SUMMARYA fast numerical algorithm to compute the local and overall responses of non-linear composite materials is developed. This alternative formulation allows us to improve the convergence of the existing method of Moulinec and Suquet (e.g. Comput. Meth. Appl. Mech. Eng. 1998; 157(1-2):69-94). In the present method, a non-linear elastic (or conducting) material is replaced by infinitely many locally linear thermoelastic materials with moduli that depend on the values of the local fields. This makes it possible to use the advantages of an algorithm developed by Eyre and Milton (Eur. Phys. J. Appl. Phys. 1999; 6(1):41-47), which has faster convergence. The method is applied to compute the local fields as well as the effective response of non-linear conducting and elastic periodic composites.
The paper addresses the problem of correlation within an array of parallel dislocations in a crystalline solid. The first two of a hierarchy of equations for the multi-point distribution functions are derived by treating the random dislocation distributions and the corresponding stress fields in an ensemble average framework. Asymptotic reasoning, applicable when dislocations are separated by small distances, provides equations that are independent of any specific kinetic law relating the velocity of a dislocation to the force acting on it. The only assumption made is that the force acting on any dislocation remains finite. The hierarchy is closed by making a standard closure approximation. For the particular case of a population of parallel screw dislocations of the same sign moving on parallel slip planes the solution for the pair distribution function is found analytically. For the dislocations having opposite signs the system of equations suggests that in ensemble average only geometrically necessary dislocations correlate, while balanced positive and negative dislocations would create dipoles or annihilate. Direct numerical simulations support this conclusion. In addition, the relation of the dislocation correlation to strain gradient theories and size effect is shown and discussed.
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