Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM)

Longitudinal free vibration analysis of axially functionally graded microbars is investigated on the basis of strain gradient elasticity theory. Functionally graded materials can be defined as nonhomogeneous composites which are obtained by combining of two different materials in order to obtain a new desired material. In this study, material properties of microbars are assumed to be smoothly varied along the axial direction. Rayleigh–Ritz solution technique is utilized to obtain an approximate solution to the free longitudinal vibration problem of strain gradient microbars for clamped–clamped and clamped-free boundary conditions. A parametric study is carried out to show the influences of additional material length scale parameters, material ratio, slenderness ratio and ratio of Young’s modulus on natural frequencies of axially functionally graded microbars.     

A nonlinear strain gradient beam formulation

In this paper, a nonlinear size-dependent Euler–Bernoulli beam model is developed based on a strain gradient theory, capable of capturing the size effect. Considering the mid-plane stretching as the source of the nonlinearity in the beam behavior, the governing nonlinear partial differential equation of motion and the corresponding classical and non-classical boundary conditions are determined using the variational method. As an example, the free-vibration response of hinged-hinged microbeams is derived analytically using the Method of Multiple Scales. Also, the nonlinear size-dependent static bending of hinged-hinged beams is evaluated numerically. The results of the new model are compared with the results based on the linear strain gradient theory, linear and nonlinear modified couple stress theory, and also the linear and non-linear classical models, noting that the couple stress and the classical theories are indeed special cases of the strain gradient theory.     

Analysis of micro-rotating disks based on the strain gradient elasticity

In this paper, the mechanical behavior of micro-rotating disks is investigated utilizing the strain gradient theory. The governing equation and boundary conditions are derived utilizing the variational method. The analytical solution for the derived equation is also presented. As a case study, some numerical results are presented to emphasize the importance of utilization of non-classical theories such as the strain gradient elasticity instead of the classical continuum theory in dealing with micro-rotating disks.     

Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory

This paper proposes to investigate the nonlinear size dependent behavior of electrically actuated carbon nanotube (CNT) based nano-actuator while including the higher-order strain gradient deformation, the geometric nonlinearity due to the von Karman nonlinear strain as well as the slack effect, and the temperature gradient effects. The assumed non-classical beam model adopts some internal material size scale parameters related to the material nanostructures and is capable of interpreting the size effect that the classical continuum beam model is unable to pronounce. The higher-order governing equations of motion and boundary conditions are derived using the so-called extended Hamilton principle. A Galerkin based reduced-order model (ROM) modal decomposition is developed to prescribe the non-classical nanotube mode shape as well as its static behavior under any applied DC actuation load. Results of the static analysis is compared with those obtained by both classical elasticity continuum and strain gradient theories. A Jacobian method is utilized to determine the variation of the natural frequencies of the nanobeam with the DC load as well as the slack level. A thorough parametric study is conducted to study the influences of the size scale dependent parameters, the geometric nonlinearity, the initial curvature, the gate voltage, and the temperature gradient effect on structural behavior of the CNT-based nano-actuator. It is found that the size effect based on the strain gradient deformation has significant influence on the fundamental nanotube natural frequency dispersion. Also, varying this size effect have revealed the offering of numerous possibilities of modes veering and crossing, all shown to be dependent of the strain gradient parameters as well as the CNT slack level.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

An assumed‐strain finite element technique for non‐linear finite deformation is presented. The weighted‐residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed‐deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual. A variety of finite element shapes fits the derived framework: four‐node tetrahedra, eight‐, 27‐, and 64‐node hexahedra are presented here. Since the assumed‐deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement‐based models; the only deviation is the consistent use of the assumed‐deformation gradient in place of the displacement‐derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Size-dependent torsion of functionally graded bars

In this paper, size-dependent static and dynamic behavior of functionally graded microbars is investigated on the basis of the modified couple stress theory. The equation of motion and corresponding boundary conditions are derived using Hamilton's principle and presented in the dimensionless form. Equivalent mechanical properties (i.e. shear modulus, density and length scale) are extracted for the functionally graded microbar based on the mechanical properties of the material constituents. In this work, it is shown that without any simplifying assumption, two equivalent length scale parameters can be defined for functionally graded bars and the size-dependent mechanical behavior of these components can be explained using these parameters. As an example, static and dynamic behavior of a functionally graded microbar with fixed-free boundary conditions is analyzed and the effect of size-dependency on mechanical behavior of this structure is discussed.     

Torsional statics of two-dimensionally functionally graded microtubes

A size-dependent governing equation is derived to investigate the torsional static behaviors of two-dimensionally functionally graded microtubes based on the modified couple stress theory. The shear modulus is assumed to vary along the tube’s length direction according to an exponential distribute function, and varies along the tube’s radius direction according to a power-law function. A generalized differential quadrature method is developed to determine the rotational angle and shear stresses. Some illustrative examples are given to investigate the effects of applied torques, the length scale parameter and various material compositions on the torsional angle and shear stresses.     

A nonlinear thick plate formulation based on the modified strain gradient theory

A geometric nonlinear first-order shear deformation theory-based formulation is presented to analyze microplates. The formulations derived herein are based on a modified strain gradient theory and the von Karman nonlinear strains. The modified strain gradient theory includes five material length scale parameters capable to capture the size effects in small scales. The governing equations of motion and the most general form of boundary conditions of an arbitrary-shaped plate are derived using the principle of virtual displacements. The analysis is general and can be reduced to the modified couple stress plate model or the classical plate model.     

Torsional oscillations of an infinite cylindrical elastic tube under large internal and external pressure

This study is concerned with the small amplitude torsional oscillations of a hyperelastic infinite circular cylindrical thick tube made of a rubber-like material subjected to a large static internal and external pressure. The material is represented by a Mooney-type strain energy relation. The governing differential equation is first solved by the Frobenius method then a variational approach, which is more suitable for numerical calculations, is developed. Several values for the natural frequencies are obtained.     

High-frequency torsional oscillations—I: Penny-shaped crack in an inhomogeneous medium

The problem of an inhomogeneous medium, whose shear modulus and density vary exponentially with radius, containing a penny-shaped crack undergoing high-frequency torsional oscillations is reduced asymptotically to Wiener-Hopf integral equation and solved by Carleman's method. Uniformly valid asymptotic results are obtained. Explicit expressions are derived for the normal displacement gradient outside the crack region, the stress-intensity factor and the energy of the crack.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories

Size dependent static and dynamic behavior of a fully clamped micro beam under electrostatic and piezoelectric actuations is investigated. The microbeam is modeled under the assumptions of Euler–Bernoulli beam theory. Viscous damping and nonlinearities due to electrostatic actuation and mid-plane stretching are considered. Residual stress and fringing field effect are taken into account as well. Governing equation of motion is derived using Hamilton’s principle along with the strain gradient theory (SGT), which is a non-classical continuum theory capable of taking size effect of elastic materials into account. Reduced order model of the partial differential equations of the system is obtained using Galerkin method. Static deflection, pull-in voltage and the primary resonance of the microbeam are examined and the effect of piezoelectric voltage and its polarization on the size dependent static and dynamic response is studied. It is found that the piezoelectric voltage can effectively change the flexural rigidity of the system which in turn affects the pull-in instability regime. The effect of material length scale parameter is examined by comparing the results of the SGT with the modified couple stress (MCST) and classical theory (CT), both of which are special cases of the former. Comparison demonstrates that the CT underestimates the stiffness and consequently the pull-in voltage and overestimates the amplitude of periodic solutions. The difference between the results of classical and non-classical theories becomes more and more as the dimensions of the system gets close to the length scale parameter. Non-classical theories predict more realistic behaviors for the micro system. The results of this paper can be used in designing microbeam based MEMS devices.     

Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory

The Eshelby tensor for a plane strain inclusion of arbitrary cross-sectional shape is first presented in a general form, which has 15 independent non-zero components (as opposed to 36 such components for a three-dimensional inclusion of arbitrary shape). It is based on a simplified strain gradient elasticity theory that involves one material length scale parameter. The Eshelby tensor for an infinitely long cylindrical inclusion is then derived using the general form, with its components obtained in explicit (closed-form) expressions for the two regions inside and outside the inclusion for the first time based on a higher-order elasticity theory. This Eshelby tensor is separated into a classical part and a gradient part. The latter depends on the position, the inclusion size, the length scale parameter, and Poisson’s ratio. As a result, the new Eshelby tensor is non-uniform even inside the cylindrical inclusion and captures the size effect. When the strain gradient effect is not considered, the gradient part vanishes and the newly obtained Eshelby tensor reduces to its counterpart based on classical elasticity. The numerical results quantitatively show that the components of the new Eshelby tensor vary with the position, the inclusion size, and the material length scale parameter, unlike their classical elasticity-based counterparts. When the inclusion radius is comparable to the material length scale parameter, it is found that the gradient part is too large to be ignored. In view of the need for homogenization analyses of fiber-reinforced composites, the volume average of the newly derived Eshelby tensor over the cylindrical inclusion is obtained in a closed form. The components of the average Eshelby tensor are observed to depend on the inclusion size: the smaller the inclusion radius, the smaller the components. However, as the inclusion size becomes sufficiently large, these components are seen to approach from below the values of their classical elasticity-based counterparts.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory

The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation.     

Dynamic finite element analysis of a gradient elastic bar with micro-inertia

We study the dynamic behavior of a first strain gradient elastic bar with micro-inertia by means of the finite element method. The partial differential equation describing the motion of the bar expressed in terms of displacement is of fourth order with respect to the spatial variable, therefore, for the standard Galerkin formulation, Hermite elements are required. Consistent mass matrices are employed. The results are validated by comparison with some special cases where the exact solutions are derivable. Dispersion relations for the longitudinal waves are also derived. The effect of micro-inertia in the dynamic response of the bar is analyzed and comparisons are made with the classical elastic case. It is found that the micro-inertia parameter considerably affects the dynamic response of the bar and the dispersion characteristics of longitudinal waves.     

The fracture of wood under torsional loading

A series of experiments have been carried out on hardwood (red lauan) and softwood (sitka spruce) test pieces using static and cyclic torsional loading under displacement control. Measurements of the applied torque, the corresponding angle of twist and the number of cycles to failure were recorded. It was found that under static torsional loading, the strength of both hardwood and softwood reduced as the grain orientation of the sample to the axis of twist increased from 0° to 90° with a corresponding decrease of elastic modulus. Hardwood is stronger than softwood. In the fatigue test, when the torsional load is plotted against cycle number, the results showed that under displacement control stress relaxation occurs. The S–N curve for softwood has a shallower gradient than that of hardwood, indicating that the torsional strength of softwood is less affected by fatigue loading than hardwood. In both static and cyclic torsional loading tests, the failure mode of hardwood is slow and incomplete, whereas, softwood fails suddenly and completely. The crack growth is along the tangential direction in the hardwood cross-section and in the radial direction in the cross-section.     

Transverse Shear and Normal Deformation Theory for Bending Analysis of Laminated and Sandwich Elastic Beams

The influence of transverse normal strain on bending analysis of cross-ply laminated and sandwich beams is presented. A higher-order shear deformation beam theory is developed. Euler-Bernoulli classical, Timoshenko first-order and simple higher-order theories have been also used in the analysis. The governing equations for a beam composed of orthotropic layers and subjected to any given mechanical load distribution are derived. Making use of Navier-like approach, exact solutions are obtained for cross-ply laminated and sandwich beams subjected to arbitrary loadings. Numerical results for beams with the simply-supported boundary conditions are presented. The effects due to transverse normal strain, transverse shear deformation and number of layers on the static response of the beams are investigated.     

Anti-plane Circular Nano-inclusion Problem with Electric Field Gradient and Strain Gradient Effects

As well known, gradient theories can describe size effects that are important in nano-scale problems. In this paper, we analyze the Eshelby-type anti-plane inclusion problem embedded in infinite dielectric body by considering both strain gradient and electric field gradient effects to account for the size effect and high-order electromechanical coupling effect. The size-dependent Eshelby and Eshelby-like tensor, strain, stress, electric field and electric displacement components are derived explicitly by means of Green's function method. Theoretical results indicate that strain and electric field are decoupled for anti-plane inclusion problem while stress field and electric displacement are coupled through strain gradient and electric field gradient. Based on the general form, the expressions for a special case of circular inclusion are obtained analytically. Numerical results reveal that when the inclusion radius becomes small, the gradient effects are significantly important and should not be ignored. The values approach asymptotically to classical solutions as increase of inclusion size. And the high-order electromechanical coupling effect in non-piezoelectric material (centrosymmetric dielectrics) can be equivalent to piezoelectricity of conditional piezoelectric materials when the inclusion size is small.