Elsevier

European Journal of Mechanics - A/Solids

Volume 47, September–October 2014, Pages 211-230
European Journal of Mechanics - A/Solids

Size-dependent functionally graded beam model based on an improved third-order shear deformation theory

https://doi.org/10.1016/j.euromechsol.2014.04.009Get rights and content

Highlights

  • This model contains both microscale and higher-order shear deformation effects.

  • Interaction between the elastic foundations and the FG microbeam is considered.

  • Length scale parameters are viewed as the function of material mixture ratio.

  • Analytical solutions are provided for simply supported boundary conditions.

Abstract

In this study, a size-dependent beam model made of functionally graded materials (FGMs) is developed. This model contains both microscale and shear deformation effects. The microscale effect is captured using the strain gradient elasticity theory, while the shear deformation effect is included using an improved third-order shear deformation theory which is based on a more rigorous kinematics of displacements. In addition, interaction between the Winkler–Pasternak elastic foundation and the FG microbeam is considered. The material properties of the FG microbeams are assumed to vary in the thickness direction and estimated through the Mori-Tanaka homogenization technique. Material length scale parameters are viewed as the function of material mixture ratio rather than a constant. The equations of motion and boundary conditions are derived from Hamilton's principle. Analytical solutions are obtained using the Navier method for bending, free vibration, and buckling problems of FG microbeams with simply supported boundary conditions. The effects of material length scale parameter, aspect ratio, various material compositions, elastic foundation parameters and shear deformation on mechanical responses of the FG microbeam are investigated in detail. Some of the present results are validated by comparing the present results to those available in literature. The results indicate that the microscale effect, elastic foundation and material compositions greatly affect the mechanical behavior of FG microbeams. The new results can be used as benchmark solutions for future researches.

Graphical abstract

Size-dependent FG microbeam resting on a Winkler-Pasternak elastic foundation.

  1. Download : Download full-size image

Introduction

Functionally graded materials (FGMs), a novel generation of composites of microscopical heterogeneity initiated by the Japanese material scientist in the early 1980s (Koizumi, 1993, Koizumi, 1997), are achieved by controlling the volume fractions, microstructure, porosity, etc. of the material constituents during manufacturing, resulting in spatial gradient of macroscopic material properties of mechanical strength and thermal conductivity. The material properties can be designed so as to improve its strength, toughness, high temperature withstanding ability, etc. Compared with conventional composite laminates, FGMs possess various advantages, such as smaller thermal stresses, stress concentrations, attenuation of stress waves and can be designed to achieve specific properties for different applications. Thus, FGMs have broad potential applications in aeronautics/astronautics manufacturing industry, nuclear power plant, etc. Recently, the application of FGMs has been widely extended in micro- and nano-scale devices and systems such as thin films (Lü et al., 2009, Ramanathan et al., 2008), atomic force microscopes (AFMS) (Rahaeifard et al., 2009), micro- and nano-electro-mechanical systems (MEMS and NEMS) (Witvrouw and Mehta, 2005). In such applications, the size-dependent deformation behavior of microscale structures has been experimentally observed by many investigators (Chong et al., 2001, Fleck et al., 1994, Lam et al., 2003, Liu et al., 2013, Liu et al., 2012 Stölken and Evans, 1998). For instance, Fleck et al. (1994) conducted torsion tests on polycrystalline copper wires and found that the normalized torque is increased sharply with by a factor of three as the wire decreased from 170 μm to 12 μm. However, their data have no enough resolution to display the elastic limit and replacing of the torsional load cell for different size specimens may cause measurement error.

Given those disadvantages, Liu et al., 2012, Liu et al., 2013 designed a kind of new experimental facility to obtain high-resolution torsion data on microscale copper wires and found abnormal Bauschinger effect in condition of cyclic loading. In the micro-bending test of thin nickel beams, Stölken and Evans (1998) observed that the normalized bending hardening had a great increase when the beam thickness decreased from 100 μm to 12.5 μm. Lam et al. (2003) showed that the bending rigidity increased about 2.4 times as the beam thickness reduced from 115 μm to 20 μm in the micro-bending testing of epoxy polymeric beams. It is evident from these experimental studies that the material properties of microscale structures are size-dependent, and hence it must be taken into account in theoretical and experimental studies.

Microbeams are the key components widely used in MEMS, NEMS and AFMS with the order of microns or submicrons, and their properties are closely related to their microstructures. Conventional beam models based on classical continuum theories (CT) do not account for such size effects due to lacking additional material length scale parameters. For this reason, various higher-order continuum theories involving additional material length scale parameters have been proposed and employed to elaborate the mechanical behavior of microscale structures. For example, classical couple stress theory proposed by some investigators, such as Mindlin and Tiersten, 1962, Toupin, 1962 and Koiter (1964), contains four material constants (two classical and two additional) for an isotropic elastic material. The classical couple stress theory has been utilized to analysis the static and dynamic problems of microbeams (Anthoine, 2000, Kang and Xi, 2007, Zhou and Li, 2001). Considering the difficulties in determining the microstructure related length scale parameters, Yang et al. (2002) first modified the classical couple stress theory and proposed the modified couple stress theory involving only one additional material length scale parameter. Based on the modified couple stress theory, the elastic bending, linear vibration, nonlinear vibration and postbuckling problems of microbeams made of homogeneous materials or layered composites were examined extensively by many investigators (Asghari et al., 2010a, Chen and Li, 2012, Ghayesh et al., 2013, Kong et al., 2008, Ma et al., 2008, Ma et al., 2010, Park and Gao, 2006, Roque et al., 2013, Xia et al., 2010). Note that the above works only studied the microbeams made of homogeneous materials. For the size-dependent FG microbeams, Asghari et al., 2010b, Asghari et al., 2011, Ke and Wang, 2011, Reddy, 2011, Ke et al., 2012a, Şimşek et al., 2013 and Akgöz and Civalek (2013a) investigated the static bending, linear free vibration, dynamic stability and nonlinear vibration of FG Euler-Bernoulli and Timoshenko microbeams. To avoid the use of the shear correction factor, Salamat-Talab et al. (2012) and Nateghi et al. (2012) utilized modified couple stress theory to investigate the static, vibration and buckling behavior of FG microbeams based on a third-order shear deformation theory. Şimşek and Reddy, 2013a, Şimşek and Reddy, 2013b proposed a unified higher-order beam theory to examine for the static bending, free vibration and buckling behaviors of FG microbeams based on the modified couple stress theory. Their results proved that neglecting Poisson's ratio alters the results significantly. FG microbeams with considering the thermal effect have been investigated by Nateghi and Salamat-talab (2013). The research works related to the modeling of size-dependent plate based on the modified couple stress theory are not reviewed here.

Another higher-order continuum theory has been developed by Mindlin (1965) in which strain energy is considered as a function of first and second-order gradients of strain tensor. In the most general, this theory involves only first-order gradient of strain tensor introduces five new constants as well as Lame's constants for an isotropic linear elastic material. By reformulating and extending the Mindlin's theory, Fleck et al. (1994) developed new type of continuum theory named as strain gradient theory in which the second-order deformation tensor separated into the stretch gradient tensor and rotation gradient tensor which leads to additional higher-order stress components compared to the couple stress theory. Lam et al. (2003) utilized the higher-order equilibrium equation suggested by Yang et al. (2002) and modified the strain gradient theory. The new theory introduces three material length scale parameters to characterize the dilatation gradient tensor, the deviatoric stretch gradient tensor and the symmetric rotation gradient tensor. The higher-order stress tensor work-conjugate to the new higher-order deformation metrics and the corresponding constitutive relations are defined. It is worth pointed out that the modified couple stress theory can be viewed as a special case of the strain gradient elasticity theory when the first two of material length scale parameters are taken to be zero. Many researchers have employed this theory to analyze the static and dynamic problems of microscale structures. For instance, Kong et al. (2009) and Wang et al. (2010) investigated static bending and free vibration behaviors of Bernoulli–Euler and Timoshenko homogeneous microbeams, respectively. The nonlinear Bernoulli–Euler and Timoshenko homogeneous microbeams have been respectively developed by Kahrobaiyan et al. (2011) and Ramezani (2012). Akgöz and Civalek, 2011a, Akgöz and Civalek, 2011b, Akgöz and Civalek, 2013b employed strain gradient elasticity and modified couple stress theories to investigate the bending, buckling and free vibration of Bernoulli–Euler microbeams. Recently, Akgöz and Civalek (2013c) proposed a size-dependent sinusoidal shear deformation beam model based on strain gradient elasticity theory. The size-dependent Kirchhoff plate model has been investigated by Wang et al. (2011) and Ashoori Movassagh and Mahmoodi, 2013 respectively. Kahrobaiyan et al. (2012) developed a FG strain gradient Euler-Bernoulli beam model. The linear and nonlinear vibration characteristics of strain gradient Timoshenko microbeams made of functionally graded materials (FGMs) were studied by Ansari et al., 2011, Ansari et al., 2012. Ansari et al. (2013) also investigate the postbuckling behavior of FG microbeams subjected to thermal loads. Tajalli et al. (2013) developed a FG Timoshenko beams with considering the variation of material length scale parameters by a power law. Sahmani and Ansari (2013) studied the free vibration behaviors of FG microplates based on strain gradient elasticity and higher-order shear deformable plate theory. Zhang et al. (2013) presented a size-dependent FG curved beam model based on nth-order shear deformation theory (Xiang and Kang, 2013). Lei et al. (2013) investigated the bending and free vibration of FG microbeam based on strain gradient elasticity and sinusoidal shear deformation theories.

In recent years, an improved third-order shear deformation theory based on a more rigorous kinematics of displacements has been developed by Shi (2007) for the problems in which the transverse shear plays an important role. Because of the kinematics of the displacement derived from an elasticity formulation rather than the hypothesis of displacement, it is interesting to using this theory for thermal buckling and free vibration analysis of FG beams in thermal environment. Shi (2007) presented orthotropic plate model based on the improved theory. Wattanasakulpong et al., 2010, Wattanasakulpong et al., 2011 investigated thermal buckling and vibration of the macroscopic FG beams under ambient temperature, and this theory presented the significant features in comparison with classical beam theory (CBT) and FSDT in particular for the thick and high flexible FG beams. However, size-dependent microbeam model based on this theory is not found in the open literature. On the other hands, beams resting on elastic foundations have been widely adopted by many researchers to model interaction between elastic medium and FG beams for various engineering problems. But studies related considering elastic foundation are still limited. We only found that Şimşek and Reddy (2013b) investigated the buckling of FG beams embedded in elastic Pasternak medium using the modified couple stress theory and an unified higher order shear deformation theory.

This paper aims to develop a size-dependent FG beam model resting on Winkler–Pasternak elastic foundation and provide the analytical solutions for the bending, buckling and free vibration problems. The microscale effect is captured using the strain gradient elastic theory, while the shear deformation effect is included using the improved third-order shear deformation theory. Material properties such as Young's modulus, Poisson's ratio, material density and material length scale parameters are all assumed to vary in the thickness direction, and estimated through the Mori-Tanaka homogenization technique. The governing equations and the relevant boundary conditions are derived by using Hamilton's principle. Analytical solutions are obtained by using Navier procedure for the simply supported boundary conditions. Comparison between results of the present work and those available in literature shows the accuracy of this model. The effects of the material length scale parameters, power law index, slenderness ratio and elasticity foundation parameters on the static deformation, buckling and free vibration behaviors are investigated in detail.

Section snippets

Strain gradient elasticity theory

The strain gradient elasticity theory introduces dilatation gradient tensor and the deviatoric stretch gradient tensor as well as the symmetric rotation gradient. In order to characterize these tensors, three independent material length scale parameters in addition to two classical material constants are introduced to analyze isotropic linear elastic materials. The strain energy for an isotropic linear elastic material occupying region Ω based on the strain gradient elasticity theory is written

The Navier solution

In this section, the analytical solutions of FG microbeam resting on elastic foundation and with simply supported boundary conditions are derived by using Navier approach. The geometry and cross-sectional shape are illustrated in Fig. 1. For simplicity, we assume that the external distributed forces fu and fϕ, and the traction forces Nu(0), Nϕ(0), Nw(0), Nw(1), Nu(1), Nϕ(1) and Nw(2) are all set to zero, and only the distributed transverse load fw and axial buckling load Qˆx are applied.

Based

Numerical results and discussion

In this section, various numerical examples are presented and discussed to verify the accuracy of present theories. The FG microbeam is composed of aluminum (AI) and silicon carbide (SiC). The material properties of aluminum are Em = 70 GPa, νm = 0.3, and ρm = 2702 kg/m3, and those of silicon carbide are Ec = 427 GPa, νc = 0.17 and ρc = 3100 kg/m3. It is known that the material length scale parameter is obtained as l = 17.6 μm for homogeneous epoxy beam (Lam et al., 2003). However, experiment

Conclusion

A size-dependent FG beam model resting on Winkler–Pasternak elastic foundation is developed for analysis the static bending, free vibration and buckling behavior of microbeam. The displacement fields of the developed model are chosen based on an improved third-order shear deformation theory, which satisfies a more rigorous kinematics of displacements. The material properties of the FG microbeams are assumed to vary in the thickness direction and are estimated through the Mori-Tanaka

Acknowledgments

This work was financially supported by the NSFC (No. 11272131 and 11072084), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20110142110039), and the Fundamental Research Funds for the Central Universities (Grants No. CXY12Q041 and No. 2013KXYQ008), HUST: Nos. 2013KXYQ008.

References (70)

  • A. Ashoori Movassagh et al.

    A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory

    Eur. J. Mech. – A/Solids

    (2013)
  • M. Aydogdu

    A new shear deformation theory for laminated composite plates

    Compos. Struct.

    (2009)
  • N. Fleck et al.

    Strain gradient plasticity: theory and experiment

    Acta Metall. Mater.

    (1994)
  • M.H. Ghayesh et al.

    Nonlinear dynamics of a microscale beam based on the modified couple stress theory

    Compos. Part B: Eng.

    (2013)
  • M. Kahrobaiyan et al.

    A nonlinear strain gradient beam formulation

    Int. J. Eng. Sci.

    (2011)
  • M. Kahrobaiyan et al.

    A strain gradient functionally graded Euler-Bernoulli beam formulation

    Int. J. Eng. Sci.

    (2012)
  • M. Karama et al.

    Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity

    Int. J. Solids Struct.

    (2003)
  • L.-L. Ke et al.

    Nonlinear free vibration of size-dependent functionally graded microbeams

    Int. J. Eng. Sci.

    (2012)
  • L.-L. Ke et al.

    Bending, buckling and vibration of size-dependent functionally graded annular microplates

    Compos. Struct.

    (2012)
  • L.L. Ke et al.

    Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

    Compos. Struct.

    (2011)
  • M. Koizumi

    FGM activities in Japan

    Compos. Part B: Eng.

    (1997)
  • S. Kong et al.

    The size-dependent natural frequency of Bernoulli–Euler micro-beams

    Int. J. Eng. Sci.

    (2008)
  • S. Kong et al.

    Static and dynamic analysis of micro beams based on strain gradient elasticity theory

    Int. J. Eng. Sci.

    (2009)
  • C. et al.

    Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory

    Int. J. Solids Struct.

    (2009)
  • D.C.C. Lam et al.

    Experiments and theory in strain gradient elasticity

    J. Mech. Phys. Solids

    (2003)
  • J. Lei et al.

    Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory

    Int. J. Eng. Sci.

    (2013)
  • D. Liu et al.

    Size effects in the torsion of microscale copper wires: experiment and analysis

    Scr. Mater.

    (2012)
  • H. Ma et al.

    A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

    J. Mech. Phys. Solids

    (2008)
  • R.D. Mindlin

    Second gradient of strain and surface-tension in linear elasticity

    Int. J. Solids Struct.

    (1965)
  • A. Nateghi et al.

    Thermal effect on size dependent behavior of functionally graded micro beams based on modified couple stress theory

    Compos. Struct.

    (2013)
  • A. Nateghi et al.

    Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory

    Appl. Math. Model.

    (2012)
  • S. Ramezani

    A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory

    Int. J. Non-Linear Mech.

    (2012)
  • J. Reddy

    Microstructure-dependent couple stress theories of functionally graded beams

    J. Mech. Phys. Solids

    (2011)
  • C.M.C. Roque et al.

    A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method

    Compos. Struct.

    (2013)
  • S. Sahmani et al.

    On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory

    Compos. Struct.

    (2013)
  • Cited by (0)

    View full text