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• Theory of lattice metamaterials under sinusoidal distributed static loads • Two basic modes of static damping of distributed pressure waves are identified • Dependence of effective shear modulus on a spatial frequency of the load • Pathway toward intelligent mechanical systems able to distinguish load patterns We discuss response properties of lattice metamaterials to sinusoidal distributed static loads applied on a material edge. Analytical displacement solutions are obtained using a Fourier domain transfer matrix for both essential and natural boundary conditions, which are valid for a state of plane strain in orthorhombic lattices, as well as for planar metasurfaces. These solutions give sinusoidal displacement profiles in the material interior of the same wavenumber (spatial frequency) as the boundary load. Their amplitudes decay in the material interior in one of two possible ways, exponential or oscillatory exponential, depending on the lattice design and on the spatial frequency of the load. There is a tendency for the oscillatory exponential behavior to occur at lower wavenumbers representing smoother boundary loads. As explained on a specific example, comparing the metamaterial's response amplitudes with those of a homogenous material also allows studying an effective, wavenumber-dependent shear modulus of the lattice, which can take both positive and negative values. [ABSTRACT FROM AUTHOR]

A class of beam-like lattice structures, or metabeams under static, sinusoidally distributed transverse loads is discussed. Their neutral axis deflects either in-phase, out-of-phase or shows no deflection, depending on the beam design parameters, and also on the spatial frequency of the static load. These outcomes contrast the behavior of continuum beams, deflecting always in-phase with the load, and they are interpreted on the basis of a positive, negative and near-infinite effective Young's modulus of the structured beams in bending. They also represent a collective effect of the behavior of multiple elements in the lattice that cannot be realized from the performance of an isolated unit cell. A long-range periodic order and nonlocality of the lattice interaction is essential for this unusual behaviors, and those are particularly pronounced at higher wavenumbers, when the load wavelength becomes comparable with the range of direct interactions in the lattice. Theoretical discussion and predictions agree well with numerical experiments performed on the basis of commonly accepted models. Practical applications could be found in advanced reinforcing materials for building foundations, deformation mitigation for lightweight structures and bridges, and in smart mechanical systems able to differentiate external stimuli and to respond selectively. [ABSTRACT FROM AUTHOR]

Highlights • Mechanics analogue of the Parseval's energy theorem is discussed, leading to a spectral density of the strain energy (in a reciprocal space). • Energy spectral density reflects the ways how an elastic medium translates static deformation patterns between two points in space. • Spectral entropy of static deformation is proposed, as a measure of information available in the material interior about surface loads or disorder introduced into elastic medium by deformation. • Both differential and discrete/numerical entropies of Shannon's type are discussed, and the differential entropy is derived analytically for isotropic solids under Gauss loads. • Approaches to numerical calculation of spectral entropy in computational solid mechanics are discussed. Abstract A broader range of analytical tools can enhance understanding of the unusual mechanical properties of metamaterials and other advanced material systems. Here, we discuss a mechanics analogue of the Parseval's energy theorem that leads to a density of the strain energy in the reciprocal space. It reflects the ways for an elastic medium to translate static deformation patterns between two points in space. A normalized spectral density also provides an information entropy of deformation at those points. Both differential and discrete (numerical) entropies of the Shannon's type are discussed. Spectral entropy is a basic measure of information available in the material interior about surface loads, or a measure of disorder introduced into elastic medium by the deformation. An exact analytical entropy function is derived for an isotropic plane solid under Gauss-distributed and point loads. Approaches to numerical calculation of spectral entropy in computational solid mechanics are also discussed. Energy spectral density and spectral entropy of an elastic continuum is shown to translate, logically, in agreement with the Saint-Venant's principle. However, it also becomes clear that microstructured media may demonstrate anomalous pathways of evolution of the strain energy spectrum, enabling interesting transformation mechanics studies of engineered material systems. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

When a usual material is loaded statically at surfaces, fine fluctuations of surface strain diminish fast in the material volume with the distance to the surface, a phenomenon widely known as the Saint-Venant edge effect. In this paper, highly nonlocal discrete lattices are explored to demonstrate structural metamaterials featuring reversal of the Saint-Venant edge effect. In these materials, certain coarse patterns of surface strain may decay faster than the finer ones. This phenomenon is shown to arise from anomalous behavior of the Fourier modes of static deformation in the material, and creates opportunities for blockage, qualitative modification and in-situ recognition of surface load patterns. Potential applications and useful practical techniques of spectral analysis of deformation, density of states and phase diagram mapping are outlined. [ABSTRACT FROM AUTHOR]

We present a method for deriving molecular dynamics boundary conditions for use in multiple scale simulations that can be applied at a planar boundary for any solid that has a periodically repeating crystal lattice. The method is based on a linearization in the vicinity of the boundary, and utilizes a Fourier and Laplace transforms in space and time to eliminate the degrees of freedom associated with atoms outside the boundary. This method is straightforward to implement numerically, and thus can be automated for a general crystal lattice. We show that this method reproduces the known kernel for a 1D linear chain, and apply the approach to obtain the damping kernel matrices for two real crystal lattices: the graphene and diamond structures of carbon. [Copyright &y& Elsevier]

We present a simple method for calculating a continuum temperature field directly from a molecular dynamics (MD) simulation. Using the idea of a projection matrix previously developed for use in the bridging scale, we derive a continuum temperature equation which only requires information that is readily available from MD simulations, namely the MD velocity, atomic masses and Boltzmann constant. As a result, the equation is valid for usage in any coupled finite element (FE)/MD simulation. In order to solve the temperature equation in the continuum where an MD solution is generally unavailable, a method is utilized in which the MD velocities are found at arbitrary coarse scale points by means of an evolution function. The evolution function is derived in closed form for a 1D lattice, and effectively describes the temporal and spatial evolution of the atomic lattice dynamics. It provides an accurate atomistic description of the kinetic energy dissipation in simulations, and its behavior depends solely on the atomic lattice geometry and the form of the MD potential. After validating the accuracy of the evolution function to calculate the MD variables in the coarse scale, two 1D examples are shown, and the temperature equation is shown to give good agreement to MD simulations. [Copyright &y& Elsevier]

• Analytical nodal displacement solutions are derived in 2D plate-like lattices • Periodic boundary conditions on side surfaces form the topological cylinder • Solutions are the superposition of Fourier mode dependent states of deformation • Analytical solutions are validated with finite element analysis software Abaqus™ Nodal displacements for two dimensional plate-like lattices under static loading are solved in exact analytical form, when periodic boundary conditions are applied to the top and bottom surfaces of the lattice forming a topological cylinder. Discrete Fourier transform converts the governing equation of static equilibrium into a set of one dimensional wavenumber dependent problems. Characteristic solutions are developed for each Fourier mode of deformation. Inverse discrete Fourier transform converts the wavenumber domain function back to the spatial domain by taking a linear combination of the one dimensional harmonic solutions. As an example, the point force is decomposed into a set of wavenumber dependent loading profiles. A linear combination of the displacement solutions for the individual harmonic loading profiles is sufficient to reproduce the overall displacement solution for the point load. This property holds for any type of the load. Nodal displacements, expressed with analytical dependence on the nodal indices n and m , match the results of commercial finite element analysis software. Overall, this paper demonstrates that the static response of discrete lattices is equivalently represented as a superposition of the wavenumber domain solutions, which is analogous to frequency domain analysis in acoustic metamaterials and phononics. [ABSTRACT FROM AUTHOR]

Abstract: Inspired by the pioneering work of Professor T.J.R. Hughes on the variational multi-scale method, this document summarizes recent developments in multiple-scale modeling using a newly developed technique called the bridging scale. The bridging scale consists of a two-scale decomposition in which the coarse scale is simulated using continuum methods, while the fine scale is simulated using atomistic approaches. The bridging scale offers unique advantages in that the coarse and fine scales evolve on separate time scales, while the high frequency waves emitted from the fine scale are eliminated using lattice impedance techniques. Recent advances in extending the bridging scale to quantum mechanical/continuum coupling are briefly described. The method capabilities are demonstrated via quasistatic nanotube bending, dynamic crack propagation and dynamic shear banding. [Copyright &y& Elsevier]