Microstructure-dependent couple stress theories of functionally graded beams

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Abstract

A microstructure-dependent nonlinear Euler–Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler–Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.

Introduction

Thin beams are commonly used in micro- and nano-scale devices and systems such as biosensors, atomic force microscopes, MEMS, and NEMS (e.g., Li et al., 2003, Pei et al., 2004). In such applications, microstructure-dependent size effects are often observed (e.g., Lam et al., 2003, McFarland and Colton, 2005). Conventional beam theories based on classical elasticity do not account for such size effects. This motivated many researchers to develop beam models using continuum theories that contain additional material constants. The model for pure bending of a circular cylinder proposed by Anthoine (2000) employs the classical couple stress elasticity theory of Koiter (1964), which includes four material constants (two classical and two additional). The higher-order Bernoulli–Euler beam model developed by Papargyri-Beskou et al. (2003) is based on the gradient elasticity theory with surface energy (see Vardoulakis and Sulem, 1995); it involves four elastic constants, two classical and two non-classical. This strain gradient beam model has been studied further by Vardoulakis and Giannakopoulos (2006) and Giannakopoulos and Stamoulis (2007). The nonlocal Bernoulli–Euler beam model by Peddieson et al. (2003), the nonlocal Timoshenko beam model by Wang et al. (2006), the nonlocal Bernoulli–Euler, Timoshenko, Reddy, and Levinson beam models formulated by Reddy, 2007a, Reddy, 2010 and Reddy and Pang (2008) are developed using a constitutive equation suggested by Eringen (1983). In view of the difficulties in determining microstructure-dependent length scale parameters (see Yang and Lakes, 1982, Lam et al., 2003, Maranganti and Sharma, 2007) and the approximate nature of beam theories, refined beam models that involve only one material length scale parameter are desirable. One such model has been developed for the Bernoulli–Euler beam by Park and Gao, 2006, Park and Gao, 2008 and for the Timoshenko beam theory by Ma et al. (2008) using a modified couple stress theory proposed by Yang et al. (2002), which contains only one material length scale parameter.

Functionally gradient materials (FGM) are a class of composite materials (Reddy, 2004) that have a gradual variation of material properties from one surface to another (see Hasselman and Youngblood, 1978, Yamanouchi et al., 1990, Koizumi, 1993). These novel materials were proposed as thermal barrier materials for applications in space planes, space structures, nuclear reactors, turbine rotors, flywheels, and gears, to name a few. In fact, the functionally gradient material characteristics are represented in most structures found in nature (e.g., sea shells, bones, etc.), and perhaps a better understanding of the highly complex form of materials in nature will help us in synthesizing new materials (the science of so called “biomimetics”). As conceived and manufactured today, these materials are isotropic and nonhomogeneous.

The objective of the current paper is to develop nonlinear Euler–Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material and moderate rotation of transverse normals through the von Kármán nonlinear strain. Extension to Reddy–Levinson third-order beam theories (Reddy, 2007a; Ma et al., 2010) is trivial and are not included here. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. Since most nanoscale devices involve beam-like elements that may be functionally graded and undergo moderately large rotations, the newly developed beam theories can be used to capture the size effects in functionally graded microbeams. Moreover, the bending–extensional coupling is captured through the von Kármán nonlinear strains. The work of Xia et al. (2010) includes the von Kármán nonlinear strains in the Euler–Bernoulli beam model, but they neglect the coupling between axial displacement and transverse deflection. Technically, it is an incorrect derivation of the geometric nonlinearity term because it neglects the coupling with the axial displacement, which can be significant in beams where the axial displacement is constrained. In the works of Yang et al. (2002), Park and Gao, 2006, Park and Gao, 2008, and Ma et al. (2008), a three-dimensional elastic constitutive relation was invoked in formulating the refined models and the effect of Poisson's ratio on the response was shown to be quite large. The papers by Asghari et al., 2010, Asghari et al., 2011 separately deal with the Timoshenko beam theory of isotropic beams and functionally graded beams, with the first one including geometric nonlinearity (but without FGM). They too neglect the effect of the axial displacement in the derived nonlinear equations. Neglect of the effect of axial displacement in the nonlinear coupling makes the beam behave stiffer than it is, and thereby loses the benefits of including both shear deformation and microstructure-dependent terms. Recent papers by Ke and Wang (2011) and Ke et al. (2011) follow the works of Asghari et al., 2010, Asghari et al., 2011 to study dynamic stability and nonlinear vibration, respectively. A positive contribution of this paper is that they do include axial displacement in their formulation.

In this study it is shown that such a model is inconsistent and results in under-prediction of deflections and over-prediction of natural frequencies and buckling loads. It is also shown that the nonlocal models considered in the works of Wang et al. (2006) and Reddy, 2007a, Reddy, 2010 and those considered in the works of Yang et al. (2002), Park and Gao (2006), Park and Gao (2008), and Ma et al. (2008) have the opposite effect on the structural response.

Section snippets

Equations of motion of microstructure-dependent beam theories

The modified couple stress theory proposed by Yang et al. (2002) results from the classical couple stress theory (see e.g., Mindlin, 1963, Koiter, 1964). As pointed out by Ma et al. (2008), the two main advantages of the modified couple stress theory over the classical couple stress theory are the inclusion of asymmetric couple stress tensor and the involvement of only one length scale parameter, which is a direct consequence of the fact that the strain energy density function depends only on

Constitutive equations

For an isotropic, linear elastic material the three-dimensional stress–strain relations are σij=2μɛij+λδijɛkkα(3λ+2μ)(TT0)δijmij=2μ2χijwhere μ and λ are the Lamé parameters (see Reddy, 2008)λ=Eν(1+ν)(12ν),2μ=E(1+ν)with E being Young's modulus and ν being Poisson's ratio, α is the coefficient of thermal expansion, and ΔT is the temperature increment from the room temperature, T0, and is the material length scale parameter. The material length scale parameter is the square root of the ratio

Euler–Bernoulli beam theory

The equations of motion in Eqs. (14), (15) can be expressed in terms of the displacements u and w by using the beam constitutive relations from Eqs. (40) (note that Qx does not appear in this model). We obtainxAxxux+12wx2+Bxx2wx2NxxT+m02ut2m13wt2xf=02x2Bxxux+12wx2+Dxx2wx2MxxT+2x2Sxy2wx2xAxxwxux+12wx2Bxxwx2wx2wxNxxT+m02wt2+m13ut2xm24wt2x2qrx=0The above equations contain, as special cases, many of the beam models based on the

Analytical solutions

Here we determine deflections, natural frequencies, and buckling loads of functionally graded beams with modified couple stress parameter when the geometric nonlinearity is omitted. The Navier solution procedure is used to determine the analytical solutions for the simply supported boundary conditions (one element of each of the pairs (u,Nxx), (w,Vx), (θx,Pxy), and (ϕx, Mxx) should be specified at x=0 and x=L)Nxx(0,t)=Nxx(L,t)=0,w(0,t)=w(L,t)=0,Mxx(0,t)=Mxx(L,t)=0Pxy(0,t)=Pxy(L,t)=0

For static

Summary

Nonlocal models for bending, natural vibration and buckling of homogeneous and functionally graded beams according to the Euler–Bernoulli and Timoshenko homogeneous are developed using a modified couple stress theory. The models contain a material length scale parameter to account for the microstructural effect, unlike that in the conventional Euler–Bernoulli and Timoshenko beam theories. Two-constituent material variation through the thickness of the beam is considered using a power-law model.

Acknowledgments

The research reported herein was carried out under National Science Foundation Research Grants CMMI-1030836 and CMMI-1000790.

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