Elsevier

Materials & Design

Volume 32, Issue 3, March 2011, Pages 1435-1443
Materials & Design

The modified couple stress functionally graded Timoshenko beam formulation

https://doi.org/10.1016/j.matdes.2010.08.046Get rights and content

Abstract

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.

Introduction

Functionally graded materials (FGMs) are produced from mixing of two different materials. This type of materials provides the specific benefits of both of the constituents. They can be defined as inhomogeneous composites which are made from a mixture of two different materials, usually a metal and a ceramic, with a desired continuous variation of properties as a function of position along certain dimension(s). The continuously compositional variation of the constituents in FGMs along different directions is the great benefit of FGMs, because this property offers a solution to the problem of appearing high magnitude shear stresses that may be induced in laminated composites, where two materials with great differences in properties are bonded. Nowadays, structures made of FGMs have a great practical role in engineering and industrial fields.

Some works have been performed by researchers on the static and dynamic behavior of beams and plates made of FGMs. Asghari et al. [1] have mentioned some instances of these works, including Refs. [2], [3], [4], [5], [6], [7]. As another instance, the thermal snapping of functionally graded plates has been investigated by Prakash et al. [8]. Also, Jomehzadeh et al. [9] presented an analytical approach for the stress analysis of functionally graded annular sector plates. Moreover, analytical modeling of thermal residual stresses in some functionally graded material systems has been presented by Bouchafa et al. [10]. It is noted that these sample works are based on the classical continuum theory, while the formulation presented in this work is based on a non-classical continuum theory, the modified couple stress theory, which is discussed in detail in the following.

In recent years, the application of FG materials has broadly been spread in micro and nano structures such as thin films in the form of shape memory alloys [11], [12], micro- and nano-electromechanical systems (MEMS and NEMS) [13], [14] and also atomic force microscopes (AFMs) [15]. Beams used in MEMS, NEMS and AFMs, have the thickness in the order of microns and sub-microns, so that the small scale effects in their behavior is considerable. The size-dependent static and vibration behavior in micro scales are experimentally validated (see for example [16], [17], [18], [19]). Considering experimental observations, it is well-known that size-dependent behavior is an inherent property of materials which appears for a beam when the characteristic size such as thickness or diameter is close to the internal material length scale parameter [20].

The classical continuum mechanics theories are not capable of prediction and explanation of the size-dependent behaviors which occur in micron- and sub-micron-scale structures. However, non-classical continuum theories such as higher-order gradient theories and the couple stress theory are acceptably able to interpret the size-dependencies.

In 1960s some researchers introduced the couple stress elasticity theory [21], [22], [23]. In the constitutive equation of this theory, some higher-order material length scale parameters appear in addition to the two classical Lame constants. Yang et al. [24] argued that in addition to the classical equilibrium equations of forces and moments of forces, another equilibrium equation should be considered for the material elements. This additional equation is the equilibrium of moments of couples. Then, they concluded that this additional equilibrium equation implies the symmetry of the couple stress tensor. Accordingly, they modified the constitutive equations of the couple stress theory and present the new constitutive equations. Utilizing the modified couple stress theory, Park and Gao [25] analyzed the static mechanical properties of an Euler–Bernoulli beam. Recently, Kong et al. [20] derived the governing equation, initial and boundary conditions of an Euler–Bernoulli beam based on the modified coupled stress theory using the Hamilton principle. Also, Asghari et al. [1] investigated the size-dependent behavior of FGM micro beams using the modified couple stress theory and the Euler–Bernoulli beam model.

The Timoshenko beam is a model for the study of behaviors of beams with less restrictive assumptions with respect to the Euler–Bernoulli beam. The normality assumption for sections in the Euler–Bernoulli beam model is discarded in the Timoshenko beam model. Hence, the Timoshenko beam is capable to capture the shear deformation in contrast to the Euler–Bernoulli beam. Although the Timoshenko beam is a complicated model with respect to the Euler–Bernoulli model, it possesses more capabilities and studying the behavior of beams based on the Timoshenko model gives closer results to the exact behavior. Recently in an interesting work, the modified couple stress theory is utilized by Ma et al. [26] in order to investigate the size-dependent behavior of a homogeneous Timoshenko beam. This work is indeed the generalization of the work of Ma et al. [26] to the FGM beams.

In this paper, considering both of bending and axial deformations, an FGM Timoshenko beam is proposed on the basis of the modified couple stress theory. In addition, generally valid closed-form analytic expressions are derived for the bending and axial deformations and also the angle of rotation of the cross sections in the static behavior. As a case study, response of a specific FGM cantilever beam subjected to a concentrated force at its free end is obtained. Further more, the natural frequency of a simply supported FGM beam is obtained and investigated in order to delineate the size-dependent vibration behavior of FGM Timoshenko beams.

Section snippets

Preliminaries

In the modified couple stress theory, the strain energy density for a linear elastic material in infinitesimal deformation is written as [24]u¯=12(σijεij+mijχij)(i,j=1,2,3),where for isotropic cases it is writtenσij=λtr(ε)δij+2μεij,εij=12(u)i+(u)iT,mij=βχij=2l2μχij,χij=12(θ)ij+θijT,in which σij, εij, mij and χij denote the components of the symmetric part of stress tensor σ, the strain tensor ε, the deviatoric part of the couple stress tensor m, and the symmetric part of the curvature

Governing equations of motion

To arrive at the governing equations of motion of the FGM modified couple stress Timoshenko beam, one can utilize (1), (7), (8), (9), (10), (11) and obtain the following expressions for the potential and kinetic energies of the beamU=120LAσijεij+mijχijdAdx=120LAE(z˜)ux+zψx2+μ(z˜)ψ+wx2+18β(z˜)ψx-2wx22dAdx,T=120LAρ(z˜)ut+zψt2+wt2dAdx,where the length and cross-section area of the FGM Timoshenko beam have been denoted by L and A. The variation of work done by external forces

Case studies

In order to delineate the derived formulation, the static and free vibration of specific Timoshenko FGM beams modeled on the basis of the modified couple stress theory are analyzed in this section. It is assumed that the Timoshenko beam is made of functionally graded material whose mechanical properties are varying through the thickness according to the simple power law as follows:ρ(z˜)=ρc+z˜hn(ρm-ρc),E(z˜)=Ec+z˜hn(Em-Ec),μ(z˜)=μc+z˜hn(μm-μc),where h is the height of the cross section. Also,

Discussion

The main objective of this work is the derivation of Eqs. (21), (22), (23), (24), (25), (26), (27), which are the governing differential equations and the corresponding boundary conditions for the motion of a functionally graded (FG) Timoshenko beam based on the modified couple stress theory. A main difference between these equations and those belonging to a homogeneous modified couple stress Timoshenko beam, presented in [26], is that the equation of the axial motion is coupled with the

Conclusion

In order to delineate the size-dependent static and free-vibration behavior of Timoshenko FG beams based on the derived equations and expressions, an specific cantilever beam and an specific simply supported beam have been considered as case studies and numerical results have been presented for them. In the case studies, it is assumed that the material properties are varying through the thickness according to the simple power law distribution with an arbitrary power index n. The numerical

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