Static analysis of functionally graded non-prismatic sandwich beams

In this paper, an accurate yet computationally efficient beam model based on hierarchical Legendre expansion functions is developed for the analysis of non-uniform or restrained torsion problems of Functionally Graded (FG) beams with solid or thin-walled section. The mechanical properties of the FG beams studied in this paper, such as Young’s modulus and shear modulus, are assumed to continuously vary along either the length or thickness direction following a power-law distribution. The proposed beam model is based on the assumption that the beam’s cross-section is infinitely rigid in its own plane. However, the longitudinal displacement field over the beam’s cross-section is enriched in an element-wise manner by the unknown longitudinal displacement parameters multiplying with hierarchical Legendre expansion functions. The proposed modeling methodology has two novel aspects: First, it allows the torsional or twisting angle, which is a priori defined as an unknown kinematic variable, to be directly captured without a post-processing recovery step, even if there exists a strong flexual-torsional coupling within the beam; second, the longitudinal warping response of the beam, triggered by stretching, bending, twisting, or the coupling between them, can be captured without the pre-determination of warping modes and at a lower level of DOFs. The strong-form governing equations of non-uniform or restrained torsion problems of the FG beam are derived based on the principle of minimum potential energy and is directly solved by the high-quality general-purpose ordinary differential equation (ODE) solver, i.e. COLSYS ODE solver. Also, a more efficient Rayleigh-Ritz energy method is applied to provide the weak-form solutions. The resulting beam model is suitable to a more general cross-section, such as solid section, branched open section, or half open-half closed section. The accuracy and efficiency of the proposed beam model are validated extensively by comparing with the previously published results in literature. Effects of the power-law index, taper ratio, and section-type on the torsional response of FG beams with material gradation along length or thickness direction are studied with various numerical examples.

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.

In this paper, the first-order shear deformation theory (FSDT) (Mindlin) for continuum incorporating surface energy is exploited to study the static behavior of ultra-thin functionally graded (FG) plates. The size-dependent mechanical response is very important while the plate thickness reduces to micro/nano scales. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Unlike the classical continuum plate models, the bulk transverse normal stress is preserved here. By incorporating the surface energies into the principle of minimum potential energy, a series of continuum governing differential equations which include intrinsic length scales are derived. The modifications over the classical continuum stiffness are also obtained. To illustrate the application of the theory, simply supported micro/nano scaled rectangular films subjected to a transverse mechanical load are investigated. Numerical examples are presented to present the effects of surface energies on the behavior of functionally graded (FG) film, whose effective elastic moduli of its bulk material are represented by the simple power law. The proposed model is then used for a comparison between the continuum analysis of FG ultra-thin plates with and without incorporating surface effects. Also, the transverse shear strain effect is studied by a comparison between the FG plate behavior based on Kirchhoff and Mindlin assumptions. In our analysis the residual surface tension under unstrained conditions and the surface Lame constants are expected to be the same for the upper and lower surfaces of the FG plate. The proposed model is verified by previous work.

A refined finite element method for bending of smart functionally graded plates

Hygro-thermal buckling of the porous FG nanobeam incorporating the surface effect is investigated. The even distribution of porosities is assumed in this paper. Various porous FG nanobeam models including classical beam theory (CBT), Timoshenko beam theory (TBT), Reddy beam theory (RBT), sinusoidal beam theory (SBT), hyperbolic beam theory (HBT) and exponential beam theory (EBT) are developed in this paper. The nonlocal strain gradient theory with material length scale and nonlocal parameters is adopted to examine the buckling behavior. The governing equations of the porous FG nanobeam are derived from principle of minimum potential energy. In the numerical examples, the effect of the nonlocal parameter, material length scale parameter, the temperature rise, the moisture concentration, surface effect, material gradient index, and porosity volume fraction on the buckling temperature and moisture are analyzed and discussed in detail. The results show that the stiffness of the beam depends on the relation of size between nonlocal parameter and length scale parameter. The paper will be helpful for the design and manufacture of the FG nanobeam under complex environments.

Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory

Lesson 1 - Principle of Minimum Potential Energy

Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory

In this article, thermal effect on free vibration behavior of composite laminated microbeams based on the modified couple stress theory is presented. The proposed anisotropic model is developed by using a variational formulation. The governing equations and boundary conditions are obtained based on a modified couple stress theory and using the principle of minimum potential energy and considering different beam theories, i.e., Euler–Bernoulli, Timoshenko and Reddy beam theories. Unlike the classical beam theories, this model contains a material length scale parameter and can capture the size effect. Free vibration of a simply supported beam is solved by utilizing Fourier series. In addition, the fundamental frequency is achieved by using the generalized differential quadrature method for four types of cross-ply laminations with clamped–clamped, clamped–hinged and hinged–hinged boundary conditions for different beam theories. For investigating different parameters including temperature changes, material length scale parameter, beam thickness, some numerical results on different cross-ply laminated beams are presented. The fundamental frequency of different thin and thick beam theories is investigated by increasing slenderness ratio and thermal loads. The results prove that the modified couple stress theory increases the natural frequency under the thermal effects for free vibration of composite laminated microbeams.

Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions

Closed-form solution in bi-Helmholtz kernel based two-phase nonlocal integral models for functionally graded Timoshenko beams

Bidirectional flexure analysis of functionally graded (FG) plate integrated with piezoelectric fiber reinforced composites (PFRC) is presented in this paper. A higher order shear and normal deformation theory (HOSNT12) is used to analyze such hybrid or smart FG plate subjected to electromechanical loading. The displacement function of the present model is approximated as Taylor’s series in the thickness coordinate, while the electro-static potential is approximated as layer wise linear through the thickness of the PFRC layer. The equations of equilibrium are obtained using principle of minimum potential energy and solution is by Navier’s technique. Elastic constants are varying exponentially along thickness (z axis) for FG material while Poisson’s ratio is kept constant. PFRC actuator attached either at top or bottom of FG plate and analyzed under mechanical and coupled mechanical and electrical loading. Comparison of present HOSNT12 is made with exact and finite element method (FEM).

ABSTRACTIn this article, the free vibration analysis of a functionally graded (FG) porous cylindrical microshell subjected to a thermal environment is investigated on the basis of the first-order shear deformation shells and the modified couple stress theories. The material properties are assumed to be temperature dependent and are graded in the thickness direction. The equations of motion and the related boundary conditions are derived using the principle of minimum potential energy and they are solved analytically. The model is validated by comparing the benchmark results with the obtained ones. The effects of material length scale parameter, temperature changes, volume fraction of the porosity, FG power index, axial and circumferential wave number, and length on the vibration behavior of the FG porous cylindrical microshell are studied. The results can have many applications such as in modeling of microrobots and biomedical microsystems.

This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-mechanical systems (MEMS/NEMS) with a wide range of aspect ratios. Based on the modified couple stress theory and the principle of minimum potential energy, the governing equations and boundary conditions for a micro-structure-dependent beam theory are derived. The present beam theory incorporates different kinds of higher-order shear assumptions as well as the two familiar beam theories, namely, the Euler-Bernoulli and Timoshenko beam theories. A numerical solution procedure, based on a generalized differential quadrature method (GDQM), is used to calculate the results of the bi-directional FG beams. The effects of the two exponential FG indexes, the higher-order shear deformations, the length scale parameter, the geometric dimensions, and the different boundary conditions on the critical buckling loads are studied in detail, by assuming that Young’s modulus obeys an exponential distribution function in both length and thickness directions. To reach the desired critical buckling load, the appropriate exponential FG indexes and geometric shape of micro-beams can be designed according to the proposed theory.

This work focus on the mechanical behaviors, which are related to the size effect, functionally graded (FG) effect and Poisson effect, of an axially functionally graded (AFG) micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction. The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory. Employing the modified couple stress theory (MCST), the components of strain, curvature, stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam. A size-dependent model related to FG effect and Poisson effect, which includes the formulations of bending stiffness, deflection, normal stress and couple stress, is developed to predict the mechanical behaviors of the AFG micro-beam by employing the principle of minimum potential energy. The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects, FG effects and Poisson effects of the AFG micro-beam. Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough. The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams. The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible, and should be also considered in the design and analysis of an AFG micro-beam. This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.