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This paper proposes computationally efficient frequency domain formulations for two well‐known multiaxial fatigue failure criteria, namely Matake’s critical plane criterion and Crossland’s criterion. For that purpose, it is shown how fatigue‐related variables involved in both criteria can be estimated from the power spectral density matrix of the local stress vector. The finite element model of an example structure is then used to illustrate the application of the proposed frequency domain approaches. It is observed that both frequency domain formulations produce consistent results when compared with those obtained in the time domain from Monte‐Carlo simulations of local stress vectors while offering tremendous computer savings. A frequency domain tool indicating whether the principal stress directions do rotate with time or not during the loading at a given location in the structure is also presented.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. International Statistical Institute (ISI) is collaborating with JSTOR to digitize, preserve and extend access to International Statistical Review / Revue Internationale de Statistique. SummaryIn crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of the Slepian model and the regression method and shows how they can be used to obtain good numerical approximations to various crossing variables. It gives a detailed account of the regression method for Gaussian processes with auxiliary variables chosen in a recursive way. It also presents a package of computer programs for the numerical calculations, and gives numerical examples on excursion lengths as well as wavelength and amplitude distributions. Further examples deal with an engineering 'jump-andbump' problem, and excursions for a X2-process.

a b s t r a c tMultivariate Laplace distribution is an important stochastic model that accounts for asymmetry and heavier than Gaussian tails, while still ensuring the existence of the second moments. A Lévy process based on this multivariate infinitely divisible distribution is known as Laplace motion, and its marginal distributions are multivariate generalized Laplace laws. We review their basic properties and discuss a construction of a class of moving average vector processes driven by multivariate Laplace motion. These stochastic models extend to vector fields, which are multivariate both in the argument and the value. They provide an attractive alternative to those based on Gaussianity, in presence of asymmetry and heavy tails in empirical data. An example from engineering shows modeling potential of this construction.

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