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.     

The continuous crack flexibility model for crack identification

The presence of a crack in a structural member introduces a local flexibility that affects its dynamic response. Moreover, the crack will open and close in time depending on the loading conditions and vibration amplitude. The changes in dynamic characteristics can be measured and lead to an identification of the structural changes which eventually might lead to the detection of a structural flaw. The results of various independent evaluations of changes in the natural frequency of vibrations of cracked structural elements are reported. A crack model of a continuous flexibility, found with fracture mechanics methods using the displacement field in the vicinity of the crack developed recently is used here. The analytical results for the cracked elements behaviour based on the continuous crack flexibility vibration theory were correlated with numerical solutions, the lumped-crack beam vibration analysis and experimental results obtained on aluminium and steel beams with open cracks.     

Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load

An analytical approach for evaluating the forced vibration response of uniform beams with an arbitrary number of open edge cracks excited by a concentrated moving load is developed in this research. For this purpose, the cracked beam is modeled using beam segments connected by rotational massless linear elastic springs with sectional flexibility, and each segment of the continuous beam is assumed to satisfy Timoshenko beam theory. In this method, the equivalent spring stiffness does not depend on the frequency of vibration and is obtained from fracture mechanics. Considering suitable compatibility requirements at cracked sections and corresponding boundary conditions, characteristic equations of free vibration response are derived. Then, forced vibration response is treated under a moving load with a constant velocity. Using the determined eigenfunctions, the forced vibration response may be obtained by the modal superposition method. Finally, some parametric studies are presented to show the effects of crack parameters and moving load velocity.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution

Based on the concepts of linear fracture mechanics, we derive analytical expressions for the determinational of natural frequencies of longitudinal and bending vibrations of beams which are rectangular in cross section, are fixed in different ways, have variable ratios of the section height to the beam length, and have transverse cracks of various types. The results of the analytical solution are compared with those obtained by the finite-element method as well as with experimental data obtained by the authors and other scientists. The analytical solution under consideration is shown to be quite simple and provide a fairly good accuracy of the results obtained. Using a cantilever beam with one or two symmetrical edge cracks or a central through crack as an example, we consider the possible dependence of the relative change in natural frequencies of vibration on the relative crack length, crack location, and the vibration mode of a beam. We discuss the possible methods of evaluating the crack size and location from the results of experimental determination of the change in natural vibration frequencies of a cracked beam. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 19–31, July–August, 1999.     

Free vibration analysis of uniform and stepped cracked beams with circular cross sections

This paper presents a novel numerical technique applicable to analyse the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse non-propagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. Modal characteristics of a cracked beam can be employed in the crack recognition process. The outcomes of the study verified that presented method is appropriate for the vibration analysis of uniform and stepped cracked beams with circular cross section.     

An investigation into the eigen-nature of cracked composite beams

It is known that the presence of cracks in composite structures introduces local flexibility associated with the changes in the dynamic characteristics of composite structures. However the nature and variations of the natural frequencies due to the presence of cracks, are still under discussion and analysis.

The present work introduces an attempt to study the variations in the eigen-nature of cracked composite beams due to different crack depths and locations. A numerical and experimental investigation has been made. The numerical finite element technique is utilized to compute the eigen pairs of laminated composite beams through several state of cracks. The model is based on elastic-plastic fracture mechanics techniques in order to consider the crack tip plasticity in the analysis. A finite element model has been developed to formulate the stiffness matrices for single edge cracked structural elements using transfer matrix theory. These matrices take into account the effects of axial, flexural and shear deformations due to crack presence. The present model has been applied to investigate the effects of state of crack, lamina code number, boundary condition on the dynamic behavior of composite beams.

The experimental tests and frequency response spectrums (FRS) is displayed on [FFT] analyzer. In experimental work the eigen pairs versus several state of cracks with various code number are measured using inductive hammering technique. The results show that the changes of the eigen parameters provide a proper indicator for detection and predication the current state of crack.     

Evaluation of a XVIII Century Steel Bar in the Cathedral of Mexico

A cracked segment from a XVIII century steel bar was removed from the Cathedral of Mexico City for analysis in hopes of determining the source of the bar, its manufacturing process and the significance of several cracks. The historical and experimental analysis showed that the bar was manufactured by a puddling process typical of English steels and one crack was caused by lack of fusion when small bars were forge welded to form the bar used in the Cathedral and another crack was a fracture that also accompanied the forge welding process. The cracks developed during manufacture and were not the result of in-service degradation.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

The improved D-M model solutions for single edge cracked plates by considering the yielding at the back side

The method of calculating the stress intensity factor (SIF) and the crack opening displacement (COD) for double edge cracks in plates under arbitrary loadings that results in solving a system of Cauchy-type singular integral equations is presented. The improved D-M model is then constructed for edge cracked plates by considering the yielding at the back side. For the cases of tension and bending, the plastic zone sizes and the crack opening displacements are calculated from the improved model solution, and the envelopes for the beginning of backside yielding and ligament yielding are obtained. The numerical results are compared with known solutions which take no account of the yielding at the back side and with experimental results.     

A closed-form solution to the equivalent through-crack model for surface cracks

Summary A closed-form solution for plates containing surface cracks is obtained by using an equivalent through-crack model, which reduces the three-dimensional problem of the surface cracked plates to a two-dimensional Hilbert problem. The effect of surface crack is replaced by a continuous distribution of forcesN(x, 0) and momentsM(x, 0) applied along the crack face of the equivalent through-crack model. A convenient form of expressing these forces and moments is by using power polynomials. Then the singular integral equation, expressing the solution of the Hilbert problem can be readily integrated.According to this model we assumed that the crack depth at extremities, where the crack intersects the free surface of the plate, is not zero. This assumption, corroborated by experimental evidence, means that a singularity exists at the extremities of the crack. The experimental evidence was achieved by using the method of caustics and photoelasticity. The results of the evaluation of the stress intensity factor for elliptical cracks were compared well with the solutions of the respective 3-D problem solved by applying the finite-element method, the line-spring model based on the Reissner plate theory, the finite-element alternating method and the benchmark estimate. The distribution of the stress intensity factor along the crack lips, as calculated in this paper, was dropping off rapidly in the surface layer and it was very close to the results given by the approximate 3-D theory.With 7 Figures     

Numerical analysis of brittle materials fractured by sharp indenters

Indentation can be used to determine the fracture properties of materials. A detailed investigation is here presented on the reliability of finite element simulations of sharp indentations on cracked specimens. Elastic analyses of the stress and deformation fields arising from Vickers, Berkovich and cube-corner indenters generated the stress intensity factor along the edge of penny-shaped or elliptical cracks. Various materials with a range of properties were analysed and the results compared with published experimental data. Additional measurements from tests on soda-lime glass provided further opportunities for experimental validation of numerical predictions. The variation of the stress intensity factor indicated trends for crack growth patterns, which were consistent with experimental observations. Particular attention was given to the evaluation of the geometry-dependent parameter appearing in the relation yielding the fracture toughness of cracked indented materials. The numerical predictions of this parameter were remarkably consistent with experimental data and results from other approximate methods.     

Fatigue crack propagation characteristics of high‐tensile steel wires for bridge cables

The viability of single edge cracked sheet test method for rapidly determining the crack propagation characteristics of steel wires was investigated. First, fatigue tests under 3 different stress ratios were conducted on the sheet specimens which were manufactured from a kind of widely used cable wires. The test data were analysed, and the crack growth rates of sheet specimens were constructed by Walker model. Then, a series of fatigue tests were performed on notched round‐bar specimens to verify the predictability of Walker model parameters. Moreover, the experimental results obtained in different studies on crack propagation characteristics of steel wires were discussed. The results show that the crack propagation characteristics of sheet specimens behave a certain dependence on depth. The sheet crack growth laws can be well used to predict the fatigue life of notched bar specimens when the mechanical heterogeneity is considered. For bridge cable steels, the rational values for the exponent parameter of Paris law, m, should be close to 3.     

贯穿裂纹管局部柔度系数的理论与试验研究

根据线性断裂力学理论和应变能释放原理,推导了考虑裂纹张开状态的圆周贯穿裂纹管的附加局部柔度方程,利用自适应的Simpson数值积分方法计算裂纹引起的局部柔度系数,并根据得到的局部柔度系数,建立了以扭转弹簧模拟裂纹行为的裂纹管模型。进行了裂纹管的动力模型试验,通过裂纹管模型固有频率的试验值与理论值对比分析验证局部柔度系数的正确性。研究结果表明:裂纹管固有频率的试验值与使用"弹簧铰"裂纹管模型求得的理论值基本吻合,验证了贯穿裂纹管局部柔度系数的合理性。     

On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 2. Experimental and calculation results

The results of experimental investigation of the relationship between the natural frequency of the first mode bending and longitudinal vibrations of titanium alloy and alloyed steel cantilever beams and the crack parameters (crack depth and location) are presented together with the data on calculation of the relative change of the vibration frequency of cracked beams, which are obtained by the proposed analytical approaches. Based on a comparative analysis, we choose fairly simple and sufficiently accurate formulas for calculating the frequencies of bending and longitudinal vibrations of a cantilever beam with an open or closing crack. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 45–53, May–June, 1999.     

Effect of Processing Parameters on Edge Cracking in Cold Rolling

Experimental and numerical investigations have been carried out to study the effect of processing parameters on edge cracks in cold rolling process of silicon steels. The onset of edge cracks has been simulated by using finite element method (FEM) based on the Gurson–Tvergaard–Needleman damage model. Factorial design and Taguchi experimental design methods have been proposed to analyze the effects of reduction ratio, tension, work roll diameter, and friction coefficient on the formation of edge cracks. Parametric studies indicated that with the increasing of reduction ratio, friction coefficient, and tension, the crack length increases. A bigger work roll is beneficial to restrict the edge crack growth as well. The rank according to the impact degrees on the crack initiation from high to low is reduction ratio, tension, work roll diameter, and friction coefficient. Meanwhile, the combined effect curve of reduction ratio and tension on crack length was obtained which was validated by the experimental results as well.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

A contour integral computation of stress intensity factors in the cracked orthotropic elastic plates

Stress intensity factors in the cracked orthotropic elastic plates are obtained numerically using a reciprocal work contour integral technique. The integral representation for calculating the intensities of elastic singular stresses at cracks, notches and corners can be extended to treat the rectilinearly anisotropic crack problems. An extension of the method can be done immediately with identification of the characteristic singular solutions and construction of the corresponding complementary elastic states. A centrally cracked tension plate with various degree of orthotropy is analyzed to assess accuracy of the computational procedures newly adopted. Two cracked orthotropic plates modelling bidirectional fiber-reinforced composites are treated: symmetric Mode I type of double edge crack tension plate and mixed mode type of cantilever plate with a single edge crack.     

复合材料正交层板基体裂纹预测

将断裂力学的能量释放率概念及A.S.D.Wang在90°层横向裂纹情况的方法推广到层板后期复杂基体裂纹情况,对[O₂/90n]s层板内的0°层纵向劈裂、0°—90°界面的两种脱层等裂纹形式进行三维有限元分析并得出其应变能释放率与裂纹尺寸的关系曲线,对每种基体开裂给出载荷的理论预测值,与实验结果比较,吻合较好。      

Tension of a cylindrical bar having an infinite row of circumferential cracks

In this paper, the crack problems in the case of a cylindrical bar having a circumferential crack and a cylindrical bar having an infinite row of circumferential cracks under tension are analyzed by the body force method. The stress field for a periodic array of ring forces in an infinite body is used to solve the problems. The solution is obtained by superposing the stress fields of ring forces in order to satisfy a given boundary condition. The stress intensity factors are calculated for various geometrical conditions. The obtained values of stress intensity factor of a single circumferential crack are considered to be more reliable than the results of other paper's. As the crack becomes very shallow, the stress intensity factor of a row of circumferential cracks approaches the value corresponding to that of a row of edge cracks in a semi-infinite plate under tension. As the crack becomes very deep, it approaches the values corresponding to that of a single deep circumferential crack.     

Effect of structural geometry and crack location on crack driving forces for cracks in welds

This paper quantifies the effect of geometry (planar or cylindrical) and crack location (internal or edge cracks; weld center or interface cracks) on crack driving force for welded joints, via systematic elastic-creep and elastic-plastic finite element (FE) analyses for welded joints. For engineering estimates of crack driving forces for mismatched welded joints, the equivalent material approach is employed. It is found that the equivalent material concept works very well only for a planar geometry with an internal crack, such as the middle cracked tension specimen. For a planar geometry with an edge crack, it works reasonably well, but tends to provide conservative results for under-matching and for interface cracks. For a cylindrical geometry with an edge crack, the results are similar to those for a planar geometry with an edge crack, but caution should be exercised for over-matching, as non-conservative estimates are possible due to gross-section yielding. For a cylindrical geometry with an internal crack, excessively conservative estimates for under-matching are found, and thus an improved estimation method is desired.