Elliptical Crack Research Articles

Size effects in the toughening of brittle materials by heterogeneities: A non-linear analysis of front deformations

Traditional computational approaches in simulating crack propagation in perfectly brittle materials rely on the estimate of stress intensity factors along the rupture front. This proves highly challenging in 3D when the crack geometry departs from very specific cases for which analytical solutions are available, like e.g. the penny-shaped crack geometry. Here, we extend the first-order theory of Gao and Rice (1987b), and predict the distribution of the mode I stress intensity factor KI along the front of a tensile coplanar crack that is slightly perturbed from a reference penny-shaped configuration, up to second order in the perturbation amplitude. Our theory is validated against analytical solutions available for embedded elliptical cracks, and its range of validity is further assessed using numerical simulations performed on cosine front perturbations of varying mode and amplitude. It is then used to develop a homogenization framework for the toughness of weakly disordered media. The effective toughness and its fluctuations are bridged quantitatively to the intensity of the toughness fluctuations and their spatial structure. Our theoretical predictions are compared to the results of ∼1 million simulations of crack propagation building on our second-order theory and Fast Fourier Transforms. We show that the impact of toughness heterogeneities is size-dependent, as they generally weaken the material when the crack size is lower or comparable to the typical heterogeneity size, but reinforces it otherwise. It results in an apparent R-curve behavior of the brittle composite at the macroscale.

Theoretical analysis and fracture process of elliptical tunnel with symmetric cracks subjected to water pressure and ground stress

The water diversion tunnel played an important role in the allocation of water resources in the water diversion project. In the process of actual use, some cracks or joints at the edge of the tunnel could produce further cracking under the action of internal water pressure and in-situ stress, which was dangerous to the safety of the tunnel. The analytical solution formulas of stress intensity factors (SIFs) of the elliptical tunnel with two cracks were derived by using the complex function method, and the external factors affecting the stability of the elliptical tunnel were analyzed. The failure process of the specimens was analyzed by using the high-speed camera and digital image correlation (DIC) method to obtain the evolution laws of crack initiation behavior and tunnel fracture modes. The results can conclude that both sides of the elliptical tunnel and the crack tip were stress concentration areas. As the inclination angle α of the elliptical tunnel is varied from 0° to 45°, tensile cracks occur at the top and bottom of the elliptical tunnel caused by tensile stress, and the compressive strength is smallest when the elliptical tunnel with α is 45°. As the inclination angle α is 75°, two penetrating cracks appear at the edge of the elliptical tunnel and crack tips. The initiation pressure increases with the inclination angle α. The SIF of mode II rises with the ellipse major semi axis and the crack length.

An inversion scheme for sizing crack from signals of the motion-induced eddy current testing method based on a new formula of signal gradient of ferromagnetic materials

In this paper, an inversion scheme for the profile reconstruction of a fatigue crack in rails from the motion-induced eddy current testing (MIECT) signals is proposed based on the conjugate gradient (CG) optimization method and the fast MIECT forward simulator developed by authors. To calculate the gradient of MIECT signals with respect to the crack parameters, a simplified analytical gradient formula is deduced, which enables a high efficiency crack reconstruction from MIECT signals for a crack in ferromagnetic material. The validity of the inversion schemes is demonstrated at first by reconstructing cracks with numerical simulated signals for ferromagnetic materials with an oblique elliptic crack model. The good reconstruction results reveal that the new signal grandient formula is efficient for reconstructing crack in ferromagnetic material with the CG algorithm. Finally, sizing of artificial cracks are carried out with use of measured MIECT signals due to cracks in plates of both non-magnetic and ferromagnetic materials. The reconstructed crack parameters are in good agreement with the true values, which experimentally proved the validity and efficiency of the proposed inversion scheme and the new signal gradient formula for MIECT inversion.

A New Equivalent Thickness Model for Fracture Prediction on Round Rods with Elliptic Surface Cracks

An equivalent thickness model is proposed to develop the fracture prediction method for round rods with elliptic surface cracks on the basis of a constraint equivalence principle and detailed three-dimensional (3D) stress field analysis near crack fronts with various elliptical shape factors t ([Formula: see text] 0.33, 0.4, 0.5, 0.67, 1) and crack depths a ([Formula: see text] 1/3, 2/3). The equivalent thickness is a function of crack depth a, elliptical shape factor t, elliptical parameter angle [Formula: see text] and radius of the round rod R. The reported fracture test data for the 20CrMo steel round rods with various elliptic surface cracks are adopted to validate the proposed equivalent thickness model. It is found that the improved 3D fracture prediction method by introducing the equivalent thickness is more effective than the traditional two-dimensional (2D) fracture prediction method. The prediction error is reduced from 10.5% to 3.1% for given round rods with elliptic surface cracks under tensile loading in this paper.

Study of stress intensity factors in sliding and tearing modes for subsurface elliptical cracks in rolling train wheels

AbstractIn the railway systems, induced contact between wheels and rails creates maximum shear stress around the contact surfaces. These shear stresses vary with the wheel's rolling and can initiate and extend subsurface cracks in the wheel and rail. The most common subsurface cracks in the rail systems have elliptical forms. Sliding and tearing modes of the fracture can be observed due to different components of the shear stresses around the contact area. Compressive normal stress has no considerable effects on in‐plane and out‐of‐plane shear modes for subsurface cracks. In this research, by proper modeling of the wheel and rail, the effects of dimensions, position, and depth of the subsurface cracks on the stress intensity factors of the sliding (KII) and tearing (KIII) modes have been investigated. Also, variations of the KII and KIII values versus wheel rolling angles and frictions have been considered. The validity and accuracy of the cracked model created by sub‐modeling technique have been verified by half‐space modeling and weight function method.

Determination of mixed-mode stress intensity factors for axial crotch corner cracks at cylindrical vessel to nozzle intersections under internal pressure and nozzle loads

In this study, Stress Intensity Factors (SIF) of quarter circle and semi elliptic cracks at the crotch corner of the intersection between a cylindrical pressure vessel and a radial nozzle have been computed for combinations of the actions of internal pressure and external forces on the nozzle. Until now, practitioners have used simplified solutions for SIFs available from the literature, such as the Newman-Raju equations for the mode I SIF produced by an internal longitudinal semi-elliptical crack in a cylinder, to approximate the SIF produced at a crotch corner crack. These equations do not take into account the effect of the opening of the cylinder for the nozzle and the resulting stress distribution along the cylinder. As a result, there is uncertainty about the validity of using these simplified solutions in engineering critical assessments and a more exact determination of the SIFs for crotch corner cracks would be beneficial.The mode I, II and III stress intensity factors have been determined for crotch corner cracks for a single size of vessel and are presented as a function of variables such as location on the crack front, crack size and aspect ratio (a/c), crack angle with respect to the axis of the vessel, and the magnitude of internal pressure and external forces. The results are presented in graphical form. The study concludes with the recommendation that the analysis be extended to a wider range of vessel and nozzle parameters.

Scattering of SH waves by cavity with arbitrary shapes and crack at arbitrary positions in anisotropic elastic half-space media

In this paper, the complex function method, Green's function method and coordinate transformation method are used to solve the scattering problem of cavity with arbitrary shape and crack at arbitrary position in the orthotropic elastic half-space under the incidence of SH waves. Firstly, the displacement field and stress field caused by cavity scattering in the orthotropic elastic half-space under the action of incident SH wave are obtained. Secondly, the solution of Green's function is derived when the plane line source load acts on the orthotropic elastic half-space with cavity. Then, the crack is constructed based on the method of crack ‘cutting.’ The total displacement field and stress field can be further obtained. Then, the analytical solutions of the dynamic stress concentration factor (DSCF) around the cavity, the surface displacement amplitude |W| and the dynamic stress intensity factor (DSIF) at the crack tip are obtained. Finally, taking the existence of elliptical cavity and crack in the orthotropic elastic half-space as an example, a large number of numerical calculations were carried out. Numerous numerical examples show that different parameters have a great influence on the DSCF around the cavity, the surface displacement amplitude |W| and the DSIF at the crack tip.

Compliance of a crack embedded in a transformed transversely isotropic material

This paper is devoted to the analytical calculation of the contribution and opening displacement tensors of an arbitrarily oriented crack in a transformed transversely isotropic (TraTI) matrix. It generalizes a recent work considering elliptically orthotropic (EO) symmetry, as EO is a particular case of TraTI. The latter allows to explore a wider range of symmetries, including non-orthotropic symmetries. The approach is based on the linear transformation between boundary value problems with TraTI and transversely isotropic (TI) bodies. A detailed analysis shows that TraTI fourth-order tensors are described by a set of 11 parameters (5 material parameters and 6 angles defining the transformation). This is a significant enrichment beyond the set of EO tensors which depends on 7 parameters (4 material parameters and 3 angles). New analytical results are obtained for elliptical and circular cracks embedded, respectively, in TraTI, orthotropic TraTI and monoclinic TraTI matrices. It is shown that the most general case leading to analytical derivations of crack contribution and opening displacement tensors is that of a TraTI matrix with a specific restriction: the initial crack must be aligned along the isotropy plane of the TI matrix. To the best of our knowledge, this corresponds to the current largest space of matrix anisotropy allowing analytical derivation of the compliance of a single elliptical crack embedded in an infinite matrix as well as an extension of analytical 3D results showing a coupling between opening and shear modes. Numerical results are presented to illustrate potential applications of the method for the case of an arbitrarily oriented elliptical crack in an infinite uniform matrix of stiffness of arbitrary anisotropy. The best-fit problem investigated in previous papers is revisited and a new algorithm providing the closest stiffness tensor for which there exists an analytical solution to the crack opening displacement tensor is developed. Numerical applications to real TI materials are finally presented.

Effect change of extra-wide concrete self-anchored suspension bridge under blast loads and its prediction considering time-dependent effect

In order to study the dynamic response of self-anchored suspension bridge with extra-wide concrete girder under explosive load, a real bridge was taken as the background, which is the widest on in China at present. Firstly, the refined finite element model (FEM) of the whole bridge was established. The reliability of numerical simulation method was validated by the measured data of the bridge at completion state and explosive test results of scaled girder specimen. The vertical displacement, reinforcement strain, acceleration and failure characteristics of the girder specimen under the explosive load of 3 kg TNT were analyzed. Through parametric simulation, the effects of different scale distance, concrete strength and reinforcement ratio on the explosion response of girder specimen were studied. Secondly, the dynamic responses of girder were studied. Finally, the change law of dynamic response at different ages was analyzed. The results show that the elliptical crack is formed on girder top plate. The maximum girder vertical displacement reaches 135.9 mm under 1000 kg, and the maximum increasing ratio of hanger force is 27%. The influence brought by the extra width of girder is significant. The girder vertical displacement changes along transverse direction and the concrete breach sizes are different. In addition, with the increase of service age, the girder displacement increases from 60.50 mm to 106.16 mm under the same explosive load. The research results can provide an important basis for the anti-explosion protection and reinforcement of similar extra-wide concrete self-anchored suspension bridge.

Dynamic Behavior Analysis of a Rotating Shaft with an Elliptical Breathing Surface Crack

PurposeIn this paper, dynamic behavior of a rotor system with an elliptical breathing crack that simulates the real shape of the crack front is investigated.MethodsA finite element model of the cracked rotor system is developed. The crack breathing mechanism is modelled based on an improved breathing model which considers the inclination of the neutral axis of the cracked element cross-section during shaft rotation. Harmonic balance method is used to solve the equations of motion of the rotor system for steady-state response characteristics. The effect of some parameters such as crack depth, crack shape factor and the spinning speed is investigated.Results and conclusionsThe results show that the unique whirl orbits behavior during passage through the subcritical speeds serve as a key indicator of crack presence in the shaft. The effects of the crack front curvature and the breathing model are revealed. The value of shape factor affects the whirl orbit characteristics such as size of the inner or outer loops and the amount by which the orbits rotate while crossing the subcritical speeds. The presented model considering the real crack front shape may contribute towards improved modelling of cracked rotors and better interpretation of measured vibration response.

Energetic approach to predict the elliptical crack growth

Energetic approach to predict the elliptical crack growth

Non-Fourier thermal shock resistance of the ceramic plate with an embedded elliptical crack

Non-Fourier thermal shock resistance of the ceramic plate with an embedded elliptical crack

Analysis and design optimization approach of vibration characteristics of an I shaped straight steel girder bridge

Bridges are subjected to environmental exposure like corrosion which may induce cracks. These cracks may cause change in natural frequency of bridge. The current study's goal is to explore the effect of crack on the vibration characteristics of bridge the design and modal analysis of bridge is conducted by ANSYS 21 simulation package. Two different designs of bridges are evaluated i.e. without crack and with crack.The effect of crack on natural frequency and mode shape is evaluated. The vibration characteristics of bridge structure changed with artificially induced transverse elliptical crack on girder.Techniques of optimization could help in determining the behaviour of the damage and extent to which it could be controlled via simulation technique without conducting tests on the actual model. Analysing the frequencies using ANSYS software, the validity of the presented numerical formulation is studied. Response spectrum methodology is used in studying the crack characteristic and optimization of crack variables. Sensitivity analysis showed the individual effect of each variable on the output parameters. Moreover, the response surface plots generated showed the peak and least deformation values of natural frequencies. Invaluable information which help practicing engineers better understand the vibration characteristics is provided.

Fracture mechanics analysis of a closure head of a PWR reactor pressure vessel LEFM-based SIF

In this study, fracture mechanics analysis of a closure head of a PWR reactor pressure vessel has been performed under accidental conditions. For the purpose, a wide range of elliptical surface cracks starting from minute cracks to worst-case cracks was postulated at the highest stress concentration locations of the head. The analyzed accidental conditions are small break loss of coolant accident (SB-LOCA) and Rancho-Seco transient (RST). The maximum linear elastic fracture mechanics-based stress intensity factors of the worst-case crack ‘a = 0.25 th; a = 0.33 c’ under SB-LOCA and RST are 104.37 MPa√m and 137.12 MPa√m respectively, which are less than the fracture toughness value of the material i.e. 155.6 MPa√m. Hence, the accidental conditions do not threaten the structural integrity of the proposed closure head.

Effective elastic properties and wave surfaces of rock materials containing multiple cavities and cracks (effective field approach)

The paper is devoted to simulation of the effective elastic properties of rock materials containing two substantially different types of defects simultaneously: random sets of ellipsoidal pores (cavities) and elliptical cracks. For solution of the homogenization problem and calculation of the effective elastic properties of such materials, the self-consistent effective field method is used. For composites with one type of heterogeneities, the method coincides with the Mori–Tanaka method. The method allows deriving analytical expressions for the effective elastic stiffness tensors of the materials containing pores and cracks of various scales. These tensors have correct symmetry with respect to tensor indices and provide physically reasonable values of the effective elastic constants in wide regions of porosity and crack density. The materials with pores and cracks of substantially different sizes and of close sizes are considered. The method predicts substantially different effective elastic constants of the materials with these microstructures. Comparison of the components of the effective elastic stiffness tensors for the materials with various values of porosity and crack density are presented. Wave surfaces of acoustical waves in the materials containing pores and cracks are constructed. It is shown that for materials with spherical pores and circular cracks, the shapes of these surfaces are close to ellipsoidal. The results of the paper can be used for determination of fracture level in damaged rock materials by acoustical methods.

Stress intensity factor assessment for reactor pressure vessel nozzles containing postulated corner cracks

Due to the structural discontinuity of geometry having higher stress concentration, the nozzles may dominate the structural integrity of reactor pressure vessel (RPV) system, even the material radiation embrittlement is insignificant. The nonmandatory Appendix G in Section XI of the ASME Boiler and Pressure Vessel code provides a fracture mechanics-based methodology using magnification factors to calculate the stress intensity factor of reference crack tips as long as the stress distribution through the nozzle section is available. This paper establishes various three-dimensional finite element models of Taiwan domestic RPV nozzles to conduct a series of stress and linear elastic fracture mechanics analyses to estimate the stress intensity factors. The circular and elliptical reference corner cracks with depths of one-fourth and one-tenth thickness of the vessel wall along the 45° path are considered. Two evaluation results using the ASME method and the finite element linear elastic fracture mechanics analysis are compared. It is found that the ASME method produces conservative values of stress intensity factor at crack tips of circular corner cracks but underestimates the elliptical corner cracks. Finally, an adjusted magnification factor formula is proposed to obtain more accurate stress intensity factors for elliptical-shaped nozzle corner cracks.

Study on the stress field concentration at the tip of elliptical cracks

Abstract The holes and defects in rock materials have a great impact on the mechanical properties and failure mechanism of rock, cracks often appear in the form of flat ellipses in natural rock mass, and the current research is still insufficient. For this purpose, based on the Particle Flow Code (PFC) of discrete element particle flow program, the numerical prefabricated fractured rock samples with the ratios of the long and short axes of the elliptical fractures being 5, 7.5, 10, 12.5, and 15 were simulated in order to obtain the strength and failure characteristics, stress concentration characteristics at tips of numerical rock samples in uniaxial compression test. The results of the numerical test simulation show that: (1) When the rock models with prefabricated elliptical crack were damaged, the initial cracks occurred at the end of the short axis of the elliptical crack, and penetrated up and down from the surface of the elliptical crack, the wing cracks occurred at both ends of the long axis, gradually formed a macro-crack, and the secondary cracks extended near the wing crack. (2) With the increase of the ratios a:b of long and short axes of the elliptical fracture, the strength and elastic modulus of the numerical rock samples gradually decreased, Poisson’s ratio gradually increased, and the total number of micro-cracks in the rock models decreased. (3) The numerical solutions of stress concentration factor k obtained by numerical simulations at the tip of the elliptical crack increased with the increase in the ratio of a:b; it was highly consistent with the variation law of the analytical solution of the stress concentration factor calculated by the theory of flat ellipse. The stress concentration is an important reason for failure of rock with elliptical cracks. Study on the crack tip will be very useful and significant.

Three‐dimensional fatigue crack growth simulation and fatigue life assessment based on finite element analysis

AbstractDamage tolerance design allows manufactures to assess fatigue crack propagation of components containing initial defects. A new modeling approach is proposed to carry out a systematic crack growth analysis based on an analytical method using reduced order models and three‐dimensional (3D) finite element (FE)‐based fatigue crack growth (FCG) analysis. 3D FE‐based FCG analysis is adopted as an advanced modeling approach for component geometries without any simplifications with respect to crack front shape or planarity of the crack path. The FE‐based FCG approach was verified and validated in two different stages: Ti‐6Al‐4V plate geometry with three different single elliptical crack configurations and Al 2024‐T3 specimens with two different multiple crack configurations. The analytical FCG approach was developed to estimate crack front evolution and crack growth life for three different elliptical cracks to verify the FE‐based approach. Experimental data of Al 2024‐T3 specimens with multi‐edge cracks was used to further validate the FE‐based FCG approach. The FE‐based FCG approach proved to be both accurate and powerful in assessing crack propagation life and crack paths of structural parts.

Damage Prediction of Pre-cracked High-Pressure Pipelines

Due to the increasing need to transport fluids, the use of industrial pipes and their sustainability is a crucial aspect to address. The motive of this work is to predict the behaviour of a pre-cracked pipeline under internal and external pressure using Finite element simulation. An intermediate range of pressure conditions has been selected to analyse the dispersal of normalized stress intensity factor along a semi elliptical crack front for a predefined crack geometry. This work also presents a comparative study of various piping material behaviour to address a relatively better choice for industrial use. Among the various alloys considered in the analysis Inconel 625 was found to be most appropriate for the pressure range.

Study of the coupling effect of elliptical cavities and cracks on tunnel stability under dynamic loads

This study to explore the coupling effect of elliptical cavities and cracks on the dynamic stability and failure modes of tunnels. A series of dynamic numerical simulations and model tests were conducted by using AUTODYN code and a modified drop hammer impact test device. A cracked tunnel model with an elliptical cavity was proposed in this study. The crack coalescence behaviour, crack initiation time and stress field around the elliptical cavity and the crack tips were calculated by using numerical simulation, and the dynamic initiation fracture toughness (DIFT) was also determined according to experimental and numerical results. According to these results, we conclude that the long and short axes of the elliptical cavity have a great influence on the crack coalescence behaviour and the stress field, and the long axis of the elliptical cavity has a greater influence than the short axis. When the inclination angle θ of the elliptical cavity is 90°, the stability of the crack is lowest, and when the inclination angle θ of the elliptical cavity is 45°, the effect of the elliptical cavity on the crack initiation behaviour is largest. DIFT was calculated and indicated that when the inclination angle of the elliptical cavity shows its smallest value, 90°, the cracked tunnel stability is lowest.