Askes H., Metrikine A.V.


Higher-order continua derived from discrete media: continualisation aspects and boundary conditions


International Journal of Solids and Structures 42, 187-202, (2005)



Summary. In this paper, the derivation of higher-order continuum models from a discrete medium is addressed, with the following aims: (i) for a given discrete model and a given coupling of discrete and continuum degrees of freedom, the continuum should be defined uniquely, (ii) the continuum is isotropic, and (iii) boundary conditions are derived consistently with the energy functional and the equations of motion. Firstly, a comparison is made between three continualisation methods, namely based on the equations of motion, on the energy functional and on the constitutive relation. The first two are shown to give identical results. The third approach deviates from the first two and must not be used. Secondly, the issue of isotropy is addressed. A new approach is developed in which two, rather than one, layers of neighbouring particles are considered. Finally, the formulation and interpretation of boundary conditions is treated. By means of the Hamilton-Ostrogradsky principle, boundary conditions are derived that are consistent with the energy functional and the equations of motion. A relation between standard stresses and higher-order stresses is derived and used to make an interpretation of the higher-order stresses. An additional result of this study is the non-uniqueness of the higher-order contributions to the energy.