New plastic collapse moment equations of defect-free and throughwall circumferentially cracked elbows subjected to combined internal pressure and in-plane bending moment

Closed-form plastic collapse moments (PCM) equations were earlier proposed for throughwall circumferentially cracked (TCC) elbow subjected to pure in-plane bending moment. However, an elbow is often subjected to combined internal pressure and bending moment in actual service condition. Therefore, the present study investigates the effect of internal pressure on the in-plane PCM of a TCC elbow. The PCM of a cracked elbow is usually expressed as a product of two parameters: PCM of a defect-free elbow multiplied by a weakening factor due to the crack. Therefore, the present study also includes analysis of defect-free elbows. Elastic-plastic finite element analysis is employed for the present analysis. A total of 396 cases of elbows with various sizes of circumferential cracks (2θ = 0-150°), different wall thickness (R/t = 5-20), different levels of normalized internal pressure (p = PR/(tσ_y) = 0-1), different elbow bend radii (R_b/R = 2,3) and two different bending modes, namely closing and opening are considered in the analysis. Elastic-perfectly plastic stress-strain response of material is assumed. The load in the elbows is split in two components: a constant internal pressure applied initially followed by in-plane bending moment monotonically increasing in definite steps. PCM are evaluated from moment—end rotation curves by twice-elastic slope method. From these results, closed-form equations are proposed to evaluate PCM of TCC and defect-free elbows subjected to combined internal pressure and in-plane closing/opening bending moment. Attempt has been made to compare the predictions of the proposed equations with the available experimental/numerical results and to rationally explain the behaviour where no experimental/numerical data is available for comparison.     

Plastic collapse moment equations of throughwall axially cracked elbows subjected to combined internal pressure and in-plane bending moment

Plastic collapse moment (PCM) equations of throughwall axially cracked (TAC) elbow subjected to in-plane closing/opening bending moment were previously proposed by the present authors. However, in actual situation, an elbow may often be subjected to combined internal pressure and bending moment loading. The present work investigates the effect of internal pressure on the in-plane plastic collapse moment of throughwall axially cracked elbows through 3-D elastic-plastic finite element analysis. Equations of un-pressurized cases are recommended where it is conservative and in other cases new equations are proposed.     

Elastic–plastic J and COD estimation schemes for 90° elbow with throughwall circumferential crack at intrados under in-plane opening moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic–plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic–plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, such schemes for elbow (or pipe bend), which is one of the important components for LBB analyses, is very meager. Recently, elastic–plastic J and COD estimation scheme has been developed for throughwall circumferentially cracked elbow subjected to closing bending moment. However, it is well known that the elbow deformation characteristics are distinctly different for closing and opening bending modes because the ovalisation patterns of elbow cross section are different under these two modes. Development of elastic–plastic J and COD estimation scheme for an elbow with throughwall circumferential crack at intrados subjected to opening bending moment forms the objective of the present paper. Experimental validation of proposed J-estimation scheme has been provided by comparing the crack initiation, unstable ductile tearing loads and crack extension at instability with the test data. The COD estimation scheme has been validated by comparing the COD of test data with the predictions of the proposed scheme.     

Elastic-plastic J and COD estimation schemes for throughwall circumferentially cracked elbow under in-plane closing moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic-plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic-plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, no such schemes are available for throughwall circumferentially cracked elbow (or pipe bend), an important component for LBB analysis. In this paper, simple J and COD estimation schemes are proposed for throughwall circumferentially cracked elbow subjected to closing bending moment. The ovalisation of elbow cross-section has a significant bearing on its fracture behavior. Therefore, unlike conventional deformation theory plasticity analysis, incremental flow theory is adopted considering both material and geometric non-linearities in the development of the proposed estimation schemes. Although it violates Ilyushin’s theorem, it has been shown that the resulting estimation schemes is still reasonably accurate for engineering purposes. Finally, experimental/numerical validation has been provided by comparing the J-integral and COD between numerical/test data and predictions of the proposed estimation schemes.     

Elastic J-integral solution for an elbow with axial through-wall crack at crown under in-plane bending

Piping elbows under bending moment are vulnerable to cracking at crown. The structural integrity assessment requires knowledge of the J-integral. The J-integral values for axially through-wall cracked thick elbows are not available in the open literature over a certain range. This paper presents a closed form expression for elastic J-integral for 90°, long radius elbows subjected to bending moment. The expression is derived, based on the results of a large number of finite element analyses covering a wide range of standard geometries. The analyses were performed using WARP3D software. The present study enables the estimation of the elastic J-integral over a range of R_m/t from 5 to 25 (R_m/t = 5, 6, 7.5, 9, 12, 15, 20 and 25) and thus extends the range of earlier solution towards the thicker elbows used in nuclear industry. The crack angles considered were 9°, 18°, 27° and 36°.     

Quantification of the yield strength-to-elastic modulus ratio effect on TES plastic loads from finite element limit analyses of elbows

This paper quantifies the effect of the yield strength-to-elastic modulus ratio on twice-elastic-slope plastic loads for 90-degree elbows under in-plane and out-of-plane bending. Based on extensive and systematic finite element limit analyses using elastic-perfectly plastic materials, simple regression equations between the yield strength-to-elastic modulus ratio and twice-elastic-slope plastic loads for elbows under in-plane closing, in-plane opening and out-of-plane bending are proposed, and validated against published experimental data. Applicability of the proposed equations to circumferential cracked elbows under in-plane bending is also investigated.     

Evaluation of postulated cross sections with ovality and thinning for 90° pipe bends with circumferential throughwall cracks subjected to in-plane closing bending

A comparative study of postulated cross sections for failure analysis of 90°structurally deformed pipe bends with critical circumferential throughwall cracks was performed. Elliptical and semi-oval cross sections were assumed to determine the collapse loads under in-plane closing bending moment. Finite element analysis was conducted based on elastic-perfectly-plastic material considering geometric nonlinearity and twice-elastic-slope method was used to obtain collapse loads for each model. The influence of pipe ratio and bend radius in the presence of ovality and thinning were also investigated for the postulated cross sections. The results indicated a pronounced effect of ovality on collapse loads of throughwall circumferentially cracked pipe bends, primarily with elliptic cross sections whereas the thinning effect is significantly lower for both the cross sections. The present study provided a better estimation of the plastic loads and elliptic cross sections were found to be ideal in the analysis of cracked pipe bends for the present geometries and across the boundary conditions considered.     

Effects of local wall thinning on plastic limit loads of elbows using geometrically linear FE limit analyses

This paper provides closed-form plastic limit load solutions for elbows under in-plane bending and internal pressure, via three-dimensional (3D), geometrically linear FE limit analyses using elastic-perfectly plastic materials. Wide ranges of elbow and thinning geometries are considered. To investigate the effect of the axial thinning length on limit loads systematically, two limiting cases are considered; a sufficiently long thinning, and the circumferential part-through surface crack. Closed-form plastic limit load solutions for wall thinning with intermediate longitudinal extents are then obtained from these two limiting cases. The effect of the axial extent of wall thinning on plastic limit loads for elbows is highlighted by comparing that for straight pipes. Although the proposed solutions are developed for the case when wall thinning exists in the center of elbows, it is also shown that they can be applied to the case when thinning exists anywhere within the elbow.     

Fatigue crack growth behavior of silica particulate reinforced epoxy resin composite

In the present work, fatigue crack growth tests of epoxy resin composite reinforced with silica particle under various R-ratios were carried out to investigate the effect of R-ratio on crack growth behavior and to discuss fatigue crack growth mechanism. Crack growth curves arranged by ΔK showed clear R-ratio dependence even under no crack closure, where the values of ΔK_th were 0.82 and 0.33 MPa √m for R = 0.1 and 0.7 respectively. However, crack growth curves arranged by K_max merged into almost one curve regardless of R-ratio, which indicated that crack growth behavior of the present composite was time-dependent. The value of K_max,th were in the range from 0.78 to 1.12 MPa √m. In situ crack growth observation revealed the crack growth mechanism: micro-cracking near the interface between silica particle and resin matrix occurs ahead of a main crack and then micro-cracks coalesce with a main crack to grow. The crack path was in the epoxy matrix, which was consistent with the time-dependent crack growth.     

Fracture experiments on through wall cracked elbows under in-plane bending moment: Test results and theoretical/numerical analyses

Fracture assessment of pipe bends or elbows with postulated through wall crack is very essential for leak-before-break qualification of primary heat transport system piping of nuclear power plants. The methodology for fracture assessment of cracked elbows is still in developing stage. Any new development in theoretical aspect requires experimental validation. However, fracture test data on cracked elbows is not so abundant as straight pipes. The earlier experiments on cracked elbows were focused mainly on the determination of limit load. Other fracture parameters e.g. crack growth, crack initiation load or crack opening displacement were not reported in the open literature. Against this backdrop, a comprehensive experimental and theoretical program on component integrity has been initiated at Reactor Safety Division (RSD) of Bhabha Atomic Research Center (BARC), India. Under this program, a number of fracture tests have been carried out on elbows with through wall circumferential/axial cracks subjected to in-plane closing/opening bending moment. These test data are then thoroughly analysed numerically through non-linear finite element analyses, analytically through limit load comparison and also through comparison of crack initiation loads by finite element and R6 methods. These test data may be utilized in future for validation of new theoretical developments in the integrity assessment of through wall cracked elbows.     

Crack closure under high load-ratio conditions for Inconel-718 near threshold behavior

Fatigue-crack-growth (FCG) rate tests were conducted on compact specimens made of an Inconel-718 alloy to study the behavior over a wide range in load ratios (0.1 ? R ? 0.95) and a constant K_max test condition. Previous research had indicated that high R (>0.7) and constant K_max test conditions near threshold conditions were suspected to be crack-closure-free and that any differences were attributed to K_max effects. During a test at a load ratio of 0.7, strain gages were placed near and ahead of the crack tip to measure crack-opening loads from local load-strain records during crack growth. In addition, a back-face strain (BFS) gage was also used to monitor crack lengths and to measure crack-opening loads from remote load-strain records during the same test. The BFS gage indicated that the crack was fully open (no crack closure), but the local load-strain records indicated significant amounts of crack closure. The crack-opening loads were increasing as the crack approached threshold conditions at R = 0.7. Based on these measurements, crack-closure-free FCG data (ΔK_eff against rate) were calculated. The ΔK_eff-rate data fell at lower ΔK values and higher rates than the constant K_max test results. In addition, constant R tests at extremely high R (0.9 and 0.95) were also performed and compared with the constant K_max test results. The constant R test results at 0.95 agreed well with the ΔK_eff-rate data, while the R = 0.9 data agreed well with constant K_max test data in the low-rate regime. These results imply that the R = 0.7 test had a significant amount of crack closure as the threshold was approached, while the R = 0.9 and K_max test results may have had a small amount of crack closure, and may not be closure free, as originally suspected. Under the high load-ratio conditions (R ? 0.7), it is suspected that the crack surfaces are developing debris-induced crack closure from contacting surfaces, which corresponded to darkening of the fatigue surfaces in the near-threshold regime. Tests at low R also showed darkening of the fatigue surfaces only in the near-threshold regime. These results suggest that the ΔK_eff against rate relation may be nearly a unique function over a wide range of R in the threshold regime.     

Evaluation of critical fracture energy parameter Gfr and assessment of its transferrability

Prediction of maximum load bearing capacity and crack growth for ductile materials using existing models like J-R curve approach has the problem of transferability and the use of micro-mechanical model (e.g. Gurson Tvergaard, and Needleman [Tvergaard V, Needleman A. Analysis of cup cone fracture in a round tensile bar. Acta Metall 1984;32:157-169]) are limited by the requirements of the huge computation time and large numbers of critical metallurgical parameters as input to analysis. Marie and Chapuliot [Marie S, Chapuliot S. Ductile tearing simulation based on local energy criterion. Fatigue Fract Engng Mater Struct 1998;21:215-227] of CEA, France, proposed a simple but convenient ductile crack growth model using critical fracture energy (G_fr) for crack growth and J_i for initiation, both of which are material parameters. They also proposed several schemes, namely, graphical and slope of modified plastic J-integral vs crack growth, J_M-pl − Δa methods for the evaluation of the value of G_fr from specimens as well as from components. In all these methods the role of non-crack displacement in the crack growth process was not considered. The necessary modifications due to non-crack displacement in the above methods to evaluate the values of G_fr was studied and published [Acharyya S, Dhar S, Chattopadhyay J. (2003). The effect of non-crack component on Critical fracture energy on ductile material. Int J Pressure Vessels Piping 2004;81:345-353] by the authors earlier. In this paper, the modified methods and formulation have been applied to evaluate the values of G_fr from experimental and FE simulated results for compact tensile (CT), three point bend (TPB) specimens and also from components like pipes and elbows. Then statistical estimation is done from these G_fr values to assess whether G_fr can be accepted as constant value material parameter. Finally, the mean value of G_fr obtained from statistical computation is used as material constant along with crack initiation toughness parameter (J_i)_SZW to consider crack growth for FE simulation of load vs load-line-displacement (LLD) and load vs crack growth curves for different specimens and components. Finite element simulated results are compared with the experimental results and good matching between the two for several components are found and maximum error in prediction of maximum load is found to be within 12%.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Size effect on fracture toughness of freestanding copper nano-films

We conducted fracture toughness experiments on freestanding copper films with thicknesses ranging from about 800 to 100 nm deposited by electron beam evaporation to elucidate the size effect on fracture toughness in the nano- or submicron-scale. It was found that initially, the crack propagated stably under loading, and then the crack propagation rate rapidly increased, resulting in unstable fracture. The fracture toughness K_C was estimated on the basis of the R-curve concept to be 7.81 ± 1.22 MPa m^1/2 for the 800-nm-thick film, 6.63 ± 1.05 MPa m^1/2 for the 500-nm-thick film and 2.34 ± 0.54 MPa m^1/2 for the 100-nm-thick film. Thus, a clear size effect was observed. The fracture surface suggested that the crack underwent large plastic deformation in the thicker 800-nm and 500-nm films, whereas it propagated with highly localized plastic deformation in the thinner 100-nm film. This size effect in fracture toughness might be related to a transition in deformation and fracture morphology near the crack tip.     

On the transfer of specimen J–R curve to piping components with throughwall circumferential flaw

Transferability of the specimen J–R/J–T curve to the component level is an important issue in the field of fracture mechanics. Towards this goal, fracture experiments have been carried out on single‐edge bend (SE(B)) and compact tension (CT) specimens and throughwall circumferentially cracked straight pipes/elbows of 200 mm nominal bore (NB) diameter. The pipe material is SA 333 Gr 6 steel (low strength and high toughness material) and specimens are machined from the pipes. Subsequently, elastic–plastic finite‐element analyses have been carried out on these cracked components/specimens in order to evaluate the stress triaxiality levels. It is found that the triaxial levels for these cracked components are similar. Hence, similar fracture behaviour is expected for these components. Consequently, one of the pipe J–R curves is used as a reference J–R curve to consider the crack growth in the analysis and the load deformation behaviour of other pipes/elbows is predicted. The load deformation behaviour of the piping components is also predicted using an extrapolated J–R curve from a specimen that exhibits the similar triaxiality level to that of the cracked components. The predicted results are in good agreement with the experiments.     

Investigation of brittle failure in transparent conductive oxide and permeation barrier oxide multilayers on flexible polymers

An oxide multilayer structure—consisting of an indium zinc oxide (IZO) conductive layer, a silicon oxide (SiO_x, x = 1.8) water vapor permeation barrier, and an aluminum oxide (Al₂O₃) interlayer—coated on polyethylene terephthalate (PET) is proposed as a transparent flexible substrate for display and photovoltaic applications. Vital properties of the multilayer, such as the low water vapor impermeability of the SiO_x barrier and the high conductance of the IZO film, degraded considerably because of the crack formation in bend geometries, attributed to the large difference between elastic properties of the oxide films and polymers. In order to suppress the crack formation, a 10-nm-thick Al₂O₃ interlayer was sputtered on Ar ion-beam treated PET surfaces prior to a SiO_x plasma-enhanced chemical vapor deposition (PECVD) process. Changes in the conductance and water vapor impermeability were investigated at different bending radii and bending cycles. It was found that the increases in resistance and water vapor transmission rate (WVTR) were significantly suppressed by the ion-beam PET pretreatment and by the sputtered Al₂O₃ interlayer. The resistance and WVTR of IZO/SiO_x/Al₂O₃/PET systems could be kept low and invariable even in severely bent states by choosing the SiO_x thickness properly. The IZO (135 nm)/SiO_x (90 nm)/Al₂O₃ (10 nm)/PET system maintained a resistance of 3.2 × 10^− 4 Ω cm and a WVTR of < 5 × 10^− 3 g m² d^− 1 after 1000 bending cycles at a bending radius of 35 mm.     

Effect of loading type on fatigue properties of high strength bearing steel in very high cycle regime

Very high cycle fatigue tests under axial loading at frequencies of 95 Hz and 20 kHz were performed to clarify the effect of loading type on fatigue properties of a high strength bearing steel in combination with experimental result of this steel under rotating bending. As a result, this steel represents the single P-S-N (probabilistic-stress-life) curve characteristics for surface-induced fracture and interior inclusion-induced fracture, just like that under rotating bending. However, fatigue strength is lower, where the run-out stress at 10⁹ cycles is evaluated to be 588 MPa, less than that under rotating bending with about 858 MPa. Occurrence probability of larger and deeper inclusion-induced fracture is much higher than that under rotating bending. Furthermore, the formation process of fine granular area (FGA) is independent of the type and frequency of loading, which is very slow and is explained as the crack nucleation process under the special dislocation mechanism. The stress intensity factor range at the front of FGA, ΔK_FGA, is approximately regarded as the threshold value controlling the stable propagation of interior crack. For the control volume of specimen under axial loading, the estimated value of fatigue limit by FGA is similar to experimental run-out stress value at 10⁹ cycles, but that by inclusion is larger. However, the corresponding estimated results under rotating bending are all conservative.     

Limit loads and approximate J estimates for axial through-wall cracked pipe bends

This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bending. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bending is proposed.     

T-Stress solutions for a semi-elliptical axial surface crack in a cylinder subjected to mode-I non-uniform stress distributions

This paper presents the T-stress solutions (T₁₁ and T₃₃) for semi-elliptical axial surface cracks in a cylinder subjected to mode-I non-uniform stress on the crack surface. Two cylindrical geometries with inner radius (R_i) to wall thickness (t) ratios R_i/t = 5 and 10 were considered. The T-stresses were applied along the crack front for normalized crack depth values a/t of 0.2, 0.4 and 0.5 and aspect ratios a/c of 0.2, 0.4, 0.6 and 1.0. Three stress distribution; uniform, linear and parabolic were applied to the crack face. In addition to these solutions, concrete formulation of the superposition principle is given for the T₃₃-stress, which is known as an elastic parameter that describes the out-of-plane crack tip constraint effect. Then, the validity of the formulation was shown through application of our T-stress solutions to the problem of an axial semi-elliptical surface crack in a cylinder subjected to internal pressure, and checking that the principle of superposition holds for the problem.