## Coarse-grained modeling of nanostructures

One of the major challenges in computational mechanics of materials remains bridging the length-scale and time-scale gaps between the computational and experimental methods to study the microstructure of materials. Traditional molecular dynamics is a powerful tool at the nanoscale, but it does not allow a simulation of such mechanical experiments as AFM microscopy dealing with sub-micrometric lengths and microsecond times. Coarse-grained (CG) modeling is one of the possible solutions to fill these gaps. CG modeling does not consider all particles in a system (such as atoms, molecules, or nanoparticles), but the beads (also known as super- or pseudo-atoms), approximating groups of real particles. This allows a significant reduction in the number of considered interactions and degrees of freedom and, consequently, saves the computational resources.

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We used rigid beads to simulate molybdenum disulfide. The interaction between the grains is based on Stillinger–Weber potential with parameters recalculated to fulfill the elastic properties of the original lattice. The model is applied to calculate the phonon spectrum and for the nanoindentation problem. It is shown that in the case of small strains the model is as accurate as regular MD simulations, but uses much fewer interatomic interactions; hence, it is much more time-efficient.

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Also, we proposed a unique method for CG modeling named ``Hierarchical Deformable and Rigid Assemblages'' (HiDRA) is proposed. The beads are chosen as assemblages of the atoms in the original crystal lattice. The method allows incorporating the beads' elasticity so that the overall strain uniformly distributes in the lattice to avoid the unexpected stress concentrations in the bonds between the beads. Beads interact via forces acting on the surface atoms. Equations of motion for the grains are derived taking the uniform strain of the grain as an independent variable. In-house code is developed to perform the simulations with elastic grains. As an application, the HiDRA method is used to study the longitudinal vibrations and full 3D dynamics of chains of one-dimensional grains. Phonon spectra show to what extent large grains can correctly describe wave propagation in molecular chains.

## Mechanical properties of artificial materials

Artificial materials include periodic structures, acoustic metamaterials, composites, etc. These materials have a wide application in mechanical engineering, aerospace technology, electronics, and other areas due to their outstanding properties based on properties of combined materials (as for nanocomposites) or on the exactingly designed structure (as for metamaterials).

A special class of materials and structures expanding in one or more directions when strained along the other direction is called ‘auxetic’. We study elastic properties of cellular lattice materials are studied using the discrete models. The models are based on a representation of the lattice as a set of interacting nodes. A potential of interaction between the nodes is calibrated such that to simulate elastic linking. As a result, the problems of analytic and computational homogenization, crack propagation, and shape-changing large strains are solved.

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Also, we studied nacre-like materials. Nacre-inspired microstructure raised the interest of scientists due to its exceptional strength. Such a brick and mortar microstructure, characterized by stiff elements of one phase with a high aspect ratio separated by thin layers of the second one, is proved to provide an efficient solution for the problem of the crack arrest. As a result, full-scale continuum modeling of both composite constituents without employing any simplifying assumptions was presented.

## Mesoscopic modeling of biomaterials and bio-inspired materials

Polymer fiber crosslinked networks represent a class of soft materials with a wide range of applications in biomaterials. Due to their mechanical properties and unique structure, random polymer materials play an important role in many fields from industrial processes to cell biology. Natural and synthetic polymer biomaterials have proven to be indispensable for a better and more accurate understanding of how cells sense their environment and are now being used to study cellular interaction.

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We develop discrete and continuum models of linear and nonlinear hyperelastic materials derived from the homogenization of random elastic fiber networks. Obtained results can be applied for specific problems in mechanobiology including modeling and simulation of ECM, hydrogels, soft tissues, and foams at various scales. Continuum model accurately describing the system behavior at a macroscopic scale will help to simplify the solution of the elastic problems for large-scale disordered media.

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## Thermal properties of graphene

Graphene is a promising two-dimensional material with a honeycomb crystal lattice. We study dynamical phenomena in graphene lattice, consisting of equal particles connected by linear and angular springs (harmonic approximation). Equations of in-plane motion for the lattice were derived. Initial conditions typical for molecular dynamic modeling were considered. In our experiments, carbon atoms had random initial velocities and zero displacements. In this case, the lattice is far from thermal equilibrium. In particular, initial kinetic and potential energies are not equal. Moreover, initial kinetic energies (and temperatures), corresponding to degrees of freedom of the unit cell, are generally different. The motion of particles leads to the equilibration of kinetic and potential energies and redistribution of kinetic energy and corresponding temperature among degrees of freedom. During equilibration, the kinetic energy performs decaying high-frequency oscillations. We showed that these oscillations are accurately described by an integral depending on the dispersion relation and polarization matrix of the lattice. At large times, kinetic and potential energies tend to equal values. Kinetic energy is partially redistributed among degrees of freedom of the unit cell. The equilibrium distribution of the kinetic energies is accurately predicted by the non-equipartition theorem. Presented results may serve for a better understanding of the approach to thermal equilibrium in graphene.

In addition, we studied the peculiarities of unsteady ballistic heat transfer in a two-dimensional graphene lattice. We showed that ballistic heat transfer in graphene is significantly anisotropic. We also showed that in the ballistic heat transfer regime each atom in graphene has two distinct temperatures corresponding to motions in zigzag and armchair directions.