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We consider the time evolution of a one dimensional $n$-gradient continuum.Our aim is to construct and analyze discrete approximations in terms ofphysically realizable mechanical systems, called microscopic because they areliving on a smaller space scale. We validate our construction by proving aconvergence theorem of the microscopic system to the given continuum, as thescale parameter goes to zero.

Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials