A mathematical model for elasticity using calculus on discrete manifolds

We propose a mathematical model to represent solid materials with discrete lattices and to analyse their behaviour by calculus on discrete manifolds. Focus is given on the mathematical derivation of the lattice elements by taking into account the stored energy associated with them. We provide a matrix formulation of the nonlinear system describing elasticity with exact kinematics, known as finite strain elasticity in continuum mechanics. This formulation is ready for software implementation and may also be used in atomic scale models as an alternative to existing empirical approach with pair and cohesive potentials. An illustrative example, analysing a local region of a node, is given to demonstrate the model performance.