Strain energy density prediction of fatigue crack growth from hole of aging aircraft structures

Fatigue crack growth rate depends not only on the load amplitude, but also on the morphology of crack path. The strain energy density theory has the ability to analyze crack growth rate. A strain energy density crack growth model is proposed. It can predict the lifetime of fatigue crack growth for mixed mode cracks while an equation for mode I crack is also obtained. The validity of the model is established with two cases: a center-crack panel and cracks emanating from the edge of a hole. The stress intensity factor expression for the former case is analytical while that of the latter is calculated numerically using finite elements. The results are compared with the testing data. Good agreement shows that the proposed model is useful.     

Mixed mode crack initiation and direction in beams with edge crack

The S-theory is applied to determine crack initiation and direction for beams with edge crack. A simple method for obtaining approximate stress intensity factors of straight cracked beams is also proposed. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. The results are in reasonable agreement with the more accurate calculations.     

Strain energy density criteria for dynamic fracture and dynamic crack branching

Dynamic extension of Sih's fracture criterion based on strain energy density factor, r_c (dW/dV), is used to analyze dynamic crack propagation and branching. Influence of the nonsingular components, which are known as the higher order terms (HOT) in the crack tip stress field, on the strain energy density distribution at a critical distance surrounding the crack tip moving at constant crack velocity is examined. This r_c (dW/dV) fracture criterion is then used to analyze available dynamic photoelastic results of crack branching and of engineering materials.     

Direction of crack growth initiation in roller contact: strain energy density criterion

Small defects or cracks near the surface of roller contact could spread and lead to failure at large. Their growth behavior depends on the rolling load, size and orientation of the initial defects, and material property in addition to friction at the contacting surfaces. Stress intensity factors K₁and K₂ are obtained for three different crack types near the surface between the roller and contacting solid. Various possible directions of crack growth initiation are obtained as the different roller loads are moved relative to the crack. The results are indicative of railway failure observed in service and are helpful to future studies on subcritical and/or critical crack growth.     

Crack energy density in arbitrary direction for piezoelectrics and its characteristic features in electromechanical mixed mode crack

The concepts of crack energy density (CED) and its derivatives in arbitrary direction were established for piezoelectric material and, keeping their application to mixed mode fracture in mind, the characteristic features of them as fracture parameters were investigated based on the approximate equations for CED and its derivatives. That is, CED and its derivatives in arbitrary direction are defined first and separation into their each mode contribution is made. Subsequently, path independent integral expressions of them are derived, and then using them, approximate equations of each mode contribution of CED are obtained concretely for the case where linear singular solution is known. The resulting equations are then used to investigate the effects of electric field and electrical boundary condition on CED and its derivatives. An infinite piezoelectric plane with a crack inclined with respect to the poling direction is considered as a numerical example. Mode I contribution of mechanical CED is mainly employed as a possible fracture parameter for the study and it was shown that applied electric field significantly influences on fracture parameters especially for the impermeable crack perpendicular to the poling direction. The effect of electric field has the tendency to decrease as crack inclination angle increases. It was also found that, even for the impermeable crack perpendicular to the poling direction, crack propagation could be deviated from self-similar direction under a strong negative electric field, and this fact is qualitatively consistent with an existing experimental observation. For the ideally sharp crack with no width, impermeable and Hao and Shen type boundary conditions are admissible showing qualitative agreement with experimental results, but exact boundary condition is not suitable and finally consistent with permeable boundary condition.     

Stress multiaxiality factor for crack growth prediction using the strain energy density theory

The multiaxiality factor defined as the ratio of the von-Misses equivalent stress to the volumetric stress has been reported to be related to the initiation and progression of failure in structures. It is demonstrated in the present paper that the location around the crack tip where the multiaxiality factor obtains minimum value is an indicator of the direction of minimum material fracture resistance for crack propagation. It is also proposed that the location along the direction of crack propagation path where multiaxiality factor obtains minimum value is considered as the critical distance away from the crack tip, where the strain energy density should be evaluated and compared to its critical value. Theoretical predictions correlate well with the test results for the investigated cases.     

Strain energy density approach to stable crack extension under net section yielding of aircraft fuselage

In this paper, the incremental theory of plasticity is used in conjunction with the strain energy density criterion to determine the stress field in 4-in. wide test specimens containing 3 holes. These specimens, made from 0.04-in. thick sheets of 2024-T3 aluminum, also contained small collinear cracks emanating from the holes. The initial crack sizes varied from 0.15 to 0.26 in. Residual strength tests conducted with these specimens revealed that stable tearing occurred before failure. Analyses were performed to predict the stable crack extension and failure by plastic collapsed. Because of the complexities involved with nonlinear stress analysis combined with subcritical crack extension, the finite element method was used with the grid pattern adjusted for each increment of stable tearing. Reasonable correlation between the experimental data and predicted results was achieved.     

Crack initiation behavior of functionally graded piezoelectric material: prediction by the strain energy density criterion

Transient response of a functionally graded piezoelectric medium is considered for a through crack under the mixed-mode in-plane mechanical and electric load. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor. This determines the direction of crack initiation. Numerical results display the effects of material constants, loading combination parameter, mechanical loading angle and material gradient parameter on the possible fracture behavior.     

Fracture instability of notched cracked plates characterized by the strain energy density theory

The fracture instability of a mechanical system is analyzed by the strain energy density theory. The local relative minima of the strain energy density function dW/dV referred to local coordinate systems at each point of the body are distinguished from the global minimum of dW/dV, G, which is referred to a fixed global coordinate system. Failure by fracture starts from the maximum of the local minima of dW/dV, L, and passes from point G. The distance l between L and G along the fracture trajectory is introduced as a length parameter to characterized the fracture instability of the system. Numerical results are obtained and discussed for a cracked plate with two symmetrical notches subjected to a monotonically rising tensile stress perpendicular to the crack axis.     

Mixed mode fatigue crack growth and the strain energy density factor

The strain energy density factor S was first proposed by Sih for the prediction of the critical of the load and failure direction under monotonic, mixed mode loading condition. It seems a natural extension to apply the same concept to fatigue crack propagation. However, a close examination of the existing theory indicates that the Strain Energy Density Factor cannot logically account for the phenomena of the R-ratio effect and crack arrest. Thus, modification is necessary before the concept can be applied successfully for the prediction of mixed mode fatigue crack propagation.Based on the concept of hysteresis energy dissipation, an effective strain energy density factor range, ΔS_p,eff, is proposed for the correlation of fatigue crack growth data. ΔS_p,eff is consistent with the concept of crack closure. Experimental investigation indicates that it could predict the crack growth rates and trajectories.     

Strain energy density bounds for linear anisotropic elastic materials

Upper and lower bounds are presented for the magnitude of the strain energy density in linear anisotropic elastic materials. One set of bounds is given in terms of the magnitude of the stress field, another in terms of the magnitude of the strain field. Explicit algebraic formulas are given for the bounds in the case of cubic, transversely isotropic, hexagonal and tetragonal symmetry. In the case of orthotropic symmetry the explicit bounds depend upon the solution of a cubic equation, and in the case of the monoclinic and triclinic symmetries, on the solution of sixth order equations.     

Failure initiation sites and instability of structural components with localized energy density

When the surface or interior of a solid undergoes curvature and/or material change, there results localized fluctuation of the energy density field depending on the type of loading. These fluctuations are related to changes in the distortion and dilatation of material elements that could lead to failure by yielding and/or fracture should their magnitude become sufficiently large. According to the strain energy density criterion, failure is assumed to initiate at site of local maximum of minimum strain energy density denoted by [(dW/dV)_min^max]_L and tends toward the global maximum of [(dW/dV)_min^max]_G. The distance l between these two stationary values of dW/dV at L and G provides an indication of failure instability. That is, large l corresponds to more wide spread failure while the opposite holds for small l.Specimens with three different geometries are analyzed; they consist of round shoulder, hole and edge notch. The loads are either bending or tension. As the severity of notch or hole curvature is varied, predicted failure path also altered from boundary to boundary or an interior point of the specimen. The narrowest section turns out to be most vulnerable. If the hole is filled with a material of higher modulus, it acts as a reinforcer such that failure site would be shifted away from the interface. In general, there prevails a trade off between l and [(dW/dV)_min^max]_L. The undesirable combination would be for l and [(dW/dV)_min^max]_L to increase simultaneously. Failure initiation and global instability would then likely occur in tandem. This corresponds to the bending of a specimen with round shoulders. A variety of other conditions are analyzed with results displayed graphically so that the ways with which load, geometry and material inhomogeneity affect the failure behavior of structural components with notches and holes could be better understood.     

Numerical investigation of ductile tearing in surface cracked pipes using line-springs

Accurate prediction of crack-driving force equations is important in any pipeline fracture assessment program. In highly ductile materials, such as pipeline steel, a considerable amount of stable crack growth can be tolerated before the failure of the structure. The existing methods use simplified analytical procedures to account for ductile tearing, and they often result in conservative critical crack sizes. Further, none of the published numerical tools for modelling crack growth is suitable for engineering applications. This work describes a simple method for simulating through-thickness ductile tearing in surface cracked pipes, using line-spring finite elements. The crack growth resistance curve is used to advance the crack front. The line-spring results are verified using crack growth simulations employing the Gurson damage model. Finally, a detailed parametric study is carried out to examine the effect of ductile tearing on crack driving force relationships in circumferentially surface cracked pipes. The results demonstrate that considering ductile tearing is important in fracture assessment procedures for pipelines.     

Mode-I crack projection procedure using the strain energy density criterion

A practice used in linear elastic fracture mechanics is the projection of a crack onto a plane normal to the principal tensile stress axes for computing the stress intensity factor K_I. The minimum strain-energy criterion is applied for different crack configurations with the introduction of a safety factor S_i which is the ratio of the strain energy density factor of the projected crack and that of the original crack. Numerous crack configurations are investigated to illustrate the degree of conservativeness of the crack projection procedure.     

The role of grain boundaries on fatigue crack initiation - An energy approach

In this paper, we construct a model for prediction of fatigue crack initiation based on the material’s microstructure. In order to do so, the energy of a persistent slip band (PSB) is monitored and an energy balance approach is taken, in which cracks initiate and the material fails due to stress concentration from a PSB (with respect to dislocation motion). These PSBs are able to traverse low-angle grain boundaries (GB), thus belonging to clusters of grains. As a consequence of the ongoing cyclic slip process, the PSBs evolve and interact with high-angle GBs, the result of which leads to dislocation pile-ups, static extrusions in the form of ledges/steps at the GB, stress concentration, and ultimately crack initiation. Hence, this fatigue model is driven by the microstructure, i.e. grain orientations, widely distributed grain sizes, precipitates, PSB-GB interactions, as well as the affect of neighboring grains. The results predict that cracks initiate near twin boundaries from PSBs spanning a single large grain with a favorable orientation or multiple grains connected by low-angle GBs. Excellent agreement is shown between model predictions and experimental data.     

Strain behavior of pressurized cracked thick-walled cylinders

External circumferential strains were measured on large thick-wall pressure vessels containing internal fatigue cracks, using bonded strain gages. When strains measured over the cracks become compressive they predict impending failure. Normalization by the Lamé strain relates them to the fraction of fatigue life consumed and provides estimates of longevity.     

Stability analysis of crack path using the strain energy density theory

Prediction of crack growth path is a pre-requisite for estimating the final shape of broken solids and structures. Crack path in broken specimens provides information for the loading conditions just before fracture. Experiments on brittle materials, pre-cracked specimens of the same geometry under similar loading conditions, however, may yield different crack trajectories at times. The existing theories for the prediction of the crack path are based on the perturbation method combining the analytical and finite elements methods. They require a knowledge of the toughness equations. Moreover, they can only be applied to specimens with simple geometry and loadings.A different approach is used in the present work. The finite element technique is used to calculate the strain energy density (SED) contours. The predicted trajectory of the crack during unstable propagation is assumed to coincide with the minimum of the strain energy density function according to the SED criterion.The degree of crack path stability depends on the sharpness of the SED oscillations. This simple method offers a reliable prediction of the crack path stability for two as well as three-dimensional problems with complex geometry structures and arbitrary loadings. To be specific, both the TPB and DCB specimens are analysed. The findings are in good agreement with the theoretical and experimental results in the literature.     

The strain energy density ratio criterion for predicting cracking direction in composite materials

The strain energy density ratio criterion for predicting cracking direction incomposite materials is proposed.The Tsai-Hill criterion and Norris criterion ofcomposite materials are extended to predict the cracking direction in composites.Thethree criteria are used to analyse the crack propagation problem of the unidirectionalfibre composite sheet with various fibre directions.The predicted results are comparedwith those of the existing normal stress ratio criterion and strain energy densitycriterion.