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In this paper, a Zr–5W alloy was fabricated via casting. In order to obtain the mechanical properties of the material, quasi-static compression tests at room temperature and split Hopkinson pressure bar tests at various temperatures were carried out. The x-ray diffraction result showed that the main components of the alloy were αZr and W2Zr, where αZr is the matrix and W2Zr is the reinforcement. The metallographic characterization results showed that there were two main forms of W2Zr in the material, namely, large particle boundary and small diffuse submicrometer precipitates. The reinforcements of both distributions have the effect of increasing the strength of the material, but the small submicrometer W2Zr precipitates would cause microcrack nucleation during the late plastic deformation stage, resulting in damage softening. In order to make theoretical calculations of the mechanical properties of materials, the Johnson–Cook (JC) constitutive model and Zerilli–Armstrong (ZAM) constitutive model of the material were obtained. It was found that the JC constitutive model had poor consistency in describing material properties. Although the consistency of the ZAM constitutive model was higher than that of the JC constitutive model, it still had obvious shortcomings. Combined with the deformation mechanism of the alloy, a modified constitutive relation was established by adding damage softening terms based on the hexagonal close-packed metal constitutive model inferred by the kinetics of heat-activated dislocations. The relative error results of all working conditions show that the correlation consistency of the improved constitutive model in this paper is significantly better than that of JC constitutive and ZAM constitutive. [ABSTRACT FROM AUTHOR]

This paper utilizes a coupled chemomechanical model that integrates chemical reactions, diffusion, and mechanical deformation to describe the high-temperature interfacial reactions and stress evolution within an enamel coating/Ni-based super-alloy system. Through finite element simulation, we conduct a comprehensive analysis of multiple reactions and processes at the interface. Experimental characterization of the concentration distribution of various elements is employed to verify the accuracy of the proposed chemomechanical model. Additionally, the simulation reveals that the interface region experiences compressive circumferential stress, which can potentially lead to interfacial failure due to buckling. On the other hand, the enamel coating undergoes tensile circumferential stress, resulting in the formation of vertical surface cracks. These findings hold practical implications for the development of advanced coatings with enhanced durability and performance. Future research can further optimize coating design and enhance the reliability of high-temperature applications based on these results. [ABSTRACT FROM AUTHOR]

In this paper, direct numerical simulations (DNSs) are performed to investigate the deformation and breakup of an elastoviscoplastic (EVP) droplet in a Newtonian matrix under simple shear flow. The two-phase interface is captured by the volume-of-fluid (VOF) method with adaptive mesh refinement technique. The Saramito model (Bingham model coupled exponential Phan-Thien–Tanner viscoelastic model) is used to characterize the rheological behavior of the droplet. The droplet deformation and conformational state are studied with different Capillary numbers Ca, Weissenberg numbers Wi, and Bingham numbers Bi, which represent the surface tension, elasticity, and yield stress of the droplet, respectively. Our results show that droplet deformation occurs at low Ca, while breakup occurs at high Ca. The droplet non-monotonically deforms with increasing Wi and Bi, while is elongated for higher Ca. In addition, three breakup modes (mid-point pinching, transitional breakup, and homogeneous breakup) are reported for EVP droplets, in which transitional breakup disappears due to the influence of high elasticity. The conformational state of the droplet intuitively demonstrates the change of breakup from horizontal shear to vertical breakup. In spite of the fact that the surface tension always inhibits the deformation of droplets, the present work indicates that Bi has little effect on the deformation with high Wi and high Ca, while the influence is obvious at low Wi and Ca. The observed elastic and plastic effects on droplet deformation and breakup are believed to have significant impacts, as yield stress fluids are widely encountered in industrial applications. [ABSTRACT FROM AUTHOR]

This paper demonstrates a novel Eulerian computational framework for modeling anisotropic elastoplastic deformations of organic crystalline energetic materials (EM) under shocks. While Eulerian formulations are advantageous for handling large deformations, constitutive laws in such formulations have been limited to isotropic elastoplastic models, which may not fully capture the shock response of crystalline EM. The present Eulerian framework for high-strain rates, large deformation material dynamics of EM incorporates anisotropic isochoric elasticity via a hypo-elastic constitutive law and visco-plastic single-crystal models. The calculations are validated against atomistic calculations and experimental data and benchmarked against Lagrangian (finite element) crystal plasticity computations for shock-propagation in a monoclinic organic crystal, octahydro-1,3,5,7-tetranitro-1,3,5,7 tetrazocine (β-HMX). The Cauchy stress components and the resolved shear stresses calculated using the present Eulerian approach are shown to be in good agreement with the Lagrangian computations for different crystal orientations. The Eulerian framework is then used for computations of shock-induced inert void collapse in β-HMX to study the effects of crystal orientations on hotspot formation under different loading intensities. The computations show that the hotspot temperature distributions and the collapse profiles are sensitive to the crystal orientations at lower impact velocities (viz., 500 m/s); when the impact velocity is increased to 1000 m/s, the collapse is predominantly hydrodynamic and the role of anisotropy is modest. The present methodology will be useful to simulate energy localization in shocked porous energetic material microstructures and other situations where large deformations of single and polycrystals govern the thermomechanical response. [ABSTRACT FROM AUTHOR]

Extensible or expandable garden hoses have become popular. However, their behavior may be surprising to physicists. In this paper, the stress–strain equations that explain this behavior are derived and compared with experiments. The results clearly demonstrate that, since deformation is very large, conventional definitions of strain and stress are not useful and so-called real definitions are needed instead. [ABSTRACT FROM AUTHOR]

Mechanical properties of nucleic acids play an important role in many biological processes that often involve physical deformations of these molecules. At sufficiently long length scales (say, above ∼ 20 − 30 base pairs), the mechanics of DNA and RNA double helices is described by a homogeneous Twistable Wormlike Chain (TWLC), a semiflexible polymer model characterized by twist and bending stiffnesses. At shorter scales, this model breaks down for two reasons: the elastic properties become sequence-dependent and the mechanical deformations at distal sites get coupled. We discuss in this paper the origin of the latter effect using the framework of a non-local Twistable Wormlike Chain (nlTWLC). We show, by comparing all-atom simulations data for DNA and RNA double helices, that the non-local couplings are of very similar nature in these two molecules: couplings between distal sites are strong for tilt and twist degrees of freedom and weak for roll. We introduce and analyze a simple double-stranded polymer model that clarifies the origin of this universal distal couplings behavior. In this model, referred to as the ladder model, a nlTWLC description emerges from the coarsening of local (atomic) degrees of freedom into angular variables that describe the twist and bending of the molecule. Different from its local counterpart, the nlTWLC is characterized by a length-scale-dependent elasticity. Our analysis predicts that nucleic acids are mechanically softer at the scale of a few base pairs and are asymptotically stiffer at longer length scales, a behavior that matches experimental data. [ABSTRACT FROM AUTHOR]

A cylindrical rubber fiber subject to a twist will also elongate: a manifestation of Poynting's effect in large strain elasticity. Here, we construct an analogous treatment for an active rubber fiber actuated via an axisymmetric pattern of spontaneous distortion. We start by constructing an exact large-deformation solution to the equations of elasticity for such fiber subject to imposed twist and stretch, which reveals spontaneous warping and twisting of the fiber cross section absent in passive rubbers. We then compute the corresponding non-linear elastic energy, which encompasses the Poynting effect but is minimized by a finite spontaneous twist and stretch. In the second half of the paper, we apply these results to understand the twist-contraction actuation of nematic elastomer fibers fabricated with director fields that encode helical patterns of contraction on heating. We first consider patterns making a constant angle with respect to the local cylindrical coordinate system (conical spiral director curves) and verify the predicted spontaneous twist, contraction, and cross-section deformation via finite elements. Second, we consider realistic director distributions for the experimentally reported fibers fabricated by cross-linking while simultaneously applying stretch and twist. Counterintuitively, we find that the maximum actuation twist is produced by applying a finite optimal twist during fabrication. Finally, we illustrate that spontaneously twisting fibers will coil into spring-like shapes on actuation if the ends are prevented from twisting relative to each other. Such a twist–torsion coupling would allow us to make a tendril-like "soft-spring" actuator with low force and high linear stroke compared to the intrinsic contraction of the elastomer itself. [ABSTRACT FROM AUTHOR]

There is a need to develop low-cost and compact frequency reconfigurable antennas with continuous frequency tuning capability. Existing approaches rely on dedicated electrical, mechanical, or MEMS-based approaches to reconfigure the antenna, which requires extra components and fabrication steps. In this paper, we propose a physics-based approach to realize tunable helical antennas based on a shape memory alloy (SMA). By utilizing the temperature-induced irreversible mechanical deformation of the SMA, the temperature-controlled antenna can continuously change its resonant frequency and lock the state when the desired frequency response is achieved. The height of the helical antenna changes in response to applied temperatures. No additional circuitry or mechanical modules are required for the reconfiguration of the antenna geometry. The fabricated helical antenna can shift resonant frequency between 150 and 195 MHz with a tuning sensitivity of 284 kHz/°C. [ABSTRACT FROM AUTHOR]

Many fish propel themselves using wave-like lateral flexion of their body and tail in the water. The undulatory body is driven by the distributed muscles, and locomotion is achieved by internal muscular stimulation and the external action of the fluid. As one of the material properties, the stiffness of the body being propelled plays an important role in the deformation process, especially for the muscle power input and phase lag. In this paper, a three-dimensional self-propelled elongated body model is employed to numerically investigate the effects of stiffness on the propulsion performance, including the forward speed, energy consumption, and energy-utilization ratio. According to various deformation characteristics and energy-utilization ratios, three deformation modes corresponding to high, medium, and low stiffness are identified. Our results indicate that a deforming body with medium stiffness has the highest efficiency, and its corresponding deformation is closest to that of fish in nature. When the stiffness of the fish body is higher than the normal level, more muscle energy is needed to sustain the tail beating of the same amplitude. A lower level of stiffness produces a more obvious phase lag in the fish body, which might lead to slow control responses. We also show that the stiffness of the fish body affects the scaling relationship between the swimming speed and the tail beating velocity. The upper and lower limits of the scaling exponent correspond to high and low levels of stiffness, respectively, and are also affected by the wavelength of the muscle contraction. [ABSTRACT FROM AUTHOR]

In large equivalent explosion tests, the method of calculating blast wave parameters by the plastic deformation angle of a cantilever beam can effectively avoid the problem of power supply, storage, communication, and low survival rate of conventional blast wave sensors. In this paper, by means of theoretical analysis and numerical simulation, the deformation response of a pure aluminum cantilever beam with different scaled distances is calculated, and the relationship between the plastic bending angle and blast wave parameters is obtained, which provides a novel and effective method for the measurement of blast wave parameters. The experimental results show that the deformation of the cantilever beam is mainly dominated by the impulse load and the error of this measurement method is within 15%. This method can provide a complementary approach to more precise but complex sensor measurement and can effectively evaluate the impulse in the explosion accident outside. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical mechanism and a general deformation law of a ferrofluid droplet suspended between the air and a liquid substrate under the action of the applied vertical uniform magnetic field are investigated by a numerical model. A generalized conservative phase-field simplified multiphase lattice Boltzmann model is adopted to solve the flow field and phase field, which has a good ability to simulate the problems of ternary fluid flows with large density ratios. Subsequently, the Maxwell equation of static magnetic field is used to solve the magnetic field using a self-correcting scheme. We used this model to numerically study the dynamic evolvement process of ferrofluid droplets placed between the air and a liquid substrate, which are exposed in a vertical uniform magnetic field. Some typical characteristics are employed to characterize the droplet shape, and it is found that the aspect ratio, the elongated velocity, and the height of the mass center of the ferrofluid droplet are related to the magnetic Bond number. Furthermore, the numerical results obtained with the model are in good agreement with the experimental results available in the literature. Finally, the quantitative power law relation between the magnetic Bond number and the aspect ratio of ferrofluid droplets is obtained by using the method of scaling law, which provides a theoretical basis for the study of the deformation mechanism and the universal laws of a ferrofluid droplet placed between the air and a liquid substrate under the action of the vertical uniform magnetic fields with different intensities. [ABSTRACT FROM AUTHOR]

In this paper, the deformation dynamics of spherical red blood cells in viscous fluid driven by ultrasound are theoretically and numerically studied. Red blood cells are considered to be composed of a homogenous cytoplasm enclosed by a biological membrane. The developed theoretical framework consists of two sets of equations, which describe the acoustic wave propagation and the time-averaged mean dynamics of the fluid-cell system, respectively. Specifically, the separated acoustics and mean responses of viscous fluid are formulated based on the acoustic perturbation method in a generalized Lagrangian framework. Considering the viscoelasticity of the membrane, the cell deformation is coupled to the fluid mean motion through the time-averaged fluid–membrane coupling boundary conditions. A computational model is established by deriving weak form formulations of the final governing equations and implementing them by the finite element method. The computational model is verified by comparing the steady-state deformation of the numerical results with previous experimental results. This model can accurately characterize the deformation of cells over time, which helps to extract the viscoelastic properties of cells. [ABSTRACT FROM AUTHOR]

Natural gas hydrate (NGH) exploitation is severely restricted by geotechnical problems. Deformation behaviors of the hydrate-bearing strata (HBS) control the occurrence and evolution of geotechnical problems during extracting natural gas from HBS. In this paper, a novel approach named Ladetes is introduced to evaluate the lateral deformation behaviors of the near-wellbore and fracture-filling regions of the HBS. The pressuremeter test and the flat dilatometer test are designed to simulate the inner boundaries of an NGH-producing well and an artificial stimulation fracture for the first time. The device can realize the in situ hydrate formation prior to the experiment and axial loading application throughout the experiment. Both the strain control mode and the stress control mode can be achieved to estimate the deformation characteristics of HBS under different downhole conditions. Prime experiments proved their adaptability and reliability. The Ladetes provides an effective and alternative way of obtaining geotechnical parameters for HBS. [ABSTRACT FROM AUTHOR]

Very recently, half-hydrogenated germanene has been achieved in an experiment. In this paper, we investigate the effects of tetracyanoquinodimethane (TCNQ) molecular adsorption and strain on the electronic properties of half-hydrogenated germanene through first-principles. As an electron-acceptor molecule, TCNQ is exploited to non-covalently functionalize the half-hydrogenated germanene. However, this physical adsorption induces a ferromagnetic semiconductor–metal transition in half-hydrogenated germanene due to charge transfer from the substrate to the TCNQ molecule. More importantly, the superstructure of half-hydrogenated germanene/TCNQ is extremely sensitive to biaxial tensile strain. Under the biaxial tensile strain of 0.25%, the ferromagnetic semiconductor–metal transition induced by molecular adsorption can surprisingly be overturned. Meanwhile, a strong p-type doping is exhibited. Remarkably, it would return from a ferromagnetic semiconductor to a metal again when the biaxial tensile strain increases to 1.5%. Our analysis based on the structural and electronic properties of half-hydrogenated germanene/TCNQ indicates that such metal–semiconductor–metal transition in half-hydrogenated germanene/TCNQ under biaxial tensile strain may originate from the strong local deformation, resulting in the energy of the valence band maximum decreasing below or increasing above the Fermi level. [ABSTRACT FROM AUTHOR]

This paper presents the constitutive modeling of the flexomagnetic (FM) effect in structures involving a non-zero magnetic field generated in the presence of inhomogeneous strain across the domain. In order to evaluate the magneto-elastic (ME) structural response, the tensorial governing equations and associated boundary conditions for the mechanical and magnetic variables are derived here using the variational principle. Following this, these differential equations are solved to determine the effect of FM over the bending response of a cantilever piezomagnetic nanobeam. Different magnetic boundary conditions are analyzed to study the effect of direct and converse FM couplings over the ME response. The significant influence of FM coupling over the ME response has been noted for thin beams with low-dimensions, which wanes as the geometric dimensions are increased. Given the significance of this size-dependent FM response in nano-structures, the theory for flexomagnetism proposed here may be utilized in the development of smart ME nano-structures with the potential for wide applications. [ABSTRACT FROM AUTHOR]

Studying the fragmentation and refinement of diamond powder as well as the diversification in the intergranular stress is crucial to produce a high-quality polycrystalline diamond. In this paper, using different micron-size diamond powders as the initial materials, the samples were compressed under different pressures at ambient temperature. The fragmentation behavior of the diamond powder was investigated by scanning electron microscopy and with a laser particle size analyzer. The results show that the fragmentation of diamond comprises three stages with increasing pressure: (i) fracturing of edges and corners, (ii) cracking of the crystal plane, and (iii) refinement of particle disorder; the particle deformation tends to become relatively stable after a certain pressure. In situ high-pressure synchrotron X-ray diffraction was used to study the intergranular stress distribution under non-hydrostatic compression to 35.1 GPa. A heterogeneous stress distribution was found in compressed diamond bulk, in which under the highest load, the maximum stress reached 69.5 GPa, whereas the minimum stress was only 18.8 GPa. [ABSTRACT FROM AUTHOR]

Soft chemo-mechanical (SCM) machines convert chemical energy into mechanical energy. They have the advantages of fast actuation and compliant deformation and possess wide applications in soft robots and flexible smart devices. Using experiments and theoretical modeling, this paper investigates the chemo-mechanical energy flows and the energy conversion cycle in the pressure and volume (P–V) plane of an SCM machine consisting of a soft membrane and cylindrical chambers. The machine generates 16.1 mJ of mechanical energy per cycle. We study heat dissipation as the key issue affecting the conversion efficiency. The states of the conversion cycle calculated from mechanical modeling fit well with those measured experimentally. The design principles and the theoretical methods can be used to guide the fabrication and characterization of SCM actuators and robots. [ABSTRACT FROM AUTHOR]

In this paper, we investigate small amplitude oscillations of a thin shape-morphing plate submerged in a quiescent, Newtonian, incompressible, viscous fluid in the vicinity of a solid wall. The plate undergoes prescribed, time-varying shape-morphing deformation in phase with the transverse oscillation, consisting of periodic curvature to an arc of a circle. As the presence of the solid wall in close proximity of the oscillating plate is known to significantly increase hydrodynamic damping on the structure due to squeeze film effects, shape-morphing is considered as a potential strategy to modulate and control hydrodynamic forces, as well as power dissipation. Via a rigorous formulation based on unsteady Stokes equations and the boundary element method, we show that the shape-morphing strategy is effective in reducing hydrodynamic forces on the structure as well as hydrodynamic power dissipation due to viscous effects. Additionally, such a reduction is enhanced as the gap between the oscillating plate and the solid wall is reduced. We present a comprehensive parametric study which investigates the effect of oscillation frequency, gap size, and imposed curvature on the hydrodynamic performance of the system and support our findings with asymptotics from inviscid flow theory and lubrication theory. These findings are of direct interest and applicability in several cantilever-based sensing and actuation systems, such as atomic force microscopy and microelectromechanical systems in liquids. [ABSTRACT FROM AUTHOR]

During compaction of large ensembles of metallic particles, the deformation and redistribution of the particles are strongly influenced by the bulk and interfacial properties of the interacting materials. Using a sharp-interface Eulerian approach, this paper studies the physics of shock compaction of clusters of particles by accurately capturing the interfacial dynamics of particle-particle interactions. The effect of disparity in bulk material properties such as impedance and yield strength and interfacial phenomena such as friction are quantified using interfacial area evolution as a measure of mixing of the materials. The results show that friction between the particles increases the mixing of the materials only for lower initial theoretical mean densities but does not significantly affect the overall dynamics of the compaction. The deformation and contact between the particles are most strongly influenced by the impedance mismatch of the materials. The difference in yield stress plays a minor role due to the predominance of pressure over deviatoric stresses. The results provide insights into the individual effects of impedance and yield stress mismatch between material pairs and also into the combined influence of mismatches in both properties. It is shown that apart from property mismatch, the average impedance of the mixture also plays a key role in mixing of the materials. [ABSTRACT FROM AUTHOR]

This paper presents a comparative analysis of the structure and photoluminescence (PL) of Si containing dislocations introduced by thermal shock or ion implantation. To study the structure of dislocation cores and their interaction with point defects, we used a high-resolution transmission mode during in situ electron irradiation in the JEM4000EX operating at 400 kV. An appropriate PL spectrum was obtained on dislocated Si after electron irradiation by an external pulse gun operating at 350 kV. This resulted only in an increase in the D2 line intensity that correlated with the formation of metastable interstitial defects on {001}, {111}, and {113} planes near various dislocation cores during in situ electron irradiation, regardless of their types. However, the D1 line relates to a core structure of a shuffle Lomer dislocation consisting of 5/7 atomic rings, which occurs when two shuffle 60° dislocations (so-called S1 type [Pizzagalli et al., Phys. Rev. Lett. 103, 065505 (2009)]) combine with each other in intersecting {111} planes in plastically deformed Si or when an extrinsic Frank partial dislocation transforms into a perfect one in an ion-implanted layer. [ABSTRACT FROM AUTHOR]

Acoustic metamaterials (AMMs) are designed with complex geometrical shapes to obtain unconventional sound-absorbing performances. As additive manufacturing is particularly suited to print complex structures in a more straightforward and controllable way, AMMs often exploit three-dimensional (3-D) printing techniques. However, when exposed to different temperature conditions, such structures can be affected by geometrical deformations, especially when they are polymer-based. This can cause a mismatch between the experimental data and the expected theoretical performance; therefore, it is important to take thermal effects into account. The present paper investigates the influence of thermal deformations on the sound absorption of three geometries: a coplanar spiral tube, a system with double coiled resonators, and a neck-embedded resonator. Measurements were performed on each 3-D printed specimen in the impedance tube after the samples had been placed in a climate chamber to modify the temperature settings (T = 10–50 °C). Numerical models, validated on the measurements, were employed to quantify the geometrical deformation of AMM structures through a multiphysics approach, highlighting the effects of thermal stress on the acoustic behavior. The main outcomes prove that the frequency shifts of sound absorption peaks depend on temperature configurations and follow exponential regressions, in accordance with previous literature on polymeric materials. [ABSTRACT FROM AUTHOR]

This paper presents a study concerning the macroscale deformation of soft stamps for nanoimprint lithography. Composite stamps consisting of two or three layers were prepared with varying process parameters, and the bow of the stamps was investigated with an optical profilometer. Deflection measurements were compared with results from a mechanical model in order to prove its validity. The composite stamps were prepared from polydimethylsiloxane (S-PDMS) and OrmoStamp® layers on different substrate materials (silicon or AF-32-glass). The influence of selected preparation parameters on the deformation was investigated, such as layer thickness, curing, and hardbake temperatures, as well as UV-exposure intensities. It is discussed how an appropriate choice of preparation parameters can reduce the deformation, too. By decreasing the curing intensity of a thin OrmoStamp layer, coated on top of a silicon wafer, from 200 mW/cm2 to 2 mW/cm2, the deformation was reduced from 6 µm to 0. It is shown that the mechanical model is capable of predicting the deformation of composite stamps consisting of OrmoStamp layers on top of a silicon substrate. Further on, deformation results for S-PDMS top layers are presented and discussed. As a promising process option for the reduction of stamp deformations, a new, room temperature curable PDMS recipe is introduced. For composite stamps prepared with this material, the deformation value was reduced below 10 µm, compared to values ranging from 10 µm up to 127 µm for thermally cured S-PDMS (50°C or 70°C). Finally, different two-layer stamps were coated with an additional third layer and measured again, in order to investigate the deflection for tri-layered cases. The research is of high relevance for the fabrication and application of substrate conformal imprint lithography stamps with minimized deflection. [ABSTRACT FROM AUTHOR]

This paper summarizes the ab-initio electronic and phonon band structures and the temperature-dependent carrier transport in layered Ti2AlC. It is found that the cylindrical Fermi surface is the origin of the anisotropic carrier effective mass (infinite effective mass along the c axis), which leads to a strong anisotropic (insulator along the c axis and metallic along the layer) carrier transport in these films. Using electronic and phonon band structure calculations, we have developed an analytical model for the carrier-phonon interaction and the in-plane carrier conductivity originating from the strong inter-valley (s → d) scattering in Ti2AlC. The density functional theory is used to calculate the average deformation potential corresponding to the acoustic phonon vibrations. The calculated deformation potential is in good agreement with the extracted deformation potential from the transport data available in the literature. The extracted deformation potential will be useful for predicting the transport quantities of these metals at elevated temperatures. [ABSTRACT FROM AUTHOR]

Interlayer shear between graphene sheets plays an important role in graphene-based materials and devices, but the effect of in-plane deformation of graphene, which may depend on the graphene size, has not been fully understood. In this paper, the size effect on interlayer shear behavior between two graphene sheets is studied based on a non-linear shear-lag model with energy barrier analysis, in which both the lattice registry effect and the elastic deformation of graphene are taken into account and molecular dynamics (MD) simulations are carried out to verify the model. Both theoretical prediction and MD simulations show that the maximum interlayer shear force of short graphene sheets increases with the graphene length and width. However, if the sheet length is beyond 20 nm, the maximum shear force cannot be further increased by increasing the graphene length due to the non-uniform relative displacement between graphene layers, which is caused by the in-plane deformation of graphene. The upper bound of the maximum shear force per unit graphene width is obtained analytically as a constant 5.6 N/m, suggesting that a small force can pull an infinite long graphene belt to slide on a graphene substrate. This study offers useful information for design and manufacture of graphene-based nano-devices and materials. [ABSTRACT FROM AUTHOR]