Crack Compliance Research Articles

Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.

The Influence of Residual Stress on Fatigue Crack Growth Rates in Stainless Steel Processed by Different Additive Manufacturing Methods

The properties and microstructure of Type 304L stainless steel produced by two additive manufacturing (AM) methods—directed energy deposition (DED) and powder bed fusion (PBF)—are evaluated and compared. Localized heating and steep temperature gradients of AM processes lead to significant residual stress and distinctive microstructures, which may be process-specific and influence mechanical behavior. Test data show that materials produced by DED and PDF have small differences in tensile strengths but clear differences in residual stress and microstructural features. Measured fatigue crack growth rates (FCGRs) for cracks propagating parallel to and perpendicular to the build directions differ between the two AM materials. To separate the influences of residual stress and microstructure, K-control test procedures with decreasing and constant stress intensity factor ranges are used to measure FCGRs in the near-threshold regime (crack growth rates ≤ 1 × 10−8 m/cycle). Residual stress is quantified by the residual stress intensity factor, Kres, measured by the online crack compliance method. Correcting the FCGR data for differences in Kres brings results for specimens of the two AM materials into agreement with each other and with results for wrought specimens, when the latter are corrected for crack closure. Differences in microstructure and tensile strength have an insignificant influence on FCGRs in these tests.

Fields of residual stresses near open assemblage holes of aircraft wing panel

The results of fatigue tests of two geometrically identical and similar in design models of the lower wing panel of a commercial aircraft are were analyzed. The panels differed in the way of installing mounting bolts, which connect the skin and stringers. Cold expansion of holes drilled both in the skin and stringer has been performed for the first panel before joining. The second panel includes no additional treatment after drilling pilot holes and final reaming. Bolts are mounted with an interference fit varying from 1.3 to 2.1% and from 2.9 to 3.2% for the first and the second panel, respectively. Changes in the interference fit are the consequence of a scatter attributed to the presence of a tolerance zone for the diameters of both bolts and mounting holes. A two-step comparison of both technologies is based on the experimental study of residual stress fields. The first stage, being a subject of the present study, includes the analysis of residual stress fields, which arise after removing bolts and separation of skin from stringers. Hole drilling and gradual crack growth were used to determine the components of residual stresses. Deformation response is measured by electronic speckle-pattern interferometry. High quality interferograms, which provide a reliable resolution of the interference fringes of ultimate density over the hole edge or directly along the notch borders, have been obtained for both ways of local removing the material. The first point-wise method based on drilling a probe hole, provides a quantitative determination of the residual stress components, starting from 1.4 mm distance from the assemblage hole edge. The second technique implements the crack compliance method of subsequent lengthening of the notch, starting directly from the mounting hole edge. This approach provides for a quantitative analysis of residual stress fields, related to different bolt mounting technologies, proceeding from the comparison of SIF values. A high level of compressive residual stresses near open holes is characteristic for both types of panels. Both experimental approaches showed the benefits of joints, where bolts are mounted into cold-expanded (reinforced) holes. For this case, the estimation of the relaxation parameters of the principal component of residual stresses in the direction of the external load is presented.

Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines, for which the rock physics model is indispensable. However, up to now, multitudinous analytical models focus primarily on the cracked rock with the isotropic background, while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth. Hence, we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy, respectively, by forty groups of numerical experiments. Based on the conclusion from the experiments, the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background. Further, we comprehensively analyzed, according to the developed model, effects of the crack dip angle, background anisotropy, filling fluid and crack density on the effective elastic properties of the cracked rock. The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P- and SH-wave velocities, whereas they have relatively small effects on the SV-wave velocity anisotropy. Moreover, the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters, while its effects on shear coefficients depend on the crack dip angle. The increasing crack density reduces velocities of the dry rock, and decreasing rates of the velocities are affected by the crack dip angle. By comparing with exact numerical results and experimental data, it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients. Moreover, the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

Quantifying effect of overload-induced residual stress behind crack tip on fatigue crack growth

Separating and quantifying effect of overload-induced residual stress (RS) behind and ahead of crack tip on fatigue behavior are inevitable for random fatigue life prediction. In this study, fatigue crack growth measurements under constant stress intensity and different stress ratios for the same overload are performed together with analyzing crack compliance curves. The results show crack closure-depended RS contribution behind crack tip to fatigue retardation results from the variation of crack opening stress intensity which can be directly obtained in fatigue test. Crack closure influence is determined by both effective RS behind crack tip and contact length of crack wake.

Magnetic field and thermal environment effects on sound transmission loss of double-walled cracked piezoelectric cylindrical nanoshells

Based on a non-local strain gradient theory (NSGT), this article analytically studies the sound transmission loss (STL) performance of a special type of double-walled partially cracked piezoelectric nanoshell structures of cylindrical shape undergoing initial electric potential and longitudinal magnetic field. The structure, which includes a specific number of axial and circumferential surface cracks, is struck by a plane wave in a thermal setting. Such cracks have minimal size compared to the nano-shell structure as a whole, minimizing the effects of curvature for this problem. The first-order shear deformation theory (FSDT) is utilized to explain the displacement components, and Hamilton’s concept is applied to obtain the coupled vibroacoustic equations that are used to generate the governing equations of the system for the two crack types, giving us an idea of the resulting structural stiffness in the presence of cracks using coefficients of crack compliance that are derived from the line spring model (LSM). The mechanical and acoustic properties are then checked to see their effect on the system response followed by a series of validation case studies. This evaluation also allows one to optimize a similar structure by selecting appropriate external parameters such as electric voltage, magnetic field intensity, different crack types, nonlocal and strain gradient parameters, variations of temperature, and incident angle of sound waves.

Effective elastic properties and S-wave anisotropy for rocks containing any oriented penny-shaped cracks in transversely isotropic background

Fractures have a significant influence on rock elastic properties. The existing elastic models for cracked medium mainly consider the crack dip angle. In this work, based on the previous transversely isotropic (TI) background and arbitrary dip crack model, a theoretical model is derived using the Bond transform matrix to quantify the effects of any oriented penny-shaped cracks on the elastic properties of rocks with a TI background. Based on this model, the effects of crack orientation on rock elastic properties are studied. Particularly, a special case with random crack orientation is investigated by using the spherical averaging of the crack compliance matrix with any orientations. Furthermore, the S-wave anisotropy for rocks containing multiple crack sets also is analyzed. The results indicate that the crack orientation has a great influence on rock elastic properties. Especially, the rock elastic properties indicate nonlinear changes with crack azimuth angles when cracks are inclined with respect to the TI background isotropic plane. The randomly oriented cracks decrease the overall rock elastic moduli but have little effect on rock anisotropy. For rocks containing multiple crack sets, parallel crack sets have great effects on the S-wave anisotropy, whereas the effects of orthogonal crack sets are negligible. Using this model, a complete workflow for the inversion of fracture properties using field data measured by acoustic logging is introduced. To explain the effect of crack azimuth angle, the crack density is inverted without considering crack orientation, only considering crack dip angle, and considering both crack dip and azimuth angles. The results indicate that the inversion results considering the crack dip and azimuth angles are in good agreement with the resistivity imaging results. This work provides the theoretical basis for fracture properties inversion for rocks with a TI background.

Validation of on-line crack compliance data analysis methods for the residual stress intensity factor

Residual stress can significantly impact the fatigue crack growth rate (FCGR) observed in standard tests. In the present study, implementation of the on-line crack compliance (OLCC) method for determining the residual stress intensity factor, Kres, in FCGR tests is supported by validation data, details of novel data analysis methods, and strategies for data reduction. Slitting measurements on C(T) specimens of quenched (but not stress relieved) and aged aluminum alloy 7050-T74 provide reference profiles of Kres as a function of crack size. The OLCC method quantifies Kres profiles from stress intensity factor control and load control FCGR tests over a range of residual stress states. Novel OLCC data reduction techniques are developed, and data analysis attributes and limitations explored, contributing to an improved methodology for standardization. The results demonstrate that the on-line crack compliance method is a valid means of quantifying Kres during fatigue crack growth rate testing.

Effects of residual stress on orientation dependent fatigue crack growth rates in additively manufactured stainless steel

Localized heating and resulting temperature gradients during additive manufacturing (AM) create significant residual stress that influences mechanical behavior, such as fatigue performance. To quantify residual stress effects on fatigue crack growth in AM materials, crack growth rates parallel and perpendicular to the build direction in directed energy deposition (DED) Type 304L austenitic stainless steel were measured. The on-line crack compliance method was used to determine the residual stress intensity factor, Kres, while simultaneously collecting fatigue crack growth rate (FCGR) data. Constant applied alternating stress intensity factor (constant ΔKapp) tests revealed the primary influence on measured FCGR is the orientation dependent Kres. Critical analysis of the compliance data from decreasing ΔKapp tests was used to quantify Kres, which was then used to correct FCGR data in the near-threshold regime. Results demonstrated that the fatigue response of DED Type 304L is inherently similar to that of annealed wrought Type 304/304L.

Stress sensitivity of multiscale elastic properties under hydrostatic loading conditions of organic shales at different thermal maturity stages

Shales serve essential functions as seals for conventional hydrocarbon reservoirs, waste storage repositories and unconventional plays. Exploration and evaluation of shales rely on the characterization of their multiscale elastic properties. However, there are limited studies on the geologic controls of elastic stress sensitivity in organic shales. This study investigated dry Lokpanta, Bakken, and Niobrara shales of different thermal maturity stages. First, we used a pulse-transmission ultrasonic setup to conduct multidirectional (0o, 45o and 90o) hydrostatic-stress sensitivity (0–27.58 MPa) tests to measure elastic waves and derive other elastic properties. Furthermore, we inverted microscopic parameters from the elastic measurements based on effective medium theories such as damage parameters (αij and βijkl), crack density (ρ), normal to tangential microcrack compliance (B), specific tangential crack compliance (snBT) and microcracks orientation anisotropy (η). Finally, we studied the effects of organic matter maturity and clay diagenesis on the multi-scaled elastic stress sensitivity. Under applied hydrostatic stress, the non-linear elastic response in the shales is consistent with the closure of aligned microcracks. Microcrack closure decreased the elastic anisotropy at lower stress (<10 MPa), while aligned clays and organic matter control the stress-insensitive anisotropy at higher stress. As organic matter matures and smectite transforms to illite, orthogonal velocity stress sensitivity increases and extrinsic contribution to elastic anisotropy increases. There is also a corresponding increase in orthogonal damage second-rank tensor (α33) and a decrease in orthogonal damage fourth-rank tensor (β3333) with organic matter-clay thermal maturity. These relationships are due to the thermally-induced development of new and/or extensional roughly-textured microcracks parallel to the laminated microfabric, which is of higher compliance in the shear than compression. Microcrack closure also decreases ρ under applied pressure. Most of the samples had B < 0.5, suggesting the presence of non-idealized roughly textured microcracks. Generally, snBT broadly decreases and η increases with clay diagenesis and organic matter maturity also because of the generation and/or enhancement of aligned microcracks resulting in higher tangential stiffness.

Elastic anisotropy of shales: The roles of crack alignment and compliance ratio

Cracks, broadly defined as compliant discontinuities, are a major cause of elastic anisotropy. However, few models are available for quantifying crack properties relevant to anisotropy. We developed a rock-physics model to quantify crack angular distribution and normal-to-tangential compliance ratio from pressure-dependent acoustic velocities measured in the laboratory. Our model uses a rectangular function of variable width and amplitude to extract the maximum dip angle of cracks, which is a direct quantification of crack alignment relative to the bedding plane. We tested the model on an organic-rich shale data set and confirmed that crack alignment and compliance ratio strongly impact Thomsen anisotropy parameters, thus demonstrating the model as a useful tool for better understanding how cracks affect elastic anisotropy.

Damage accumulation near the cold-expanded hole due to high-cycle fatigue by crack compliance method

The novel destructive method is implemented for quantitative assessment of fatigue damage accumulation in the stress concentration zone accompanied by residual stress due to cold expansion of the through-thickness hole. Damage accumulation is reached by preliminary cyclic loading of plane specimens with cold-expanded holes. Narrow notches, emanating from the hole edge at different stages of high-cycle fatigue, serve to manifest a damage level. These notches are inserted without applying external load. Deformation response to local material removing, caused by pure residual stress influence, is measured by electronic speckle pattern interferometry (ESPI) in terms of in-plane displacement components. Normalized values of the notch mouth open displacement (NMOD), in-plane displacement component at the initial point of the notch acting in the notch direction (U0), in-plane displacement component at the final point of the notch acting in the notch direction (U1) and the stress intensity factor (SIF) are used as current damage indicators. Numerical integration of curves, describing an evolution of each fracture mechanics parameter over lifetime, produces the damage accumulation function in an explicit form. It is established that all four fracture mechanics parameters give very close results.

Low-cycle fatigue damage accumulation near the cold-expanded hole by crack compliance data

Novel destructive method is implemented for quantitative assessment of fatigue damage accumulation in the stress concentration zone near the cold-expanded hole, which is characterized by residual stress influence. Damage accumulation is reached by preliminary push–pull loading of plane specimens with cold-expanded holes. Artificially introduced narrow notches, emanating from the hole edge at different stages of low-cycle fatigue, serve to manifest a damage level. Deformation response to local material removing is measured by electronic speckle pattern interferometry (ESPI) in terms of in-plane displacement components. Notches are inserted under constant external load to increase the measurement accuracy by reaching optimal fringe density. Normalized values of the notch mouth open displacement (NMOD), in-plane displacement component acting in the notch direction (U), the stress intensity factor (SIF), the T-stress and the biaxiality ratio are used as current damage indicator. Numerical integration of curves, describing an evolution of each fracture mechanics parameter, produces the damage accumulation function in an explicit form. It is established that all five fracture mechanics parameters give very close results. Developed approach is the essential link in predictable joints design to reach maximal beneficial effect of cold hole expansion on the fatigue life improvement of advanced aerospace structures.

Prediction and Measurement of Quenching and Pre-Stretching Stress in 7050 Aluminum Alloy Thick Plate

The article established a modelling that predicted and measured the quenching and pre-stretching stress in 7050 aluminum alloy thick plate in order to make the alloy suitable enough for manufacturing in the aerospace industries. The results show that both the rolling direction component stress and the transverse direction component stress have a “M” distribution along the plate thickness of 7050 aluminum alloy thick plate, but the level is slightly different, but the level is slightly different. The maximum value of quenching residual stress is about 187MPa in the 80mm thick of 7050 plate. Pre-stretching of the heat treated and quenched alloy to 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, 3.0% and 3.5% to reduce the residual stresses. It has been recorded that more than 90% reduction of residual stresses were observed in a pre-stretching rage of 2-3%. Measuring the surface residual stresses of the alloy by ultrasonic methods and measuring the internal residual stresses of the alloy by crack compliance methods. The two methods are used to verify the model. The difference between the measurement and the simulation is within 40MPa. The experimental data of the ultrasonic and the crack compliance methods were found to correspond well with the simulated models. This shows that the simulation model can effectively predict the evolution of residual stress in 7050 aluminum alloy thick plate, and the simulation model can also be reconstructed and inspected through the experimentally measurement.

Residual stresses near cold-expanded hole at different stages of high-cycle fatigue by crack compliance data

Experimental method for a characterization of high-cycle fatigue evolution of residual stress near cold-expanded hole is developed and implemented. The technique is based on simultaneous measurements of deformation response to narrow notch, inserted in residual stress field, on opposite specimen’s faces by electronic speckle-pattern interferometry (ESPI). Two-side measurements of notch opening displacements are performed when a single notch, emanating from cold-expanded hole edge, is inserted. The transition from in-plane displacement component to residual stress intensity factor (SIF) values follows from the relationships of modified version of the crack compliance method. The approach provides a difference in residual stress values referred to mandrel entrance and exit surface. Notches are inserted at different stages of low-cycle fatigue without applying external load. The results obtained describe fine nuances of residual stress evolution, which cannot be considered as monotonic relaxation.

Influence of stress ratio on residual stress evolution near cold-expanded hole due to low-cycle fatigue by crack compliance data

Modified version of the crack compliance method is used for determination of stress intensity factor (SIF) related to narrow notches emanating from cold-expanded holes. These notches are inserted at different stages of low-cycle fatigue under constant external load. It is shown how residual SIF values, generated by residual stress field influence, can be separated from total experimental SIF values. Residual SIF values, obtained at different stage of low-cycle fatigue with the same stress range Δσ = 350 MPa but different stress R = –0.4 and R = –1.0, provide quantitative description of residual stress evolution near cold-expanded hole. It shown that maximal residual stress relaxation of order 20 per cent occurs at 95 lifetime per cent for both loading programs.

Measuring internal residual stress in Al-Cu alloy forgings by crack compliance method with optimized parameters

Measuring the internal stress of Al alloy forgings accurately is critical for controlling the deformation during the subsequent machine process. In this work, the crack compliance method was used to calculate the internal residual stress of Al-Cu high strength alloys, and the effect of various model parameters of crack compliance method on the calculated precision was studied by combining the numerical simulation and experimental method. The results show that the precision first increased and then decreased with increasing the crack range. The decreased precision when using a high crack range was due to the strain fluctuation during the machining process, and the optimized crack range was 71% of the thickness of forgings. Low orders of Legendre polynomial can result in residual stress curve more smooth, while high orders led to the occurrence of distortion. The Tikhonov regularization method effectively suppressed the distortion of residual stress caused by the fluctuation of strain data, which significantly improved the precision. In addition, the crack compliance method with optimized parameters was used to measure the residual stress of Al-Cu alloy with different quenching methods. The calculated results demonstrated that the distribution of residual stress was obtained accurately.

Experimental and Numerical Investigation of Residual Stress in Coatings on Steel

Abstract In the present work, an alumina–titania (13 %) coating has been deposited by the air plasma spray process on a mild steel substrate by preheating the substrate to different temperatures. Coating thickness was measured by a scanning electron microscope, the hardness of coatings was measured, and morphology was also checked. The crack compliance technique has been used to determine the residual stress, as it is a very simple method, and also, it is compatible with conventional machining equipment and wire electric discharge machining equipment. Specimens were cut by the diamond cutter up to certain depth in microns, and the strain relieved was recorded. For finding out the residual stress, these recorded values of strain have been used by doing a finite element analysis (FEA) using the ANSYS 15.0 (Ansys, Inc., Canonsburg, PA) software. For validation of the model, uniaxial tension was applied on a sample by using the INSTRON universal testing machine (Instron Inc., Norwood, MA), and strain values at both the coating side and the substrate side were recorded and matched with the model under the same value of the load. After verifying the model, an equation has been developed using the Minitab software (Minitab Inc, Bengaluru, India), and the strains previously recorded were used as an input to the equation obtained, and the value of residual stress was calculated. The maximum value of residual stress was found at the coating–substrate interface, and the values of residual stress for sample preheated at 250°C, 200°C, and 150°C were found to be compressive, with the magnitude of −176.28 MPa, −243.54 MPa, and −255.42 MPa, respectively.

Evolution of the fracture mechanics parameters in the vicinity of the hole in conditions of low-cycle fatigue according to the data of modeling a crack with narrow notches

We propose a new method for studying the effect of low-cycle fatigue on the evolution of the fracture mechanics parameters in conditions of loading plane specimens with stress concentrators. Three programs of loading with a constant value of the stress range and different values of the stress ratio, as well as two programs with a constant value of the stress ratio but different stress ranges are considered. One program is common for both cases. All the programs include uniaxial tension-compression. Each program was implemented by testing a batch of the same specimens from seven to nine in each. One specimen from each batch was assigned to assess the durability. The other specimens were brought to various stages of low-cycle fatigue in each program. Experimental data were obtained for cracks of different length which were modeled by a sequence of three narrow notches launched from a through hole in a rectangular specimen. Each notch was exposed to constant external load of the same level. The deformation response to a small increment in the notch length at a constant external load was measured at different stages of low-cycle fatigue using electronic speckle pattern interferometry. The interference fringe patterns used as initial experimental data provided determination of the tangential components of in-plane displacements directly on the notch sides and the values of notch opening were thus determined from the results of direct measurements. The transition from measured displacement components to the values of the stress intensity factor and T-stress was performed using the relationships of a modified version of the crack compliance method based on Williams formulation. Distributions of the fracture mechanics parameters along the notches were obtained at various stages of cyclic loading. The dependences of the crack mouth opening displacement, the stress intensity factor and the T -stress on the number of loading cycles are constructed for the notches of a fixed length at different stages of low-cycle fatigue. It is shown that experimental distributions of the stress intensity factor values over the life time practically coincide for all four combinations of the loading cycle parameters.

Evaluation of the incremental compliances of non-elliptical contacts by treating them as external cracks

The incremental normal compliance of a contact of “irregular” (non-elliptical) shape between two elastic half spaces is considered. The contact is modeled as external crack. An approximate expression for the stress intensity factor (SIFs) around its boundary, in conjunction with Rice (1975) theorem that interrelates SIFs and crack compliances, are used to evaluate the crack compliance. The results cover a broad class of contact shapes, including – but not limited to - all convex ones. Bounds, similar to Hill's comparison theorem in micromechanics, are formulated for the compliances of contacts. The cross-property elasticity-conductivity relation yields similar results for contact conductance. The methodology is illustrated on several examples.