Size-dependent effect on thermo-electro-mechanical responses of heated nano-sized piezoelectric plate

The booming development of nanotechnology motivates the widespread applications of piezoelectric nanomaterials (e.g. ZnO, ZnS, GaN) and their nanostructures (e.g. nanobelts, nanorings nanowires). It is noted that the coupled field analysis of nano-sized piezoelectric structure under non-uniform temperature in-service environment is of great importance for the fabrication and exploitation of nanoelectromechanical devices. In such situation, spatial size effect of heat conduction is necessary to be taken into account due to its important significance in characterizing the nonlocal feature of heat transport in nanosystems. In this study, thermal nonlocal effect is introduced into the thermo-electro-mechanical model based on nonlocal elasticity theory to further shed light on the size-dependent coupling behavior of thermal, electric, and elastic fields. The coupled field equations involving size-dependent parameters are derived. The solutions can be obtained using Laplace transformation methods. Parametric studies are conducted to evaluate the influences of thermal as well as elastic nonlocal parameters on the transient responses. The results indicate that the piezoelectric performance of the nanoplate is greatly improved in the presence of thermal nonlocal effect.     

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.     

Effects of two-temperature parameter and thermal nonlocal parameter on transient responses of a half-space subjected to ramp-type heating

Based upon the coupled thermoelasticity and Green and Lindsay theory, the new governing equations of two-temperature thermoelastic theory with thermal nonlocal parameter is formulated. To more realistically model thermal loading of a half-space surface, a linear temperature ramping function is adopted. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Specific attention is paid to study the effect of thermal nonlocal parameter, ramping time, and two-temperature parameter on the distributions of temperature, displacement and stress distribution.     

Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory

In this paper, the viscoelastic wave propagation in an embedded viscoelastic single-walled carbon nanotube (SWCNT) is studied based on the nonlocal strain gradient theory. The characteristic equation for the viscoelastic wave in SWCNTs is derived. The emphasis is placed on the influence of the tube diameter on the viscoelastic wave dispersion. A blocking diameter is observed, above which the wave could not propagate in SWCNTs. The results show that the blocking diameter is greatly dependent on the damping coefficient, the nonlocal and the strain gradient length scale parameters, as well as the Winkler modulus of the surrounding elastic medium. These findings may provide a prospective application of SWCNTs in nanodevices and nanocomposites.     

Early transient responses of multi-span stepped single-walled carbon nanotubes using the method of reverberation ray matrix

The early transient responses of multi-span stepped single walled carbon nanotubes (SWCNTs) under impact loadings are studied by the method of reverberation ray matrix (MRRM). The dynamics model of the carbon nanotubes is established in the Fourier phase space on the basis of the nonlocal Timoshenko beam model. The wave solutions of SWCNTs with arbitrary boundary conditions are obtained by the wave method. The reverberation ray matrix of the multi-span stepped SWCNTs is the product of scattering, phase and permutation matrices, which can be determined by the impact loadings, continuous conditions and boundary conditions. The early transient responses can be calculated by the inverse Fourier transform of the sum of initial ray groups. It can be found that the early transient displacement response in the very short time subjected to the impact loading is very small, while the transient transverse shear strain becomes large in the very short time. The influences of nanotubes span number, nanotubes type and boundary conditions on the early transient responses of multi-span stepped SWCNTs are investigated.     

Nonlocal plate model for nonlinear bending of single-layer graphene sheets subjected to transverse loads in thermal environments

This paper investigates the nonlinear bending behavior of a single-layer rectangular graphene sheet subjected to a transverse uniform load in thermal environments. The single-layer graphene sheet (SLGS) is modeled as a nonlocal orthotropic plate which contains small scale effect. Geometric nonlinearity in the von Kármán sense is adopted. The thermal effects are included and the material properties are assumed to be size dependent and temperature dependent, and are obtained from molecular dynamics (MD) simulations. The small scale parameter e ₀ a is estimated by matching the deflections of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The numerical results show that the temperature change as well as the aspect ratio has a significant effect on the nonlinear bending behavior of SLGSs. The results reveal that the small scale parameter reduces the static large deflections of SLGSs, and the small scale effect also plays an important role in the nonlinear bending of SLGSs.     

Wave dispersion in viscoelastic single walled carbon nanotubes based on the nonlocal strain gradient Timoshenko beam model

Based on the nonlocal strain gradient theory and Timoshenko beam model, the properties of wave propagation in a viscoelastic single-walled carbon nanotube (SWCNT) are investigated. The characteristic equations for flexural and shear waves in visco-SWCNTs are established. The influence of the tube size on the wave dispersion is clarified. For a low damping coefficient, threshold diameter for shear wave (SW) is observed, below which the phase velocity of SW is equal to zero, whilst flexural wave (FW) always exists. For a high damping coefficient, SW is absolutely constrained, and blocking diameter for FW is observed, above which the wave propagation is blocked. The effects of the wave number, nonlocal and strain gradient length scale parameters on the threshold and blocking diameters are discussed in detail.     

Scale effects on thermal buckling properties of carbon nanotube

In this Letter, the thermal buckling properties of carbon nanotube with small scale effects are studied. Based on the nonlocal continuum theory and the Timoshenko beam model, the governing equation is derived and the nondimensional critical buckling temperature is presented. The influences of the scale coefficients, the ratio of the length to the diameter, the transverse shear deformation and rotary inertia are discussed. It can be observed that the small scale effects are significant and should be considered for thermal analysis of carbon nanotube. The nondimensional critical buckling temperature becomes higher with the ratio of length to diameter increasing. Furthermore, for smaller ratios of the length to the diameter and higher mode numbers, the transverse shear deformation and rotary inertia have remarkable influences on the thermal buckling behaviors.     

On the determination of the volume dependence of Grüneisen parameters in cubic and non-cubic solids

For a material of orthorhombic symmetry under hydrostatic stress, three generalized Grüneisen parameters relative respectively to the three principal directions of the deformation can be defined. Finite strain expressions for these parameters are derived in terms of the Lagrangian strain tensor and the frame-indifferent analogue of the Eulerian strain tensor. The expressions require for their evaluation the thermal expansion coefficients, the elastic moduli and their pressure and temperature derivatives, and the specific heat of the material, so that there are no arbitrary constants or “curve fitting.” In the case of cubic materials, it is possible to determine the unique Grüneisen parameter as a function of volume directly. For non-cubic materials, the volume dependence of the Grüneisen parameters is calculated using a quasi-harmonic finite strain model of equation of state. Numerical applications are given for some cubic and hexagonal metals and the results are discussed.     

Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium

In the present work, thermal buckling of single-layered graphene sheets lying on an elastic medium is analyzed. For this purpose, the sinusoidal shear deformation plate theory in tandem with the nonlocal continuum theory, which takes the small scale effects into account, is employed. The non-linear stability equations, which contain the reaction of Winkler–Pasternak elastic substrate medium, are derived and then solved analytically for a plate with various boundary conditions and based on various plate theories. Closed form solutions are formulated for three types of thermal loadings as uniform, linear and nonlinear temperature rise through the thickness of the plate. A number of examples are presented to illustrate the numerical results concerned with the buckling temperature response of nanoplates resting on two-parameter elastic foundations. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all investigated.     

Viscoelastic dynamics of polymer thin films and surfaces

The strain relaxation behavior in a viscoelastic material, such as a polymer melt, may be strongly affected by the proximity of a free surface or mobile interface. In this paper, the viscoelastic surface modes of the material are discussed with respect to their possible influence on the freezing temperature and dewetting morphology of thin polymer films. In particular, the mode spectrum is connected with mode coupling theory assuming memory effects in the melt. Based on the idea that the polymer freezes due to these memory effects, surface melting is predicted. As a consequence, the substantial shift of the glass transition temperature of thin polymer films with respect to the bulk is naturally explanied. The experimental findings of several independent groups can be accounted for quantitatively, with the elastic modulus at the glass transition temperature as the only fitting parameter. Finally, a simple model is put forward which accounts for the occurrence of certain generic dewetting morphologies in thin liquid polymer films. It demonstrates that by taking into account the viscoelastic properties of the film, a morphological phase diagram may be derived which describes the observed structures of dewetting fronts. It is demonstrated that dewetting morphologies may also serve to determine nanoscale rheological properties of liquids.Received: 1 January 2003, Published online: 14 October 2003PACS: 47.50. + d Non-Newtonian fluid flows - 68.47.Mn Polymer surfaces - 68.60.Dv Thermal stability; thermal effects     

Dynamic characteristics of curved nanobeams using nonlocal higher-order curved beam theory

Here, an analytical approach for the dynamic analysis, viz., free and forced vibrations, of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through thickness stretching is investigated. The formulation is general in the sense that it can be deduced to analyse the effect of various structural theories pertaining to curved nanobeams. It also includes inplane, rotary and coupling inertia terms. The governing equations derived, using Hamiltons principle, are solved in conjunction with Naviers solutions. The free vibration results are obtained employing the standard eigenvalue analysis whereas the displacement/stress responses in time domain for the curved nanobeams subjected to rectangular pulse loading are evaluated based on Newmarks time integration scheme. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loading pulse duration, and nonlocal scale parameter are brought out on the dynamic behaviours of curved nanobeams.     

Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.     

Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.     

Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate

The generalized coupled thermoelasticity based on the Lord-Shulman theory is considered to study the transient thermoelastic response of functionally graded rectangular plates. The state equations of functionally graded rectangular plate subjected to time-dependent thermal loads were established by using of state space approach, in which three displacement components, three stress components, the temperature and the heat flux were chosen as state variables. By giving simply supported boundary conditions and assuming that the material properties of the plate have an exponential law distribution along the thickness-coordinate, the equations were solved by the numerical Laplace transformations and shooting methods for transient thermal responses of a three dimensional functionally graded rectangular plate due to a thermal shock on its top surface. Effects of the volume fraction distributions of material constituents on the thermal responses, including the temperature change, the displacement and the stresses distributions were investigated.     

Polymer thin films and surfaces: Possible effects of capillary waves

It is discussed how the proximity of a free surface or mobile interface may affect the strain relaxation behavior in a viscoelastic material, such as a polymer melt. The eigenmodes of a viscoelastic film are thus derived, and applied in an attempt to explain the experimentally observed substantial shift of the glass transition temperature of sufficiently thin polymer films with respect to the bulk. Based on the idea that the polymer freezes due to memory effects in the material, and exploiting results from mode-coupling theory, the experimental findings of several independent groups can be accounted for quantitatively, with the elastic modulus at the glass transition temperature as the only fitting parameter. The model is finally applied discussing the possibility of polymer surface melting. A surface molten layer is predicted to exist, with a thickness diverging as the inverse of the reduced temperature. A simple model of thin polymer film freezing emerges which accounts for all features observed experimentally so far. Received 8 August 2001     

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.     

Thermal shock fracture of piezoelectric materials

This paper deals with a case study for the piezoelectric materials suddenly exposed to an environmental medium of different temperature. The problem is idealized to a plate containing an edge crack or an embedded crack. The stress and electric displacement histories in an uncracked plate are calculated. These stresses and electric displacements are then added to the crack surface tractions and electric displacements with opposite sign to formulate a mixed boundary value problem. The cracking problem is thus reduced to a singular integral equation of Cauchy type, which is then solved numerically. Both impermeable crack assumption and permeable crack assumption are considered. The results for stress and electric displacement intensity factors are computed as a function of normalized time and crack size. Lower bound solutions are obtained for the maximum thermal shock that the material can sustain without catastrophic failure according to the two distinct criteria: (i) The maximum local tensile stress equals the tensile strength of the medium. (ii) The maximum stress intensity factor for the pre-existing representative crack equals the fracture toughness of the medium. The parameters that control the transient thermal stress and electric displacement are also identified. The method can be used to explore susceptibility to thermal fracture in piezoelectric materials containing pre-cracks.     

Numerical simulation of flat-tip micro-indentation of glassy polymers: Influence of loading speed and thermodynamic state

Flat-tip micro-indentation tests were performed on quenched and annealed polymer glasses at various loading speeds. The results were analyzed using an elasto-viscoplastic constitutive model that captures the intrinsic deformation characteristics of a polymer glass: a strain-rate dependent yield stress, strain softening and strain hardening. The advantage of this model is that changes in yield stress due to physical aging are captured in a single parameter. The two materials studied (polycarbonate (PC) and poly(methyl methacrylate) (PMMA)) were both selected for the specific rate-dependence of the yield stress that they display at room temperature. Within the range of strain rates experimentally covered, the yield stress of PC increases linearly with the logarithm of strain rate, whereas, for PMMA, a characteristic change in slope can be observed at higher strain rates. We demonstrate that, given the proper definition of the viscosity function, the flat-tip indentation response at different indentation speeds can be described accurately for both materials. Moreover, it is shown that the model captures the mechanical response on the microscopic scale (indentation) as well as on the macroscopic scale with the same parameter set. This offers promising possibilities of extracting mechanical properties of polymer glasses directly from indentation experiments.     

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.