Non-classical Continuum Theory Research Articles

Prediction of nonlocal elasticity parameters using high-throughput molecular dynamics simulations and machine learning

Higher-order elasticity theories are able to model the mechanics at nanoscale. However, these theories have length-scale parameters that need to be evaluated either through experiments or molecular dynamics (MD) simulations. In the current literature, these length-scale parameters are assumed to be constant for a chosen material. However, this is not true, as shown in this paper through a set of high-throughput MD simulations on a variety of boundary value problems (BVP). For carbon and boron nitride nanotubes, the length-scale parameter in the modified strain gradient theory is found to vary with the absolute dimensions, type of the BVP, and the deformation level. This complex dependence makes prediction of the length-scale parameter l0 challenging. To address this issue, a supervised machine learning (ML) based framework is developed here. In this new framework, MD, continuum formulation, and ML are used in tandem to predict the length-scale parameter for a given material, dimension, and boundary condition. It is a three-step process. First, a number of MD simulations are performed for varying chirality, length, and boundary conditions. Then, the values of l0 are found using these MD output in the continuum formulation. Finally, an ML-based regression model is used to capture the variability of l0 and subsequently predict its value for any chirality, length, and boundary condition. Here three regression models – Gaussian process regression, support vector machine, and neural network – are used to make the predictions.This predictive tool eliminates need for expensive MD simulations for further continuum scale analysis. In a broader context, this novel three-step framework opens an accurate yet inexpensive door for applying non-classical continuum theories to nanoscale mechanics problems. The accuracy is imposed by the MD simulations, while the computational cost reduction is achieved by machine learning.

Size-dependent coupled bending-torsional analysis of piezoelectric micro beams

Nowadays, piezoelectric micro-beam structures are used in various industries, and knowing and analysis of these smart microstructures is important. However, in the modeling of these electromechanical microbeams, the torsion during bending is generally omitted for simplicity. Considering that in some cases, twisting and bending cannot be omitted and must be modeled simultaneously. Therefore, this article presents a coupled bending-torsion model for the first time for static deflection and free vibration analysis of piezoelectric microbeams with nonclassical continuum theories. The governing equations of the motion and boundary conditions of the Timoshenko beam at the micro-scale have been obtained using size-dependent electromechanical theory and Hamilton’s principle. In addition, the Laplace transform is used to solve the static equations, while the generalized differential quadratic (GDQ) method is used to discretize and solve the free vibration equations. In the following, the influence of various parameters, such as the effect of size and geometric conditions, on the static displacements and the natural frequency of the microbeam is discussed.

On the Advances of Computational Nonclassical Continuum Theories of Elasticity for Bending Analyses of Small-Sized Plate-Based Structures: A Review

On the Advances of Computational Nonclassical Continuum Theories of Elasticity for Bending Analyses of Small-Sized Plate-Based Structures: A Review

Finite element modelling of exponentially graded composites with microstructure

Composite material properties are dependent on their microstructure. To adequately model these microstructural effects, a formulation of elasticity that accounts for microstructural effects must be considered. Size-dependent behaviour is an inherent property of such materials, resulting in a need for non-classical continuum theories for adequate characterization. A three-dimensional model for an exponentially graded composite with microstructural effects is developed under the framework of Cosserat elasticity. The mixed boundary value problem is formulated and solved through the finite element method, and implemented in the commercial software Abaqus through two user developed elements. Recovery of the classical limit for the Dirichlet problem is discussed as an initial model validation until complete experimental measurements can be conducted.

The strain gradient-based torsional vibration analysis of micro-rotors with nonlinear flexural-torsional coupling

The powerful non-classical continuum theory of strain gradient elasticity is capable of effectively capturing small-scale effects in micro-structures. Based on this theory, a formulation is developed to investigate the coupled torsional-flexural vibrations of micro-rotors in the presence of inertia nonlinearities arising from the eccentricity and gyroscopic motion of the rotors. With the aid of a weighted-residual technique, coupled nonlinear partial differential equations of motion are truncated into a discrete model. Then, the resonance of the fundamental mode of the torsional vibration excited by the flexural vibration is investigated by utilizing the perturbation method of multiple scales. Moreover, a numerical simulation approach is conducted to confirm the validity of the analytical solution proposed by the multiple scales perturbation method. Obtained results for the resonant frequency and amplitude of the fundamental torsional mode indicate that the strain gradient theory can predict far more reliable results than the classical continuum mechanics for micro-rotors with very thin shafts.

Nonclassical continuum theories for fluent media incorporating rotation rates and their thermodynamic consistency

AbstractIn this paper, we present three micropolar nonclassical continuum theories (NCCT) for fluent medium in which a material point always has velocities as degrees of freedom, additionally, we consider (i) in the first NCCT, we consider classical or internal rotations rates (known) due to skew‐symmetric part of velocity gradient tensor; (ii) in the second NCCT, we consider rotation rates and Cosserat or microrotation rates (unknown degrees of freedom); (iii) in the third NCCT, we consider Cosserat or microrotation rates only, hence are neglected. Conservation and balance laws (CBL) and the constitutive theories are derived for all three NCCT. We examine consistent choice of kinematic variables, modification of the CBL of classical continuum theories (CCT) due to new physics and determine if the new physics of rotation rates requires additional balance law. All three NCCT derived here are examined for thermodynamic and/or mathematical consistency and are compared with published works. Model problem studies are also presented.

Size-dependent wave propagation in two-dimensional functionally graded lattice materials

Although the size-dependency of small-scale structures has limited the application of classic continuum theories to model their various mechanical properties, it has opened new avenues for the introduction of nonclassical continuum theories with the ability to capture the size effects. In this work, the modified strain gradient theory, for the first time, is used to model two-dimensional planar lattices of hexagonal and triangular architectures made of functionally graded materials. Using the finite element method and Bloch's theorem, the size-dependent wave propagation in such micro-lattices is investigated and the effect of the functionally graded distribution of materials on the wave-propagation and wave-filtering performances of such structures is studied. The results show that using the modified strain gradient theory, the dispersion curves and stop-bands are predicted at higher frequencies compared to the classical and modified couple stress theories. Additionally, by implementing functionally graded constituting elements, the location and width of stop-bands can be tuned.

Impacts of Length Scale Parameter on Material Dependent Thermoelastic Damping in Micro/nanoplates Applying Modified Coupled Stress Theory

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and need tobe alleviated in resonators inorder to enhance its performance parameters by improving its thermoelastic dampinglimited qualityfactor, QTED. The maximum energy dissipation is also interrelated with critical length (???????? ) of theplates and by optimizing the dimensions the peaking of energy dissipation can be diminished. As the size of thedevices is scaled down, classical continuum theories are not able to explain the size effect related mechanicalbehavior at micron or submicron levels and as a result non-classical continuum theories are pioneered with theinception of internal length scale parameters. In this paper, analysis of isotropic rectangular micro-plates based onKirchhoff model applying Modified Coupled Stress Theory is used toanalyzethe size-dependent thermoelasticdamping and its impact on quality factor and critical dimensions.Hamilton principle is adapted to derive thegoverning equations of motion and the coupled heat conduction equation is employed to formulate the thermoelasticdamping limited quality factor of the plates. Five different structural materials (PolySi, Diamond,Si, GaAs andSiC)are used for optimizing QTED which depends on the materialperformance index parameters. ThermoelasticDamping Index [TDI] and thermal diffusion length, lT. According to this work, the maximum QTED is attained forPolySi with the lowest TDI and Lcmax is obtained for SiC which is having the lowest lT. The impact of lengthscaleparameters (l), vibration modes, boundary conditions (Clamped–Clamped and Simply Supported), and operatingtemperatures on QTED and Lcare also investigated. It is concluded that QTED is further maximized by selecting lowtemperatures and higher internal length scale parameters (l).The prior knowledge of QTED and Lchelp the designers tocome out with high performance low loss resonators.

Nonlinear Vibration Movements of the Mid-Supported Micro-Beam

This study analyzes the vibration movements of multi-support micro beams placed in an electrically smooth area using the modified couple stress theory. It has been assumed that the potential voltage that creates the electrical field strength varies harmonically. Large number of experiments in recent years have indicated that classical continuum theory is unable to predict the mechanical behavior of microstructure with small size. However, nonclassical continuum theory should be used to accurately design and analyze the microstructures. Modified couple stress theory models the micro and nanomechanical systems with higher accuracy because they employ additional material parameters to the equation considering size dependent behavior. The most general nonlinear motion equations for multi-support microbeams have been obtained by considering the material size parameter, the number of support and support positions, damping effect, axial stresses, electrical field strength, and nonlinear effects resulting from elongations. The nonlinear equations of motion are obtained according to the Hamilton method using the modified couple stress theory (MCST). The resulting equations of motion are nondimensionalized. In this way, the mathematical model has been made independent of the type and geometric structure of the material. Approximate solutions of the obtained dimensionless motion equation are obtained by the multi-scale method, which is one of the perturbation methods. As a result, an increase occurs in the first mode frequencies ([Formula: see text]) and nonlinear correction effect parameters ([Formula: see text]) with the progress of the center support position gradually towards [Formula: see text] and the increase of the microbeam elasticity coefficient ([Formula: see text]).

Porous flexoelectric cylindrical nanoshell based on the non-classical continuum theory

Porous flexoelectric cylindrical nanoshell based on the non-classical continuum theory

State-of-the-Art of Vibration Analysis of Small-Sized Structures by using Nonclassical Continuum Theories of Elasticity

State-of-the-Art of Vibration Analysis of Small-Sized Structures by using Nonclassical Continuum Theories of Elasticity

Thermoelastic Damping Limited Quality Factor Enhancement and Energy Dissipation Analysis of Rectangular Plate Resonators Using Nonclassical Elasticity Theory

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and needs to be alleviated in vibrating resonators to mitigate parameters by improving the thermoelastic damping limited quality factor, QTED. The maximum energy dissipation is also interrelated with the critical dimension h c of the plates, and by optimizing the dimensions, the peaking of energy dissipation can be diminished. As the size of the devices is scaled down, classical continuum theories become incompetent to explain the size-effect related mechanical nature at the micron and submicron levels, and, as a result, nonclassical continuum theories have been pioneered with the inception of internal length scale parameters. In this work, an analysis of isotropic rectangular microplates based on the Kirchhoff model and a higher order theory like Modified Couple Stress Theory is utilized to study size-dependent thermoelastic damping and its impact on the quality factor and critical dimensions. The Hamilton principle is adapted to derive the governing equations of motion, and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the plates. Five different structural materials (PolySi, diamond, Si, GaAs, and SiC) are used for optimizing QTED and hc, which depends on two material performance index parameters: the thermoelastic damping index (TDI) and the material thermal diffusion length, l T . According to this work, the maximum QTED is attained for PolySi with the lowest TDI, and hcmax is obtained for Si with the maximum l T . The impacts of the dimensionless length-scale parameters (l/h), vibration modes, and boundary conditions (clamped-clamped and simply supported) on QTED and hc are also investigated. From the current analysis, QTED can be further enhanced by selecting higher vibration modes and clamped-clamped boundary conditions. QTED can be maximized by fixing the internal length scale parameter (l) and making the thickness of the beam equal to l. The analytical study is numerically simulated by using MATLAB 2015 software. Prior knowledge of QTED and hc will help designers to produce high-performance and low-loss resonators for the futuristic technological applications.

Size-dependent vibration of multi-scale sandwich micro-beams: An experimental study and theoretical analysis

A series of multi-scale sandwich cantilever micro-beams are prepared to investigate the size-dependent phenomenon in this paper. The sandwich micro-beam samples consist of a titanium substrate with a thickness of 1.996μm and two symmetrical nickel coatings with thickness between 106.6 and 362.6 nm by employing a physical vapor deposition method. The vibration responses of these composite micro-beams are excited by an acoustic exciter and measured by a laser Doppler vibration measurement system. It is revealed that with the decreasing of coating thickness, the dimensionless bending rigidity of nickel coating increases, and the dimensionless natural frequency of micro-beam increases first then decreases. Based on the non-classical continuum theories, the theoretical model of the sandwich micro-beam is established. Then the material length scale parameters are derived. The applicability of the non-classical continuum theories is verified when the material characteristic size of the structure is smaller than the material length scale parameter. The results are of great significance to the design and optimization of micro-/nano-electro mechanical systems.

Rotational Inertial Physics in Non-Classical Thermoviscous Fluent Continua Incorporating Internal Rotation Rates

In this paper, we derive non-classical continuum theory for physics of compressible and incompressible thermoviscous non-classical fluent continua using the conservation and balance laws (CBL) by incorporating additional physics of internal rotation rates arising from the velocity gradient tensor as well as their time varying rates and the rotational inertial effects. In this non-classical continuum theory time dependent deformation of fluent continua results in time varying rotation rates i.e., angular velocities and angular accelerations at material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor; (ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular accelerations to demonstrate the influence of the new physics. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and principle of equipresence when appropriate. Constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity (complete basis). It is shown that the mathematical model consisting of the conservation and balance laws and constitutive theories presented in this paper has closure. Influence of new physics in the conservation and balance laws on compressible and incompressible thermoviscous fluent continua is demonstrated due to presence of angular velocities and angular accelerations arising from time varying rotation rates when the deforming fluent continua offer rotational inertial resistance. The fluent continua are considered homogeneous and isotropic. Model problem studies are considered in a follow-up paper.

Design of an integral super twisting sliding mode controller for position control and extending the traveling range of initially curved micro-beams

Based on the super twisting concept, this article develops an integral sliding mode controller with nonlinear disturbance observer for position control and extending the traveling range of an initially curved micro-beam. The nonlinear damped model of the curved micro-beam is modeled based on the non-classical continuum theory. The single-mode assumption is used to transform the nonlinear governing PDE of the system into a nonlinear state-space form. The robust controller is designed to overcome the position control problem in the presence of non-parametric uncertainties and unknown non-symmetric input saturation due to the existing constraints in the electrostatic actuation. The effects of the uncertainties are considered as a compound disturbance term which can be estimated in finite time. Furthermore, due to the estimation properties of the observer, knowledge about the bounds of the uncertainties is not required. Extensive simulation results clearly verified the effectiveness of the controller in position tracking and extending the traveling range of the initially curved micro-beam to the unstable zones under smooth control effort.

On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations

The topic presented in this research is the calibration of small-scale parameters of non-classical continuum theories such as nonlocal strain gradient theory, strain gradient theory, stress-driven nonlocal elasticity, and strain-driven nonlocal elasticity. Governing equations of vibrational behavior of circular nanoplate and associated boundary conditions for each method derived using Hamilton's principle. Obtained governing differential equations from non-classical methods were solved using the general differential quadrature rule (GDQR). Then, the first natural frequencies for different radiuses and different size parameters were obtained. On the other hand, the first natural frequencies of circular nanoplates are calculated using molecular dynamics simulation based on AIREBO and Tersoff potentials for different radiuses. Fast Fourier transform (FFT) was utilized to calculate natural frequencies based on the molecular dynamics simulation. Using the accurate size parameter is an important point in the application of non-classical continuum theories. To obtain the size parameters related to different non-classical methods, the results of molecular dynamics compared to those of nan-classical methods and simulated annealing (SA) algorithm optimization technique was utilized. Results show that stress-driven nonlocal, strain-driven nonlocal, and strain gradient methods cannot predict the behavior predicted by molecular dynamics for all ranges of radius. In other words, the responses of these three methods for any value of the size parameters (in some interval radius) and results of the molecular dynamics method are not equal for a few numbers of studied radii. In contrast to these three methods, the nonlocal strain gradient method predicts the results obtained by molecular dynamics well for all radii. The results of this paper are very useful for researchers in the field of non-classical continuum mechanics.

Onset of chaos in nano-resonators based on strain gradient theory: Numerical analysis

In recent years, experimental and theoretical studies have shown that size effect is non-negligible in mechanical micro- and nano-structures. As a result, classical continuum theory cannot model these structures. To accurately predict the behavior of micro- and nano-structures, non-classical continuum theories should be used. In this paper, the effect of size on the chaotic region of resonators is studied by comparing the results of classical and strain gradient theories. A bifurcation diagram and Lyapunov exponent are used for detecting chaos in nano-resonators.It is shown that the effect of size on the chaotic region in bifurcation diagram varies significantly depending on the bias voltage. Lyapunov exponent is utilized to validate the chaotic region and it is in best agreement with bifurcation diagram. For some DC voltage values, size effect can eliminate chaotic vibrations while in higher DC voltages, it can result in chaos at much lower amplitudes than those predicted by classical theories. Dynamic behaviors of nano-resonators are categorized based on their bias voltages and their chaotic behaviors are analyzed based on classical and non-classical theories in detail. The obtained numerical results are compared with Melnikov and Maximum Velocity methods and it is shown that all methods predict the lower critical actuation amplitude for strain gradient theory.

A micro/nano-scale Timoshenko-Ehrenfest beam model for bending, buckling and vibration analyses based on doublet mechanics theory

A micro-nano-scale Timoshenko-Ehrenfest beam model is investigated using doublet mechanics theory in the present study. The governing equations and all possible boundary conditions are obtained based on doublet mechanics model. The static bending, buckling and vibration problems of Timoshenko microbeams are examined in detail. Deflection, rotation, critical buckling loads and natural frequencies predicted by the present doublet mechanics model are obtained for simply supported micro-scale Timoshenko beams by the Navier solution method. The obtained results are compared to other classical and non-classical continuum theories. To illustrate the present doublet mechanics model, the influences of thickness to length scale parameter ratio of the considered material and slenderness ratio on static bending, buckling and vibration problems are investigated. It is shown that there are two frequency spectrums in the vibration of nanobeams similar to macro Timoshenko beams. It is interesting to note that acceptable physical frequencies (mode numbers) have an upper bound due to Van Hove singularity depending on geometrical and material properties of the beam. That fact is observed first time in the open literature by using scale dependent theories.

On the dynamics of a laminated annular piezoelectric microsystem on a viscoelastic substrate within modified couple stress elasticity

This article deals with the vibrational characteristics of a laminated composite annular microplate using a non-classical continuum theory called modified couple stress theory (MCST). Also, the structure is patched with the piezoelectric layer and covered with a viscoelastic foundation that simulated via the Kelvin-Voight model. The non-classical governing equations and boundary conditions of the size-dependent annular microplate are derived by adding the symmetric rotation gradient and higher-order stress tensors to the strain energy. The current non-classical model is capable of capturing the size-dependency in the annular microplate by using only one material length scale parameter. Moreover, the mathematical formulation of annular microplate based on the classical model can be recovered from the present model by neglecting the material length scale parameter. Finally, the non-classical governing equations are solved using the generalized differential quadrature method (GDQM) for various boundary conditions. Afterward, a parametric study is carried out to investigate the effects of the length scale parameter, piezoelectric layer, radius ratio, circumferential and radial mode number, geometry of laminated layer, and boundary conditions on the frequency responses of the annular microplate by considering MCST. The results show that in the grater elastic properties ratio () factor, increasing the damping parameter () cannot make any changes in the frequency of the disk.

Vibrational characteristics of a higher-order laminated composite viscoelastic annular microplate via modified couple stress theory

This article deals with the vibrational characteristics of a laminated composite annular microplate using a non-classical continuum theory called modified couple stress theory (MCST). The structure is covered with a viscoelastic foundation that is simulated via the Kelvin-Voight model. The non-classical governing equations and boundary conditions of the size-dependent annular microplate are derived by adding the symmetric rotation gradient and higher-order stress tensors to the strain energy. The current non-classical model is capable of capturing the size-dependency in the annular microplate by using only one material length scale parameter; moreover, the mathematical formulation of annular microplate via the classical model can be recovered from the present model by neglecting the material length scale parameter. Finally, the non-classical governing equations are solved using the generalized differential quadrature method (GDQM) for various boundary conditions. Afterward, a parametric study is carried out to investigate the effects of the length scale parameter, radius ratio, circumferential and radial mode number, geometry of the laminated layer, and boundary conditions on the frequency responses of the annular microplate by considering MCST. The results show that in the grater elastic properties ratio (E1/E2) factor, increasing the viscoelastic parameter cannot make any changes in the frequency of the disk.