The aim of this paper is the development of equilibrium and non-equilibrium extensions of the quasicontinuum (QC) method. We first use variational mean-field theory and the maximum-entropy formalism for deriving approximate probability distribution and partition functions for the system. The resulting probability distribution depends locally on atomic temperatures defined for every atom and the corresponding thermodynamic potentials are explicit and local in nature. The method requires an interatomic potential as the sole empirical input. Numerical validation is performed by simulating thermal equilibrium properties of selected materials using the LennardJones pair potential and the EAM potential and comparing with molecular dynamics results as well as experimental data. The max-ent variational approach is then taken as a basis for developing a three-dimensional non-equilibrium finite temperature extension of the quasicontinuum method. This extension is accomplished by coupling the local temperature-dependent free energy furnished by the max-ent approximation scheme to the heat equation in a joint thermo-mechanical variational setting. Results for finite-temperature nanoindentation tests demonstrate the ability of the method to capture non-equilibrium transport properties and differentiate between slow and fast indentation.

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In this paper, we propose a computational method for coarse graining the atomistic description at finite temperature using formal asymptotics. The method is based on the ansatz that there exists a separation of scales between the time scale of the atomic fluctuations and that of the thermodynamic processes, such as thermal expansion. We use the WKB method to propose an averaging scheme for treating the thermal degrees of freedom and deriving an effective Hamiltonian for the atomistic system. This energy functional is incorporated into the quasicontinuum framework to achieve a seamless coarse graining on the spatial scale. Numerical validation is performed by computing the thermal equilibrium properties of selected materials. The scope of the method based on the use of perturbation theory is discussed, and its capability is illustrated by way of simulating dislocation nucleation under a nanoindenter.

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