From molecular systems to continuum solids: A multiscale structure and dynamics

The heterogeneous multiscale method applied to inelastic polymer mechanics

Mechanisms emerging across multiple scales are ubiquitous in physics and methods designed to investigate them are becoming essential. The heterogeneous multiscale method (HMM) is one of these, concurrently simulating the different scales while keeping them separate. Owing to the significant computational expense, developments of HMM remain mostly theoretical and applications to physical problems are scarce. However, HMM is highly scalable and is well suited for high performance computing. With the wide availability of multi-petaflop infrastructures, HMM applications are becoming practical. Rare applications to mechanics of materials at low loading amplitudes exist, but are generally confined to the elastic regime. Beyond that, where history-dependent, irreversible or nonlinear mechanisms occur, not only computational cost but also data management issues arise. The micro-scale description loses generality, developing a specific microstructure based on the deformation history, which implies inter alia that as many microscopic models as discrete locations in the macroscopic description must be simulated and stored. Here, we present a detailed description of the application of HMM to inelastic mechanics of materials, with emphasis on the efficiency and accuracy of the scale-bridging methodology. The method is well suited to the estimation of macroscopic properties of polymers (and derived nanocomposites) starting from knowledge of their atomistic chemical structure. Through application of the resulting workflow to polymer fracture mechanics, we demonstrate deviation in the predicted fracture toughness relative to a single-scale molecular dynamics approach, thus illustrating the need for such concurrent multiscale methods in the predictive estimation of macroscopic properties. This article is part of the theme issue 'Multiscale modelling, simulation and computing: from the desktop to the exascale'.

Continuum balances from extended Hamiltonian dynamics

The classical procedure devised by Irving and Kirkwood in 1950 and completed slightly later by Noll produces counterparts of the basic balance laws of standard continuum mechanics starting from an ordinary Hamiltonian description of the dynamics of a system of material points. Post-1980 molecular dynamics simulations of the time evolution of such systems use extended Hamiltonians such as those introduced by Andersen, Nosé, and Parrinello and Rahman. The additional terms present in these extensions affect the statistical properties of the system so as to capture certain target phenomenologies that would otherwise be beyond reach. We here propose a physically consistent application of the Irving-Kirkwood-Noll procedure to the extended Hamiltonian systems of material points. Our procedure produces balance equations at the continuum level featuring non-standard terms because the presence of auxiliary degrees of freedom gives rise to additional fluxes and sources that influence the thermodynamic and transport properties of the continuum model. Being aware of the additional contributions may prove crucial when designing multiscale computational schemes in which information is exchanged between the atomistic and continuum levels.