Failure modes of specimens containing surface flaws under cyclic torsion

Turbine-generator shafts are often subjected to a complex transient torsional loading. Such a torsional loading may initiate yielding at the outer radius of the shaft or in the fillets. The methods for predicting turbine-shaft fatigue life due to transient loading depent upon the mode of crack growth from an undetected crack. The most common location for the existence of a crack is the fillets or shoulders of the shaft. Specimens were designed from AISI 4340 steel with two diametrically opposed flat surfaces. Initial defect orientations of 0 deg, 45 deg and 90 deg with respect to the sepcimen axis on the fillet were studied. The specimens were subjected to cyclic torsion with zero mean torque and with a torque amplitude necessary to cause yielding at the outer radius of the specimen. When initial defects were aligned with a plane of maximum shear stress (0 deg and 90 deg), the cracks propagated along that plane. For 45-deg defects (aligned to a plane of maximum tensile stress) the crack still propagated along the plane of maximum shear. However, the number of cycles to initiate and to propagate the crack to failure for 45-deg defects were (two to three times) larger than those for 0-deg and 90-deg defects. Mode II and Mode III crack-growth rates were measured from specimens containing 0-deg and 90-deg defects. It was found that the crack-growth rate in Mode II was higher than in Mode III. However, all the specimens failed due to reduction of the net cross section, mostly attributed to Mode III crack growth. Similar results were obtained from specimens of turbine-shaft material (A469 steel), and 2024 aluminum with different rolling directions. Fatigue-crack-growth rates in Mode III were measured from circumferentially notched bar. They were found to be a unique function of ΔK _III alternating stress intensity in Mode III. It was found that the mechanism of crack growth is produced by the formation and linkage of elongated cavities at the crack tip.     

Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.     

Modeling of fatigue crack growth from a notch

When a fatigue crack is nucleated and propagates into the vicinity of the notch, the crack growth rate is generally higher than that can be expected by using the stress intensity factor concept. The current study attempted to describe the crack growth at notches quantitatively with a detailed consideration of the cyclic plasticity of the material. An elastic–plastic finite element analysis was conducted to obtain the stress and strain histories of the notched component. A single multiaxial fatigue criterion was used to determine the crack initiation from the notch and the subsequent crack growth. Round compact specimens made of 1070 steel were subjected to Mode I cyclic loading with different R-ratios at room temperature. The approach developed was able to quantitatively capture the crack growth behavior near the notch. When the R-ratio was positive, the crack growth near a notch was mainly influenced by the plasticity created by the notch and the resulted fatigue damage during crack initiation. When the R-ratio was negative, the contact of the cracked surfaces during a part of a loading cycle reduced the cyclic plasticity of the material near the crack tip. The combined effect of notch plasticity and possible contact of cracked surface were responsible for the observed crack growth phenomenon near a notch.     

Torsion of a thick-walled cylinder with an external crack: boundary collocation method

The boundary collocation method is used to obtain the torsional rigidity and Mode III stress intensity factor of a thick-walled cylinder with an external radial crack. When the internal radius of the cylinder is very small, the results agree well with those obtained previously from other methods for an edge crack in a solid cylindrical bar. The present method is shown to be expedient when applied to obtain results for different ratios of the internal and external radii of the cracked cylinder.     

Dynamic instability and steady-state response of an elliptical cracked shaft

An elliptical front crack has been found to be more accurate and realistic for modeling the transverse surface crack in rotating machinery compared with the widely used straight front crack. When the shaft rotates, the elliptical crack opens and closes alternatively, due to gravity, and thus, a “breathing effect” occurs. This variance in shaft stiffness is time-periodic, and hence, a parametrically excited system is expected. Therefore, the dynamic instability and steady-state response of a rotating shaft containing an elliptical front crack are studied in the paper. The local flexibility due to the crack is derived, and the governing equations of the crack shaft system are established using the assumed modes method. Utilizing the Bolotin’s method and harmonic balance method, the boundaries of two typical instability regions and maximum response amplitude of the cracked shaft could be computed numerically. The elliptical crack parameters (depth, shape factor and position) and damping are, respectively, considered and discussed for their effects on the dynamic behavior of the elliptical cracked shaft. Some research results might be helpful for the crack detection in rotating machinery.     

Impact failure prediction of Mode III crack in orthotropic functionally graded strip

Mode III impact of a crack in an orthotropic functionally graded strip is investigated. The shear moduli in two directions of the material are assumed to vary proportionately with gradient. Laplace transform and Fourier cosine transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Energy density factor criterion is applied to obtain the maximum of minimum energy density and direction of crack initiation. Numerical results are given graphically. The effects of orthotropy, nonhomogeneity and height of the strip on the energy density factor are discussed.     

Free torsional vibration of cracked nanobeams incorporating surface energy effects

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

Finite plane and anti-plane elastostatic fields with discontinuous deformation gradients near the tip of a crack

In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains.The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures.The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The materials is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations.for both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves (elastostatic shocks) issuing from the crack-tip across which displacement gradients and stresses are discontinuous.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research.     

Stress and strain field near tip of mode III growing crack in materials with creep behavior

Using a proposed constitutive relation for materials with creep behavior, the stress and strain distribution near the tip of a Mode III growing crack is examined. Asymptotic equations of the crack tip field are derived and solved numerically. The stresses remain finite at the crack tip. Obtained qualitatively is the crack tip velocity and the local autonomy of the near tip field solution is discussed.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio η introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.     

Arrest characteristics of an impacted crack dissipating plastic energy

When a structural member is accidentally struck, an initial defect may grow and then arrest. An estimate of its size increase is made by considering the geometry of a centrally cracked panel subjected to a step function load in time. Two load amplitudes differing by a factor of five are considered. Under impact, the crack accelerates and then decelerates prior to arrest. The dynamic characteristics depend on the interaction of the elastic-plastic stress waves intervening with the physical boundaries. This effect is assessed quantitatively by computing for the energy stored in a unit volume of material and by incorporating sliding nodes in the finite element method. The energy dissipated by plastic deformation must be accounted for as it is no longer available for creating new macrocrack surface. Obtained are the near tip normal stresses that are found to change from compression to tension. Their magnitude is considerably larger than the corresponding static values. Increase in crack length changes from 0.87% to 20.6% when the magnitude of the impact load is raised five times. The rate of change of the strain energy density factor ΔS with crack growth Δa is found to be governed by the condition ΔS/Δa = const. during loading while dynamic relaxation corresponded to a nonlinear behavior. The physical implication of this remains to be clarified in view of the fact that plasticity theory may not adequately explain the near tip crack behavior.     

Mode II dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

An asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic—perfectly-plastic solid (plane strain). It is shown that, like for Modes I and III (Gao and Nemat-Nasser, 1983), the complete dynamic solution for Mode II predicts a logarithmic singularity for the strain field, but unlike for those modes which involve no elastic unloading, the pure Mode II solution includes two elastic sectors next to the stress-free crack surfaces. This is in contradiction to the quasi-static solution which predicts a small central plastic zone, followed by two large elastic zones, and then two very small plastic zones adjacent to the stress-free crack faces. The stress field for the complete dynamic solution varies throughout the entire crack tip neighborhood, admitting finite jumps at two shock fronts within the central plastic sector. This dynamic stress field is consistent with that of the stationary crack solution, and indeed reduces to it as the crack growth speed becomes zero.     

Effect of strain rate on crack growth in aluminum alloy 1100-0

Stress and damage analysis are performed to analyze the Mode I crack growth behavior of a central crack panel made of aluminum alloy 1100-0. On account of the highly nonhomogeneous stress state, each material element would experience a different strain rate depending on the location and loading rate. A data bank of uniaxial stress and strain curves is provided to cover the range local strain rates depending on the load time history. Such a approach is referred to as the strain rate dependent model in contrast to plasticity that utilizes a single constitutive relation.The strain energy density criterion is applied to determine the onset of crack initiation, stable crack growth and final termination. A unique feature of the approach is that the same criterion could describe the foregoing three distinct events of fracture behavior. Results are obtained for applied loads with different strain rates and compared with those obtained from the classical theory of plasticity, which is unconservative.     

Strain energy density prediction of crack initiation and direction in cracked T-beams and pipes

In this paper, the S-theory is applied to determine crack initiation and direction for cracked T-beams and circumferentially cracked pipes. It makes use of a parameter called strain energy density factor, S, which is a function of the stress intensity factors. The strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions in beams and pipes are studied.     

压电体中考虑表面效应时开裂孔洞的断裂特征

研究了反平面机械载荷和面内电载荷作用下压电体中考虑表面效应时孔边双裂纹问题的断裂特征。基于Gurtin-Murdoch表面理论模型,通过构造映射函数,利用复势电弹理论获得了应力场和电位移场的闭合解答。给出了裂纹尖端应力强度因子、电位移场强因子和能量释放率的解析解。讨论了开裂孔洞几何参数和施加力电载荷对电弹场强因子和能量释放率的影响。     

Mode III fracture problem of an arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane

This paper studies the Mode III electric-elastic field of a cracked functionally graded piezoelectric strip bonded to a homogeneous piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties of the strip vary along the strip thickness in exponential forms. By using the Fourier transform, the problem can be formulated to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects come from the edge, crack orientations and the nonhomogeneous material parameter on intensity factors are discussed graphically.     

Free-edge stresses and progressive cracking in surface coatings of circular torsion bars

This paper considers the explicit solutions of free-edge stresses near circumferential cracks in surface coatings of circular torsion bars and their application in determining the progressive cracking density in the coating layers. The problem was formulated within the framework of linear elastic fracture mechanics (LEFM). The free-edge stresses near crack tip and the shear stresses in the cross-section of the torsion bar were approached in explicit forms based on the variational principle of complementary strain energy. Criterion for progressive cracking in the coating layer was established in sense of strain energy conservation, and the crack density is thereby estimated. Effects of external torque, aspect ratio, and elastic properties on the density of progressive cracking were examined numerically. The present study shows that, in the sense of inducing a given crack density, compliant coating layer with lower modulus has much higher critical torque than that of a stiffer one with the same geometries and substrate material, i.e., compliant coating layer has greater cracking tolerance. Meanwhile, the study also indicates that thicker surface coating layer is more pliant to cracking than the thinner ones. The present model can be used for analyzing the damage mechanism and cracking tolerance of surface coatings of torsion shafts and for data reduction of torsional fracture test of brittle surface coatings, etc.     

Propagating semi-infinite crack in fluid-saturated porous medium of finite height

Derived in this work are the Mode I stress intensity factor results for a constant velocity semi-infinite crack moving in a fluid-saturated porous medium with finite height. Two limiting cases are discussed; they correspond to a low and high speed crack propagation. To be expected is that the crack front stress intensification would increase as the medium height is reduced in relation to the segment length in which mechanical pressure is applied. Moreover, the stress intensity factor for the high speed crack is larger than the low speed crack, the magnification of which depends on the material. Dissatisfaction of the crack surface and tip boundary condition is found in the present solution which calls possibly for the additional consideration of a local boundary layer as discussed by other authors.