This paper details a procedure to determine lower bounds on the size of representative volume elements (RVEs) by which the size of the RVE can be quantified objectively for random heterogeneous materials. Here, attention is focused on granular materials with various distributions of inclusion size and volume fraction of inclusions. An extensive analysis of the RVE size dependence on the various parameters is performed. Both deterministic and stochastic parameters are analysed. Also, the effects of loading mode and the parameter of interest are studied. As the RVE size is a function of the material, some material properties such as Young's modulus and Poisson's ratio are analysed as factors that influence the RVE size. The lower bound of RVE size is found as a function of the stochastically distributed volume fraction of inclusions; thus the stochastic stability of the obtained results is assessed. To this end a newly defined concept of stochastic stability (DH-stability) is introduced by which stochastic effects can be included in the stability considerations. DH-stability can be seen as an extension of classical Lyapunov stability. As is shown, DH-stability provides an objective tool to establish the lower bound nature of RVEs for fluctuations in stochastic parameters.
A new methodology to obtain metallic functional materials with predefined sets of strength properties has been developed. It has been shown that in order to accurately estimate set of material properties at the macro‐level, information from the micro‐level needs to be taken into account. As a result a two‐level estimation model, based on the theory of fuzzy sets, has been proposed. To demonstrate the developed methodology, a reinforcing steel has been analysed. Using microstructural information, derived from an available set of experimentally obtained digital images of material microsections under different heat treatment conditions, macroscopic strength properties of reinforcing steel have been determined.
The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of 'mistakes' in coefficients and delays on solutions' behavior of the delay differential neutral systemThis topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a 'real' system with uncertain coefficients and/or delays and corresponding 'model' system. We develop the so-called Azbelev W-transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W-transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a 'model' used in W-transform is 'close' to a given 'real' system. In this paper we choose, as the 'models' , systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient 'models' . We use the W-transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions.Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and theNew tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.
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