Probabilistic models for defect initiated fracture in ceramics

Fracture tests on hot-pressed silicon nitride containing voids and several types of inclusion have been conducted. Fracture models pertinent to each defect type have been proposed and correlated with the data. The specificity of the fracture models is emphasized, and the various trends with defect size that result from the models are described. The resultant fracture probability relations are one of the key inputs to accept/reject decisions for nondestructive failure prediction.     

Probabilistic fracture assessment of surface cracked pipes using strain-based approach

Simplified strain-based fracture mechanics equations, established for external surface cracked pipelines subjected to an external bending load, are presented and used in probabilistic assessment of a pipeline girth weld. The model takes into account several parameters, such as variation in crack depth, crack length, internal pressure and material hardening. The critical strain from ductile tearing in the cracked pipeline is found by using the tangency criterion. The reliability problem is solved using first and second order reliability methods for different pipe dimensions and load levels.     

Probabilistic dose calculations and sensitivity analyses using analytic models

Newly developed analytic models mimic numerical models for radionuclide transport and dose calculations for a deep repository for spent nuclear fuel, reducing computation times more than three orders of magnitude. In this paper, the analytic models are used to extend preliminary probabilistic dose calculations reported in a recent performance assessment for a deep repository in Sweden. It is demonstrated that the analytic models are useful for gaining insights into the probabilistic properties of the system concerning, e.g. the importance of input variable correlations and various properties of input distributions. Regarding sensitivity analyses, the analytic model is used for screening out nuclides which do not influence the calculation end-point, for demonstrating monotonicity and for developing tailored regression models with non-linear expressions.     

Probabilistic cell modelling of cleavage fracture

Various aspects on probabilistic modelling of cleavage fracture are discussed. The investigation involves consideration of a unit cell with an explicitly modelled void. The results from this model are compared with results for the case when the void content is accounted for in the sense of a Gurson-Tvergaard law. It is found that explicit modelling of the void can give substantially different results for the fracture probability. The effect depends on the exponent in the assumed Weibull distribution, the threshold stress, the constraint conditions and the hardening of the material. This revised version was published online in August 2006 with corrections to the Cover Date.     

The analysis of dynamic fracture using lumped mass-spring models

It is shown that many dynamic effects in fracture can be usefully modelled using lumped mass-spring models of the fracturing speciemens, since many effects are not a consequence of stress waves but of the dynamic behaviour of the specimen and loading system. When the energy release rate, G, is calculated for specimens with harmonic motions present, it is found that multiple solutions occur which, if analysed using quasi-static calibration factors, can give apparent scatter and limiting speeds in G _c . This simple approach can be used to interpret several types of experimental data.

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Résumé On montre que de nombreux effets dynamiques en rupture peuvent être utilement modélisés en ayant recours à des masses et des ressorts pour simuler des éprouvettes en cours de rupture, dans la mesure où plusieurs de ces effets ne résultent pas des ondes de contraintes, mais plutôt du comportement dynamique de l'éprouvette et de son système de charge.Lorsque l'on calcule la vitesse G de relaxation de l'énergie pour des éprouvettes où sont présents des mouvements parmoniques, on trouve que se présente une multitude de solutions qui, lorsqu'on les analyse en utilisant des facteurs d'étalonnage quasi-statiques, peuvent donner une apparente dispersion et des vitesses limites pour Gc.Cette approche simple est utilisable pour l'interprétation de divers types de données expérimentales.

     

Probabilistic aspects of fracture energy of concrete

A test method to determine fracture energy and strain-softening in direct tension is described. Experimental results on cylinders of equal diameter and varying length are reported. It is found that the tensile strength decreases with increasing volume while the fracture energy remains constant within the observed volume range. By means of numerical simulation, it is shown that in a direct tension test several fracture process zones appear in the initial states of cracking and that final rupture is induced by the development of only one of these fracture zones. This phenomenon has been observed experimentally by other authors. A comparatively large number (44) of identical samples were tested by using the wedge-splitting test. Half the specimens were grooved. The fracture energy of the grooved and ungrooved specimens turned out to be the same within the given range of accuracy. It was observed experimentally and simulated numerically that in grooved specimens the crack is forced to follow a ragged fracture surface which is statistically not the weakest one. In an ungrooved specimen the crack path generally diverts from the centre line and advances through weaker zones. For the formation of these skew cracks, however, more energy is consumed due to aggregate interlock. In addition, the fracture process zone observed in ungrooved specimens is generally wider.     

Numerical and experimental validation of computational models for mode I composite fracture failure

This work compares the numerical determination of the strain energy release rate in mode I for the interlaminar fracture of composite laminates by means of two different models: the virtual crack closure technique (VCCT) and the two-step extension method. Results were compared with empirical data obtained from double cantilever beam (DCB) tests carried out on unidirectional AS4/8552 carbon/epoxy laminates. The study showed that, in a pure mode I state, results obtained via the two-step method were in agreement with a straightforward calculation of the elastic energy variation in the system. Results from VCCT calculations were slightly lower than those obtained through the two-step method using four node elements in plane strain. As expected, results from both models converge as element length decreases.     

Probabilistic failure prediction for FRP composites

Results are presented from a theoretical study on failure-locus prediction for unidirectional FRP laminae under complex in-plane loading, taking into account statistical aspects of the basic strengths of the material. The purpose of the analysis is to establish simple rules and methodologies for failure prediction under specific reliability requirements. Therefore, the problem one is faced with is the definition of the cumulative distribution function (CDF) of the failure condition if the respective CDFs of the basic material strengths are known by experiment. To this end, two analytical approaches, namely a functional expansion technique and the introduction of Pearson’s semi-empirical distribution function, were developed and implemented in software. Both methods were shown to predict satisfactory results compared with Monte Carlo simulated ones and experimental data, wherever available. Finally, a semi-deterministic approach was examined according to which the failure criterion is used in the usual deterministic way but with basic strength values of a certain reliability level. Results from this simple and fast method were found in good agreement with those derived by expensive pure statistical methods or numerically simulated ones and experimental data.     

Probabilistic model of fatigue fracture of carbon nanostructures

Presented evaluation of risk of fatigue fracture of carbon nanostructures (CN) is based on: (a) the criterion of the threshold of the dissipated energy of tensile stresses, and (b) simulation of stochastic character of fracture by the failures in queueing system. The assessments of probability of fatigue fracture of CN in conditions of uniaxial tension, plane torsion and uniaxial compression are done.     

Probabilistic failure assessment of ferritic steels using the master curve approach including constraint effects

The present study is concerned with an enhancement of the master curve approach for the probabilistic failure assessment of ferritic steel structures considering constraint effects. In an experimental program based on a variety of different specimens, distinct effects of the specimen geometry on the reference temperature T₀ are observed. The experimental data base is examined in terms of the K-T_stress-, J-A₂-, J-Q- and J-h-concepts. Based on the results, a constraint enhancement for the master curve concept is suggested consisting of a constraint dependent temperature shift or an alternative constraint dependent scaling of the stress intensity factor.     

Probabilistic fracture mechanics of 2D carbon-carbon composites

The fracture toughness of two types of carbon fabric reinforced carbon composite (KKARB^®, Types A and C) is evaluated, the mechanisms of crack propagation resistance are identified and both are related to microstructural differences.The two composites have the same constituents, i.e. fibers, yarns, fabric weaving and matrix precursor. However, different processing cycles result in apparent differences in microstructure (e.g. different number and length distribution of microcracks, crimp angle) and toughness.The crack diffusion model (CDM) is invoked to parameterize the fluctuating strength field of the composite in terms of an average fracture energy , a minimum fracture energy 179-1 and a shape parameter . The values of 179-1 are in direct correlation with the average size of microcracks in each composite, and is found to correlate with the scatter in fracture toughness.     

Mode I fracture of adhesive joints using tailored cohesive zone models

Cohesive zone models are explored in order to study cleavage fracture in adhesive bonded joints. A mode I cohesive model is defined which correlates the tensile traction and the displacement jump (crack faces opening) along the fracture process zone. In order to determine the traction-separation relation, the main fracture parameters, namely the cohesive strength and the fracture energy, as well as its shape, must be specified. However, owing to the difficulties associated to the direct measurement of the fracture parameters, very often they are obtained by comparing a measured fracture property with numerical predictions based on an idealized traction separation relation. Likewise in this paper the cohesive strength of an adhesive layer sandwiched between elastic substrates is adjusted to achieve a match between simulations and experiments. To this aim, the fracture energy and the load-displacement curve are adopted as input in the simulations. In order to assess whether or not the shape of the cohesive model may have an influence on the results, three representative cohesive zone models have been investigated, i.e. exponential, bilinear and trapezoidal. A good agreement between experiments and simulations has been generally observed. However, a slight difference in predicting the loads for damage onset has been found using the different cohesive models.     

Reliability study of two engineering models using LU decomposition

This paper presents two different methods — LU decomposition and Runge–Kutta — for reliability engineering assessment. These methods are used to calculate the steady-state probabilities and frequencies of two different engineering models. The effect of the different methods is shown on the simple numerical examples by comparing the steady-state probabilities and frequencies of both models. The paper concludes that the LU decomposition method is useful to practicing engineers and students of reliability concepts. Also, the final results for both methods is almost the same.     

Probabilistic fracture mechanics by using Monte Carlo simulation and the scaled boundary finite element method

A numerical technique to model the effect of uncertainties in the crack geometry on the reliability of cracked structures is presented. The shape sensitivity analysis of stress intensity factors to the crack size and orientation is performed by using the scaled boundary finite element method (SBFEM). Only a single boundary mesh is required. The varying crack size and orientation are represented by simply moving the scaling center and without the need for remeshing. The reliability assessment is performed by Monte Carlo simulations. Numerical examples are analyzed to verify the accuracy and demonstrate the efficiency and simplicity of the proposed technique.     

Probabilistic fracture mechanics by the boundary element method

In this paper, a new boundary element formulation is presented for probabilistic analysis of crack problems in context of linear elastic fracture mechanics. The method involves an implicit differentiation method for calculating fracture response derivatives with respect to random parameters and applies first-order reliability method to predicting the reliability of cracked structures. Direct differentiation is used to obtain the derivatives of fracture parameters with respect to crack size which is required for probabilistic analysis. Numerical examples are presented to illustrate the proposed method. The accuracy of the proposed method is demonstrated by comparing with either analytical solution or other established method.     

Probabilistic models of fatigue crack growth—II

The history dependence of the statistical evolution of fatigue crack growth is demonstrated for one set of data. A generalized B-model which includes this dependence is presented whose parameters are estimated via first and second moments including covariance. Implications of any dependence of modeling of fatigue crack growth are discussed, motivated by application to problems in reliability and maintainability.     

Self-similar solutions for axisymmetric jets using two turbulence models

Similarity solutions for incompressible axisymmetric jets using a k–? and a constant eddy diffusivity turbulence models are considered. For the k–? model, the governing equations are very complex. Therefore, a transformation that simplifies these equations and makes them amenable to efficient numerical solution is used. Results for the velocity, turbulent kinetic energy and dissipation rate are obtained. Also, velocity decay rate, growth rate, entrainment and kinetic energy decay rate are determined. A comparison with experimental data and other works utilizing a parabolic marching asymptotic solution to the full partial differential equations is made. This comparison shows that similarity solutions are more accurate than solutions using numerical marching procedures.     

Nonlinear failure analysis of carbon nanotubes by using molecular-mechanics based models

Equivalent nonlinear fracture models for pristine and reconstructed one- and two-atom vacancy defected single wall carbon nanotubes are developed by using the molecular mechanics based models where the initial reconstructed nanotube models are obtained by using molecular dynamic simulations and nonlinear characteristic of the covalent bonds are obtained by using the modified Morse potential. As a result of analyses, it is concluded that fractures of all types of nanotubes are brittle, armchair nanotubes are stiffer than zigzag nanotubes and vacancy defects significantly affect the mechanical behavior of nanotubes. In brief, fracture stress and strain values of pristine armchair nanotubes are respectively 30% and 32% larger than those of pristine zigzag nanotubes, and predicted failure stress and strain values of vacancy defected nanotubes are respectively 27% and 52% smaller than those of pristine ones. It is shown that large deformation and nonlinear geometric effects are important on fracture behavior of nanotubes. Comparisons are made with the failure stress and strain results reported in literature that show good agreement with our results.     

A finite element ductile failure simulation method using stress-modified fracture strain model

This paper proposes a new method to simulate ductile failure using finite element analysis based on the stress-modified fracture strain model. A procedure is given to determine the stress-modified fracture strain as a function of the stress triaxiality from smooth and notched bar tensile tests with FE analyses. For validation, simulated results using the proposed method are compared with experimental data for cracked bar (tensile and bend) tests, extracted from API X65 pipes, and for full-scale burst test of gouged pipes, showing overall good agreements. Advantages in the use of the proposed method for practical structural integrity assessment are discussed.     

Probabilistic initial failure strength of hybrid and non-hybrid laminates

This paper deals with the initial failure of unidirectional hybrid laminates and [±θ/90]_s non-hybrid laminates. The initial failure strength or strain at the failure of low elongation layers is analysed by the statistical approach based upon the weakest link model. First ply failure is adopted as a criterion for the initial failure of the laminate. An expression for determining the first ply failure strength has been derived and this expression can be reduced to a volumetric relation as a special case. It is also shown that the initial failure strength or strain is greater in composites composed of low elongation and high elongation materials than in the pure low elongation composite. This is the result of the “size effect”, that is the failure probability is lower in the composite with the smaller size of low elongation material. A good agreement between the theoretical and experimental results is achieved. On leave from the Institute of Interdisciplinary Research, The University of Tokyo, Japan.