Stochastic modeling of fatigue crack propagation

This paper presents a stochastic model of fatigue-induced crack propagation in metallic materials. The crack growth rate predicted by the model is guaranteed to be non-negative. The model structure is built upon the underlying principle of Karhunen–Loève expansion and does not require solutions of stochastic differential equations in either Wiener integral or Itô integral setting. As such this crack propagation model can be readily adapted to damage monitoring and remaining life prediction of stressed structures. The model results have been verified by comparison with experimental data of time-dependent fatigue crack statistics for 2024-T3 and 7075-T6 aluminum alloys.     

Probabilistic approach for break-up and coalescence in bubbly-flow and coupling with CFD codes

In this paper, a CFD model for bubbly-flow including break-up and coalescence is discussed. The probabilistic method, developed according to this model, is tested for an industrial configuration: numerical bubble-size distributions are compared with those derived by experiments.     

Probabilistic simulation of offshore collisions

Collisions between tankers and offshore structures may result in serious problems such as oil pollution. It is therefore of interest to estimate the probability of a collision. This is done here by means of a simulation model using the GASP IV simulation program. The new method is based on a statistical prediction of the ship's path after some kind of critical failure affecting the manoeuvrability of the tanker has taken place.     

Probabilistic modeling of multiperiod service levels

This study investigates multiperiod service level (MSL) policies in supply chains facing a stochastic customer demand. The objective of the supply chains is to construct integrated replenishment plans that satisfy strict stockout-oriented performance measures which apply across a multiperiod planning horizon. We formulate the stochastic service level constraints for the fill rate, ready rate, and conditional expected stockout MSL policies. The modeling approach is based on the concept of service level trajectory and provides reformulations of the stochastic planning problems associated with each MSL policy that can be efficiently solved with off-the-shelf optimization solvers. The approach enables the handling of correlated and non-stationary random variables, and is flexible enough to accommodate the implementation of fair service level policies, the assignment of differentiated priority levels per products, or the introduction of response time requirements. We use an earthquake disaster management case study to show the applicability of the approach and derive practical implications about service level policies.     

Probabilistic modeling of computer system availability

System availability is becoming an increasingly important factor in evaluating the behavior of commercial computer systems. This is due to the increased dependence of enterprises on continuously operating computer systems and to the emphasis on fault-tolerant designs. Thus, we expect availability modeling to be of increasing interest to computer system analysts and for performance models and availability models to be used to evaluate combined performance/availability (performability) measures. Since commercial computer systems are repairable, availability measures are of greater interest than reliability measures. Reliability measures are typically used to evaluate nonrepairable systems such as occur in military and aerospace applications. We will discuss system aspects which should be represented in an availability model; however, our main focus is a state of the art summary of analytical and numerical methods used to solve computer system availability models. We will consider both transient and steady-state availability measures and for transient measures, both expected values and distributions. We are developing a program package for system availability modeling and intend to incorporate the best solution methods.     

Thermo-mechanical modeling of crack propagation in dynamically loaded elastomer specimens using a scaled boundary finite element approach

Recently, a scaled boundary finite element (SBFE) formulation for geometrically and physically nonlinear materials has been developed using the scaled boundary finite element method (SBFEM). The SBFE formulation has been employed to describe plane stress problems of notched and unnotched hyperelastic elastomer specimens. In this contribution, the derived SBFE formulation is extended to nonlinear time- and temperature-dependent material behavior. Subsequently, the SBFE formulation is incorporated into a crack propagation scheme to model crack propagation in cyclically loaded elastomer specimens of the so-called tear fatigue analyzer (TFA). (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the steady propagation of a semi-infinite crack

We consider the rectilinear propagation of a semi-infinite crack with constant velocity in a crystal structure. We obtain the solutions of homogeneous boundary-value problems for the corresponding difference-differential operators in spaces of one and two dimensions. We give a justification of the computational aspect of the problem. Bibliography: 8 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 127–153.     

Supersonic crack propagation in blocks of polymethyl methacrylate

It is shown that a giant laser pulse can cause supersonic crack propagation in PMMA. The results of the experiments and a microstructural analysis of the fracture surface are presented. A study of the microstructure shows that supersonic crack propagation is associated with the propagation of a shock wave in the focal region.Institute of Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1027–1029, November–December, 1971.     

多重点源自由面波运动规律数值模拟

基于带耗散源格林函数去奇异积分数值方法,建立了均匀流浸没下双点源自由面波高计算模型,研究在不同耗散系数、弗劳德数及不同点源距离下自由面波高的变化规律以及自由面波形等高线的变化规律.数值模拟结果与前人的数值结果作验证比对,吻合较好.     

Probabilistic modeling of random variables with inconsistent data

The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.     

A computational framework of three dimensional configurational-force-driven crack propagation

A variational formulation of quasi-static brittle fracture is considered and a new finite-element-based computational framework is developed for propagation of cracks in three-dimensional bodies. We outline a consistent thermodynamical framework for crack propagation in elastic solids and show that the crack propagation direction associated with the classical Griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front. The evolving crack discontinuity is realized by the doubling of critical nodes and triangular interface facets of the tetrahedral mesh. The crucial step for the success of the procedure is its embedding into an r-adaptive crack-facet reorientation procedure based on configurational-force-based indicators in conjunction with crack front constraints. We further propose a staggered algorithm which minimizes the stored energy at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive definite subproblems with successively decreasing overall stiffness, providing a very robust algorithmic setting in the postcritical range. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of crack propagation in 3-dimensional anisotropic solids

This contribution presents ideas, how crack propagation in three-dimensional solids composed of anisotropic materials can be predicted using the Griffith energy principle. Since the work of Irwin the change of potential energy caused by a straight elongation of a crack in an isotropic two-dimensional homogeneous structure can be expressed in quadratic terms of the stress intensities at the crack tip. This result was generalized in the last decades using methods of asymptotic analysis by many authors [1] to more complicated geometries, to anisotropic and inhomogeneous materials. With the energy release rate at hand, quasi-static scenarios of crack propagation can be simulated for plane problems [2], but this is still a complicated task for three-dimensional problems [3]. We show an idea how the change of energy caused by propagation of a crack surface in a fully three-dimensional solid of nearly arbitrary shape can be computed in anisotropic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An exact and efficient X-FEM-based reanalysis algorithm for quasi-static crack propagation

This study suggests a specific reanalysis algorithm termed decomposed updating reanalysis (DUR) for quasi-static linear crack propagation based on the extended finite element method (X-FEM). It is well known that the number of iterative steps is usually very large during X-FEM simulation procedures because a small crack increment is required to improve the accuracy of the simulation. However, according to the features of the X-FEM, the small crack increment only influences the nearby elements and only leads the local change of the stiffness matrix at each iterative step. Therefore, the DUR method is proposed to accelerate the X-FEM solving process by only calculating the changed part of the equilibrium equations. Moreover, the local updating strategy can efficiently update the modified stiffness matrix and the Cholesky factorization. Compared with other reanalysis algorithms, such as combined approximations (CA), the DUR method is more accurate. Numerical examples demonstrate that the DUR method improves the efficiency of the X-FEM significantly with a high accuracy.     

Simulation of short crack propagation using a hybrid boundary element technique

Service life of cyclically loaded components is often determined by the propagation of short fatigue cracks, which is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of such stage I-cracks is presented. The crack is discretised by dislocation discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient simulation of wave propagation in heterogeneous materials

In this contribution we explain the core idea of the recently proposed spectral cell method, which combines a fictitious domain approach with finite elements of high order and noticeably relieves the burden of mesh generation. Moreover, it employs mass lumping techniques and significantly reduces the computational expenditure. Our studies show that the spectral cell method leads to similar results yet with less computational effort as compared to standard techniques. These properties turn the method to a viable tool for the wave propagation analysis of structures that obey a complicated geometry. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical modeling of electromagnetic wave propagation in nonlinear electrodynamics

The eikonal method for an electromagnetic wave propagating according to the laws of non-linear electrodynamics in vacuum in external electromagnetic and gravitational fields is developed. A mathematical model of the propagation of electromagnetic signals in the parameterized post-Maxwellian electrodynamics in vacuum is constructed. As an example of using the proposed method, the angles of the nonlinear electrodynamical and gravitational curvature of the normal wave rays propagating in the field of a charged collapsar are calculated.