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A practical framework for generating cross correlated random fields with a specified marginal distribution function, an autocorrelation function and cross correlation coefficients is presented in the paper. The approach relies on well-known series expansion methods for simulation of a Gaussian random field. The proposed method requires all cross correlated fields over the domain to share an identical autocorrelation function and the cross correlation structure between each pair of simulated fields to be simply defined by a cross correlation coefficient. Such relations result in specific properties of eigenvectors of covariance matrices of discretized field over the domain. These properties are used to decompose the eigenproblem, which must normally be solved in computing the series expansion, into two smaller eigenproblems. Such a decomposition represents a significant reduction of computational effort.Non-Gaussian components of a multivariate random field are proposed to be simulated via memoryless transformation of underlying Gaussian random fields for which the Nataf model is employed to modify the correlation structure. In this method, the autocorrelation structure of each field is fulfilled exactly while the cross correlation is only approximated. The associated errors can be computed before performing simulations and it is shown that the errors happen only in the cross correlation between distant points and that they are negligibly small in practical situations.Some comments on available techniques for simulation of underlying random variables in connection with the accuracy of basic fields' statistics at a given sample size are made. For this purpose a simple error assessment procedure is presented.Simulated random fields can be used both for representation of spatially correlated properties of structure or random load in the stochastic finite element method (SFEM). An example of this application is related to size effect studies in the nonlinear fracture mechanics of concrete, and is used to illustrate the method.

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We attempt the identification, study and modeling of possible sources of size effects in concrete structures acting both separately and together. We are particularly motivated by the interplay of several identified scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material law to show that a large portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the complex size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments is explained by the presence of a weak surface layer of constant thickness (caused e.g., by drying, surface damage, aggregate size limitation at the boundary, or other irregularities). All three named sources (deterministic-energetic, statistical size effects, and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complex quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very difficult for quasibrittle materials such as concrete.

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The present study addresses the influence of variations in material properties along the multi-filament yarn on the overall response in the tensile test. In Part I (Chudoba, Vořechovský and Konrad, 2006), we have described the applied model and studied the influence of scatter of material characteristics varying in the cross-section with no variations along the filaments. In particular, we analyzed the influence of varying cross-sectional area, filament length and delayed activation. Inclusion of these effects has lead to a better interpretation of the experimental data, especially with respect to the gradual stiffness activation, post-peak behavior and some form of size effect. In the present paper, the lengthrelated distributions of local stiffness and strength are included in terms of theoretical considerations and by applying the Monte Carlo type simulation of random fields. Such an approach allows us (1) to demonstrate the strong need for including length scale to random fluctuation of strength along the filaments and (2) to combine several sources of randomness in a single analysis so that their significance can be evaluated from the tensile test response.

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