Influence of Stress Singularities on Scaling of Fracture of Metal-Composite Hybrid Structures

It has been recently shown that the nominal structural strength of metal-composite structures depends on the structure size, and such dependence is strongly influenced by the stress singularities. Nevertheless, previous studies only focused on structures that exhibit very strong stress singularities, which are close to the crack-like stress singularity. In the actual engineering designs, due to the mismatch of material properties and complex structural geometries, many metalcomposite structures may contain stress singularities that are much weaker than the crack-like stress singularity. This paper presents a numerical study on the size dependence of scaling of fracture of metal-composite hybrid structures for a wide range of stress singularities. The numerical examples include a series of metalcomposite hybrid beams with a V-notch under three-point bending with different notch angles, which lead to various magnitudes of stress singularities. By assuming that the bimaterial interface is weaker than both metal and composite, we use a mixed-mode cohesive element model to simulate the fracture behavior of these hybrid beams. It is shown that the resulting size effect curves strongly depend on the magnitude of stress singularities. The simulation results agree well with a recently developed energetic-statistical scaling model.     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

Energetic–statistical size effect simulated by SFEM with stratified sampling and crack band model

The paper presents a model that extends the stochastic finite element method to the modelling of transitional energetic–statistical size effect in unnotched quasibrittle structures of positive geometry (i.e. failing at the start of macro‐crack growth), and to the low probability tail of structural strength distribution, important for safe design. For small structures, the model captures the energetic (deterministic) part of size effect and, for large structures, it converges to Weibull statistical size effect required by the weakest‐link model of extreme value statistics. Prediction of the tail of extremely low probability such as one in a million, which needs to be known for safe design, is made feasible by the fact that the form of the cumulative distribution function (cdf) of a quasibrittle structure of any size has been established analytically in previous work. Thus, it is not necessary to turn to sophisticated methods such as importance sampling and it suffices to calibrate only the mean and variance of this cdf. Two kinds of stratified sampling of strength in a finite element code are studied. One is the Latin hypercube sampling of the strength of each element considered as an independent random variable, and the other is the Latin square design in which the strength of each element is sampled from one overall cdf of random material strength. The former is found to give a closer estimate of variance, while the latter gives a cdf with smaller scatter and a better mean for the same number of simulations. For large structures, the number of simulations required to obtain the mean size effect is greatly reduced by adopting the previously proposed method of random property blocks. Each block is assumed to have a homogeneous random material strength, the mean and variance of which are scaled down according to the block size using the weakest‐link model for a finite number of links. To check whether the theoretical cdf is followed at least up to tail beginning at the failure probability of about 0.01, a hybrid of stratified sampling and Monte Carlo simulations in the lowest probability stratum is used. With the present method, the probability distribution of strength of quasibrittle structures of positive geometry can be easily estimated for any structure size. Copyright © 2007 John Wiley & Sons, Ltd.     

THE SIZE EFFECT OF MICROSTRUCTURAL IMPLICATIONS OF THE WEAKEST LINK MODEL

Abstract— CT specimens made of a reactor pressure vessel steel were loaded at—40°C until final failure occurred by cleavage fracture. The samples of J _lcl values obtained in these tests are analysed using the weakest link model. The size effect observed with specimens of different thicknesses is compared with the predictions of the weakest link model. A formula is derived for the distribution of the locations of fracture origins which have been determined for almost all specimens with a scanning electron microscope. The distribution of the size of the "weak spots" is calculated from the distribution of the fracture origins using two different models for the stress field ahead of the crack tip. These fractographic results and the J _lcl data confirm the basic ideas of the weakest link model. The deviations observed between the quantitative predictions of the weakest link model and the data can partly be explained by the change in the stress state ahead of the crack tip caused by a change in the specimen thickness.     

Stress singularities in bimaterial bodies of revolution

An eigenfunction expansion solution is first developed for determining the stress singularities of bimaterial bodies of revolution by directly solving the equilibrium equations of three-dimensional elasticity in terms of displacement functions. The characteristic equations are explicitly given for determining the stress singularities in the vicinity of the interface corner of two intersecting bodies of revolution having a sharp corner with free boundary conditions along the corner. The characteristic equations are found to be equivalent to a combination of the characteristic equations for plane elasticity problems and St. Venant torsion problems. The strength of stress singularities varying with the interface angles is also investigated. The first known asymptotic solutions for the displacement and stress fields are also explicitly shown for some interface angles. The present results will be useful not only for understanding the singularity behaviors of stresses in the vicinity of a revolution interface corner, but also for developing accurate numerical solutions with fast convergence for stress or vibration analysis of a body of revolution having an interface corner.     

Three-dimensional asymptotic stress fields at the front of a trimaterial junction

A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the junction corner front of an infinite pie-shaped trimaterial wedge, of finite thickness, formed as a result of bimaterial (matrix plus reaction product or contaminant) deposit over a substrate or reinforcement. The wedge is subjected to extension/bending (mode I), inplane shear/twisting (mode II) and antiplane shear (mode III) far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 (substrate) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I, II, III eigenvalues (or stress singularities) with various moduli ratios as well as the wedge aperture angle of the material 1 (reaction product/contaminant), are also presented. Hitherto unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric exponentially decaying distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.     

MODELLING OF 3-D MIXED-MODE FRACTURES NEAR PLANAR BIMATERIAL INTERFACES USING SURFACE INTEGRALS

Mixed-mode fractures of arbitrary orientation with respect to a planar bimaterial interface have been effectively modelled using a surface integral approach. By requiring only that the surface of the fracture be discretized, the surface integral method circumvents the practical difficulties associated with having to mesh the interacting dual singularities in stress along the three-dimensional (3-D) crack front and at the interface. The key elements of this numerical capability are discussed in detail. These include: the derivation of the fundamental solutions for a generalized fracture event near a planar bimaterial interface, formulation of the governing integral equation including its decomposition into singular and non-singular terms, development of analytical and numerical techniques for performing the singular integrations, and efficient numerical integration of the non-singular terms using non-dimensionalized surface approximations of the dipole solutions. The problem of a pressurized planar crack near a bimaterial interface was used to assess convergence. The effect of material contrast and crack shape on tendencies for crack growth were also examined.     

Non‐proportional size scaling of strength of concrete in uniaxial and biaxial loading conditions

A procedure for non‐proportional size scaling of the strength of concrete based on the weakest‐link statistics is proposed to synchronize strength data from specimens of different geometries and different loading modes. The procedure relies on proportional size scaling of strength to determine the parameters of the statistical model and often on finite element analysis to calculate the coefficient of the equivalent strength. The approach for non‐proportional size scaling is capable to synchronize the uniaxial strength data of concrete from uniaxial tensile specimens and 3‐point bending specimens, or the biaxial tensile strength data of circular plates in different loading mode. The non‐transference of the uniaxial strength data to the biaxial strength data is unclear in its mechanism but possibly due to the variation of statistical distribution of microcracks with stress states in different specimens.     

The effect of surface treatments on interfacial fatigue crack initiation in aluminum/epoxy bonds

The effect of different substrate surface pre-treatments on the initiation of interfacial fatigue cracks was studied for adhesive bonds. Aluminum-epoxy bimaterial specimens were used to investigate how surface pre-treatment affects resistance to fatigue crack initiation at the interface corner. A stress singularity approach was utilized to assess the effect of four different treatments; P2 etch, phosphoric acid anodization (PAA), sulfuric acid anodization (SAA), and sol-gel. A bimaterial system with a 90° epoxy wedge was tested under sinusoidal cyclic loading. The surface treatment effect was rather significant on the resistance to fatigue crack initiation at the interface. Results show that PAA generated the strongest interface, while SAA led to the weakest for the material system studied here.     

Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks

The random distribution of microcracks in terms of their size, shape, orientation and spatial location has direct impact on the cumulative probability of brittle fracture induced failure, with the effect of spatial distribution being rarely explored. Recently, two weakest link theory‐based formulations for the cumulative probability of brittle fracture induced failure have been proposed for the spatial distribution of microcracks obeying the Poisson postulates and the uniform distribution, respectively. This work compares these two new formulations with the currently commonly adopted one built on the Poisson postulates under both the uniform and the non‐uniform uniaxial loading conditions. It is concluded that under general loading conditions involving non‐uniform stress states, the existing formulation is equivalent to or closely approximate to neither of the two new formulations thus should be discarded, because of its inaccurate derivation. The new formulations are featured with unique symmetry or self‐similarity in their expressions. Their capability in revealing the size effect or the scaling law of failure is highlighted and validated by a set of published uniaxial and biaxial flexural strength data of brittle material.     

Indirect determination of size effect on strength of asphalt mixtures at low temperatures

Low temperature cracking of asphalt pavements is a major distress in cold regions. Accurate assessment of strength of asphalt mixtures at low temperatures is of great importance for ensuring the structural integrity of asphalt pavements. It has been shown that asphalt mixtures behave in a quasibrittle manner at low temperatures and consequently its nominal strength strongly depends on the structure size. The size effect on the strength of asphalt mixtures can be directly measured by testing geometrically similar specimens with a sufficiently large size range. Recent studies have shown in theory that for quasibrittle structures, which fail at the macrocrack initiation from one representative volume element, the mean size effect curve can also be derived from the scaling of strength statistics based on the finite weakest link model. This paper presents a comprehensive experimental investigation on the strength statistics as well as the size effect on the mean strength of asphalt mixtures at ?24 °C. It is shown that the size effect on mean structural strength can be obtained by strength histogram testing on specimens of a single size. The present study also indicates that the three-parameter Weibull distribution is not applicable for asphalt mixtures.     

Conducting cracks in dissimilar piezoelectric media

Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of real-valued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.     

A probabilistic model for cleavage fracture with a length scale--parameter estimation and predictions of stationary crack experiments

This study presents a large experimental investigation in the transition temperature region on a modified A508 steel. Tests were carried out on single-edge-notch-bend specimens with three different crack depth over specimen width ratios to capture the strong constraint effect on fracture toughness. Three test temperatures were considered, covering a range of 85 °C. All specimens failed by cleavage fracture prior to ductile tearing. A recently proposed probabilistic model for the cumulative failure by cleavage was applied to the comprehensive sets of experimental data. This modified weakest link model incorporates a length scale, which together with a threshold stress reduce the scatter in predicted toughness distributions as well as introduces a fracture toughness threshold value. Model parameters were estimated by a robust procedure, which is crucial in applications of probabilistic models to real structures. The conformity between predicted and experimental toughness distributions, respectively, were notable at all the test temperatures.     

Effect of vertex singularities on stress intensities near plate free surfaces

The stress intensity factor (SIF) distribution along the front of a through‐the‐thickness crack is significantly influenced by the presence of the 3D corner (vertex) singularities. All past 3D finite element studies indicated that for mode I, SIF rapidly decreases near the free surface and for mode II, it sharply increases. From the previous numerical simulations, it is unclear what the limiting values of SIF near the surface are and whether these values are infinite or bounded at the vertex point. In this paper, we conduct a careful finite element study and propose a theoretical equation, which describes the SIF behaviour near the vertex. We demonstrate that the asymptotic behaviour of SIF near the surface is governed by the difference in the strength of the corner and edge singularities. Furthermore, we validate our numerical approach and calculations by utilising the invariant properties of J‐integral.     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Prediction of failure probabilities for cleavage fracture from the scatter of crack geometry and of fracture toughness using the weakest link model

The weakest link model proposed by Landes and Shaffer[1] is generalized in order to predict the failure probabilities of real structures with surface cracks and to include the scatter in crack size and crack geometry. The failure probabilities and the distributions of crack depth, of depth to length ratio and of fracture toughness of the failed structures are calculated in this model and compared with the results of the usual probabilistic calculations.     

Failure analysis of martensitic stainless steel bridge roller bearings

The paper is aimed at finding the likely failure mechanism of a bridge roller bearing made of high strength martensitic stainless steel. Spectroscopy and finite element stress analysis of the roller indicated that an initial radial surface crack, found at an end face of the roller and close to the contact region, was induced by stress corrosion cracking (SCC). The initial crack subsequently changed shape and increased in size under growth through fatigue and finally formed a quarter-circle radial crack centred on the end face corner of the roller. Numerically computed stress intensity factors for the final crack showed that crack loading was predominantly in Mode II. For a crack size as observed on the fracture surface, the maximum service load, as specified by the manufacturer, enhanced by a certain roller bearing misalignment effect, was sufficient for failure through fracture.     

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.This work has been supported by ONR, with Dr. Y. Rajapakse as the program official.     

A treatment of interfacial cracks in the presence of friction

Frictional sliding on interface crack surfaces results in weak crack tip stress singularity and zero strain energy release rate. A fracture criterion based on finite extension strain energy release rate, is proposed to capture the intrinsic fracture toughness. The finite extension strain energy release rate is shown to represent the magnitude of the singular stress field. Numerical simulations of a center crack in a bimaterial infinite medium under remote shear as well as fiber pull-out and push-out in composite materials are presented to illustrate the frictional effect in both small and large scale contacts near the crack tip.