Optimum strain gage location for evaluating stress intensity factors in single and double ended cracked configurations

A general methodology has been proposed for calculation of optimum radial strain gage locations for measurement of the stress intensity factors using strain gage technique. The upper bound (r_max) of strain gage locations for complex single ended and double ended cracked configurations has been determined using the proposed method. Further, dependency of the r_max on the crack length to width ratio and on the state of stress is investigated. Numerical results obtained from the present investigation are observed to be in accordance with the theoretical predictions. Using the proposed approach the correctness of strain gage locations used by the earlier researchers is also verified.     

Crack growth in cylindrical aluminum shells with inner reinforcing foam layer

The occurrence of cracks in aging aircraft fuselage is major problem in the airline industry. The remaining life of the aircraft is strongly dependent on the residual strength of its structure. Residual strength is affected by crack sizes and their growth rates. In the case of a longitudinal crack in a pressurized cylinder (as in the case of an aircraft fuselage), the geometry and loading conditions cause the edges of the crack to bulge out generating a complex stress field around the crack tips; this is known as the ‘bulging effect’. The geometry of the shell, crack size and pressure contribute to this phenomenon. A proposed solution to reduce the effect of bulging for this type of crack is to apply a layer of polyisocyanurate (PIR) foam to the inner side of the fuselage near the crack site. This layer will bond to the shell and has the effect of reducing the bulge and consequently, the Stress Intensity Factor (SIF) at the crack tips. PIR foam is a lightweight material that adheres well to the shell and provides additional stiffness around the crack area. In the present study the effect of applying a PIR foam layer to a longitudinal crack in a pressurized cylindrical shell is assessed. Nonlinear Finite Element Analysis (FEA) is used in conjunction with the Modified Crack Closure Integral technique (MCCI) in order to evaluate the effect of bulging on the crack’s SIF. Parameters considered in this study include shell radius, shell thickness, crack length, foam thickness and pressure. Numerical results are compared with existing experimental data and the effect of foam thickness for several shell configurations is presented. Results indicate that the bulge factor (BF) could be reduced by as much as 45% depending on shell configuration, foam thickness and pressure.     

Optimum location of a three strain gauge rosette for measuring mixed mode stress intensity factors

This paper analyzes the errors inherent to the determination of mixed mode stress intensity factors from data obtained by using a three strain gauge rosette. The analysis shows that the errors are mainly due the third characteristic value (3/2) and its corresponding coefficients. It is also shown that the errors do not depend on the orientation angle of the rosette, the angle between the strain gauges and the material properties. The error mainly depends on its location (radius, angle), being linear in the radius. For pure mode I, an angle of 90° will completely eliminate the error due to the angle, while for pure mode II, a 0° angle will minimize it. The normalized variation of the errors with the angle at any radius is shown for different ratios of the corresponding coefficients of the third characteristic value. The analytical results are applied to a numerical example of an edge crack subjected to mixed mode loading. From the numerical example, it is recommended to use two strain gauge rosettes at the same angle, and linearly extrapolate their results, if errors less than 15% for a mixed mode field are desired.     

A numerical analysis of cracks emanating from a square hole in a rectangular plate under biaxial loads

This note concerns with stress intensity factors of cracks emanating from a square hole in rectangular plate under biaxial loads by means of the boundary element method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results illustrate that the present approach is very effective and accurate for calculating stress intensity factors of complicated cracks in a finite plate and can reveal the effect of the biaxial load and the cracked body geometry on stress intensity factors.     

Theoretical simulations of a propagating crack subjected to in-plane stress wave loading by caustic method

The optical method of caustics for measuring the dynamic stress intensity factor in a transient process is investigated in this study. The transient full-field solutions of a propagating crack contained in an infinite medium subjected to step-stress wave and ramp-stress wave loadings are used to establish the exact equations of the initial and caustic curves. The results of the stress intensity factor obtained from the caustic method are compared with theoretical predictions and some experiments. The results demonstrate that a significant deviation can occur in the determination of the dynamic stress intensity factor from shadow spot measurements. The factors, such as screen distance, magnitude of loading, crack speed and rising time which can influence the accuracy of the experimental measurements are discussed in detail. In addition, the valid region of the dynamic stress singular field for the propagating crack is discussed in detail and it gives a better understanding of the appropriate region of measurements for investigators. This revised version was published online in July 2006 with corrections to the Cover Date.     

Radial locations of strain gages for accurate measurement of mode I stress intensity factor

Strain gage methods are popular in experimental determination of stress intensity factors (SIFs). Radial location of gages with respect to the crack tip plays an important role in accuracy of strain measurements and thus accurate determination of SIFs. The present work proposes a finite element based simple, accurate and consistent method for determination of the limiting value of the radial distance (r_max) of a strain gage. This parameter is in turn useful in deciding the valid strain gage location for accurate measurement of opening mode SIF. The results obtained from the present investigation agree well with the theoretical predictions and could be used for experimental determination of SIFs for both single ended and double ended cracked specimens. The r_max values of center cracked and edge cracked plates with different crack length to width ratio are estimated. The results of the present investigation show that the relative size of the crack length and net ligament length strongly influences the r_max value and the effect of Poisson’s ratio is marginal on the r_max value.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

Stress intensity factors for surface cracks at countersunk holes

Fatigue crack growth from countersunk fastener holes loaded in remote tensile loading was studied using the transparent polymer PMMA. A single edge corner crack at the bottom of the plate and a single internal surface crack at the sharp intersection between the bore and the countersink were induced in the PMMA specimens by pre-cracking. The specimens were then fatigue tested under constant amplitude remote tensile loading and the ‘back-calculation’ method was used to determine stress intensity factors at several crack front locations. When variations in fatigue crack closure were taken into account, the experimental stress intensity factors agreed well with the computational results at selected crack fronts.     

Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a nonhomogeneous functionally graded interphase was formulated for singular integral equations with Cauchy kernels, which were integrated using the Lobatto–Chebyshev collocation technique. Mixed-Mode Stress Intensity Factors (SIFs) and Strain Energy Release Rates were calculated. The SIFs were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various nonhomogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.     

Analysis of strain rate dependence of tensile elongation for a mechanical milling Al–1.1Mg–1.2Cu alloy tested at 748 K from a dislocation dynamics viewpoint

Tensile deformation was carried out for a mechanically milled and thermo-mechanically treated Al–1.1Mg–1.2Cu (at.%) alloy at 748 K and three nominal strain rates of 10⁻³, 10⁰, and 10² s⁻¹. Despite the prevailing belief that superplasticity occurs by grain boundary sliding which requires slow strain rates at high temperatures, the maximum elongation was observed at the intermediate strain rate of 10⁰ s⁻¹, neither at the lowest nor the highest strain rates. In order to explain this phenomenon, the true stress–true strain behaviors at these three nominal strain rates were analyzed from a viewpoint of dislocation dynamics by computer-simulation with four variables of the thermal stress component σ*, dislocation immobilization rate U, re-mobilization probability of unlocked, immobile dislocations Ω and dislocation density at yielding ρ₀. It can then be concluded that the large elongation (>400% in nominal strain) at the intermediate strain rate is produced by a combination of a very large Ω and a moderate U, resulting in a large strain rate sensitivity m value.     

A quantitative evaluation of fatigue crack shielding forces using photoelasticity

The mechanisms controlling the phenomenon of plasticity-induced shielding during fatigue are investigated and quantified by fitting a recently developed model to photoelastic data. The model derives from the Muskhelishvili approach and includes additional terms to describe the effect of plasticity on the elastic stress field around the crack tip. The photoelastic technique used a polycarbonate CT specimen containing a naturally propagating fatigue crack from which full-field data was obtained digitally using the phase-stepping method. The model was fitted to approximately 1000 data values for the isochromatic fringe order around the crack tip and generated values for the stress intensity factor and T-stress plus an interfacial shear stress intensity factor and a retardation intensity factor which together characterize the influence of plasticity on crack growth.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

The effect of second principal stress on the fatigue propagation of mode I surface crack in Al2O3/Al alloy composites

Using a plate made of A2017-T6 metal matrix composites reinforced with 10 volume % and 20 volume % Al₂O₃ particles and Al alloy possesses the same composition as matrix alloy, the crack propagation rate da/dN of a mode I surface crack by the simultaneous action of plane bending and cyclic torsion are studied. And the effects of crack tip opening stress σ_top, crack opening displacement COD, biaxial stress ratio C (=second principal stress/first principal stress) and the surface roughness of crack section are examined. When stress intensity factor range ΔK is lower than the specific level, da/dN decreases with the increase of volume fraction of Al₂O₃ in C=0 and C=−0.55. But, da/dN of Al alloy becomes minimum in C=−1 and the effect of Al₂O₃ particles disappears. σ_top rises with the increase of volume fraction of Al₂O₃ particles and the decline of C. On the other hand, COD doesn’t always rise with the decline of C. These phenomena can be explained by the residual compressive stress formed at the surface layer of the specimen by the fatigue test and the surface roughness of crack section.     

界面具有垂直裂纹的热障涂层的失效模拟

采用有限元分析软件ANSYS构造了一个在热生长氧化层（TGO）与陶瓷层界面具有一个垂直裂纹的纳米结构热障涂层的有限元模型。并计算了在热震过程中裂纹处的应力分布图,及裂纹尖端的应力场强度因子K1变化图。计算结果表明：裂纹处存在应力集中现象,且裂纹尖端的应力场强度因子K1在热障涂层热循环的冷却过程中随着时间的延长而减小,且在冷却最开始阶段,温度梯度变化最大,K1值也变化最大,裂纹在冷却的初始具有最大的扩展可能性。且涂层最有可能发生开裂失效。     

A modified Kachanov method for analysis of solids with multiple cracks

Kachanov proposed an approximate method for the analysis of multiple cracks by assuming that traction in each crack can be represented as a sum of a uniform component and a non-uniform component, and the interaction among the cracks are only due to the uniform components. These assumptions simplify considerably the mathematics and allow ‘closed-form’ solutions to be obtained for some cases. However, it is noted that the assumptions may not be valid when the cracks are very close. Therefore, an improved method of elastic solids with closely spaced multiple cracks is proposed. Unlike the Kachanov method, traction in a crack is decomposed into a linearly varying component and a non-uniform component so that the sum of the two components to be equal to the traction along the crack length. It is further assumed that the interaction effect due to the non-uniform component can be neglected, and therefore, only the effect of the linearly varying component has to be considered. The accuracy of the present method is validated by comparing the results of two and three collinear open cracks obtained by the present method with those of the exact solutions and the original Kachanov method. Applications of the approach in solving non-collinear parallel crack and friction crack problems are also presented to demonstrate the versatility and accuracy of the method.     

Fracture and fatigue analysis of an agitator shaft with a circumferential notch

This paper is concerned with the fracture analysis of an agitator shaft of a large vessel and predicting its high cycle fatigue life. The agitator shaft has a circumferential notch around it and is subjected to remote bending and torque created by the mixing operation. The problem is comprised (i) the analyses of the bending force and torque acting on the agitator by using the analytical method, (ii) calculation of stress intensity factors under mode I and III loading conditions by using finite element method and, (iii) fatigue analysis of the agitator shaft failed in service.An agitator model is set up and data obtained from the agitator are processed to make more realistic approximations for bending forces, since they form a base for stress analysis, in which mode I stress intensity factors are evaluated. Mode I stress intensity factors obtained by finite element analysis are compared with the results provided by using the body force method.     

Stress intensity factor for center-cracked plate with crack surface interference

This paper presents a simple and physically acceptable analysis of stress intensity factor (SIF) for the center-cracked infinite and finite-width plates. The analysis includes the effect of crack surface interference (i.e., the upper and lower crack surfaces are not allowed to overlap) that influences both the SIF at the tension-side crack tip and the crack opening displacement (COD) profile. For an infinite plate, exact solutions are obtained by superimposing the classical (overlapping) solutions. For a finite-width plate, where the SIF solutions cannot be found in closed form, the solutions are carried out numerically. The overlapping SIF solutions from the weight function method are used. An example is given for the case of a finite-width plate under bending. It was found that the overlapping solutions underestimate the stress intensity factor at the tension-side crack tip up to 15%. The analysis results are also compared with the finite element solutions for verification purpose.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.