Finite Fracture Mechanics Research Articles

Numerical Analysis of Brittle Crack Initiation at Stress Concentrations in Composites

As potential locations for failure initiation, stress concentrations require sophisticated analysis models. In composites, not only geometrical discontinuities may cause high local stresses, but also material discontinuities or even a combination of both. A universally applicable hybrid approach utilizing finite fracture mechanics is employed in this study to model crack initiation at stress concentrations. It overcomes deficiencies of classical criteria. The model is implemented in a versatile finite element procedure. Examples demonstrate the capability of the model to cover relevant effects for simple configurations as well as for glass-ceramic sealing joints in solid oxide fuel cells subject to thermomechanical loading.

A comparison of critical distance methods for fracture prediction

This paper is concerned with four methods which are used to predict the failure of bodies containing notches and other stress concentration features. Two of these methods, which we call the line method (LM) and point method (PM), use parameters taken from the elastic stress field ahead of the notch. The other two methods make use of linear elastic fracture mechanics (LEFM): we call these the imaginary crack method (ICM) and finite fracture mechanics (FFM). A common feature of all the methods is the use of a material constant with the units of length, which we call the critical distance. In this work we test the hypothesis that these four methods all give similar predictions. Firstly, we show analytically that, for the simple case of a straight, through-thickness crack in an infinite body, predictions using the LM are identical to those of the ICM and FFM and, in addition, there is a very simple relationship between the critical distances for the three methods. For notches no precise relationship exists; we used both closed-form solutions and finite element analyses (FEA) to compare predictions of failure from common types of notches, such as circular holes, edge notches and slots, in large (essentially infinite) bodies. We modelled both isotropic and anisotropic materials. In all cases, we found that predictions from the four different methods were of similar magnitude, always falling within an error band of ±10%. However, large differences emerge in the case of finite bodies when the remaining width is similar in size to the critical distance; then predictions become asymptotic in different ways. These findings have practical consequences for the use of these methods in engineering design. The results also promote some discussion about the theoretical basis of critical distance approaches.

Evaluation of toughness by finite fracture mechanics from crack onset strain of brittle coatings on polymers

Crack onset strain measurements of a confined layer in tension provide the means for layer toughness estimation. The procedure can be simplified if steady-state conditions prevail starting from the commencement of crack propagation, an assumption frequently employed in energy release rate evaluation. It is demonstrated, by numerical analysis of experimental data, that an estimate of the defect size in the film is needed in order to reliably evaluate its fracture toughness from the crack onset strain. Only if microcracks of sufficient size are present in the brittle layer, the steady-state energy release rate at the crack onset strain can be identified with layer toughness. Otherwise, the toughness estimate obtained by such a procedure is likely to be non-conservative.

Illustrations of a microdamage model for laminates under oxidizing thermal cycling

Degradations initiated near the edges of a laminate can have a significant effect on its state of degradation, even at the core. Indeed, results from the literature show that laminates which have the same stress state at the core can have completely different states of degradation, even far away from the edges. The paper discusses the influence of the edge effect on damage initiation and propagation for a specific example. A computational micromechanical approach to the degradation of laminated composites was developed recently at LMT-Cachan. This is a hybrid approach in which, depending on the scale, the mechanisms are described using continuous damage mechanics or finite fracture mechanics. Initially developed for static loading, this technique is being extended to fatigue and environmental effects. The aim of this paper is to illustrate the capability of such an approach to take into account major observations during cyclic loading in an oxidizing atmosphere, even when edge effects are significant.

Solution Methods for General Stress Concentrators in Anisotropic Heterogeneous Media

The increasing use of fibre-reinforced composites in high performance structures has brought a renewed interest in the analysis of cracks, wedges, and multi-material wedges in anisotropic materials. This paper will address three crucial stages of the general stress concentrator analysis: i) numerical procedures for the determination of eigenvalues and eigenvectors in Williams-like asymptotic expansion for multi-material wedge; ii) approaches to an accurate calculation of the near crack tip fields – the application of so-called two-state (or mutual) conservation integrals; iii) application of fracture criteria for the assessment of fracture inception at the general stress concentrators - concept of the so called finite fracture mechanics.

Sickle‐shaped crack in a round bar under complex Mode I loading

ABSTRACTA sickle‐shaped surface crack in a round bar under complex Mode I loading is considered. First, the stress‐intensity factor (SIF) along the front of the flaw is numerically determined for five elementary Mode I stress distributions (constant, linear, quadratic, cubic and quartic) directly applied on the crack faces. The finite element method and linear elastic fracture mechanics concepts are employed. Then, a numerical procedure to calculate approximate values of SIF for a complex Mode I stress distribution on the crack faces is proposed based on both the power series expansion of the function describing such a stress distribution and the superposition principle. In order to validate the results obtained through the above procedure, a comparison with numerical data available in the literature is made.

A finite fracture mechanics approach to structures with sharp V-notches

Criteria assuming that failure of quasi-brittle materials is affected by the stresses acting over a finite distance from the crack tip are widely used inside the scientific community. For instance, they have been applied to predict the failure load of specimens containing sharp V-notches, assuming as a critical parameter the average stress ahead the notch tip. However, this kind of approaches disregards energy balance considerations, which, as well known, are the basis of linear elastic fracture mechanics (LEFM). In order to overcome these drawbacks, the present paper uses a recently introduced finite fracture mechanics (FFM) criterion, i.e. a fracture criterion assuming that crack grows by finite steps. The length of this finite extension is determined by a condition of consistency of both energy and stress requirements; as a consequence, the crack advancement is not a material constant but a structural parameter. The criterion is applied to structures with sharp V-notches. The expression of the generalized fracture toughness, which is a function of material tensile strength, fracture toughness and notch opening angle, is given analytically. Finally, we provide comparisons with: (i) the experimental data we obtained from testing Polystyrene specimens under three point bending; (ii) some experimental data available in the literature. The agreement between theoretical predictions and experimental results is generally satisfactory and, for most of the cases analyzed, the FFM predictions are better than the ones provided by the simple average stress approach.

Crack simulation in pneumatic tyres using the finite element method

The study is concerned with the durability problem of pneumatic tyres, in particular the simulation of belt edge separation. The finite element method and fracture mechanics concepts are combined to study the initiation and growth of a rectangular crack at the belt edge. Implicit steady state formulation of rolling and the global-local modelling technique are used in the analysis. J-integral is calculated as part of the fatigue crack growth in the circumferential and meridional directions. The reliability and accuracy of J-integral calculations for crack models with sharp corners is studied by comparing the results with smoother crack models as well as more robust methods of calculating energy release rate.

Prediction of crack initiation at blunt notches and cavities – size effects

Crack initiation at corners, V-notches and other situations such as interfaces breaking a free surface (delamination initiation) cannot be correctly predicted by the usual brittle fracture criteria (either Griffith or maximum stress). They give contradictory results and neither one nor the other agrees with the experiments. An additional characteristic length is required to define a satisfying criterion giving rise to the so-called “Finite fracture mechanics”. The crack is supposed to jump this length which depends both on the material properties and the local geometry of the structure; it is not a material parameter. In most cases this crack increment is small. The size effect arises with the interaction between the crack increment and another length characterising a microstructure such as a pore diameter, a notch root radius or an interface layer thickness. The remote load at failure depends on the actual value of this microstructure parameter whereas it was not expected in all cases. Assuming that the two interacting lengths remain small compared to the size of the global structure, an asymptotic procedure allows bringing into evidence the change in the apparent resistance of the structure due to this phenomenon. Results are compared with experiments in various domains: polymers, ceramics and rocks.

Modelling crack initiation by means of finite fracture mechanics

AbstractIn this study, a generally applicable hybrid criterion for brittle crack initiation overcoming deficiencies of classical approaches is applied to regular and singular stress concentrations. Possible formulations of the resulting optimisation problem for the failure load are discussed. Examples demonstrate the capability of the model to cover relevant effects as well as a versatile finite element analysis procedure for notch problems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite fracture mechanics: A coupled stress and energy failure criterion

The aim of the present paper is to introduce a new failure criterion in the framework of Finite Fracture Mechanics. Criteria assuming that failure of quasi-brittle materials is affected by stress or energy flux acting on a finite distance in front of the crack tip are widely used inside the scientific community. Generally, this distance is assumed to be small compared to a characteristic size of the structure, i.e. to any length describing the macroscale. A key point of the present paper is to analyse what happens if the smallness assumption does not hold true. The proposed approach relies on the assumption that the finite distance is not a material constant but a structural parameter. Its value is determined by a condition of consistency of both energetic and stress approaches. The model is general. In order to check its soundness, an application to the strength prediction for three point bending tests of various relative crack depths and of different sizes is performed. It is seen that, for the un-notched specimens, the present model predicts the same trend as the Multi-Fractal Scaling Law (MFSL). Finally, a comparison with experimental data available in the literature on high strength concrete three point bending specimens is performed, showing an excellent agreement. It is remarkable to observe that the method presented herein is able to provide the fracture toughness using test data from un-notched specimens, as long as the range of specimen sizes is broad enough.

A hybrid method to assess interface debonding by finite fracture mechanics

Adhesive connections are potentially weak locations in many kinds of engineering structures. Since adhesive joints can be regarded locally as bimaterial notches, the assessment of the hazard of crack nucleation, initiation and propagation in the vicinity of bimaterial notches and the reliability of the junctions is an important problem. An essential requirement in this context is a sufficient criterion for crack nucleation. The present contribution proposes a modified approach based on Leguillon’s hypothesis in order to provide a feasible criterion. A crack at a notch is assumed to be initiated and to grow if and only if both the released energy and the local stresses exceed critical values. Thus, simulating virtual crack growth along an interface of two dissimilar bonded materials, the integrity of the bond is revisable. The approach enables the determination of characteristic lengths for freshly nucleated cracks forming the base for any further integrity assessment. As an example, the concept is applied to the analysis of an adhesive bond of metallic and ceramic materials under severe thermal loading conditions as they occur, among other examples, in high temperature fuel cell technology. It is shown that the failure hazard of the adhesive joint can be reduced significantly by an appropriate local design.

Stress Corrosion Cracking Life Estimation of Hold-Down Spring Screw for Nuclear Fuel Assembly

Hold-down spring screw fractures due to primary water stress corrosion cracking were observed in nuclear fuel assemblies. The screw fastens hold-down springs that are required to maintain the nuclear fuel assembly in contact with upper core plate and permit thermal and irradiation-induced length changes. In order to investigate the primary causes of the screw fractures, the finite element stress analysis and fracture mechanics analysis were performed on the hold-down spring assembly. The elastic-plastic finite element analysis showed that the local stresses at the critical regions of head-shank fillet and thread root significantly exceeded the yield strength of the screw material, resulting in local plastic deformation. Preloading on the screw applied for tightening had beneficial effects on the screw strength by reducing the stress level at the critical regions, compared to the screw without preload. Calculated deflections and strains at the hold-down springs using the finite element analysis were in very close agreements with the experimentally measured deflections and strains. Primary water stress corrosion cracking (PWSCC) life of the Inconel 600 screw was predicted by integrating the Scott’s model and resulted in a life of 1.42years, which was fairly close to the field experience. Cracks were expected to originate at the threaded region of the screw and propagated to the opposite side of the spring, which was confirmed by the fractographic analysis of the fractured screws.

A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials

In engineering applications, it is difficult to avoid the non-singular stress concentrations that often play an important role in structure designs. The simplest engineering strength criteria are in general not appropriate due to their incapacity in dealing with important size effect induced by stress gradients. In this paper, we present first a simple experimentation, which consists of plates with a central hole under uniaxial tensile loading, showing important size effect. Second, numerous criteria, including commonly used engineering criteria, crack initiation criteria based on the finite fracture mechanics, or cohesive criteria, were adapted to fit the experimental results. We found that most of these criteria, including criteria with a single material parameter and those with two material parameters, are not suitable for fracture prediction of materials under non-singular stress concentrations. It seems that three material parameters would be the minimum to establish an adequate fracture criterion for arbitrary stress concentrations. By analysing the energy dissipation of micro-crack bands under different stress concentrations, we proposed a new fracture criterion with three material parameters based on the finite fracture mechanics. It is shown that this criterion can provide accurate critical remote loads comparing with experimental data. We believe that the three parameter concept is physically reasonable and can be used in establishing fracture criteria in both the cases of singular and non-singular stress concentrations.

Spot-welded joints subjected to shear-tension loading. Part 2: Mixed-mode fracture parameters

Fracture parameters that can be used in the fatigue analysis of spot-welded joints subjected to shear-tension loading, with local restraints, are presented. Finite element analysis has been used to obtain stress intensities in the vicinity of three different types of fatigue crack which, from practical experience, are introduced into a spot-welded joint, i.e. assumed notch crack, through-thickness crack, and through-nugget crack. A combination of elastic-plastic finite element and linear elastic fracture mechanics finite element analysis has been used and the results are presented in the form of fatigue and fracture parameters (i.e. structure configuration factor, notch configuration factor, crack configuration factor, and J-integral value) for a wide range of specimen and crack geometric parameters. These parameters are chosen to be representative of typical practical situations and have been determined from evidence presented in the open literature. The extensive range of structure, notch, and crack configuration factors obtained from the finite element analyses are then used to obtain equivalent prediction equations using a statistical multiple non-linear regression model. The accuracy of this model is measured using a multiple coefficient of determination, R2, where 0 ≤ R2 ≤ 1. This coefficient is found to be greater than or equal to 0.98 for all cases considered in this study, demonstrating the quality of the model fit to the data. Predictive equations for stress intensity factors and J-integral values, which are based on the elastic stress concentration factor, are also developed. Some approximate equations for crack configuration factors have been published in the past, all based on analytical solutions. All equations were derived in terms of two sets of non-dimensional parameters (i.e. the bending stress-membrane stress ratio at the nugget and the geometric ratio). These equations have to be coupled with a shell-beam finite element analysis in order to predict bending and membrane stresses at the weld nugget. Therefore, the accuracy of the results depends on the accuracy of the shell-beam finite element models used. In the present study, all prediction equations are directly related to the geometric ratios, which provides a simple and independent calculation procedure. Discrepancies between the results from the present study and those from the literature are discussed.

The fracture mechanics of finite crack extension

This paper describes a modification to the traditional Griffith energy balance as used in linear elastic fracture mechanics (LEFM). The modification involves using a finite amount of crack extension (Δ a) instead of an infinitesimal extension (d a) when calculating the energy release rate. We propose to call this method finite fracture mechanics (FFM). This leads to a change in the Griffith equation for brittle fracture, introducing a new term Δ a/2: we denote this length as L and assume that it is a material constant. This modification is extremely useful because it allows LEFM to be used to make predictions in two situations in which it is normally invalid: short cracks and notches. It is shown that accurate predictions can be made of both brittle fracture and fatigue behaviour for short cracks and notches in a range of different materials. The value of L can be expressed as a function of two other material constants: the fracture toughness K c (or threshold Δ K th in the case of fatigue) and an inherent strength parameter σ 0. For the particular cases of fatigue-limit prediction in metals and brittle fracture in ceramics, it is shown that σ 0 coincides directly with the ultimate tensile strength (or, in fatigue, the fatigue limit), as measured on plain, unnotched specimens. For brittle fracture in polymers and metals, in which larger amounts of plasticity precede fracture, the approach can still be used but σ 0 takes on a different value, higher than the plain-specimen strength, which can be found from experimental data. Predictions can be made very easily for any problem in which the stress intensity factor, K is known as a function of crack length. Furthermore, it is shown that the predictions of this method, FFM, are similar to those of a method known as the line method (LM) in which failure is predicted based on the average stress along a line drawn ahead of the crack or notch.

Fracture mechanics of composites with residual stresses, imperfect interfaces, and traction-loaded cracks

By partitioning the total stresses in a damaged composite into either mechanical and residual stresses or into initial and perturbation stresses, it was possible to derive two exact results for the energy release rate due to crack growth. These general results automatically include the effects of residual stresses, traction-loaded cracks, and imperfect interfaces. These effects are normally not needed in the fracture mechanics of homogeneous materials, but they are commonly needed for the fracture mechanics of composites. The general results were used to consider mode I fracture in composites, fracture and thermal cracking for two-phase, isotropic composites, and interfacial fracture in the microbond and single-fiber, pull-out tests. The analysis of interfacial fracture illustrates the importance of including friction effects in the energy release rate and not as part of the toughness of the composite. Many composite damage modes consist of a series of events instead of stable crack propagation. A new analysis method, referred to as finite fracture mechanics, is proposed which predicts that the next event occurs when the total energy released by that event exceeds some critical value or toughness for that type of event. A finite fracture mechanics model for microcracking that can correlate the results from many laminates is described.

Fracture mechanics analysis of coating/substrate systems: Part I: Analysis of tensile and bending experiments

A finite fracture mechanics model is used to predict the development of multiple cracks in the coating layer of coating/substrate systems. The stresses in a cracked coating are evaluated by a variational mechanics approach. These stresses are then used to calculate the total energy released due to the formation of a complete crack in the coating layer. The analysis can handle tensile loads or bending loads and includes the effect of residual thermal stresses. By assuming the next coating crack forms when the energy released due to the formation of a complete microcrack equals the in situ fracture toughness of coating, it is proposed that one can predict the number of coating cracks as a function of applied strain. Alternatively, it is proposed that experimental data for number of cracks vs. strain can be fit to the fracture analysis and be used to determine an in situ coating fracture toughness.

Plasma immersion ion implantation for SOI synthesis: SIMOX and ion-cut

We have demonstrated feasibility to form silicon-on-insulator (SOI) substrates using plasma immersion ion implantation (PIII) for both separation by implantation of oxygen and ion-cut. This high throughput technique can substantially lower the high cost of SOI substrates due to the simpler implanter design as well as ease of maintenance. For separation by plasma implantation of oxygen wafers, secondary ion mass spectrometry analysis and cross-sectional transmission electron micrographs show continuous buried oxide formation under a single-crystal silicon overlayer with sharp Si/SiO2 interfaces after oxygen plasma implantation and high-temperature (1300°C) annealing. Ion-cut SOI wafer fabrication technique is implemented for the first time using PIII. The hydrogen plasma can be optimized so that only one ion species is dominant in concentration and there are minimal effects by other residual ions on the ion-cut process. The physical mechanism of hydrogen induced silicon surface layer cleavage has been investigated. An ideal gas law model of the microcavity internal pressure combined with a two-dimensional finite element fracture mechanics model is used to approximate the fracture driving force which is sufficient to overcome the silicon fracture resistance.

Strength of friction welded ceramic–metal joints

In preceding theoretical examinations, the friction welding process of ceramics to metals was investigated by the finite element method and statistical fracture mechanics. The irifluence of residual stresses on the resulting joint strength was of special interest. It was found that, besides material combination, interface geometry does strongly influence the residual stress distributions and therefore has a strong influence on joint strength. In the framework of this examination, these theoretical findings have been experimentally verified. Furthermore, the irifluence of surface preparation and welding parameters have been examined.