The elastic moduli of single layer graphene sheet (SLGS) have been a subject of intensive research in recent years. Calculations of these effective properties range from molecular dynamic simulations to use of structural mechanical models. On the basis of mathematical models and calculation methods, several different results have been obtained and these are available in the literature. Existing mechanical models employ Euler-Bernoulli beams rigidly jointed to the lattice atoms. In this paper we propose truss-type analytical models and an approach based on cellular material mechanics theory to describe the in-plane linear elastic properties of the single layer graphene sheets. In the cellular material model, the C-C bonds are represented by equivalent mechanical beams having full stretching, hinging, bending and deep shear beam deformation mechanisms. Closed form expressions for Young's modulus, the shear modulus and Poisson's ratio for the graphene sheets are derived in terms of the equivalent mechanical C-C bond properties. The models presented provide not only quantitative information about the mechanical properties of SLGS, but also insight into the equivalent mechanical deformation mechanisms when the SLGS undergoes small strain uniaxial and pure shear loading. The analytical and numerical results from finite element simulations show good agreement with existing numerical values in the open literature. A peculiar marked auxetic behaviour for the C-C bonds is identified for single graphene sheets under pure shear loading.

Plane wave propagation in infinite two-dimensional periodic lattices is investigated using Floquet-Bloch principles. Frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies: hexagonal honeycomb, Kagomé lattice, triangular honeycomb, and the square honeycomb. These topologies exhibit dramatic differences in their long-wavelength deformation properties. Long-wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices. The slenderness ratio of the constituent beams of the lattice (or relative density) has a significant influence on the band structure. The techniques developed in this work can be used to design lattices with a desired band structure. The observed spatial filtering effects due to anisotropy at high frequencies (short wavelengths) of wave propagation are consistent with the lattice symmetries.

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A redhead: Surface‐grafted hydrophilic polymer brushes (see picture) with high molecular weight and graft density caused reversible bending and stretching of soft polymeric substrates on a macroscale. The shape change of the substrate was tuned to respond to different stimuli including humidity, temperature, and pH.

Periodic composites such as acoustic metamaterials use local resonance phenomenon in designing low frequency sub-Bragg bandgaps. These bandgaps emerge from a resonant scattering interaction between a propagating wave and periodically arranged resonators. This paper develops a receptance coupling technique to combine the dynamics of the resonator with the unit cell dynamics of the background medium to analyze flexural wave transmission in a periodic structure, involving a single degree of freedom coupling between the medium and the resonator. Receptance techniques allow for a straightforward extension to higher dimensional systems with multiple degrees of freedom coupling and for easier experimental measurements. Closed-form expressions for the location and width of sub-Bragg bandgaps are obtained. Rigid body modes of the unit cell of the background medium are shown to set the bounding frequencies for local resonance bandgaps. Results from the receptance analysis compare well with Bloch wave analysis and experiments performed on a finite structural beam with periodic masses and resonators. Stronger coupling and inertia of the resonator increase the local resonance bandgap width. Two-fold periodicity widens the Bragg bandgap, narrowed by local resonators, thus expanding the design space and highlighting the advantages of hierarchical periodicity.

A systematic molecular dynamics simulation study is performed to assess the effects of temperature and free edges on the ultimate tensile strength and Young's modulus of a single-layer graphene sheet. It is observed that graphene sheets at higher temperatures fail at lower strains, due to the high kinetic energy of atoms. A numerical model, based on kinetic analysis, is used to predict the ultimate strength of the graphene under various temperatures and strain rates. As the width of a graphene reduces, the excess edge energy associated with free edge atoms induces an initial strain on the relaxed configuration of the sheets. This initial strain has a greater influence on the Young's modulus of the zigzag sheet compared with that of the armchair sheets. The simulations reveal that the carbon–carbon bond length and amplitude of intrinsic ripples of the graphene increases with temperature. The initial out-of-plane displacement of carbon atoms is necessary to simulate the physical behaviour of a graphene when the Nosé–Hoover or Berendsen thermostat is used.

Brush-like structures emerge from stretching of long polymer chains, densely grafted on to the surface of an impermeable substrate. They arise due to the competition between conformational entropic elasticity of polymer chains and excluded volume interactions from the intra and interchain monomer repulsions. Recently, stimuli responsive polymer brush based soft materials have been developed to produce controllable and reversible large deformations of the host substrate. To understand these systems, and improve their functional properties, we study elastic stress distribution and surface stress-curvature relations of a neutral polymer brush grafted on to an elastic beam, made of a soft material. In the strongly stretched brush regime, we combine mean field theory from polymer physics with a continuum mechanics model and show that the residual stress variation is a quartic function of distance from the grafting surface, with maximum stress occurring at the grafted surface. Idealizing the brush as a continuum elastic surface layer with residual stress, we derive a closed form expression for surface stress and the surface elasticity of the layer as a function of brush parameters, such as graft density and molecular weight. The generalized continuum beam model accounts for the Young-Laplace and Ogden-Steigman curvature elasticity correction terms, and yields a surface stress-curvature relation, which contains existing relations in the literature as special cases. Further, we report experiments on a thermoresponsive random copolymer brush, Poly(N-isopropylacrylamide)-co-Poly(N,N-Dimethylacrylamide) (PNIPAm-co-PDMA) brush, grafted on one side of a plasticized poly(vinyl chloride) (pPVC) thin film. Estimated surface stress from measured curvature is on the order of −10 N/m, and it decreases gradually, and reversibly, with increasing ambient temperature from 15 • C to 55 • C.