Cracked Bar Research Articles

Failure analysis of brass rotor bars

Cracked rotor bars with characteristic chemical composition of 90Cu–10Zn are investigated in this paper. A brazing technique was employed to join these rotor bars with rings in a motor assembly factory. However, frequent edge cracks were observed after this joining process. Intergranular facets of these cracked bars revealed brittle fracture. Mechanical tests of the bars showed lack of ductility and strength, and the bars were found to have a higher lead (Pb) content. In addition, the season cracking susceptibility of copper alloy that can result in intergranular brittle fracture was assessed. The metallurgical structure revealed a very large grain size (150 μm) in the copper alloy. Non-uniform thermal stresses in the heating process triggered cracking of the bars owing to an insufficiency of material strength. The large grain size accounted for the failures.

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.

Sandwiched‐Beam Procedure for Precracking Brittle Materials

A technique for precracking brittle materials is presented. This procedure, which is called the sandwiched‐beam (SB) technique, allows the production of sharp through‐thickness cracks with predetermined length in specimens with a rectangular section. A bar, in which an initial notch is produced by using a conventional saw, is inserted between two supporting beams and the sandwich assembly is loaded in three‐point bending. Conditions can be defined that allow the stable propagation of a sharp flaw from the notch as the applied load is increased. Then, the cracked bar can be used to determine the fracture toughness. The SB technique is applied to different brittle materials, including soda‐lime‐silica glass, alumina, Si3N4, a SiCw‐Si3N4 composite, graphite, a Ti‐Al intermetallic, and Carrara marble.

Longitudinal vibration of a continuous cracked bar

A continuous cracked bar vibration theory is developed for longitudinal vibration of rods with an edge crack. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked bar as a one-dimensional continuum. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack found with fracture mechanics methods. The results of three independent evaluations of the lowest natural frequency of longitudinal vibrations of a bar with a single edge crack are presented: the continuous cracked bar vibration theory, the lumped crack bar vibration analysis, and experimental results obtained on aluminum bars with fatigue cracks. Experimental results fall between the values predicted by the two analytical methods. Moreover, the continuous bar theory agrees better with the experimental results than the lumped crack flexibility theory for small cracks. For larger cracks, a/ h>0.4, experimentation was difficult due to the co-existence of several coupled modes and no reliable results could be obtained.

Longitudinal vibration of a bar with a breathing crack

The dynamics of a cracked fixed-free bar with a breathing crack in longitudinal vibration is investigated. The Hu–Washizu–Barr variational formulation was used to develop the equation of motion and the boundary conditions of the cracked bar as a one-dimensional continuum. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack found with fracture mechanics methods. The eigenfrequency changes due to a single open-edge breathing crack, are shown to depend on the bilinear character of the system. The associated linear problems are solved over their respective domain of definition and then the solutions are matched through the initial conditions. These changes are smaller than the ones caused by open cracks. The method has been tested for different bar configurations corresponding to crack location, crack depths, cross-section dimensions, and Poisson’s ratio. The natural frequencies obtained from this model agree well with experimental results.

Dual boundary element analysis for cracked bars under torsion

A dual integral formulation for a cracked bar under torsion is derived, and a dual boundary element method is implemented. It is shown that as the thickness of the crack becomes thinner, the ill‐posedness for the linear algebraic matrix becomes more serious if the conventional BEM is used. Numerical experiments for solution instability due to ill‐posedness are shown. To deal with this difficulty, the hypersingular equation of the dual boundary integral formulation is employed to obtain an independent constraint equation for the boundary unknowns. For the sake of computational efficiency, the area integral for the torsion rigidity is transformed into two boundary integrals by using Green’s second identity and divergence theorem. Finally, the torsion rigidities for cracks with different lengths and orientations are solved by using the dual BEM, and the results compare well with the analytical solutions and FEM results.

Vibration of a Cracked Cantilever Beam

A continuous cracked bar vibration model is developed for the lateral vibration of a cracked Euler-Bernoulli cantilevered beam with an edge crack. The Hu-Washizu-Barr variational formulation was used to develop the differential equation and the boundary conditions for the cracked beam as an one-dimensional continuum. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack found with fracture mechanics methods. The results of three independent evaluations of the lowest natural frequency of lateral vibrations of an aluminum cantilever beam with a single-edge crack are presented: the continuous cracked beam vibration model, the lumped crack model vibration analysis, and experimental results. Experimental results fall very close to the values predicted by the continuous crack formulation. Moreover, the continuous cracked beam theory agrees better with the experimental results than the lumped crack flexibility theory.

DETECTION OF FATIGUE CRACKS IN FLEXIBLE GEOMETRICALLY NON-LINEAR BARS BY VIBRATION MONITORING

The results of mathematical simulation of flexural vibrations of a geometrically non-linear cracked bar under external harmonic excitation are presented. It is shown, that thanks to the influence of elastic non-linearity of the crack, new non-linear properties which are impossible in the initial undamaged structure, appear in the system (self-excitation of subharmonic regimes, appearance of even-numbered harmonic components in frequency spectrum of 3/1 order superharmonic regime). By the utilization of these non-linear effects a new method for detection of cracks is developed. Unlike traditional procedures, the proposed method permits recognition of the legitimated response to non-linearity of the crack through interfering signals from the geometrical non-linearity of a bar.

A CONSISTENT CRACKED BAR VIBRATION THEORY

A consistent continuous cracked bar vibration theory is developed. The stress and displacement field about the crack was used to modify the stress and displacement field throughout the bar, and reduction to one spatial dimension was achieved by integrating the stress and displacement fields throughout the bar cross-sections so that the total displacement would be exact. The resulting linear differential equation with variable coefficients has the modified displacement field due to the crack imbedded in it. Any number of cracks can be introduced into the differential equation as modifications of the displacement field. A numerical solution and a first order perturbation solution are presented for the prediction of changes in longitudinal vibration natural frequencies of a fixed-free bar with a single open-edge transverse crack. To assess the validity of the assumptions made experiments on aluminium bars with fatigue cracks were performed. The analytical results correlate very closely with experimental results with better correlation than the local flexibility solution.

Vibration of cracked structures: A state of the art review

The presence of a crack in a structural member introduces a local flexibility that affects its vibration response. Moreover, the crack will open and close in time depending on the rotation and vibration amplitude. In this case the system is nonlinear. Furthermore, if general motion is considered, the local stiffness matrix description of the cracked section of the shaft leads to a coupled system, while for an uncracked shaft the system is decoupled. This means that the crack introduces new harmonics in the spectrum. In fact, in addition to the second harmonic of rotation and the subharmonic of the critical speed, two more families of harmonics are observed: 1. (1) higher harmonics of the rotating speed due to the nonlinearity of the closing crack, and 2. (2) longitudinal and torsional harmonics are present in the start-up lateral vibration spectrum due to the coupling. Over 500 papers on the subject were published in the past 10 yrs. A wealth of analytical, numerical and experimental investigations now exists. However, a consistent cracked bar vibration theory is yet to be developed. There are still many unanswered questions, especially in the area of closing cracks in rotating shafts.

Wave propagation and localization in a rate-dependent cracked medium—model formulation and one-dimensional examples

A smeared-crack model is proposed in which the stress after cracking is not only a function of the crack strain (softening function), but also of the crack strain rate. This rate dependence is shown to introduce an internal length scale into initial value problems. As a result, solutions of initial value problems that involve smeared cracking remain well posed. The width of the localization zone. where high strain rates are present, is proven to be independent of the finite element discretization. Convergence upon mesh refinement is also obtained for the consumption of energy in a cracked bar and the extent of wave reflection on a cracked zone. This is because in a rate-dependent medium, waves are dispersive and propagate with a real wave speed. Numerical results have been obtained for one-dimensional localization problems to substantiate these analytical findings.

Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending

The strain energy release rate for a straight-fronted edge crack in a bar of circular cross section subjected to pure bending is determined. The cracked bar is modelled with two-dimensional plane-stress finite elements and strain energy release rates, determined from this model, are shown to be in close agreement with existing results for a bar subjected to three-point bending in which strain energy release rates were determined by measuring the compliance of the bar experimentally. The strain energy release rates for a crack in the circular cross section bar are found to be lower than those in a rectangular cross section bar having the same relative crack length and subjected to the same bending moment. Previously determined results for uniform tension are superimposed to obtain strain energy release rates for a circular cross section bar which is subjected simultaneously to a tensile load and a bending moment.

Third stress intensity factors of torsion crack bar with ring sections

Torsion crack problems of a bar with ring sections are discussed. The procedure utilized consists of the following steps: 1. (1) Using the partitioning plan and conformal mapping technique 2. (2) Using the harmonic function continuation technique 3. (3) Using the compliance method. The single-valued condition must be considered, since the cracked ring section is multiply connected. The procedure used belongs to an indirect one with respect to calculation of K 3 values. Numerical results concerning torsional rigidity and K 3 are shown in Tables 1 and 2.

Fatigue of cold worked ribbed reinforcing bar—a fracture mechanics approach

A theoretical method for the prediction of fatigue life of structural members containing external stress raisers has been developed. The method is based on a linear elastic fracture mechanics approach to fatigue. The theory has been applied to fatigue of Torbar steel in concrete beams. The required parameters for such application were obtained from fatigue tests on samples and finite element analysis of cracked bars. Comparisons between the theory and experimental data taken from other investigations have shown that the theory predicts a reasonable lower limit fatigue life of Torbar in concrete beams. The theory has also successfully predicted the effect of the minimum stress on the fatigue life of this type of bar.

A boundary collocation technique for evaluating the torsional rigidity and third stress intensity factors of cracked bars

A boundary collocation technique for evaluating the torsional rigidity and third stress intensity factors of cracked bars

Requirements for a one parameter characterization of crack tip fields by the HRR singularity

Detailed crack tip stress and strain fields are generated for the edge crack bar subjected to bending and the center cracked panel and single cracked panel subjected to tensile loads. These fields are compared with the singular fields, due to Hutchinson and Rice and Rosengren, at the same level of applied J. The comparisons show that the size R of the region at the crack tip dominated by the HRR singularity is substantially larger in the bend specimen than in the center-cracked panel for contained and large scale plasticity. Hardening properties also have a strong influence on R. The implications of these results on the minimum size requirement essential to a one parameter fracture criterion based on the J-integral or crack tip opening displacement are discussed.

Finding elasto-plastic stress and strain in cracked bars under torsion and shear by assembling screw dislocations

A plastic strain distribution in a cracked body is found from a solution of a complex, non-linear, integro-differential equation by successive approximation. The solution is repeatedly corrected to account for the boundary conditions except on the flanks of the crack where the traction-free condition is incorporated analytically in the formulation. In this numerical procedure, the continuum plastic deformation is simulated by an array of discrete screw dislocation dipoles. The dislocations perturb the stress and strain field so as to satisfy the governing equations for deformation plasticity. Results compare favorably with the corresponding analytical solutions.

環状予き裂材のねじりクリープ破断に対する実験的および解析的検討

The creep crack growth behavior in a bar with a circumferential hyperbolic crack was analysed by means of a method proposed in previous papers. This method combines the stress and strain analysis with a generalized creep damage hypothesis applied to the point located ahead of the current crack by a characteristic length ρs. The effect of several parameters on the creep crack growth behavior was discussed. In parallel with this analysis creep rupture tests of deeply precracked round bars, which simulate the bars with a hyperbolic crack, were carried out on S15CK low carbon steel at 450°C.The results obtained are summarized as follows:(1) The analytical predictions as to the creep rupture lives of deeply cracked bars agreed well with the experimental data.(2) It was reconfirmed by the experimental data that notch-weakening prevails under torsional creep in the absence of metallurgical strengthening mechanisms.(3) It was found that rupture life of a precracked specimen can be successfully correlated with the initial value of modified J-integral, which is a measure of stress and strain fields in the vicinity of a precrack under the steady-state creep conditions.

A Direct Numerical Method for Stress and Stress-Intensity Factor in Arbitrary, Cracked, Elastic Bars Under Torsion and Longitudinal Shear

A numerical method is proposed for a direct local determination of stresses in a general twisted and sheared cracked bar. Conceptually, the method employs the integral approach of the potential theory but is modified to yield the stress components without numerical differentiation or interpolation from a potential function. A general expression for the stress-intensity factor at the crack tip is formulated. It checks favorably with the corresponding theoretical solutions. Computer running time and storage requirements for typical problems indicate the comparative efficiency of the method in addition to its relatively high accuracy near the crack tip.

On the Fatigue Strength of Press-Fitted Assemblies

In press-fitting an axle into the hub of a wheel the part of the axle not yet in the hub must bear the total pressing force, while in the hub, the force is distributed along the fitting surface. As a result the amount of the axial contraction of the axle, due to pressing force, becomes different from that of the bore of the hub and, consequently, when the force is removed, large axial tensile stress will be induced on the surface of the axle near the portion just inside of the hub face. This stress will lower the fatigue strength of press-fitted assemblies. In this paper the effect of the stress stated above on the fatigue strength of press-fitted bars has been confirmed by calculating the said stress, as well as by measuring the fatigue strength of cracked bars with fillet or with a press-fitted part.