Non-classical Continuum Mechanics Research Articles

Investigation on the thermoelastic response of a nanobeam in modified couple stress theory considering size-dependent and memory-dependent effects

For nano/micro-scale structures, the classical continuum mechanics can’t properly depict the size-dependent effect arising in them anymore. In addition, Fourier’s law is limited in describing the heat transfer process in nano/microstructures due to its nature of predicting infinite heat speed. To amend such defects, the non-classical continuum mechanics based on the nonlocal elasticity theory and the modified couple stress theory incorporating the non-Fourier heat transfer model with memory-dependent effect is formulated to characterize the thermoelastic behaviors in micro/nano-scale system in the present work. Then, the model is used to study the dynamic response of an Euler-Bernoulli nanobeam subjected to a ramp-type heating. The corresponding governing equations are derived and then solved by Laplace transform method and its numerical inversion. The distributions of the non-dimensional transverse deflection, temperature and displacement are obtained and illustrated graphically. In calculation, the influences of the ramp heating time parameter, the material length parameter, the nonlocal parameter, the time-delay factor and the kernel functions on the considered physical quantities are examined and discussed in detail.

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.

Dynamic analysis of FG nanobeam reinforced by carbon nanotubes and resting on elastic foundation under moving load

In the context of nonclassical continuum mechanics, the nonlocal strain gradient theory is employed to develop a nonclassical size dependent model to investigate the dynamic behavior of a CNTs reinforced composite beam resting on two parameters elastic foundations under a moving load. The governing dynamic equations of motion are derived based on third-order shear deformation theory using Hamilton’s principle. An analytical solution methodology is developed using Navier’s procedure is developed to derive the analytical solution for the equations of motion. The developed methodology is checked and compared. Parametric studies are conducted to demonstrate the applicability of the developed procedure to investigate the dynamic behavior of CNTs beams under moving load. Effects of the elastic foundation parameters, volume fraction, CNTs configuration, the nonclassical parameters, and the moving load velocity parameter on the dynamic behavior of CNTs beams under moving load are investigated and analyzed. The obtained results are supportive for design and manufacturing of composite CNTs beams.

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.

Strain gradient enhanced chemo-mechanical modeling of fracture in cathode materials for lithium-ion batteries

Next generation lithium-ion batteries (LIBs) cathodes are composed of polycrystalline microstructures where spatial heterogeneity such as grain size is often comparable to the geometric dimension of the cathodes. Incorporating the influence of the aforementioned heterogeneity in a chemo-mechanical continuum description is necessary to address the diffusion induced stress field during electrochemical cycling. Therefore, in the present study, the micro-structural size-dependent characteristic length of the cathode is embedded through a non-classical continuum mechanics approach known as the strain gradient elasticity (SGE) theory. Accordingly, a thermodynamically consistent multi-physics framework is developed where nonlinear diffusion kinetics is included along with the energetic interaction between the lithium-ions and host lattice of the electrode. The coupled governing equations are derived for the spherical cathode, and the generalized differential quadrature (GDQ) method is utilized to build the numerical procedure. The role of material heterogeneity length scale is addressed for strongly coupled chemo-mechanical process. Subsequently, the growth of potential surface flaws is examined under multiple length scales, particle size, and charge rate conditions. Ultimately, a comprehensive fracture map is established to illustrate the regime of flaw insensitivity, partial breakage and complete disintegration. Thus, the present findings can provide a suitable design perspective for fail-safe particulate type composite cathodes.

Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory

In the present manuscript, based on the nonlocal strain gradient theory, a nonclassical dynamic finite element model is developed to study and analyze the dynamic behavior of perforated nanobeam structures under moving mass/load. In the context of nonclassical continuum mechanics and Timoshenko beam theory, dynamic equations of motion of perforated nanobeams are derived including both size scale (nonlocal) and microstructure (strain gradient) effects. The modification of the geometrical parameters due to the perforation process is included in the equations of motion for squared holes arranged in the arrayed form. The effect of the moving mass (the inertia, Coriolis and centripetal forces, and the gravity force) or moving load are included in the proposed model. To remove shear locking problem in slender nanobeams, finite element model on nonclassical shape function basis is developed. Elements stiffness and mass matrices and force vector including the nonlocal and strain gradient effects are derived. The proposed model is verified and checked with previous works. Impacts of perforation, mass/load velocities, inertia of mass, microstructure parameter and nonlocal size scale effects on the dynamic and vibration responses of perforated nanobeam structures have been investigated in a wide context. The following model is beneficial for the design of MEMS/NEMS structures such as frequency filters, resonators, relay switches, accelerometers, and mass flow sensors, with perforation.

Nonlocal elasticity and boundary condition paradoxes: a review

Nonclassical continuum mechanics theories have seen a rise in implementation over the past several years due to the increased research into micro-/nanoelectromechanical systems (MEMS/NEMS), micro-/nanoresonators, carbon nanotubes (CNTs), etc. Typically, these systems exist in the range of several nanometers to the micro-scale. There are several available theories that can capture phenomena inherent to nanoscale structures. Of the available theories, researchers utilize Eringen’s nonlocal theory most frequently because of its ease of implementation and seemingly accurate results for specific loading conditions and boundary conditions. Eringen’s integral nonlocal theory, which leads to a set of integro-partial differential equations, is difficult to solve; therefore, the integral form was reduced to a set of singular partial differential equations using a Green-type attenuation function. However, a so-called paradox has arisen between the integral and differential formulations of Eringen’s nonlocal elasticity. For certain boundary and loading conditions, instead of the expected softening effect inherent in nonlocal particle interactions, some researchers have found a stiffening effect. Still, others have found no variation from those results found using classical theories. As such, the discrepancies between the integral and differential forms have been the subject of debate for nearly two decades, with several proposed resolutions published in recent years. This paper serves to review and consolidate existing theories in nonlocal elasticity along with selected theories in nonclassical continuum mechanics, the utilization of Eringen’s nonlocal elasticity in beams, shells, and plates, the existing discrepancies and proposed solutions, and recommendations for future work.

Thermodynamically consistent thermoelastic plate and shell formulations for small deformation and small strain using non-classical continuum mechanics incorporating internal rotations

The work presented in a recent paper by the authors [35] for a thermodynamically consistent and kinematic assumption free plate and shell formulation for small deformation and small strain based on the conservation and balance laws of classical continuum mechanics (CCM) is extended here for non-classical continuum mechanics (NCCM). This formulation incorporates additional physics due to internal rotations that arise due to the deformation gradient tensor. This physics is neglected in CCM, hence is absent in the plate and shell formulation of reference [35]. Consideration of this new physics requires modifications of the current balance laws as well as consideration of a new balance law “balance of moment of moments” (BMM) [2, 3]. Cauchy stress tensor becomes non-symmetric. Cauchy moment tensor is conjugate to the symmetric part of the rotation gradient tensor which exists now due to new physics. Balance of angular momenta yields additional three differential equations as part of the mathematical model. The new balance law (BMM) establishes symmetry of the Cauchy moment tensor. The new physics considered here exists in all deforming solid continua as it is due to the deformation gradient tensor, but is ignored in CCM. The consequence of this new physics is additional stiffness, hence additional strain energy storage and change in the time history of displacements and stress field compared to formulations based on CCM. The basic mathematical model for the plate and shell deformation consists of conservation and balance laws in $$\mathbb {R}^3$$ based on NCCM incorporating internal rotations. The associated finite element formulations for obtaining the solution of the mathematical model consists of : (i) geometry of the plate or shell described by the flat or curved middle surface (as done conventionally) and nodal vectors locating the top and bottom faces of the plate/shell (ii) the displacement field approximation that is p-version hierarchical in the plane as well as in the transverse direction (iii) integral form is constructed using Galerkin Method with Weak Form (GM/WF) and the corresponding element equations. The formulation presented here remains valid and accurate for thin as well as thick plate/shell and naturally reduces to the formulation of reference [35] based on classical continuum mechanics. Model problem studies and comparisons with the studies based on CCM formulation [35] will be presented in a follow up paper.

Emerging Challenges for Numerical Simulations of Quasi-Static Collision Experiments on Laser-Welded Thin-Walled Steel Structures

This paper re-evaluates recently published quasi-static tests on laser-welded thin-walled steel structures in order to discuss the fundamental challenges in collision simulations based on finite element analysis. Clamped square panels were considered, with spherical indenter positioned at the mid-span of the stiffeners and moved along this centerline in order to change the load-carrying mechanism of the panels. Furthermore, the use of panels with single-sided flat bar stiffening and web-core sandwich panels enabled the investigation of the effect of structural topology on structural behavior and strength. The changes in loading position and panel topology resulted in different loading, structural and material gradients. In web-core panels, these three gradients occur at the same locations making the panel global responses sensitive for statistical variations and the failure process time-dependent. In stiffened panel with reduced structural gradient, this sensitivity and time-dependency in failure process is not observed. These observations set challenges to numerical simulations due to spatial and temporal discretization as well as the observed microrotation, which is beyond the currently used assumptions of classical continuum mechanics. Therefore, finally, we discuss the potential of non-classical continuum mechanics as remedy to deal with these phenomena and provide a base for necessary development for future.

Preface to the special issue “Recent advances in nonlocal and non-classical continuum mechanics: in honor of Professor J. N. Reddy” on the occasion of his 75th Birthday

Preface to the special issue “Recent advances in nonlocal and non-classical continuum mechanics: in honor of Professor J. N. Reddy” on the occasion of his 75th Birthday

A Review on Strain Gradient Plasticity Approaches in Simulation of Manufacturing Processes

Predicting the performances of a manufactured part is extremely important, especially for industries in which there is almost no room for uncertainties, such as aeronautical or automotive. Simulations performed by means of numerical methods such as Finite Element Methods represent a powerful instrument in achieving high level of predictability. However, some particular combinations of manufactured materials and manufacturing processes might lead to unfavorable conditions in which the classical mathematical models used to predict the behavior of the continuum are not anymore able to deliver predictions that are in good agreement with experimental evidence. Since the first evidences of the shortcomings of the classical model were highlighted, many non-classical continuum mechanics theories have been developed, and most of them introduce dependencies at different levels with the Plastic Strain Gradient. This manuscript aims at gathering the milestone contributions among the Strain Gradient Plasticity Theories developed so far, with the object of exploring the way they interface with the requirements posed by the challenges in simulating manufacturing operations. Finally, the most relevant examples of the applications of Strain Gradient Plasticity Theories for manufacturing simulations have been reported from literature.

Dynamic behavior of thermoelastic solid continua using mathematical model derived based on non-classical continuum mechanics with internal rotations

This paper considers dynamic behavior of non-classical thermoelastic solid continua. The mathematical model consists of the conservation and balance laws of non-classical continuum mechanics that incorporates additional physics of internal rotations arising due to deformation gradient tensor. We consider plane stress behavior with small deformation, small strain physics only. Galerkin Method with Weak Form (GM/WF) in space is considered to construct a space–time decoupled finite element formulation giving rise to ordinary differential equations (ODEs) in time containing mass matrix, stiffness matrix due to classical as well as non-classical physics and acceleration and displacement associated with nodal degrees of freedom. This formulation is utilized to: (1) study natural undamped modes of vibration (2) study transient dynamic response by time integrating the ODEs in time (3) study the transient dynamic response by transforming the ODEs in time to modal basis using eigenvectors of the undamped natural modes. The ODEs in modal basis are used to construct transient dynamic response by time integrating them as well as by considering their analytical solutions. The solutions of the model problem obtained using the mathematical model based on non-classical continuum mechanics with internal rotations are presented and are compared with those obtained using the mathematical model based on classical continuum mechanics to demonstrate the influence of new physics due to internal rotations on the dynamic response of solid continua.

Two types of scale effects on the nonlinear forced vibration of axially moving nanobeams

Various non-classical continuum mechanics models appearing in previous studies cannot perfectly explain the mechanical properties of micro- and nanomaterials. Establishing a reasonable continuum mechanics model that comprehensively reflects the scale effect on material deformation is of great practical significance for objectively explaining the variation law of mechanical properties of micro- and nanomaterials under the combined action of different scale effects. Based on nonlocal strain gradient theory, a new scale-dependent model is proposed for axially moving nanobeams. In this study, an asymptotic expansion is performed using the multiscale time method to obtain the amplitude-frequency response curve of the equilibrium solutions for the forced vibration problem. Afterwards, the effects of various system parameters, especially the scale parameters, on the resonance curve are examined. Finally, the effects of nonlocal parameters and material characteristic length parameters on the amplitude-frequency response curves are investigated through typical numerical examples. The numerical results show that the nonlocal parameters promote the emergence of the main resonance, whereas the material characteristic length parameters suppress the emergence of the main resonance. Moreover, these parameters also affect the response amplitude and the skewness and jumping point of the amplitude-frequency characteristic curve.

Nonlinear Vibration and Tip Tracking of Cantilever Flexoelectric Nanoactuators

This paper examines the nonlinear vibration and tracking of cantilever nanoactuators made of isotropic nanodielectric materials with flexoelectric effect. The nonlinear governing equation of Euler–Bernoulli nanoactuators is derived based on non-classical continuum mechanics making use of material length scale parameters. By employing a higher-order curvature relation, the governing nonlinear partial deferential equations of motion are obtained by using the Hamilton’s principle. Incorporating the Galerkin method, the nonlinear partial deferential equation is reducing into a set of nonlinear ordinary differential equations. The obtained reduced-order model is solved by a perturbation method for free vibration response in semi-closed form. By introducing a new set of variables, the state space model of nanobeam is derived. The sliding mode control algorithm is employed to achieve a desired output for tip tracking, and Lyapunov stability theory is used to prove convergence in finite time. The effectiveness of the proposed control algorithm and input voltage is illustrated by numerical simulations. Regarding to the finding of this paper, it can be found that the sliding mode controller has better performance than linear controller, e.g., fuzzy controller.

A review on non-classical continuum mechanics with applications in marine engineering

Marine structures are advanced material and structural assemblies that span over different length scales. The classical structural design approach is to separate these length scales. The used structural models are based on classical continuum mechanics. There are multiple situations where the classical theory breaks down. Non-classical effects tend arise when the size of the smallest repeating unit of a periodic structure is of the same order as the full structure itself. The aim of the present paper is to discuss representative problems from different length scales of ship structural design.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Limit State Analyses in Design of Thin-Walled Marine Structures—Some Aspects on Length-Scales

AbstractThis paper gives an overview of the factors that affect the strength and structural design of advanced thin-walled marine structures with reduced plate thickness or alternative topologies to those used today in the marine industry. Due to production-induced initial deformations and resulting geometrical non-linearity, the classical division between primary, secondary, and tertiary responses becomes strongly coupled. Volume-averaged, non-linear response of structural element can be used to define the structural stress–strain relation that enables analysis at the next, larger, length scale. This, today’s standard homogenization process, needs to be complemented with localization, where the stresses are assessed at the details, such as welds for fatigue analysis. Due to this, the production-induced initial distortions need to be considered with high accuracy. Another key question is the length-scale interaction in terms of continuum description. Non-classical continuum mechanics are needed when consecutive scales are close. Strain-gradients are used to increase the accuracy of the kinematical description of beams, plates, and shells. The paper presents examples of stiffened and sandwich panels covering limit states such as fatigue, non-linear buckling, and fracture.

Nonlinear size dependent modeling and performance analysis of flexoelectric energy harvesters

In this study, the effect of flexoelectricity on rectangular prismatic beams is analytically investigated when considering various boundary conditions including fixed–fixed, fixed–hinged, and hinged–hinged. The dynamical system undergoes harmonic loading and has the von Karman strain nonlinearity due to the mid-plane stretching. The beams are assumed to be long and slender, allowing for Euler–Bernoulli beam analysis to be conducted. To account for the size dependent effects of the material, Eringen’s nonlocal elasticity theory is employed. After determining the exact mode shapes, the Galerkin discretization is utilized to derive the nonlinear reduced-order model. The fixed–hinged model is the only model that can harvest power due to its lack of boundary condition symmetry and is the model that is heavily examined. The harvested power from the macro to the micro scales is small, as is expected; however, at the nanoscale the softening effect due to the nonlocality increases the nonlinear hardening behavior and decreases the system’s natural frequency. This allows for lower excitation frequencies to harvest energy, with a broader range of frequencies being able to harvest energy. Micro and macro scales do not exhibit any variation based on nonlocal effects, nor can they harvest significant amounts of power through flexoelectricity alone. These nano-devices can be developed and can produce significantly more energy than previously thought; but they require more advanced modeling techniques that necessitate a more thorough understanding of non-classical continuum mechanics and its coupling to nonlinear dynamics.

Thermodynamic consistency of beam theories in the context of classical and non-classical continuum mechanics and a thermodynamically consistent new formulation

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

Lyapunov-Based Boundary Control of Strain Gradient Microscale Beams With Exponential Decay Rate

In nonclassical microbeams, the governing partial differential equation (PDE) of the system and corresponding boundary conditions are obtained based on the nonclassical continuum mechanics. In this study, exponential decay rate of a vibrating nonclassical microscale Euler–Bernoulli beam is investigated using a linear boundary control law and by implementing a proper Lyapunov functional. To illustrate the performance of the designed controllers, the closed-loop PDE model of the system is simulated via finite element method (FEM). To this end, new nonclassical beam element stiffness and mass matrices are developed based on the strain gradient theory and verification of this new beam element is accomplished in this work.