Shear Deformable Model Research Articles

Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams

Abstract In this paper, thermal and shear deformation effects on the vibrational response of non-homogeneous microbeams made of functionally graded (FG) materials are carried out. It is assumed that the temperature-dependent material properties of FG microbeams change smoothly and gradually throughout the height according to the classical rule of mixture. The governing differential equations and related boundary conditions are derived by implementing Hamilton's principle on the basis of hyperbolic shear deformation beam and modified couple stress theories and they are analytically solved. The results are given together with other beam theories. A detailed parametric study is performed to indicate the influences of slenderness ratio, material length scale parameter, gradient index, shear correction factors and temperature rise on natural frequencies of FG microbeams. It is revealed that the use of modified shear correction factor can provide more accurate and valid results for first-order shear deformable microbeam model.

Buckling analysis of orthotropic protein microtubules under axial and radial compression based on couple stress theory

Protein microtubules (MTs) are one of the important intercellular components and have a vital role in the stability and strength of the cells. Due to applied external loads, protein microtubules may be involved buckling phenomenon. Due to impact of protein microtubules in cell reactions, it is important to determine their critical buckling load. Considering nature of protein microtubules, various parameters are effective on microtubules buckling. The small size of microtubules and also lack of uniformity of MTs properties in different directions caused the necessity of accuracy in the analysis of these bio-structure. In fact, microtubules must be considered as a size dependent cylinder, which behave as an orthotropic material. Hence, in the present work using first-order shear deformation model (FSDT), the buckling equations of anisotropic MTs are derived based on new modified couple stress theory (NMCST). After solving the stability equations, the influences of various parameters are measured on the MTs critical buckling load.

A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

AbstractA homogeneous shear deformable isotropic elastic plate model is addressed in which the normal transverse fibers are allowed to rotate and to warp in a physically consistent manner specified by a fixed value of a real non‐negative warping parameter ω. On letting ω vary continuously (at fixed load and boundary conditions), a continuous family of shear deformable plates is generated, which spans from the Kirchhoff plate at the lower limit , to the Mindlin plate at the upper limit ; for , identifies with the third‐order Reddy plate. The boundary‐value problem for the generic plate is addressed in the case of quasi‐static loads, for which a principle of minimum total potential energy is given. Taking the Kirchhoff deflection wK and the shear angle potentials as basic unknown fields, the problem turns out to be governed by three uncoupled fourth‐order PDEs, of which one coincides with the classical PDE for the Kirchhoff plate, the other two are PDEs of Helmholtz type. A closed‐form representation of the general solution for the plate family is given in terms of to within an arbitrary biharmonic function and an arbitrary pair of conjugate harmonic functions. These arbitrary functions are available to enforce the inherent boundary conditions. The connections between the present theory and other existing theories are pointed out. An application to a family of simply supported circular plates under uniform load is analytically worked out.

A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells

The present study addresses the size dependency in nonlinear instability of functionally graded multilayer graphene platelet-reinforced composite (GPLRC) nanoshells under hydrostatic pressure including jointly the nonlocal elastic and strain gradients stress fields. For this objective, the new unconventional continuum theory namely as nonlocal strain gradient theory of elasticity is utilized within the framework of a refined hyperbolic shear deformation shell theory. Via stacking up a number of individual layers, the graphene platelet (GPL) nanofillers are distributed uniformly and three different functionally graded patterns based upon a layerwise change of the GPL weight fraction through the shell thickness direction. The effective material properties corresponding to uniform (U-GPLRC) and X-GPLRC, O-GPLRC, A-GPLRC functionally graded patterns of dispersion are extracted by Halpin-Tsai micromechanical scheme. By employing jointly the boundary layer theory of shell buckling and a two-stepped perturbation technique, explicit analytical expressions are achieved for nonlocal strain gradient stability curves of functionally graded multilayer GPLRC nanoshells. It is indicated that by increasing the value of GPL weight fraction for the U-GPLRC and O-GPLRC nanoshells, the significance of the both nonlocal and strain gradient size dependencies reduces, while for the X-GPLRC and A-GPLRC nanoshells, it increases.

Size-dependent bending, buckling and vibration of higher-order shear deformable magneto-electro-thermo-elastic rectangular nanoplates

The aim of the present study is to propose a unified size-dependent higher-order shear deformable plate model for magneto-electro-thermo-elastic (METE) rectangular nanoplates by adopting the nonlocal elasticity theory to capture the size effect, and by utilizing a generalized shape function to consider the effects of transverse shear deformation and rotary inertia. By considering various shape functions, the proposed plate model can be reduced to the nonlocal plate model based upon the Kirchhoff, Mindlin and Reddy plate theories, as well as the parabolic, trigonometric, hyperbolic and exponential shear deformation plate theories. The governing equations of motion and corresponding boundary conditions of METE nanoplates subjected to external in-plane, transverse loads as well as magnetic, electric and thermal loadings, are obtained using Hamilton’s principle. Then, as in some case studies, the static bending, buckling, and free vibration characteristics of simply-supported METE rectangular nanoplates are investigated based upon the Navier solution approach. Numerical results are provided in order to investigate the influences of various parameters including the nondimensional nonlocal parameter, type of transverse loading, temperature change, applied voltage, and external magnetic potential on the mechanical behaviors of METE nanoplates. Furthermore, comparisons are made between the results predicted by different nonlocal plate models by utilizing the developed unified nonlocal plate model and selecting the associated shape functions. It is illustrated that by using the presented unified nonlocal plate model, the development of a nonlocal plate model based upon any existing higher-order shear deformable plate theory is a simple task.

Calibration of developed nonlocal anisotropic shear deformable plate model for uniaxial instability of 3D metallic carbon nanosheets using MD simulations

In comparison with semiconducting carbon structures, the metallic carbon exhibits more novel properties due to their high density of states as well as superconductivity. As a result, prediction of mechanical properties associated with three-dimensional (3D) carbon structures is very attractive. In this paper, the size-dependent nonlinear uniaxial instability of 3D metallic carbon nanosheets containing interlocking hexagonal carbon bonds is investigated based on a novel calibrated nonlocal anisotropic plate model. To accomplish this purpose, Eringen’s nonlocal continuum elasticity is incorporated to an exponential shear deformation plate theory. To extract the proper value of nonlocal parameter, the critical buckling loads and associated postbuckling equilibrium curves obtained by the established size-dependent plate model are compared with those of molecular dynamics (MD) simulations corresponding to various nanosheet side lengths and different boundary conditions. It is displayed that the calibrated nonlocal plate model with the proposed proper value of nonlocal parameter equal to 1.12 nm for fully simply supported edge supports and 1.26 nm for fully clamped edge supports has a very good capability to anticipate accurately both of the buckling and postbuckling characteristics of the uniaxially loaded 3D metallic carbon nanosheets.

A higher order shear deformation model of a periodically sectioned plate

This talk develops a higher order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton’s principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the...

Static deflection and dynamic behavior of higher-order hyperbolic shear deformable compositionally graded beams

In this work we introduce a higher-order hyperbolic shear deformation model for bending and frees vibration analysis of functionally graded beams. In this theory and by making a further supposition, the axial displacement accounts for a refined hyperbolic distribution, and the transverse shear stress satisfies the traction-free boundary conditions on the beam boundary surfaces, so no need of any shear correction factors (SCFs). The material properties are continuously varied through the beam thickness by the power-law distribution of the volume fraction of the constituents. Based on the present refined hyperbolic shear deformation beam model, the governing equations of motion are obtained from the Hamilton\'s principle. Analytical solutions for simply-supported beams are developed to solve the problem. To verify the precision and validity of the present theory some numerical results are compared with the existing ones in the literature and a good agreement is showed.

Nonlinear Free Vibration Analysis of Laminated Carbon/Epoxy Curved Panels

Nonlinear frequency responses of the laminated carbon/epoxy composite curved shell panels have been investigated numerically and validated with in-house experimentation. The nonlinear responses have been computed numerically via customised computer code developed in MATLAB environment with the help of current mathematical model in conjunction with the direct iterative method. The mathematical model of the layered composite structure derived using various shear deformable kinematic models (two higher-order theories) in association with Green-Lagrange nonlinear strains. The current model includes all the nonlinear higher-order strain terms in the formulation to achieve generality. Further, the modal test has been conducted experimentally to evaluate the desired frequency values and are extracted via the transformed signals using fast Fourier transform technique. In addition, the results are computed using the simulation model developed in commercial finite element package (ANSYS) via batch input technique. Finally, numerical examples are solved for different geometrical configurations and discussed the effects of other design parameters (thickness ratio, curvature ratio and constraint condition) on the fundamental linear and nonlinear frequency responses in details.

A unified nonlocal nonlinear higher-order shear deformable plate model for postbuckling analysis of piezoelectric-piezomagnetic rectangular nanoplates with various edge supports

By considering the small-scale effect based on the nonlocal elasticity theory, the nonlinear postbuckling of thick and moderately thick rectangular piezoelectric-piezomagnetic nanoplates with various edge supports subjected to the magneto-electro-thermo-mechanical loading is investigated. For this objective, a unified nonlinear higher-order shear deformable plate model is proposed. By adopting the nonlocal theory to capture the small-scale effect and utilizing a generalized displacement field to consider the influence of transverse shear deformation, unified size-dependent nonlinear governing equations and related boundary conditions are derived based on the virtual work principle in conjunction with von Kármán geometric nonlinearity. By choosing appropriate shape functions, the developed plate model can be reduced to the size-dependent Kirchhoff, Mindlin, Reddy, parabolic, trigonometric, hyperbolic and exponential shear deformable plate models. The nonlinear governing equations and boundary conditions are discretized using the generalized differential quadrature method first. Then, the pseudo arc-length continuation technique is used to solve the discretized equations and obtain the secondary equilibrium path of nanoplate in the postbuckling regime. In addition to providing significant guidelines for accurate prediction of the stability conditions nanoplates, extracting various plate models on the basis of any existing shear deformable plate theory becomes readily attainable by utilizing the proposed unified plate model.

Size-Dependent Buckling and Postbuckling Analyses of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

This paper presents a nonlocal nonlinear first-order shear deformable plate model for investigating the buckling and postbuckling of magneto-electro-thermo elastic (METE) nanoplates under magneto-electro-thermo-mechanical loadings. The nonlocal elasticity theory within the framework of the first-order shear deformation plate theory along with the von Kármán-type geometrical nonlinearity is used to derive the size-dependent nonlinear governing partial differential equations and associated boundary conditions, in which the effects of shear deformation, small scale parameter and thermo-electro-magneto-mechanical loadings are incorporated. The generalized differential quadrature (GDQ) method and pseudo arc-length continuation algorithm are used to determine the critical buckling loads and postbuckling equilibrium paths of the METE nanoplates with various boundary conditions. Finally, the influences of the nonlocal parameter, boundary conditions, temperature rise, external electric voltage and external magnetic potential on the critical buckling load and postbuckling response are studied.

Effect of temperature on crack kinking and jumping in a cross-ply laminated beam

To understand the mechanisms of crack kinking and jumping that occur within the 90° layer of a cross-ply fiber reinforced plastic laminated plate, double cantilever beam tests were performed at several temperatures for laminated plates with two types of stacking sequences. The crack kink angles were calculated using a bi-layer shear-deformable beam model. Furthermore, the interlaminar shear stresses were calculated using finite-element models to clarify the mechanism of the repeated jumps. The following results were obtained from these experiment and analysis; (i) a crack at the center tended to propagate in a self-similar manner more stably at higher temperatures, (ii) the applied load at which value the crack at the center started propagation decreased as the 90° layer became thicker, and (iii) the crack along a 0°/90° interface jumped to the other interface because the shear force along the 0°/90° interface due to the thermal stress decreased as the crack propagated along the interface.

Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory

The hygro-thermo-mechanical bending behavior of sigmoid functionally graded material (S-FGM) plate resting on variable two-parameter elastic foundations is discussed using a four-variable refined plate theory. The material characteristics are distributed within the thickness direction according to the two power law variation in terms of volume fractions of the constituents of the material. By employing a four variable refined plate model, both a trigonometric distribution of the transverse shear strains within the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are respected without utilizing shear correction factors. The number of independent variables of the current formulation is four, as against five in other shear deformation models. The governing equations are deduced based on the four-variable refined plate theory incorporating the external load and hygro-thermal influences. The results of this work are compared with those of other shear deformation models. Various numerical examples introducing the influence of power-law index, plate aspect ratio, temperature difference, elastic foundation parameters, and side-to-thickness ratio on the static behavior of S-FGM plates are investigated.

‘One-half order shear deformation theory’ as a new naming for the transverse, but not in-plane rotational, shear deformable structural models

Based on Hamilton’s principle and using method of hypotheses, governing equations for the existing typical thin-walled structure analysis models are first derived systematically in addition to those for the author’s alternative beam and plate theories with the deformation concept that the total deflection w can be assumed as the sum of the bending and shearing deflections wb and ws, and in addition formal expressions of shear forces for some beam and plate models are described on the basis of the equilibrium condition as a solid. Two specific static beam boundary value analyses are then carried out based on the alternative, Euler-Bernoulli and Timoshenko beam theories for which the respective calculated results turn out to differ. Through the discussions, it is shown that in the alternative beam and plate theories only the half categories of shear deformations, which involve also in-plane rotational shearing in addition to transverse shearing, can be expressed explicitly in terms of fundamental variables, in contrast to the traditional first-order shear deformation theories. Thus, it is proposed to call the alternative theory ‘one-half order shear deformation theory’ as a new naming for the moderately thick structural models with the superimposed total deflection concept as w=wb+ws.

Nonlocal nonlinear first-order shear deformable beam model for postbuckling analysis of magneto-electro-thermo elastic nanobeams

In this study, the size-dependent postbuckling behavior of magneto-electro-thermo-elastic (METE) nanobeams with different edge supports is investigated. Based on the nonlocal first-order shear deformation beam theory and considering the von Karman hypothesis, a size-dependent nonlinear METE nanobeam model is developed, in which the effects of small scale parameter and thermo-electro-magnetic-mechanical loadings are incorporated. A numerical solution procedure based on the generalized differential quadrature (GDQ) and pseudo arc-length continuation methods is utilized to describe the size-dependent postbuckling behavior of METE nanobeams under various boundary conditions. The effects of different parameters such as nonlocal parameter, external electric voltage, external magnetic potential and temperature rise on the postbuckling path of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential and temperature rise decrease the critical buckling load and post-buckling load-carrying capacity of METE nanobeams, whilean increase in the negative voltage, positive magnetic potential lead to a considerable increase of critical buckling load as well as postbuckling strength of the METE nanobeams.

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

ABSTRACTSmall scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.

The Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models

The unified solution is studied for a beam of rectangular cross section. With the rotation defined in the average sense over the cross section, the kinematics with higher-order shear deformation models in axial displacement is first expressed in a unified form by using the fundamental higher-order term with some properties. The shear correction factor is then derived and discussed for the four commonly used higher-order shear deformation models including the third-order model, the sine model, the hyperbolic sine model and the exponential model. The unified solution is finally obtained for a beam subjected to an arbitrarily distributed load. The relation with that from the conventional beam theory is established, and therefore the difference is reasonably explained. A very good agreement with the elasticity theory validates the present solution.

Thermo-Mechanical Vibration of Size Dependent Shear Deformable Functionally Graded Conical Nanoshell Resting On Elastic Foundation

In this paper, the size-dependent shear deformable conical shell formulation is derived based on the modified couple stress theory and first order shear deformation model to investigate the free vibration of functionally graded conical shell embedded in an elastic Pasternak medium and subjected to thermal environment. The material properties are considered temperature-dependent and graded in thickness direction according to power law distribution. The governing equations and boundary conditions are derived using Hamilton’s principle. The size effect is taken into account using the modified couple stress theory, and, the free vibration of simply supported FG truncated conical nanoshell is investigated as a special case. The effects of different parameters such as dimensionless length scale parameter, temperature change and distribution of the nanoshell components on the natural frequency are investigated based on the modified couple stress theory and classical continuum theory.

A size-dependent nonlinear third-order shear-deformable dynamic model for a microplate on an elastic medium

This paper develops a size-dependent nonlinear third-order shear-deformable model for the dynamic analysis of microplates. Taking into account in-plane and out-of-plane displacements and inertia as well as rotations (via using the third-order shear deformation theory) and the modified couple stress theory, the Lagrange equations are employed to derive the equations of motion. An assumed-mode technique is applied to the expressions for the elastic strain energy of the microplate, the elastic potential energy due to the translational springs, the kinetic energy of the microplate, the energy dissipation function due to damping, and the work of a harmonically varying external loading on the microplate; these expressions are then inserted in the Lagrange equations in order to obtain the discretised equations of motion as nonlinear coupled functions of generalized coordinates. The pseudo-arclength continuation technique and a direct time-integration are employed to solve these equations and to obtain the generalized coordinates, hence system responses, numerically. Apart from the nonlinear analysis, a linear analysis is conducted by means of an eigenvalue analysis. The motion behaviour of the system is analysed and the importance of employing the modified couple stress theory, rather than the classical continuum theory, is discussed.

Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

Presented herein is a comprehensive study on the size-dependent coupled longitudinal-transverse-rotational free vibration behavior of post-buckled functionally graded (FG) micro- and nano-beams based on the most general Mindlin's strain gradient theory. The current model enables us to incorporate size effects via introducing material length scale parameters and is developed in the framework of the first-order shear deformable beam model and the von Karman geometric nonlinearity. The FG micro- and nano-beams, whose volume fraction is expressed by using a power law function, are assumed to be made of a mixture of metals and ceramics. By using Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro- and nano-beams in the postbuckling domain. Afterwards, the governing equations and boundary conditions are discretized using the generalized differential quadrature (GDQ) method in conjunction with a direct approach without linearization, before solving numerically by Newton's method. The effects of length scale parameter, length-to-thickness ratio, material gradient index and boundary conditions on the postbuckling path and frequency of FG micro- and nano-beams are carefully investigated. Finally, numerical results obtained from both the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) are compared.