar, M.D. German and C.F. Shih, EPRI NP-1931, Project 1237-1, Technical Report (1981)] for strain hardening exponent n = 3, 5,7,10,13,16,20 and two different Ramberg-Osgood coefficients. The results indicate that the proposed model could well correlate CTODs from the Dugdale model and the J- integral , thus it is an alternative tool to take account of the strain hardening effect in engineering applications.

Experiments on the transferability of creep crack growth laws to tubes

The present study served to investigate creep crack growth experiments for a tube with a 180° surface crack in circumferential direction. The tube had an outer diameter of 197 mm and a wall thickness of 23.5 mm. It was loaded with four-point bending at 700°C. No solutions for calculation of the C ∗- integral were available in the literature for this testing geometry, so new approaches had to be found. An approximation method, the so-called limit analysis technique was used for this purpose. It requires the plastic limit load of a specimen and the displacement rate of the load application points as the experimental input parameters. Before the limit analysis technique was applied to the tube under bending, results from earlier creep crack growth experiments were used to verify this technique. The C ∗ values determined from the limit analysis were compared with values based on the numerical calculations of the EPRI handbook. A good agreement was found for both calculation methods. It therefore seems justified to apply the limit load procedure also to the tube loaded in bending. The calculation on the basis of the limit analysis was performed in two ways, which differed by the type of crack front modelling. The simpler model yielded results that were too conservative in comparison with the experimental results of CT specimens. The application of the second model, which better covers the true crack front, provides values for da/ dt versus C ∗ which are actually within the scatter band of CT specimen results. This procedure therefore permits the transfer of standard specimen results to components.

J-Integral Estimation With the Stiffness Gradient Method for Power Law Hardening Materials

A simplified method for determining the J-integral for elastic-plastic conditions using the stiffness gradient method is described. Power law hardening material is shown to simplify the J-integral estimation, rendering elastic and fully plastic J solution terms. Numerical results are obtained for an interior complete circumferential crack in a pipe and compared to the analogous EPRI handbook solution. Because the stiffness gradient method is based on external displacement and forces, and not directly dependent on the stresses, it is sufficient to use relatively coarse finite element meshes to detemine the J-integral instead of highly refined and costly meshes required in most conventional J-integral computations.

Engineering application of elastic-plastic fracture assessment methods: an exploratory study

Three aspects of the J -integral concept and its engineering applicability have been investigated experimentally and numerically for a structural steel: • - determination of single specimen J R -curves by the DC potential drop- and Key Curve methods; • - use of the EPRI Handbook and finite element computations for crack growth behaviour predictions of (SENB and CNT) specimens; • - prediction of crack extension initiation of a three-dimensional plate model geometry, simulating a nozzle corner crack.

Treatment of material laws for approximate [formula omitted] estimation methods

In order to estimate the load and elastic-plastic deformation behaviour of cracked structures, methods must take into account the influence of arbitrarily shaped real material laws on the J integral and displacement behaviour, at least in an approximate form in the sense of an engineering approach. Such methods are generated and discussed in this paper. They are based on the well-known formulas in the EPRI Handbook, which are valid only for specific power laws. An extensive study has been carried out using idealised material laws with two distinct strain-hardening regions. The influence on J has been evaluated by a systematic variation of the parameters of these idealised material laws in an unambiguous way. The approximation methods generated have been compared with finite element (FE) calculations, and their accuracy with respect to load- J curves is discussed.