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Mathematical and numerical multi-scale modelling of multiphysics problems

Schrefler B. A., Boso D. P., Pesavento F., Gawin D., Lefik M.

Computer Assisted Mechanics and Engineering Sciences

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2011

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Vol. 18, no. 1-2

91-113

EN

In this paper we discuss two multi-scale procedures, both of mathematical nature as opposed to purelynumerical ones. Examples are shown for the two cases. Attention is also devoted to thermodynamical aspects such as thermodynamic consistency and non-equilibrium thermodynamics. Advances for the firstaspect are obtained by adopting the thermodynamically constrained averaging theory TCAT as shown in the case of a stress tensor for multi-component media. The second aspect has allowed to solve numerically,with relative ease, the case of non-isothermal leaching. The absence of proofs of thermodynamic consistencyin case of asymptotic theory of homogenization with ?nite size of the unit cell is also pointed out.

2

Recent developments in numerical homogenization

Boso D. P., Lefik M., Schrefler B. A.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

161-183

EN

This paper deals with homogenization of non linear fibre-reinforced composites in the coupled thermo-mechanical field. For this kind of structures, i.e. inclusions randomly dispersed in a matrix, the self consistent methods are particularly suitable to describe the problem. Usually, in the framework of the self consistent scheme the homogenized material behaviour is obtained with a symbolic approach. For the non linear case, that method may become tedious. This paper presents a different, fully numerical procedure. The effective properties are determined by minimizing a functional expressing the difference (in some chosen norm) between the solution of the heterogeneous problem and the equivalent homogenous one. The heterogeneous problem is solved with the Finite Element method, while the second one has its analytical solution. The two solutions are written as a function of the (unknown) effective parameters, so that the final global solution is found by iterating between the two single solutions. Further, it is shown that the considered homogenization scheme can be seen as an inverse problem and Artificial Neural Networks are used to solve it.

3

Modelling damage processes of concrete at high temperature with thermodynamics of multi-phase porous media

Gawin D., Pesavento F., Schrefler B. A.

Journal of Theoretical and Applied Mechanics

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2006

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Vol. 44 nr 3

505-532

EN

In this paper, the authors present a mathematical description of a multi-phase model of concrete based on the second law of thermodynamics. Exploitation of this law allowed researchers to obtain definitions and constitutive relationships for several important physical quantities, like capillary pressure, disjoining pressure, effective stress, considering also the effect of thin films of water. A mathematical model of hygro-thermo-chemo-mechanical phenomena in heated concrete, treated as a multi-phase porous material, has been formulated. Shrinkage strains were determined using thermodynamic relationships via capillary pressure and area fraction coefficients, while thermo-chemical strains were related to thermo-chemical damage. In the model, a classical thermal creep formulation has been modified and introduced into the model. Results of numerical simulations based on experimental tests carried out at NIST laboratories for two types of concrete confirmed the usefulness of the model in the prediction of the time range, during which the effect of concrete spalling may occur.

PL

Autorzy pracy prezentują równania modelu wielofazowego betonu wyprowadzone na podstawie II zasady termodynamiki. Zastosowanie tej zasady pozwoliło badaczom na sformułowanie definicji i związków konstytutywnych dla kilku ważnych wielkości fizycznych, takich jak: ciśnienie kapilarne, ciśnienie rozklinowywujące, naprężenie efektywne oraz na uwzględnienie efektu wywołanego przez cienkie warstwy wody obecnej w betonie. Przedstawiono model matematyczny higro-termo-chemo-mechanicznych zjawisk zachodzących w podgrzewanym betonie traktowanym jako ośrodek porowaty. Skurcz betonu określono za pomocą równań termodynamiki opisujących ciśnienie kapilarne i współczynniki udziału powierzchniowego, podczas gdy odkształcenia termochemiczne skorelowano z parametrem zniszczenia chemicznego. Do rozważań wprowadzono zmodyfikowaną teorię pełzania opartą na sformułowaniu klasycznym. Wyniki przeprowadzonych symulacji numerycznych, bazujących na badaniach doświadczalnych zrealizowanych w laboratoriach NIST na dwóch typach betonu, potwierdziły użyteczność zaprezentowanego modelu w przewidywaniu przedziału czasu, w którym może dojść do termicznego odpryskiwania betonu.