A Method of integration of molecular dynamics and continuum mechanics for solids

In this paper a formal system called collection of dynamical systems with dimensional reduction is considered. This is a multiscale method of mathematical description which allows to consider molecular dynamics and continuum mechanics within one theoretical framework. Transition between molecular dynamics and continuum mechanics is realized by means of the dimensional reduction procedure. In order to realize such a procedure the formulation of continuum mechanics is modified. This modification consists in incorporation scale of averaging for properties of processes considered during modelling into this formulation. As a result we introduce finite-dimensional fields on continuum only. All fundamental terms of continuum mechanics are now joined with an elementary dynamical system. In such a case continuum mechanics can be obtained by means of the dimensional reduction procedure applied to the elementary dynamical system. A numerical example of vibrating chain of material points is realized in order to show how in practice the dimensional reduction can be carried out. In this example decomposition of processes into slowly and quickly varying parts is accomplished. To this end a finite element representation of averaged fields is applied. Solutions of equations of the elementary dynamical system and the dimensionally reduced dynamical system are compared.