Material Length Scale Research Articles

Size-Dependent Rigid–Flexible Coupling Dynamics of Functionally Graded Rotating Moderately Thick Microplates

Micro air vehicles, which are typical small-sized rotating-motion systems, have seen major advancements in recent years. To provide some theoretical basis for developing and producing micro air vehicles, this study establishes a novel rigid–flexible coupling dynamic model for functionally graded (FG) moderately thick rectangular microplates attached to a central rotating rigid hub based on the modified couple stress theory and first-order shear deformation theory. The proposed model incorporates nonlinear coupling term of in-plane deformation to reflect the dynamic stiffening effect caused by rotational motion. Material characteristics of the FG microplate have a linear power-law distribution along the thickness axis. Further, the discrete form dimensionless coupling dynamic equations and their numerical solutions are obtained by combining the Euler–Lagrange equation and the Chebyshev–Ritz method. Convergence and comparative studies are carried out to demonstrate the accuracy and validity of the proposed model. Thereafter, the influence of material length scale parameter, rotational speed, gradient index, and aspect ratio on the frequency of the microplates is investigated. Numerical results reveal that couple stress and dynamic stiffening effects both enhance the rigidity of the microplates, whereas the gradient index decreases the rigidity. Nonlinear coupling term which leads to significant differences in frequency value and trace line can’t be ignored for rotative structure. In-plane motion and its coupling terms play a significant function for the moderately thick or thick microplates. The increase of rotational speed and gradient index will reduce the size dependency of the microplate. Furthermore, the frequency trajectory steering and corresponding mode transition phenomenon are graphically represented.

Strain gradient plasticity phenomenon in surface treated plain carbon steel

Surface engineering is essential for materials being used for functional and structural applications. There have been various approaches and mechanisms used for improving, mechanical, tribological and surface properties of materials over the years. One phenomenon that require much attention is indentation size effect (ISE). Mechanistic and phenomenological theories have been developed to explain the observed size dependence of mechanical properties in materials. However, the dislocations microstructure evolution, deformation mechanisms and the contributions of length scale parameters are not fully understood. This paper explores the effects of surface carbo-nitriding on material length scale and dislocation microstructure mechanisms of 1045 steel. The material was carbo-nitrided at 500 °C and 900 °C, to induce nitrogen and carbon diffusion respectively, prior to, microstructural and mechanical (micro/nano-indentation) characterization. The statistically stored and geometrically necessary dislocation densities increased with increasing carbo-nitriding temperature. A bi-linear relationship was also established based on the indentation depth and hardness measurements. Three regimes of dislocation behaviour (dislocations source–limited/starvation, discrete dislocations, and continuum dislocations) are observed. The observed trends are explained using dislocation densities that were estimated using dislocation theories. The implications of the results are also discussed for the carbonitrided steels.

Free vibration of axially loaded microbeam in MEMS based on sinusoidal shear deformation theory

Free vibration of an axially loaded silicon microbeam in MEMS under electrostatic actuation is studied on the basis of a sinusoidal shear deformation theory. Based on the modified couple stress theory (MCST) and von Kármán geometric nonlinearity, a nonlinear finite element formulation is derived and employed to construct the discrete nonlinear governing equations for the microbeam. The deflection of the microbeam at a given DC voltage is firstly calculated using the Newton Raphson based iterative procedure, and used to evaluate the tangent stiffness matrix. The natural frequencies of the microbeam corresponding the DC voltage are then computed. The influence of the applied voltage, the axial force as well as the material length scale parameter on the frequencies of the microbeam is studied in detail and highlighted. The dependence of the pull-in voltage upon the material length scale and the axial force is also examined and discussed.

Numerical study of large deflection of nanobeam according to the MCST and SE

In the current study, the large deflection of nanobeam according to the modified couple stress theory (MCST) with surface effects (SE) was investigated by applying finite element method (FEM). For modeling of the nanobeam, we applying the Timoshenko beam theory (TBT), and for the solution of the problem, we used a total lagrangian finite element method. Also, shear deformation and axial displacements of nanobeams have been considered in this model. The FEM has been used for the solution of the problem. With the present model, small to large deflection of nanobeam with arbitrary loadings and boundary conditions can be studied. The numerical results for the small deflection of nanobeam achieved by the present model are well matched with results reported in the previous researches. Effects of the ratio of material length scale parameter to thickness on the large deflection of nanobeams have been investigated. Also, the influence of considering different beam constitutive theories in the large deflection analysis of nanobeams has been examined. The results achieved in this study show the relative importance of considering surfaces and size effects in the analysis of nanobeam.

Size-dependent constitutive model for shape memory alloys based on couple stress elastoplasticity

A constitutive model for shape memory alloys considering size effects is developed based on thermodynamic framework. Couple stress elastoplasticity is utilized to simulate size effects. Three material length scale parameters, two for elastic size effect of the austenite and martensite phases and one for crystal structure transformation, are introduced as additional material properties. The developed model is formulated with three-dimensional equations and can be used for the analysis of structures having arbitrary shapes by being adopted into the finite element method. Two benchmark problems (cantilever and bulge test) are solved using the developed model, which shows significant increase in structural stiffness as the size of the structure decreases. The developed size-dependent constitutive model is expected to be utilized for the analysis of micro/nano-scale structures composed of shape memory alloys.

A thermo-electro-mechanically coupled cohesive zone formulation for predicting interfacial damage

Material interfaces can occur at different material length scales. Understanding the properties and the behaviour of interfaces is of utmost importance because interfaces can significantly influence the effective constitutive response of the material under consideration. Based on the fundamentals of electro-mechanically coupled cohesive zone formulations for electrical conductors and on the associated finite element framework proposed in Kaiser and Menzel(2021), a thermo-electro-mechanically coupled cohesive zone formulation is established in this article. To this end, the governing equations of continuum thermodynamics for materials with interfaces under combined mechanical, thermal, and electrical loads are derived. A damage variable is introduced to account for the evolution of interface damage in a thermodynamically consistent way. Motivated by deformation-induced property changes, the effective thermal and electrical conductivities are moreover assumed to be functions of the damage variable. Finally, analytical solutions are derived to validate the finite element formulation, and representative boundary value problems are studied so as to reveal key properties of the proposed cohesive-zone framework.

Free vibration of size-dependent FGM Mindlin microplates in viscous fluid

This paper studies the free vibration of size-dependent functionally graded material (FGM) microplates in contact with viscous fluid. The Mori-Tanaka model is applied to formulate the continuous gradual variation of material properties of FGM microplates along thickness direction. A non-classical microplate model is established based on the modified couple stress theory, which considers the size effect by introducing the material length scale parameter. A physical neutral plane is introduced to eliminate the stretching-bending coupling effect. The motion of viscous fluid is defined by Navier-Stokes equations, with which the hydrodynamic loading on microplates is determined with consideration of inertial effect and viscous damping effect. The governing equations for FGM microplates in contact with viscous fluid are derived using the Hamilton’s principle and solved by differential quadrature method. Numerical results are obtained to discuss the influences of the aspect ratio, fluid depth, slenderness ratio, fluid viscosity, gradient index, fluid density, and size parameter on the vibration behaviors of microplates in contact with viscous fluid.

New closed-form solutions for flexural vibration of microplates based on a modified strain gradient theory

In this work, the flexural vibration of size-dependent Kirchhoff microplates has been investigated analytically following a modified strain gradient theory. A displacement-based variational formulation is established to derive the governing differential equations and all possible boundary conditions, including both essential and natural ones, for general curved boundaries. In particular, the natural boundary conditions are expressed in terms of force quantities in a concise form.The direct separation of variables method (DSVM) is developed to handle the eigenvalue problems of rectangular microplates with any combination of simply supported and clamped boundaries. Especially for the cases wherein two adjacent sides are clamped and the remaining sides are either simply supported or clamped, the closed-form solutions which used to be recognized as inexistent, are successfully obtained in the current work for the first time. These newly obtained results can serve as benchmarks for other relevant researches. Additionally, a parametric analysis is conducted to assess the influences of material length scale parameters and boundary conditions on size-dependent behavior of microplates.

Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory

In this article, the nonlinear postbuckling behavior of the magneto-electro-thermo-elastic (METE) laminated microbeams is presented. According to the modified couple stress theory (MCST) and Reddy's three-order shear deformation theory (RTSDT), in conjunctions with the von Karman geometric nonlinearity, the nonlinear static model of METE laminated microbeam is established. The nonlinear governing equations and the corresponding boundary conditions are derived by using the principle of virtual work principle. Afterwards, using an analytical method, the nonlinear postbuckling behavior of METE laminated microbeam with simple supported and clamped boundary conditions at the both ends is described. Moreover, numerical examples are exhibited to reveal the effects of the material length scale parameter, slenderness ratio, temperature rise, magneto-electric potential on the critical buckling load and the nonlinear postbuckling response. Also the distribution law of magneto-electric potential through the thickness direction of the microbeam with various lay-up modes is discussed.

Three-Dimensional Vibrational Characteristics of Functionally Graded Rotating Microbeams Based on the Higher-Order Refined Shear Deformation Theory

This study establishes a novel three-dimensional dynamic model for functionally graded rotating microbeams simultaneously considering flapwise and chordwise shear effects based on a refined shear deformation theory and the modified couple stress theory. Five kinematic variables are introduced to describe the rotating microbeam’s axial, flapwise, and chordwise motions using a higher-order refined shear deformation theory. The related strain energy and kinetic energy functionals are discretized by the assumed mode method. Applying the Euler–Lagrange variational formulation, the governing equations for the rotating microbeam are obtained. Selective examples are provided to demonstrate the convergence and effectiveness of the proposed model. Finally, the impacts of slenderness ratio, rotation speed, material index, material length scale parameter (MLSP), and Poisson’s effect on the vibrational characteristics of the microbeam are examined. Numerical results indicate that the mode shape transition and frequency loci veering phenomena could easily occur under slight changes in the above factors when dense modes appear in the rotating microbeam; moreover, there is a positive/negative competitive relationship among the effects of rotation speed, MLSP, and shear deformation; Poisson’s effect seriously affects the coupling between axial and non-axial motions of the rotating microbeam. The results obtained are expected to provide data accumulation and methodological basis for the failure analysis and vibration shape determination of micro air vehicles.

Forced Vibration Analysis of Bimodular Microbeams Based on the Modified Couple Stress Theory

The forced vibration model of bimodular microbeams is studied in this paper based on the modified couple stress theory. The effects of the bimodular characteristic and the size effect are considered in this model. The forced vibration governing equation, boundary conditions, and initial conditions are obtained by Hamilton’s principle. The influence of the bimodular ratio and the material length scale parameters on the time history response are investigated based on the analytical method. Results show that the larger the two parameters, the higher the frequency and the smaller amplitude.

Dynamic finite element analysis of microbeams under a moving mass based on refined third-order shear deformation theory

The dynamic behavior of microbeams with rectangular cross section under action of a moving mass is studied in the present paper in the framework of a refined higher-order shear deformation beam theory. The modified couple stress theory (MCST) with only one additional scale parameter is adopted to describe the influence of the microsize effect on the dynamic response of the microbeams. A finite element formulation with ten degrees of freedom is formulated and used to establish the discretized equation of motion for the microbeams. The formulation, taking into account the influence of the inertial effect, the Coriolis and centrifugal forces resulted from the mass moving, is derived from the expressions of the elastic and kinetic energies of the microbeams. The accuracy and efficiency of the derived formulation are confirmed by comparing the result obtained in the present work with the published data. The dynamic response of the microbeams with simply supported ends, such as the curves for dimensionless mid-span deflection-moving time relationship, the dynamic magnification factors (DMFs) and the thickness distribution of stresses are assessed by using an implicit Newmark method. The obtained numerical results reveal that the material length scale parameter which is introduced in the MCST has an important role on the dynamic behavior of the microbeams, and the DMF obtained from the theory incoporating the MCST is considerably lower than that using the conventional beam theory. It is also shown that the amplitude of both the axial stress and shear stress is considerably decreased by the increase of the material length scale parameter. A numerical study is carried out to highlight the influence of various parameters such as the moving mass velocity and the the ratio of the total beam length to its height on the dynamic response of the microbeams.

Vibration analysis of Bi-directional FG-GNPs reinforced rotating micro-beam under Thermo-mechanical loading

Functionally graded metal matrix and graphene nano-platelets (GNPs) reinforced composites have wide usage in present days. Such materials have high strength-to-weight ratios, good energy efficiency and absorption ability, high thermal conductivity. Due to these properties, they are used widely in micro-electromechanical systems, aerospace, automotive, and biomedical applications. Further, micro-beams are used in micro-sensors, micro turbines and rotors, which are subjected to severe fatigue loads. Adding nano-fillers as reinforcement to metal foams improves the stiffness to weight ratio. Present work deals with vibration analysis of functionally graded GNPs reinforced metallic core rotating micro-beam of taper section. The micro-beam is exposed to thermal environment under centrifugal loads due to rotation. Material modelling and dynamic formulation of the beam are first derived using Halpin-Tsai model and variational approach using modified couple stress theory. The material length scale parameter (MLSP) and functionally grading parameters in thickness and length directions are considered. The solutions are obtained using Ritz approximation technique. A validation study is performed to check the accuracy of present model. The natural frequencies are obtained as a function of length scale parameter, rotational speed, inhomogeneity index, hub and taper parameter. It is found that FG-A and FG-X distribution of GNPs give maximum and minimum values of natural frequency, respectively.

Synchronous/Asynchronous Buckling of Double-Layered Microplate Systems

This study develops a linear buckling model for double-layered microplate system filled with an elastic medium between layers using the modified couple stress theory and a two-variable higher-order shear deformation theory. The governing differential equations of system buckling are derived from the Euler-Lagrange equation. By applying the Navier method, the synchronous and asynchronous buckling solutions are analytically solved for the case of both upper and lower plates being simply supported on four edges. The influence of each parameter on the buckling characteristics of the system is discussed by selective examples. Numerical results showed that the asynchronous buckling characteristics of the system depend on the material length scale parameter, aspect ratio and elastic medium moduli, while the synchronous buckling characteristics depend on the first two only; the asynchronous buckling critical load is noticeable greater than that of the synchronous buckling case; the Pasternak modulus has a more significant effect on the buckling characteristics of the system than the Winkler modulus.

Vibration of electrically actuated MEMS Timoshenko microbeams based on a hierarchical beam element

In this paper, vibration of Timoshenko microbeams with an axial force in micro-electromechanical systems (MEMS) is studied for the first time by using a nonlinear finite element procedure. Based on the von Kármán geometric nonlinearity and the modified couple stress theory (MCST), a beam element is formulated by employing hierarchical functions to interpolate the displacement field. Using the derived element, the discretized equation of motion for the microbeam is constructed and then solved by the Newton-Raphson iterative procedure in conjunction with the Newmark method. The natural frequencies, pull-in voltages and dynamic deflections are computed for a clamped-clamped microbeam under electrostatic actuation of a given direct current (DC) voltage. The numerical result reveals that the axial force and the microsize effect have a significant influence on the vibration, and the fundamental frequency of the microbeams is underestimated by ignoring the size effect. The effects of the axial force, the applied voltage and the material length scale parameter on the vibration of the beam are studied in detail and highlighted.

Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution

This paper presents for the first time a closed-form solution of the dynamic response of sigmoid bidirectional functionally graded (SBDFG) microbeams under moving harmonic load and thermal environmental conditions. The formulation is established in the context of the modified couple stress theory to integrate the effects of microstructure. On the basis of the elasticity theory, nonclassical governing equations are derived by using Hamilton’s principle in combination with the parabolic higher-order shear deformation theory considering the physical neutral plane concept. Sigmoid distribution functions are used to describe the temperature-dependent thermomechanical material of bulk continuums of the beam in both the axial and thickness directions, and the gradation of the material length scale parameter is also considered. Linear and nonlinear temperature profiles are considered to present the environmental thermal loads. The Laplace transform is exploited for the first time to evaluate the closed-form solution of the proposed model for a simply supported (SS) boundary condition. The solution is verified by comparing the predicted fundamental frequency and dynamic response with the previously published results. A parametric study is conducted to explore the impacts of gradient indices in both directions, graded material length scale parameters, thermal loads, and moving speed of the acted load on the dynamic response of microbeams. The results can serve as a principle for evaluating the multi-functional and optimal design of microbeams acted upon by a moving load.

Accurate mechanical buckling analysis of couple stress-based skew thick microplates

We investigate, for the first time, the elastic buckling of skew thick microplates under combined in-plane shear and compressive loading within the framework of modified couple stress theory. The displacement field of the microplates is described by a two-variable refined shear deformation theory, wherein the transverse deflection is partitioned into bending and shear components, resulting in a simple and universal elastic buckling model. The Euler-Lagrange equation is used to obtain the equations governing the motion of the skew thick microplates. Obtaining an analytical buckling solution for general boundary supported skew microplates is challenging; therefore, a C1-type four-node 32-DOF differential quadrature finite element is developed. The element stiffness and geometric stiffness matrices are derived according to the minimum potential energy principle. Significant parametric studies are conducted with different geometrical dimensions, boundary edges, in-plane loadings, and material length scale parameters. The buckling characteristics of skew microplates are investigated using a combination of unequal biaxial compression loads and in-plane compression and shear loads. Numerical results imply that the buckling modes are influenced by the combination of size effects and skew angle and not by the buckling loads alone.

Analysis of Nonlinear Vibration of Functionally Graded Simply Supported Fluid-Conveying Microtubes Subjected to Transverse Excitation Loads.

This paper presents a nonlinear vibration analysis of functionally graded simply supported fluid-conveying microtubes subjected to transverse excitation loads. The development of the nonlinear equation of motion is based on the Euler-Bernoulli theory, Hamilton principle and Strain gradient theory. The nonlinear equation of motion is reduced to a second-order nonlinear ordinary differential equation by the Galerkin method. The Runge-Kutta method is adapted to solve the equation, and the effects of the dimensionless microscale parameters, the amplitude and frequency of excitation loads on the stability of the microtubes system are analyzed. It is found that when the microtube diameter is equal to the material length scale parameter, the microtube movement pattern is quasi-periodic. With the increase of the dimensionless microscale parameter, the microtube movement changes from quasi-periodic to chaos. The smaller the power-law index of volume fraction, the smaller the vibration displacement of microtubes and the better the stability. The larger the amplitude of excitation loads is, the larger the vibration displacement of the microtubes will be. When the frequency of excitation loads is equal to the natural frequency of the microtubes, it will have resonance and the vibration displacement will increase significantly.

Periodic motion of microscale cantilevered fluid-conveying pipes with symmetric breaking on the cross-section

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of cantilevered fluid-conveying micropipes with asymmetric cross-section. Fluid-conveying microscale pipes with ring-shaped cross-sections of O(2) symmetry are widely used in engineering. However, their cross-sections usually have a small symmetric breaking owing to manufacturing errors, which can be described by a small parameter. Hence, the dynamic behaviors of such geometrically imperfect micropipes are extremely required to be explored. This study mainly investigates the effects of the symmetric breaking parameter, flow velocity and dimensionless material length scale parameter on the dynamic behaviors of microscale cantilevered fluid-conveying micropipes. Based on the modified couple stress theory (MCST) and Hamilton's principle, the governing equation is derived, where the effects of the asymmetry of cross-section on the equation are reflected. Using the center manifold theory and normal form method, the original governing equation is rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system, and the averaging equation of the reduced system is obtained by combining the averaging method. The existence and stability of the equilibrium point of the averaging equation are studied to obtain the results of the existence and stability of periodic motions of the original system. The results demonstrate that the types of periodic motions and stability when the pipe loses stability from different places on the curve of the critical flow velocity versus mass ratio (critical flow velocity curve) may differ. Meanwhile, the effects of various parameters on these periodic motions are presented. The increase in the dimensionless material length scale parameter increases the section corresponding to stable plane periodic motion and reduces the section corresponding to stable spatial periodic motion on the critical flow velocity curve. When dimensionless material length scale parameter is given, the existence and stability of various periodic motions of the system depend on the ratio of the increment of flow velocity around the critical value to the symmetric breaking parameter. In addition, some new phenomena are found in this paper. Stable 0 planar periodic motion of the macropipe can occur when pipe loses its stability from the hysteresis part of the critical flow velocity curve. Stable 0 planar periodic motion of microscale pipes can occur when pipe loses its stability from the non-hysteresis part of the critical flow velocity curve, and then stable 0 planar periodic motion and stable π planar periodic motion can coexist. Furthermore, stable planar and stable spatial periodic motions of pipes with symmetric breaking can coexist, and stable torus motion has also been observed; this is different from the dynamics of pipes with O(2) symmetry.

An asymptotic consistent couple stress theory for the three‐dimensional free vibration analysis of functionally graded microplates resting on an elastic medium

Based on the three‐dimensional (3D) consistent couple stress theory (CCST), we develop an asymptotic theory for the free vibration analysis of functionally graded (FG) microplates resting on an elastic medium. The material properties of the FG microplate are assumed to obey a power‐law distribution of the volume fractions of the constituents in the thickness direction, for which the effective material properties are estimated using the rule of mixtures. The interactions between the FG microplate and its foundation medium are simulated using a Winkler–Pasternak foundation. Carrying out the asymptotic expansion method, we obtain recursive sets of motion equations for various order problems. The CCST‐based classical plate theory is derived as a first‐order approximation of the 3D CCST. The asymptotic theory is shown to converge rapidly and to be in excellent agreement with the exact solutions for FG macroplates reported in the literature when the value of the material length scale parameter is assigned to be zero.