Stress intensity factors by enriched finite elements

The finite element method is extended to direct calculation of combined modes I and II stress intensity factors for axisymmetric and planar structures of arbitrary geometry and loading. High order conventional isoparametric elements are combined with a fracture mechanics enrichment of the same element so that a corner node is made to correspond to a crack-tip. The development is explained in detail and necessary modifications to a standard finite element computer program are identified. Stress intensity factors as well as a complete stress analysis are obtained directly from the computer printout. Few elements are generally required, minimizing engineering costs and eliminating the need for data generators. Example problems demonstrate the ease of using the method, the high accuracy to be expected from the method, and the fact that accuracy is relatively insensitive to variations of the element mesh in the vicinity of crack tips. Important applications include complex geometries for which other methods fail and situations wherein multiple crack tips may interfere with one another.     

FRACTURE MECHANICS ASSESSMENT OF PLANAR BRANCHED CRACKS IN AN EQUIBIAXIAL STRESS FIELD

Abstract A simplified fracture mechanics assessment is presented of branched planar cracks in an equibiaxial stress state. In linear-elastic fracture mechanics the stress intensity factors which characterize the load at the crack tips depend, for a given external load, only on the crack geometry. The stress intensity factors of a large number of branched cracks were evaluated using the Boundary Element method, and correlations between the stress intensity factors and the crack geometry were investigated. Formulae are presented which assign an individual effective crack length to each crack tip of a branched crack and hence allow approximate stress intensity factors to be determined for very complicated crack geometries. An algorithm is used for the stochastic simulation of an irregular crack pattern formation in thermal fatigue.     

A COMPARATIVE STUDY OF METHODS FOR ESTIMATING BOLT FATIGUE LIMITS

The approach to predicting fatigue limits from calculated stress concentration factors, using thread load distributions obtained from analytical theories is examined. In particular combinations of the methods of Shigley, Otaki, Heywood, Birger, Bluhm and Flanagan, Sopwith and a modification to Sopwith's theory are assessed against fatigue test data and photoelastic results for a range of bolt and nut geometries. The Snow-Langer-Cook, the Goodman and the Gerber methods of allowing for the effect of mean stress were also examined. It was concluded that mean stress effects are significant, that none of the theories are sufficiently sensitive to changes in geometry, and that they seriously overestimate the fatigue limit when the geometry is more severe than the standard. The use of Sopwith's theory of load distribution combined with Heywood's method and the Snow-Langer-Cook technique was found to be the safest for design purposes, but not the most accurate.     

Crack modelling: A novel technique for the prediction of fatigue failure in the presence of stress concentrations

Finite element (FE) analysis and other computational methods have developed rapidly in recent years, allowing accurate predictions of elastic stresses in components of complex geometry. However, the prediction of fatigue failure in these components is still a non-trivial problem; one reason for this is the difficulty of assessing stress concentrations and regions of high stress-gradient. This paper describes a new technique, called “crack modelling”, which addresses the problem through a modification of linear-elastic fracture mechanics (LEFM). LEFM is designed to deal with cracks in nominally elastic stress fields, using elastic analysis to derive a characteristic stress intensity, K or, for cyclic loading, a range ΔK. This methodology is modified in two ways. Firstly it is shown that LEFM can be extended to predict the fatigue behaviour of bodies containing notches of standard geometry, instead of cracks. Secondly, FE analysis is used in conjunction with a modelling exercise in order to extend the method to include bodies of arbitrary shape subjected to any set of loads. The method was first tested using standard notch geometries (blunt and sharp notches in beams), where accurate predictions of fatigue limit could be achieved. It was then applied to an industrial problem, giving a prediction of high-cycle fatigue behaviour for an automotive crankshaft. The method requires only simple mechanical-property data (the material fatigue limit and stress-intensity threshold) and uses only linear-elastic FE modelling. It allows fracture mechanics theory to be used without the need to specifically model the presence of a crack and uses far-field elastic stresses to infer behaviour in the region of a stress concentration.     

Constant stress intensity factors through closed-loop control

A new way of obtaining a constant stress intensity factor is achieved for any test specimen geometry subjected to closed-loop control loading. In contrast to using only load or displacement control the method draws on combined feedback from both displacement and load sensing, reducing the variation in the stress intensity factor by two decades compared to that if tested under constant load or displacement. A change in the signal mix ratio for a rectangular compact tension specimen is equivalent to changing the angle of a tapered compact tension specimen. This method can eliminate the need for the use of ‘complex’ geometries or for geometries in which measured crack lengths are used in conjunction with a computer or some other means to adjust the loading continuously for achieving a constant stress intensity factor.     

Evaluation of the IITRI compression test method for stiffness and strength determination

The objective of this paper is to evaluate the IITRI Compression Test Method for measuring the axial compressive modulus and strength of composite materials possessing a high degree of anisotropy in the axial direction. The influence of material anisotropy and test specimen geometry on the data reduction scheme used to characterize the axial compressive modulus and ultimate strength is examined. The findings demonstrate that ASTM D3410 specimen geometries, recommended for the IITRI and Celanese compression test methods, may not always be appropriate for materials that are highly anisotropic in nature. The significance of the findings on appropriate specimen design for thick section laminates is also discussed. An error investigation is included which demonstrates to what extent the experimentally determined compressive modulus may be affected if careful consideration of the specimen geometry is not made. This investigation utilizes the finite element method and an elasticity solution based on Saint-Venant's principle for an upper bound estimate on stress decay length in a parametric study involving combinations of specimen geometries with varying material anisotropy. A methodology is presented for sizing specimen geometries as a function of material anisotropy that ensures accurate determination of the linear elastic axial compressive modulus and provides an estimate of the maximum amount of axial strain that can be tolerated prior to Euler buckling induced failure. Based on the degree of material anisotropy and anticipated failure strain, the appropriateness of the test method for a given material system can be evaluated.     

Treatment of residual stress in failure assessment procedure

The effect of residual stress on component failure has been investigated using the distributions from current failure assessment procedures, and a residual stress profile simple to apply with less conservatism has been proposed for the weld geometries of T-plate and tubular T-joint. The stress intensity factors (SIFs) in the two weld geometries under various types of loads have been calculated using the Green’s function method. The Green’s functions were determined not only for the T-plate but also for the tubular T-joint with the built-in ends. The use of a linear (bending) stress profile, derived from an analysis of measured residual stress distributions in T-plate and tubular T-joints, has been examined. The profile was validated with experimentally measured residual stress distributions in two materials, a high strength and medium strength ferritic steel and two geometries, a T-plate joint and a tubular T-joint for crack lengths up to half the plate or pipe thickness. Whereas the recommended residual stress distributions are geometry and material specific, it is shown that a simplified linear bending profile provides a possible guideline, applicable to a range of materials and geometries, where detailed information on weld procedures or residual stress profiles are unavailable.     

Determination of crack-tip stress intensity factors in complex geometries by the composition of constituent weight function solutions

Linear elastic fracture mechanics (LEFM) is the science frequently used to understand the stable and progressive fatigue crack growth that often occurs in engineering components under varying applied stress. The stress intensity factor (SIF) is its basis and describes the stress state at the crack tip. This can be used with the appropriate material properties to calculate the rate at which the crack will propagate in a linear elastic manner. Unfortunately, the SIF is difficult to compute or measure, particularly if the crack is situated in a complex three‐dimensional geometry or subjected to a non‐simple stress state. This is because the SIF is not only a function of the crack and component geometry but is also dependent on the applied stress field. In the last 20 years, the SIF weight function has gained prominence as a method for calculating and presenting SIFs independent of applied stress. This paper demonstrates that the real promise of the SIF weight Function lies in its use to rapidly generate SIF solutions for cracks in complex geometries by simple composition of geometric influences from reference constituent solutions.     

Finite element models of the application of stress-relieving dielectrics

The addition of ferroelectric fillers to a polymer can produce a material with electric field-dependent permittivity properties, which can be used for the purposes of electrical stress relief for applications under AC fields. The paper illustrates the electrical stress-relieving properties of these composite polymers through finite element models. The electric field-dependent permittivity characteristics of the stress-relieving dielectric have been based on a barium titanate filler in an acrylic resin, for a range of filler levels. Two sets of models are considered. The first set of models considers the case of a needle-point plane geometry and the electrical stress distribution in the stress-relieving dielectric for needles of different geometries. The second set of models considers the stress distribution in a coaxial cable using the stress-relieving dielectric. The effectiveness of the stress-relieving dielectric in each of these geometries is clearly illustrated     

Variable wall thickness scroll geometry modeling with use of a control volume approach

In a scroll-type compressor, compression is achieved through relative contact between two spiral curves. Since the scroll invention by Leon Creux (1905), multiple methods have been developed for calculating scroll geometry. What can generally be considered the most classical method, is defining each scroll curve as the involute of a circle. Gravesen and Henriksen (2001) introduced a new method to calculate scroll geometry by deriving each scroll curve from the radius of curvature parameterized with involute angle. This allows a wide range of involute geometries to be considered not included in the classical method. In this paper, Gravesen's method is extended to the tip region to include all tip geometries involved in a two arc configuration resulting in a more comprehensive scroll geometry definition. Lastly, with parametric representation of all scroll geometry, the pocket volume can be easily solved using a derived control volume approach.     

Correlation of fatigue crack propagation in polyethylene pipe specimens of different geometries

Correlation in mechanisms and kinetics of step-wise fatigue crack propagation in polyethylene pipe specimens of different geometries is studied experimentally. It is shown that crack propagation in a non-standard specimen cut from a real pipe and conserving the pipe geometry can be effectively simulated using a standard compact tension specimen. Good correlation in both kinetics of step-wise crack propagation and fractography between the specimens is achieved if experimental conditions are chosen to assure equal values of (a) stress intensity factor and (b) stress intensity factor gradient at the initial notch tips. These results extend previous technique of fatigue accelerating slow crack growth used to predict lifetime of polyethylene pipes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approximate elastic-plastic J estimates of cylinders with off-centred circumferential through-wall cracks

This paper provides approximate J estimates for off-centred, circumferential through-wall cracks in cylinders under bending and under combined tension and bending. The proposed method is based on the reference stress approach, where the dependence of elastic and plastic influence functions of J on the cylinder/crack geometry, the off-centred angle and strain hardening is minimised through the use of a proper normalising load. Based on published limited FE results for off-centred, circumferential through-wall cracks under bending, such normalising load is found, based on which the reference stress based J estimates are proposed for more general cases, such as for a different cylinder geometry and for combined loading. Comparison of the estimated J with extensive FE J results shows overall good agreements for different crack/cylinder geometries and for combined tension and bending, which provides sufficient confidence in the use of the proposed method for fracture mechanics analyses of off-centred circumferential cracks. Furthermore, the proposed method is simple to use, giving significant merits in practice.     

Low cycle fatigue life prediction in shot‐peened components of different geometries—part I: residual stress relaxation

In this study, the residual stress relaxation behaviour occurring during low‐cycle fatigue in shot‐peened specimens with either a flat or a notched geometry has been studied. A representative low‐pressure steam turbine material, FV448, was used. The residual stress and strain hardening profiles caused by shot peening were measured experimentally and were then incorporated into a finite element model. By allowing for both effects of shot peening, the residual stress relaxation behaviour was successfully simulated using this model and correlated well with the experimental data. Although more modelling work may be required to simulate the interaction between shot peening effects and external loads in a range of notched geometries, the model predictions are consistent with the specimens tested in the current study. The novelty of this study lies in the development of such a modelling approach which can be used to effectively simulate the complex interaction between shot peening effects and external loads in notched regions. Compared with the un‐notched geometry, the notched geometry was found to be more effective in retaining the improvement in fatigue life resulting from shot peening, by restricting the compressive residual stress relaxation during fatigue loading.     

Intralaminar toughness characterisation of unbalanced hybrid plain weave laminates

A numerical and experimental investigation on the mode-I intralaminar toughness of a hybrid plain weave composite laminate manufactured using resin infusion under flexible tooling (RIFT) process is presented in this paper. The pre-cracked geometries consisted of overheight compact tension (OCT), double edge notch (DEN) and centrally cracked four-point-bending (4PBT) test specimens. The position as well as the strain field ahead of the crack tip during the loading stage was determined using a digital speckle photogrammetry system. The limitation on the applicability of the standard data reduction schemes for the determination of intralaminar toughness of composite materials is presented and discussed. A methodology based on the numerical evaluation of the strain energy release rate using the J-integral method is proposed to derive new geometric correction functions for the determination of the stress intensity factor for composites. The method accounts for material anisotropy and finite specimen dimension effects regardless of the geometry. The approach has been validated for alternative non-standard specimen geometries. A comparison between different methods currently available for computing the intralaminar fracture toughness in composite laminates is presented and a good agreement between numerical and experimental results using the proposed methodology was obtained.     

The Effect of Notch Geometry on Tensile Strength at Low and Intermediate Strain Rates

Abstract: Mechanical behaviour of notched components can significantly be influenced by notch geometry. In this work, triangular, circular and square notches are examined by experiment. Two materials including a st52 steel and brass are employed for the investigation. The experiments are conducted at low and intermediate strain rates. It is observed that: (i) the yield stress of the brass and the specimens with circular and square notches decreases as the notch length increases such that for 8 mm notches, the value of yield stress converges to the yield stress obtained for plain specimens at the same strain rate. (ii) For triangular notches the yield stress rises with the increase of notch length up to a minimum and then begins to decline. (iii) The effect of notch geometry on yield stress is significant such that for triangular notches, in particular for higher notch lengths, the increase of yield stress is much more profound than the other two notch geometries.     

Geometries of inhomogeneities with minimum field concentration

This paper is devoted to the study of geometries of inhomogeneities with minimum strain or stress concentration. The solutions are achieved by the indirect method of first deriving lower bounds and then constructing the geometries to attain the lower bounds. In particular, we show that a new class of geometries, namely, E-inclusions and periodic E-inclusions, are the optimal geometries with minimum field concentrations. We also obtain the explicit relation between the shape matrix of E-inclusion and remote applied strain which will be convenient for engineering applications of these new geometries.     

Routine FE determination of stress intensity factors at curved crack fronts using an influence function technique

A remarkably simple and accurate one-step application of the finite element (FE) method is suggested as a means for the engineer's routine determination of stress intensity factors in linear fracture mechanics for complicated non-symmetric geometries with three-dimensional states of stress and curved crack fronts. The vector-valued influence functions (Green functions) used here are a special kind of weight functions. Mode separation is inherent to the present procedure. Numerical examples demonstrate the versatility of the method. Accuracies within 1% are easily achieved. Detailed guidance to the design of the FE mesh at the crack front is given. Any standard FE code can be used, without requirements for special finite or boundary elements. In retrospect, the present method can be seen as a rather trivial calculation technique which has been made feasible and attractive by the capabilities of today's computers and softwares.     

Using a variable value of the fictitious radius to estimate fatigue life of notched elements

The article discusses the problem related to estimating fatigue life of elements containing stress concentrators. The algorithm for estimating fatigue life presented in this work uses a fictitious ray based on the Neuber method. The proposed algorithm takes into account the variability of microstructural length, which depends on the number of failure cycles. The function thus obtained can be used to select the appropriate value for this length, which simplifies the calculation procedure needed to estimate fatigue life. In addition, this work includes an analysis of the impact of variable cutout geometry on the value of the support coefficient. As a result, it was possible to extend the application of this concept to lower fatigue life values and to structural elements with different notch geometries. The method was verified by performing experiments using elements made of three steel grades. A good correlation between calculation and experimental results was obtained.     

A phase-field method for modeling cracks with frictional contact

We introduce a phase-field method for continuous modeling of cracks with frictional contacts. Compared with standard discrete methods for frictional contacts, the phase-field method has two attractive features: (i) it can represent arbitrary crack geometry without an explicit function or basis enrichment, and (ii) it does not require an algorithm for imposing contact constraints. The first feature, which is common in phase-field models of fracture, is attained by regularizing a sharp interface geometry using a surface density functional. The second feature, which is a unique advantage for contact problems, is achieved by a new approach that calculates the stress tensor in the regularized interface region depending on the contact condition of the interface. Particularly, under a slip condition, this approach updates stress components in the slip direction using a standard contact constitutive law, while making other stress components compatible with stress in the bulk region to ensure nonpenetrating deformation in other directions. We verify the proposed phase-field method using stationary interface problems simulated by discrete methods in the literature. Subsequently, by allowing the phase field to evolve according to brittle fracture theory, we demonstrate the proposed method's capability for modeling crack growth with frictional contact.