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Nonlocal constitutive laws generated by matrix functions: Lattice Dynamics Models and their Continuum Limits

We analyze one-dimensional discrete and quasi-continuous linear chains of $N>>1$ equidistant and identical mass points with periodic boundary conditions and generalized nonlocal interparticle interactions in the harmonic approximation. We introduce elastic potentials which define by Hamilton's principle discrete "Laplacian operators" ("Laplacian matrices") which are operator functions ($N\times N$-matrix functions) of the Laplacian of the Born-von-Karman linear chain with next neighbor interactions. The non-locality of the constitutive law of the present model is a natural consequence of the {\it non-diagonality} of these Laplacian matrix functions in the $N$ dimensional vector space of particle displacement fields where the periodic boundary conditions (cyclic boundary conditions) and as a consequence the (Bloch-) eigenvectors of the linear chain are maintained. In the quasi-continuum limit (long-wave limit) the Laplacian matrices yield "Laplacian convolution kernels" (and the related elastic modulus kernels) of the non-local constitutive law. The elastic stability is guaranteed by the positiveness of the elastic potentials. We establish criteria for "weak" and "strong" nonlocality of the constitutive behavior which can be controlled by scaling behavior of material constants in the continuum limit when the interparticle spacing $h\rightarrow 0$. The approach provides a general method to generate physically admissible (elastically stable) {\it non-local constitutive laws} by means of "simple" Laplacian matrix functions. The model can be generalized to model non-locality in $n=2,3,..$ dimensions of the physical space.