Multiple Edge Cracks Research Articles

Multiple edge cracks in a coated semi-infinite medium due to non-Fourier thermal shock

In this wok, based on the non-Fourier C-V model, the problem for multiple edge cracks is investigated for a coating-substrate pair subjected to a sudden temperature drop at the surface of coating. The temperature and resulting thermal stresses without cracks are obtained by the method of Laplace transform. By numerically solving the singular integral equations, the thermal stress intensity factors (SIFs) are evaluated. The results from the Fourier model and C-V heat conduction model are presented for comparison. Two dimensionless quantities are proposed to account for the coupling effects of heat conduction parameters (i.e., thermal conductivity, thermal diffusivity, and thermal relaxation time). Numerical results are presented for the thermal SIF as a function of normalized quantities such as time, crack depth, material constants and crack spacing. The findings in this work are expected to provide references for maintaining the integrity of the coating-substrate pair in the extreme heat transfer applications.

Fracture behavior of superconductor REBCO tapes with multiple edge cracks under electromagnetic force

It is often necessary to narrow wide tapes through a slitting process in applications of the second-generation high-temperature superconductor REBa2Cu3O7-x (REBCO, RE = rare earth) tapes. However, this results in some random length edge cracks appearing on the slit edge and extending at certain angles towards the center of the tape. In this paper, we proposed a randomly distributed multiple-edge crack model to investigate the fracture mechanism of the superconducting tape under electromagnetic force due to high magnetic fields and current density (∼10 T and ∼ 1010A/m2). The electromagnetic body force distributions within the superconducting tape, incorporating multiple edge cracks, are simulated based on the H-formulation. Then a fracture analysis is conducted under the combined loading of tensile and electromagnetic forces, with the numerical calculation of stress intensity factors (SIFs) using the J-integral method. To validate the model's accuracy, we compare the body force distribution caused by an edge crack and the SIFs of edge cracks under uniform tension with existing results. Detailed analyses are undertaken to explore the impacts of electromagnetic body force non-uniformity, interactions between neighboring cracks, and the stochastic nature of geometrical parameters (length, angle, and spacing) on the fracture behavior under an alternating magnetic field. Numerical results indicate that non-uniform electromagnetic body forces arising from defects are a critical factor in crack propagation. Furthermore, the randomness of the geometrical parameters amplifies the interactions between some of the neighboring cracks, causing the tendency of the tape containing random edge cracks to fracture. Therefore, it is advisable to avoid random cracks with substantial length, angle, and spacing during the mechanical slitting process.

Effects of multiple edge cracks, shear force, elastic foundation, and boundary conditions on bucking of small-scale pillars

The buckling instability of micro- and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with small-scale beam-like particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a well-posed stress-driven nonlocal model is developed, which allows the calculation of the critical loads and buckling configurations of the miniaturized pillars on an elastic foundation and with arbitrary numbers of edge cracks. The discontinuities in bending slopes and deflection at the damaged cross-sections due to the edge cracks are captured through the incorporation of both rotational and translational springs. A comprehensive analysis is conducted to investigate the instability of pillars containing a range of one to four cracks. This analysis reveals interesting effects regarding the influence of crack location, nonlocality, and elastic foundation on the initial and subsequent critical loads and associated buckling configurations. The main findings are: (i) the shielding and amplification effects related to a system of cracks become more significant as the dimensions of pillars reduce, (ii) the influence of the shear force at the damaged cross-section related to the translational spring must not be neglected when dealing with higher modes of buckling and long cracks, (iii) an elastic foundation decreases the effects of the cracks and size dependency on the buckling loads, and (iv) the effects of the edge cracks on the critical loads and buckling configurations of the miniaturized pillars are highly dependent on the boundary conditions.

Transient analysis of a cracked functionally graded magneto‐electro‐elastic rectangular finite plane under anti‐plane mechanical and in‐plane electric and magnetic impacts

AbstractThe transient anti‐plane problem of a functionally graded magneto‐electro‐elastic (FGMEE) rectangular plane weakened by several embedded and edge cracks under various boundary conditions is solved. First, the solutions to the dynamic magneto‐electro‐elastic dislocation in the finite rectangular domain are derived by employing Laplace and finite Fourier cosine transforms. The solutions can be utilized to construct the singular integral equations in Laplace transform domain for FGMEE rectangular plane with multiple embedded and edge cracks. The integral equations arising from dislocation solutions have Cauchy‐type singularities with unknown dislocation density functions. These unknown functions may be obtained by reducing the integral equations into a system of linear algebraic equations, hereby, the dynamic stress intensity factors (DSIFs) for several arbitrarily embedded and edge cracks are obtained. The overshoots of dynamic stress intensity factors are shown to be intensely affected by the rectangular plane dimensions, length and position of the cracks, loading parameter, and negative or positive gradient index.

Numerical analysis of the mechanical and electrical properties of (RE)BCO tapes with multiple edge cracks

One of the leading causes of critical current degradation in rare-earth barium–copper-oxide tapes is the micro-cracks produced by mechanical slitting. These cracks are scattered near the edge of the tape and vary in length and angle. In this work, a tape model with multiple edge cracks is established. Under tensile loading, the effects of the Poisson ratio, crack length, crack angle, crack spacing, and geometric mutation between cracks on the stress intensity factor are investigated using the extended finite element method (XFEM). Tensile experiments were conducted at room temperature to investigate the crack propagation behavior of tapes with multiple edge cracks. The results show that the stress intensity factor obtained using XFEM is more informative than the analytical solution, which ignores the Poisson effect. The stress intensity factor is sensitive to crack length and angle variations and exhibits an evident jump characteristic when a geometric mutation occurs. The jump level strongly depends on the geometric difference. The jump location is the initiation site for crack propagation, which is consistent with the experiment results. The strain analysis of the tape implies that high-strain regions exist at the crack tip before the tensile strain reaches the irreversible strain limit. The critical strain of crack propagation is closely related to the form of crack distribution. It dominates the irreversible strain limit of critical current degradation, which facilitates understanding the early degradation of critical current. Finally, some engineering suggestions are given.

The Fracture Behavior of REBCO Tape with Multiple Oblique Edge Cracks

The Fracture Behavior of REBCO Tape with Multiple Oblique Edge Cracks

Effect of edge cracks on critical current degradation in REBCO tapes under tensile stress

The slitting process for manufacturing REBa2Cu3O7-δ (REBCO, RE = Rare earth) tapes of required width significantly improves the production efficiency and reduces production costs. However, edge cracks induced by the slitting process of wide REBCO tapes may cause premature degradation under high tensile stress in high-field magnets. Therefore, it is necessary to evaluate the effect of edge cracks of REBCO tapes on the critical current (Ic) degradation. Firstly, Ic degradation under artificial cracks was measured to validate the applicability of linear elastic fracture mechanics for the REBCO layer. The maximum circumferential stress criterion was used to derive the mixed-mode stress intensity factor of multiple oblique edge cracks. A semi-analytical model considering edge crack properties such as angle β, spacing d, and length a, was built to evaluate the critical load and critical crack. We found that when the stress intensity factor at the crack tip is below KIC=2.3 MPam, edge cracks did not propagate. We examined commercial REBCO tapes manufactured by two different processes, concluding that edge cracks in these tapes will not cause premature degradation.

Nonlinear vibration analysis of a cantilever beam with multiple breathing edge cracks

The nonlinear vibration analysis of cracked structures could be a suitable indicator for diagnosing defects. When breathing cracks appear on the beam, its behavior can be modeled as a bilinear oscillator. In this paper, the nonlinear vibration behavior of a cantilever beam with multiple breathing cracks is investigated. For this purpose, the multi-cracked beam’s stiffness is calculated by introducing the effective length for the local stiffness reduction in the vicinity of cracks. Furthermore, owing to the importance of the damping variation in a cracked beam, its exact value is evaluated through theoretical expressions. A continuous polynomial function with the Weierstrass approximation is exploited to simulate the bilinear behavior of breathing cracks. By performing nonlinear analysis using the perturbation technique, the cracked beam’s dynamic behavior is studied. Moreover, the sensitivity of the response to the crack parameters in the primary resonance of the structure is analyzed. Finally, the sensitivity of the beam’s nonlinear response to the different number of breathing cracks and various crack depths is investigated. It is shown that the beam with a higher number of cracks or deeper ones has obvious softening behavior, and hence a more significant jumping can appear in the response.

Insights on the crack modeling and effectiveness of piezoelectric energy harvesters

A damage model for investigating the performance of cracked piezoelectric vibrational energy harvesters (VEHs) is developed. The crack model adopted does not alter the piezoelectric properties of the lead zirconate titanate patch but modifies the structural stiffness at the crack location following the Griffith’s strain release formulation. Multiple VEH configurations are considered to determine how the performance of various sized piezoelectric patches are impacted by multiple edge cracks. It is demonstrated that thicker piezoelectric patches are more severely impacted by cracks than thinner patches. The number of cracks, proximity of the cracks to the base of the VEH, and crack depth are shown to affect VEH performance. It is shown that severe cracks have a pronounced influence on the stiffness of the energy harvesting system, thus causing a deviation in the resonance region and amplitudes of the damaged system. The increased flexibility of the cracked piezoelectric patches increases the tip deflection and decreases the resonant frequency which can shift the initial optimal resistance to some higher value thus, greatly affecting the efficiency of piezoelectric energy harvesters.

Nonlinear modeling and performance analysis of cracked beam microgyroscopes

Cracks constitute a common structural defect in microelectromechanical systems that may arise during manufacturing, mechanical fatigue, or shock loading. Areas weakened by cracks increase the flexibility of the microstructure. Depending on the crack severity, the increased flexibility can cause dramatic changes in the static and dynamic behaviors of electrically actuated MEMS. In this work, a numerical investigation of the depth and location of multiple edge cracks on the performance of beam microgyroscopes is conducted. Applying the Griffith strain energy release theorem for an edge crack, the vibrational characteristics of microgyroscopes with multiple cracks including the static deflection, natural frequencies, and mode shapes are studied analytically. Then, the differential quadrature method is employed to model the nonlinear dynamic behaviors of the cracked gyroscope system. Cracks initiated along the driving or sensing directions of the microgyroscope are found to have differing effects on its nonlinear dynamic response. This numerical study reveals that the onset of severe single cracks or a series of multiple cracks can significantly affect the sensitivity of the microgyroscope to base rotations and therefore leads to a degradation in the performance of the damaged MEMS sensor. This study also shows that a potential design approach for broadband microgyroscope systems manufactured with cracks can be proposed.

Evaluation of surface energy for formation of multiple edge cracks using Medg-integral

A novel contour integral approach termed Medg is introduced for computation of the surface energy required for the formation of multiple edge cracks. The method is developed by reinterpretation of the conventional M-integral with deliberate delimitation of integration contour and selection of coordinate origin. Due to path independence, this method is efficient, easy to implement by using finite elements, and does not require a complicated mesh around the crack tips for good accuracy. Attention is also addressed to discussion of the size effects on proper interpretation of its physical meaning. The adequacy of the numerical results computed for the finite size corrections has been validated by using some of the available empirical formulations. It is observed that the size effect can be neglected when the crack size remains under one-tenth of the structure size. The results of a specific multi-cracked geotechnical structure suggest that, the damage state such as degradation of the structural stiffness due to the presence of edge cracks can be properly inspected by using Medg.

Damage Localization in Laminated Composite Beams with Multiple Edge Cracks Based on Vibration Measurements

This study presents identification and localization of open transverse edge cracks in laminated composite beams by using vibration measurements. To this aim, a composite cantilever beam is considered under six damage scenarios. Natural frequencies and mode shapes of the beam for each case are obtained numerically by using a shear-deformable finite element model as well as ANSYS® software. Experimental measurements are then performed to validate the numerically calculated results. To show consistency among the calculated and measured results, modal assurance criterion values are obtained for each case. For determination of the crack locations on the beam, changes in the mode shape curvature and the modal flexibility are employed based on the experimental and numerical modal data. Comparisons showed that the methods considered give reasonable results, and they can be efficiently used for damage localization in laminated composite beams including multiple edge cracks.

Stochastic MMSIF of multiple edge cracks FGMs plates subjected to combined loading using XFEM

Abstract The second order statistics of multiple edge crack functionally graded materials (FGMs) under tensile, shear and combined loading assuming uncertain system parameters is presented in this paper. The uncertain parameters used under the present study are the material properties, and crack parameters such as crack length and crack angle. In this present analysis extended finite element method (XFEM) is used. The stochastic analysis is carried out using second order perturbation technique (SOPT) for the evaluation of mean and coefficient of variance (COV) of mixed mode stress intensity factor (MMSIF).

Geometrically non-linear forced vibrations of fully clamped functionally graded beams with multi-cracksresting on intermediate simple supports

The objective of this work is to investigate the non-linear forced dynamic response at large vibration amplitudes of fully clamped functionally graded beams containing a multiple open edge cracks and resting on intermediate simple supports. The theoretical model is based on the Euler-Bernoulli beam theory and the crack rotational spring model. The functionally graded beam properties are supposed to vary continuously through the beam thickness. A homogenization procedure based on the neutral surface approach taking into account the presence of the crack is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. In the non-linear analysis, harmonic motion is assumed and the discretized expressions for the beam total strain and kinetic energies are derived by applying Hamilton’s Principle, so as to reduce the problem to a non-linear algebraic system solved using an approximate explicit method previously applied to various non-linear structural vibration problems. Considering the forced vibration case, the non-linear frequency response functions are obtained numerically in the neighbourhood of the predominant non-linear mode shape using a single mode approach. The effects of crack, the beam material gradient and the applied harmonic force are presented and discussed.

Parametric Vibrations of Axially Moving Beams with Multiple Edge Cracks

Nonlinear transverse vibrations of axially moving beams with multiple cracks is handled studied. Assuming that the beam moves with mean velocity having harmonically variation, influence of the edge crack on the moving continua are investigated in this study. Due to existence of the crack in the transverse direction, the healthily beam is divided into parts. The translational and rotational springs are replaced between these parts so that high stressed regions around the crack tips are redefined with the springs' energies. Thus, the problem is converted to an axially moving spring-beam system. The equations of motion and its corresponding conditions are obtained by means of the Hamilton Principle. In numerical analysis, the natural frequencies and responses of the spring-beam system are investigated for principal parametric resonance in detail. Some important results are obtained; the natural frequencies decreases with increasing crack depth. In case of the beam travelling with high velocities, the effects of crack's depth on natural frequencies seems to be vanished.

Free vibrations of laminated composite beams with multiple edge cracks: Numerical model and experimental validation

Free vibration analysis of laminated composite beams including open transverse cracks is presented by using a shear-deformable thirteen degrees-of-freedom finite element model, which considers extension-twist, bending-twist and bending-extension couplings, and the Poisson's effect. Lagrange's equation is employed to derive the equations of motion. Trial functions are selected as cubic Lagrange polynomials for deflections while quadratic for the others in derivation of element mass and stiffness matrices. Damage is introduced into the beam by reducing stiffness of elements within damaged zone. Comparisons are made with the available literature to show the present element's accuracy. Experimental validation is also performed with through ambient vibration tests on a cantilever laminated composite beam under six damage scenarios. According to the results, the methodology presented in this study provides a simple and quick way in a sufficient accuracy to determine the natural frequencies and mode shapes of laminated composite beams with open transverse non-propagating edge cracks.

Fracture Parameters for Cracked Cylincal Shells

In this paper, 2D boundary element stress analysis is carried out to obtain the T-stress for multiple internal edge cracks in thick-walled cylinders for a wide range of cylinder radius ratios and relative crack depth. The T-stress, together with the stress intensity factor K, provides amore reliable two-parameter prediction of fracture in linear elastic fracture mechanics. T-stress weight functions are then derived from the T-stress solutions for two reference load conditions corresponding to the cases when the cracked cylinder is subject to a uniform and to a linear applied stress variation on the crack faces. The derived weight functions are then verified for several non-linear load conditions. Using the BEM results as reference T-stress solutions; the T-stress weight functions for thick-walled cylinder have also been derived. Excellent agreements between the BEM results and weight function predictions are obtained. The weight functions derived are suitable for obtaining T-stress solutions for the corresponding cracked thick-walled cylinder under any complex stress fields. Results of the study show that the two dimensional BEM analysis, together with weight function method, can be used to provide a quick and accurate estimate of T-stress for 2-D crack problems.

Stress analysis in a sheet with multiple cracks

• Three- dimensional formulation of elasticity is used to derive dislocation solution in an elastic sheet. • Integral equations are constructed for multiple interacting cracks. • Numerical solution for the integral equations are carried out for a single and two Interacting cracks. • Contrary to two-dimensional problems modes II and III fracture occur simultaneously. • The stress intensity factors vanishes at a crack corners. The stress field due to the presence of a Volterra dislocation in an isotropic elastic sheet is obtained. The stress components exhibit the familiar Cauchy type singularity at dislocation location. The solution is utilized to construct integral equations for elastic sheets weakened by multiple embedded or edge cracks. The cracks are perpendicular to the sheet boundary and applied traction is such that crack closing may not occur. The integral equations are solved numerically and stress intensity factors (SIFs) are determined on a crack edges.

Magneto-electro-elastic analysis of a strip containing multiple embedded and edge cracks under transient loading

This paper deals with the dynamic behavior of a magneto-electro-elastic strip weakened by multiple horizontal, vertical, and edge cracks within the framework of linear magneto-electro-elasticity. The analysis is based on stress and the magneto-electrical fields caused by horizontal and vertical Volterra-type screw dislocation in a medium. The problem was formulated through Fourier and Laplace transforms into singular integral equations in which the unknown variables are the jumps of displacement and magneto-electrical potential across the crack surface. The dislocation densities and the numerical Laplace inversion are then employed in order to derive the dynamic field intensity factors at the crack tips for both permeable and impermeable cracks. The effects of length and position of the cracks on the dynamic field intensity factors and interaction between the two cracks are investigated. Furthermore, the results show that, for a fixed value of mechanical load, the dynamic field intensity factor at the crack tips depends on the magnitude and direction of the applied magneto-electrical load.

A transfer matrix method for in-plane bending vibrations of tapered beams with axial force and multiple edge cracks

This paper proposes a transfer matrix method for the bending vibration of two types of tapered beams subjected to axial force, and it is applied to analyze tapered beams with an edge or multiple edge open cracks. One beam type is assumed to be reduced linearly in the cross-section height along the beam length. The other type is a tapered beam in which the cross-section height and width with the same taper ratio is linearly reduced simultaneously. Each crack is modeled as two sub-elements connected by a rotational spring, and the method can evaluate the effect of cracking on the desired number of eigenfrequencies using a minimum number of subdivisions. Among the power series available for the solutions, the roots of the differential equation are computed using the Frobenius method. The computed results confirm the accuracy of the method and are compared with previously reported results. The effectiveness of the proposed methods is demonstrated by examining specific examples, and the effects of cracking and axial loading are carefully examined by a comparison of the single and double tapered beam results.