Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

Wave propagation responses of porous bi-directional functionally graded magneto-electro-elastic nanoshells via nonlocal strain gradient theory

This study examines the wave propagation characteristics for a bi-directional functional grading of barium titanate(BaTiO_(3)) and cobalt ferrite(CoFe_(2)O_(4)) porous nanoshells,the porosity distribution of which is simulated by the honeycomb-shaped symmetrical and asymmetrical distribution functions.The nonlocal strain gradient theory(NSGT) and first-order shear deformation theory are used to determine the size effect and shear deformation,respectively.Nonlocal governing equations are derived for the nanoshells by Hamilton's principle.The resulting dimensionless differential equations are solved by means of an analytical solution of the combined exponential function after dimensionless treatment.Finally,extensive parametric surveys are conducted to investigate the influence of diverse parameters,such as dimensionless scale parameters,radiusto-thickness ratios,bi-directional functionally graded(FG) indices,porosity coefficients,and dimensionless electromagnetic potentials on the wave propagation characteristics.Based on the analysis results,the effect of the dimensionless scale parameters on the dispersion relationship is found to be related to the ratio of the scale parameters.The wave propagation characteristics of nanoshells in the presence of a magnetoelectric field depend on the bi-directional FG indices.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

Spalling characteristics of high-temperature treated granitic rock at different strain rates

The dynamic spalling characteristics of rock are important for stability analysis in rock engineering.This paper presented an experimental investigation on the dynamic spalling characteristics of granite with different temperatures and strain rates.A series of dynamic spalling tests with different impact velocities were conducted on thermally treated granite at different temperatures.The dynamic spalling strengths of granite with different temperatures and strain rates were determined.A model was proposed to correlate the dynamic spalling strength of granite,high temperature and strain rate.The results show that the spalling strength of granite decreases with increasing temperature.Moreover,the spalling strength of granite with a higher strain rate is larger than that with a lower strain rate.The proposed model can describe the relationship among dynamic spalling strength of granite,high temperature and strain rate.

Experimental and numerical study on dynamic mechanical behaviors of shale under true triaxial compression at high strain rate

High-energy gas fracturing of shale is a novel,high efficacy and eco-friendly mining technique,which is a typical dynamic perturbing behavior.To effectively extract shale gas,it is important to understand the dynamic mechanical properties of shale.Dynamic experiments on shale subjected to true triaxial compression at different strain rates are first conducted in this research.The dynamic stress-strain curves,peak strain,peak stress and failure modes of shale are investigated.The results of the study indicate that the intermediate principal stress and the minor principal stress have the significant influence on the dynamic mechanical behaviors,although this effect decreases as the strain rate increases.The characteristics of compression-shear failure primarily occur in shale subjected to triaxial compression at high strain rates,which distinguishes it from the fragmentation characteristics observed in shale under dynamic uniaxial compression.Additionally,a numerical three-dimensional Split Hopkinson Pressure Bar(3D-SHPB),which is established by coupling PFC3D and FLAC3D methods,is validated to replicate the laboratory characteristics of shale.The dynamic mechanical characteristics of shale subjected to different confining stresses are systematically investigated by the coupling PFC3D and FLAC3D method.The numerical results are in good agreement with the experimental data.

Review on dynamic analysis of road pavements under moving vehicles and plane strain conditions

This paper reviews works on the dynamic analysis of flexible and rigid pavements under moving vehicles on the basis of continuum-based plane strain models and linear theories.The purpose of this review is to provide in-formation about the existing works on the subject,critically discuss them and make suggestions for further research.The reviewed papers are presented on the basis of the various models for pavement-vehicle systems and the various methods for dynamically analyzing these systems.Flexible pavements are modeled by a homogeneous or layered half-plane with isotropic or anisotropic and linear elastic,viscoelastic or poroelastic material behavior.Rigid pavements are modeled by a beam or plate on a homogeneous or layered half-plane with material properties like the ones for flexible pavements.The vehicles are modeled as concentrated or distributed over a finite area loads moving with constant or time dependent speed.The above pavement-vehicle models are dynamically analyzed by analytical,analytical/numerical or purely numerical methods working in the time or frequency domain.Representative examples are presented to illustrate the models and methods of analysis,demonstrate their merits and assess the effects of the various parameters on pavement response.The paper closes with con-clusions and suggestions for further research in the area.The significance of this research effort has to do with the presentation of the existing literature on the subject in a critical and easy to understand way with the aid of representative examples and the identification of new research areas.

Deformation characteristics and novel strain criteria of strainbursts induced by low-frequency cyclic disturbance

Strainbursts induced by cyclic disturbance with low frequency(termed as cyclicinduced strainbursts)are major dynamic disasters during deep excavation and mining.There is currently no quantitative criterion available for the prediction of such disastrous events.In this study,based on true triaxial experiments,we analyzed the deformation characteristics,established two novel strain criteria for the cyclic-induced strainbursts,and explained the physical meaning of these criteria.Characteristic strains for the cyclic-induced strainbursts were defined,including the control strain ε_(ctr),the strain caused by the combined dynamic and static loading ε_(sd),and the ultimate strain ε_(u) after strainbursts.As indicated by the results,the deformation evolution of the cyclic-induced strainbursts shows remarkable fatigue characteristics,which resemble that of rock subjected to cyclic loading and unloading.In other words,there are three stages during deformation evolution,namely,initial rapid growth,uniform velocity growth after several periods of disturbance,and sudden sharp growth preceding the burst.The ultimate strain ε_(u) is insensitive to the tangential static stress and disturbance amplitude,but it changes nonlinearly with disturbance frequency.From the perspective of deformation,the occurrence of a cyclic-induced strainburst is controlled by the control strainε_(ctr).Thus,a control strain criterion is proposed;that is,when the stain ε_(sd) is larger than the control strain ε_(ctr),a strainburst will be induced by cyclic disturbance.Moreover,based on the statistical results,a strain ratio criterion is proposed;that is,when the strain ratio ε_(sd)/ε_(u) is greater than 30%,a cyclic-induced strainburst will be induced.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Generalizing J_2 flow theory: Fundamental issues in strain gradient plasticity

It has not been a simple matter to obtain a sound extension of the classical J2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of classical J2 theory have been proposed： one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhans and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rateindependent J2 flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a comparable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J2 flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J2 deformation theory of gradient plasticity for deformation histories characterized by proportional straining.

Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory

In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations （kine-matic equations） are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory （SGT） is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton＇s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes （MWCNTs） are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton＇s principle and Rayleigh＇s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes （SWCNTs） and double-walled carbon nanotubes （DWCNTs） with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

Transition sets of bifurcations of dynamical systems with two statevariables with constraints

Bifurcation of periodic solutions widely exists in nonlinear dynamical systems. In this paper, categories of bifurcations of systems with two state variables with different types of constraints are discussed, where some new types of transition sets are added. Additionally, the bifurcation properties of two-dimensionM systems without constraints are compared with the ones with constraints. The results obtained in this paper can be used by engineers for the choice of the structural parameters of the systems.

NEW STRAIN GRADIENT THEORY AND ANALYSIS

A new strain gradient theory which is based on energy nonlocal model is proposed in this paper, and the theory is applied to investigate the size effects in thin metallic wire torsion, ultra-thin beam bending and micro-indentation of polycrystalline copper. First, an energy nonlocal model is suggested. Second, based on the model, a new strain gradient theory is derived. Third, the new theory is applied to analyze three representative experiments.

Valence band structure and density of states effective mass model of biaxial tensile strained silicon based on k·p theory

After constructing a stress and strain model, the valence bands of in-plane biaxial tensile strained Si is calculated by k·p method. In the paper we calculate the accurate anisotropy valance bands and the splitting energy between light and heavy hole bands. The results show that the valance bands are highly distorted, and the anisotropy is more obvious. To obtain the density of states （DOS） effective mass, which is a very important parameter for device modeling, a DOS effective mass model of biaxial tensile strained Si is constructed based on the valance band calculation. This model can be directly used in the device model of metal-oxide semiconductor field effect transistor （MOSFET）. It also a provides valuable reference for biaxial tensile strained silicon MOSFET design.

A Nonlinear Dynamical Theory of Non-Classical Plates

In ths paper. a new nonlinear formulation of plates. including shear and rotatory inertia and transverse normal stress effects, is developed by means of general assumptions, of which the von Karman-type formulation and some thick plate theories are special cases. To keep the formulation fairly general, the problem addressed in this paper simultaneously includes: the effects of shear deformation according to the geometric deformation similarity of the crosssection, the rotatory inertia, and the transverse normal stress. The three-dimensional compatible equations are applied to derive the basic equations. Numerical results are given for linear and non-linear analysis of plates.

Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory

This manuscript presents the comprehensive study of thickness stretching effects on the free vibration,static stability and bending of multilayer functionally graded(FG)carbon nanotubes reinforced composite(CNTRC)nanoplates.The nanoscale and microstructure influences are considered through a modified nonlocal strain gradient continuum model.Based on power-law functions,four different patterns of CNTs distribution are considered in this analysis,a uniform distribution UD,FG-V CNTRC,FG-X CNTRC,and FG-O CNTRC.A 3D kinematic shear deformation theory is proposed to include the stretching influence,which is neglected in classical theories.Hamilton's principle is applied to derive the governing equations of motion and associated boundary conditions.Analytical solutions are developed based on Galerkin method to solve the governing equilibrium equations based on the generalized higher-order shear deformation theory and the nonlocal strain gradient theory and get the static bending,buckling loads,and natural frequencies of nanoplates.Verification with previous works is presented.A detailed parametric analysis is carried out to highlight the impact of thickness stretching,length scale parameter(nonlocal),material scale parameter(gradient),CNTs distribution pattern,geometry of the plate,various boundary conditions and the total number of layers on the stresses,deformation,critical buckling loads and vibration frequencies.Many new results are also reported in the current study,which will serve as a benchmark for future research.

Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

We present a study on the dynamic stability of porous functionally graded(PFG)beams under hygro-thermal loading.The variations of the properties of the beams across the beam thicknesses are described by the power-law model.Unlike most studies on this topic,we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent,simultaneously,by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory(NSGIT)which are strictly equipped with a set of constitutive boundary conditions(CBCs),and through which both the stiffness-hardening and stiffness-softening effects of the structures can be observed with the length-scale parameters changed.All the variables presented in the differential problem formulation are discretized.The numerical solution of the dynamic instability region(DIR)of various bounded beams is then developed via the generalized differential quadrature method(GDQM).After verifying the present formulation and results,we examine the effects of different parameters such as the nonlocal/gradient length-scale parameters,the static force factor,the functionally graded(FG)parameter,and the porosity parameter on the DIR.Furthermore,the influence of considering the size-dependent hygro-thermal load is also presented.

THE ELASTO-PLASTIC STRAIN ANALYSIS OF NOTCHED PLATE UNDER CYCLIC LOADING—AN APPLICATION OF PLASTIC ANISOTROPIC HARDENING THEORY

In order to predict the life of engineering structures, it is necessary to investigate the strain distribution in notched members. In gineral, the Uauschinger Effect of materials under cyclic loading is not negligible, and so the anisolropic hardening model has been suggested. From the comparison between the calculated and experimental results in this paper, we can see that even the linear kinematic hardening model is quite suitable for strain analysis under cyclic loading.

Strain effects on Li^(+) diffusion in solid electrolyte interphases:A molecular dynamics study

Dilithium ethylene dicarbonate(Li_(2)EDC) and dilithium butylene dicarbonate(Li_(2)BDC) are the common organic compositions of the solid electrolyte interphase(SEI) layers in rechargeable lithium-ion batteries.The Li^(+) diffusion in the amorphous and ordered phases of Li_(2) EDC and Li_(2) BDC under various strains has been investigated by using molecular dynamics simulations.It is found that different strains lead to diverse changes in Li^(+) diffusivity.The tensile strain makes the Li+diffusion coefficients increase in amorphous and ordered Li_(2)EDC or Li_(2) BDC,and the compressive strain makes the Li+diffusion coefficients decrease in them.The average Li+coordination number calculation,ion conductivity calculation and the calculation of the residence autocorrelation function in amorphous and ordered Li_(2)EDC or Li_(2)BDC are performed to further analyze the strain effects on Li^(+) transport in them.The factors influencing Li^(+) diffusion in amorphous and ordered Li_(2)EDC or Li_(2) BDC under the strain are discussed.