Traction-free Boundary Conditions Research Articles

Analytic solution of angle-ply laminated plates under extension, bending, and torsion

This paper develops a Hamiltonian state space approach for analytic determination of deformation and stress fields in multilayered monoclinic angle-ply laminates under the combined action of extension, bending, and torsion. The present solution satisfies the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the four edge surfaces of the rectangular section, and the interfacial continuity conditions in multilayered laminates. The proposed method only requires the solutions of matrix and eigen equations, regardless of the number or lamination of the layers. The finite element analyses are used to validate the accuracy of the analysis. The analytical solution and the numerical solutions are in excellent agreement.

Nonlocal free and forced vibration of multilayered two-dimensional quasicrystal nanoplates

A three-dimensional plate model capturing the size effect is developed for free and forced vibration of a layered two-dimensional quasicrystal nanoplate. The simply supported nanoplate subjected a top surface harmonic excitation is assumed for forced vibration analysis, and the traction free boundary conditions are considered for free vibration analysis. The exact solution for layered two-dimensional quasicrystal nanoplates is obtained by using the pseudo-Stroh formalism, Eringen’s nonlocal theory, and a dual variable and position method. Numerical examples show the effects of nonlocal parameter, stacking sequence and aspect ratio on natural frequency as well as forcing frequency on harmonic response.

FE modeling for geometrically nonlinear analysis of laminated plates using a new plate theory

The aim of the present work is to study the nonlinear behavior of the laminated composite plates under transverse sinusoidal loading using a new inverse trigonometric shear deformation theory, where geometric nonlinearity in the Von-Karman sense is taken into account. In the present theory, in-plane displacements use an inverse trigonometric shape function to account the effect of transverse shear deformation. The theory satisfies the traction free boundary conditions and violates the need of shear correction factor. The governing equations of equilibrium and boundary conditions associated with present theory are obtained by using the principle of minimum potential energy. These governing equations are solved by eight nodded serendipity element having five degree of freedom per node. A square laminated composite plate is considered for the geometrically linear and nonlinear formulation. The numerical results are obtained for central deflections, in-plane stresses and transverse shear stresses. Finite element Codes are developed using MATLAB. The present results are compared with previously published results. It is concluded that the geometrically linear and nonlinear response of laminated composite plates predicted by using the present inverse trigonometric shape function is in excellent agreement with previously published results.

Cell- and Node-Based Smoothing MITC3-Finite Elements for Static and Free Vibration Analysis of Laminated Composite and Functionally Graded Plates

Cell- and node-based smoothing MITC3-finite elements for static and free vibration analysis of laminated composite and functionally graded plates are proposed in this paper. It is based on a higher-order shear deformation theory which accounts for a higher-order variation of in-plane displacements and traction-free boundary conditions on top and bottom surfaces of transverse shear stresses. Novel CS-MITC3 and NS-MITC3 finite elements are proposed in which an MITC3 three-node triangle element with seven degrees-of-freedom per nodes that only requires the [Formula: see text]-type continuity is developed, and then cell- and node-based smoothing techniques are applied. Numerical results for the laminated composite and functionally graded plates are proposed to validate the accuracy of developed theory and to investigate the effects of material distribution and side-to-thickness ratio on the deflection, stresses and natural frequencies of the plates.

Buckling and Post-Buckling Analysis of Geometrically Imperfect FGM Pin-Ended Tubes Surrounded by Nonlinear Elastic Medium Under Compressive and Thermal Loads

The present study aims to analyze the buckling and post-buckling behavior of the geometrically imperfect functionally graded pin-ended tube. Imperfect FGM tube is surrounded by nonlinear elastic medium and is subjected to the axial compression or various thermal loads. Pinned-pinned boundary conditions are movable or immovable for the FGM tube under axial compression or thermal loads, respectively. In thermal analysis, different types of thermal loads such as uniform temperature rise, linear temperature distribution, and heat conduction are analyzed and contrasted. Displacement field of the FGM tube satisfies the tangential traction-free boundary conditions on the inner and outer surfaces. Properties of the FGM tube are assumed to be temperature-dependent and are distributed through the radial direction of tube using a power law function. The governing equilibrium equations of the FGM tube are obtained by means of the virtual displacement principle. These are nonlinear coupled differential equations based on a higher order shear deformation tube theory and the von Kármán nonlinear assumption. The coupled nonlinear dimensionless differential equations are solved using the two-step perturbation method. These asymptotic solutions are as explicit functions of the axial compression or different types of thermal load. Numerical results are provided to explore the effects of the linear and nonlinear spring stiffness of elastic medium and imperfection parameter of the tube. The effects of the volume fraction index and two geometrical parameters of the FGM tube are also included.

Direct extraction of stochastic stress intensity factors using generalized polynomial chaos

The stochastic approximation of the Stress Intensity Factors (SIFs) for cracked domains with traction free boundary conditions on the crack edges having stochastic material property (stochastic variable) is presented explicitly using the generalized Polynomial Chaos (gPC). Using a set of deterministic SIF extracted from finite element model according to the stochastic properties of the material property, we compute the stochastic approximation of the SIF as a set of orthogonal stochastic polynomials, multiplied by a deterministic combination of the extracted SIFs. The method provides a direct stochastic SIF approximation and does not require a stochastic representation of the stresses or the displacement. Numerical examples are presented in which we consider either a stochastic Young modulus or stochastic Poisson ratio with normal distribution as a random variable. The example problems are compared with the indirect gPC SIF extraction method and Monte-Carlo results. The results demonstrate the efficiency and accuracy of the proposed method.

Large amplitude free vibration analysis of shear deformable FGM shallow arches on nonlinear elastic foundation

Current investigation deals with the small and large amplitude free vibrations of a curved beam (arch) resting on a nonlinear elastic foundation. The arch is made of a functionally graded material, where the properties are graded across the thickness. Uniform temperature elevation in the arch is also considered and material properties are assumed to be temperature dependent. The governing motion equations of the arch are established using a higher order arch theory which satisfies the traction free boundary conditions and the von Kármán type of non-linearity. The governing equations of the arch are solved for the case of an immovable pinned arch using the two step perturbation technique. Closed form expressions are given to estimate the nonlinear frequencies of the arch as a function of the mid-span deflection. Numerical results of this study are validated for the case of flat FGM beams on elastic foundation. Afterwards, novel numerical results are given to explore the influences of power law index, elastic foundation coefficients, length to thickness ratio, length to radius ratio, and the temperature effects.

Effects of mode couplings on the vibration characteristics of partially electroded thin-film bulk acoustic wave resonators

Mode coupling vibrations in thin-film bulk acoustic wave resonators (FBARs) were systematically analyzed in the series of our work. Previously, the two-dimensional theory of multi-layered FBAR plate has been established and used for the analysis of both free and forced vibrations of a fully electroded plate. In this paper, coupling vibrations in a partially electroded FBAR were taken into account, where the top electrode only covers the central part of the film. The two-dimensional equations were written into matrix form based on the state-vector approach. Transfer matrix functions of the global structure were obtained when continuous and traction-free boundary conditions were satisfied. Aspect ratios of both electroded and unelectroded regions were varied to research the influence of them on the coupling effects. The admittance versus frequency response was also obtained through forced vibration analysis to investigate the coupling effects. Results in this paper show the existence of energy-trapping behavior and the importance of structure size on mode coupling effects, which is important to the design of real FBAR devices.

Analysis of cracked functionally graded material plates using XIGA based on generalized higher-order shear deformation theory

A generalized higher-order shear deformation theory (GHSDT) along with extended isogeometric analysis (XIGA) is presented for the analysis of cracked functionally graded material (FGM) plates. The material properties of the FGM vary exponentially along the thickness of the plate. The GHSDT ensures a traction-free boundary conditions at the surfaces of the plate without a need of shear correction factor (SCF). To model a crack, the isogeometric analysis (IGA) approximation is enriched with partition of unity (PU) enrichment functions. The C1 continuity requirement of the GHSDT is easily achieved in IGA through Non-uniform rational B-splines (NURBS). The stress intensity factor (SIF) at any location along the thickness of the plate is computed using domain based interaction integral approach. In the interaction integral, 2-D asymptotic crack-tip fields obtained from the LEFM provide a straightforward and simple solution scheme for SIF along the thickness of the FGM plate. The accuracy of the proposed approach is demonstrated by solving various problems. The effect of various parameters such as loading, boundary conditions, crack length and plate thickness is investigated on the SIF. The solutions of several cracked FGM plate problems obtained by GHSDT based XIGA (GHSDT-XIGA) are compared with FSDT based XIGA (FSDT-XIGA) and reference solutions.

Surface Shear Waves in a Half-Plane with Depth-Variant Structure

We consider the propagation of surface shear waves in a half-plane, whose shear modulus and density depend continuously on the depth coordinate. The problem amounts to studying the parametric Sturm–Liouville equation on a half-line with frequency and wave number as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum in the wave number/frequency plane for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of a finite number of surface wave solutions; the number grows and tends to infinity with the wave number. The most intriguing result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behaviour of the shear modulus and density close to infinite depth.

The response of a partially loaded composite half-space weakened by local defects

The elastic field in a fiber-reinforced half-space which consists of distinct constituents and localized internal defects is predicted by employing a multiscale analysis. A normal distributed uniform loading is applied over a rectangular region of the half-space surface, while the other parts are kept traction-free. The effect of internal defects and the half-space traction-free boundary conditions are accounted to by incorporating a damage tensor in the constitutive relations which describe the half-space phases. In the micro level of the micro-to-macro analysis, the effective behavior of the composite half-space is determined by employing a micromechanics analysis. This is followed by a macromechanical analysis in which the triple discrete Fourier transform is applied in the domain of which the formulated problem is solved. The damaged composite half-space response is obtained by inverting the Fourier transform, in conjunction with an iterative procedure. The predicted elastic field in the half-space which is based on the present approach is validated by comparisons with exact and finite element solutions. The offered method is applied to generate the elastic field in a fiber-reinforced half-space with a broken fiber, lost fiber, debonded fiber, missing fiber and a matrix void. Applications of numerous defects of these and other types are discussed.

Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation

• Closed form expressions are used to estimate the deflection of curved FGM tubes under thermal load . • Different types of thermal loading are considered and temperature dependency is considered. • The solution method is based on the two step perturbation technique . • Interaction of the tube with hardening/softening nonlinear elastic foundation and initial imperfection are taken into account. • The developed formulation is based on a higher order tube theory which satisfies the traction free boundary conditions. Present research deals with the thermal buckling and post-buckling analysis of the geometrically imperfect functionally graded tubes on nonlinear elastic foundation. Imperfect FGM tube with immovable clamped–clamped end conditions is subjected to thermal environments. Tube under different types of thermal loads, such as heat conduction, linear temperature change, and uniform temperature rise is analyzed. Material properties of the FGM tube are assumed to be temperature dependent and are distributed through the radial direction. Displacement field satisfies the tangential traction free boundary conditions on the inner and outer surfaces of the FGM tube. The nonlinear governing equations of the FGM tube are obtained by means of the virtual displacement principle. The equilibrium equations are based on the nonlinear von Kármán assumption and higher order shear deformation circular tube theory. These coupled differential equations are solved using the two-step perturbation method. Approximate solutions are provided to estimate the thermal post-buckling response of the perfect/imperfect FGM tube as explicit functions of the various thermal loads. Numerical results are provided to explore the effects of different geometrical parameters of the FGM tube subjected to different types of thermal loads. The effects of power law index, springs stiffness of elastic foundation, and geometrical imperfection parameter of tube are also included.

The anti-plane shear elasto-static fields near a crack terminating at an isotropic hyperelastic bi-material interface

In the framework of hyperelasticity, we treat the configuration of a terminated crack at the interface of an incompressible full plane composite. Considering the traction-free boundary conditions, three particular cases are discussed when a cylinder is subjected to an anti-plane shear transformation. Taking all these conditions into account, an asymptotic analysis is performed to identify the sufficient orders contributing to the singular form of the Cauchy stress static fields. Adding to that, an inquiry about the presence of logarithmic singularities was achieved using the approach of Dempsey and Sinclair.

A refined sinusoidal theory for laminated composite and sandwich plates

This paper proposes a new zig-zag function to develop the refined sinusoidal theory. A refined sinusoidal shear deformation theory can be obtained by including the proposed zig-zag function and meeting the traction-free boundary conditions on the top and the bottom surfaces. Governing equations and boundary conditions are obtained by using the principle of virtual displacement. Using Navier’s method, analytical solutions are presented by solving the boundary value problem. By investigating the typical bending problems of laminated composite and sandwich plates, the effect of various parameters like aspect ratio, thickness ratio and material properties on the bending response of plate is studied in detail. Comparison of the present results with the three-dimensional elasticity solutions and the results calculated using other models published recently in the literature shows that the proposed theory can yield accurately the displacements and stresses of multilayered composite and sandwich plates. Moreover, the accuracy of the displacement and stresses are not impacted by the number of layers and the material properties.

Vibration of thermo lemv composite multilayered hollow pipes

The present paper is concerned the effects of thermo elastic waves in a composite multilayered hollow pipes which contain inner and outer zinc layers bonded by Linear Elastic Material with Voids (LEMV) is considered. The equation of movement is derived by means of the constitutive equations of linear thermo elasticity. The equation of movement and heat conduction models are decoupled by the displacement of potentials which are constructed based on equilibrium equations of elasticity. The dispersion equation are acquired by means of traction free boundary conditions and are numerically analyzed for Zinc material. The enumerated frequency, phase velocity are presented graphically for LEMV and CFRP layers.

Nonpolynomial Zigzag Theories for Random Static Analysis of Laminated-Composite and Sandwich Plates

The present paper aims to study the random static behavior of laminated composite and sandwich plates using nonpolynomial zigzag theories. The nonpolynomial zigzag theories are proposed earlier by the authors, which satisfy the transverse shear-stress continuity conditions at the layer interfaces including the traction-free boundary conditions on the top and bottom surfaces of the plate. These theories incorporate the realistic nonlinear distribution of transverse shear stresses across the thickness of the plate. A probabilistic procedure is developed using a stochastic finite element method that is in the conjunction of finite element method with a mean-centered first-order perturbation technique, to obtain the second-order statistics of deflections of the laminate. The problem is analyzed considering the material properties of laminated composite and sandwich structures to be random in nature while the other system properties are assumed to be deterministic. The stochastic results in terms of mean as well as standard deviation of the responses have been obtained for various combinations of geometric parameters, stacking sequences, span–thickness ratios, and boundary conditions and compared with those existing in the literature and Monte Carlo simulation approach.

Vibration response of a functionally graded graphene nanoplatelet reinforced composite beam under two successive moving masses

The vibration response of a functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) beam under two successive moving masses is investigated in the current study. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary continuously and smoothly in the thickness direction. The modified Halpin–Tsai micromechanics model is used to evaluate the effective Young’s modulus of the FG-GPLRC beam. A new high order shear deformation theory (HSDT) is proposed for the structural behavior analysis of FG-GPLRC beams. The proposed theory assumes nonlinear distribution of transverse shear stresses and satisfies the traction-free boundary conditions at the bottom and top surfaces of the beam and hence, a shear correction factor is not required. A generalized closed-form solution methodology of the Navier-type is implemented to ensure the validity and efficiency of the present theory for the free vibration and dynamic response of the beam. The Newmark-β method is adopted to obtain the response of the tube in time domain. Two successive moving masses are considered as the moving load acting on the beam. A parametric study is performed to investigate the effects of GPL weight fraction, distribution pattern, and distance between the two masses on the vibration responses of the beam.

Numerical solutions for cracks in an elastic half-plane

The behavior of the stress intensity factor at the tips of cracks subjected to uniaxial tension $$\sigma ^{\infty }_{x}=p$$ with traction-free boundary condition in half-plane elasticity is investigated. The problem is formulated into singular integral equations with the distribution dislocation function as unknown. In the formulation, we make used of a modified complex potential. Based on the appropriate quadrature formulas together with a suitable choice of collocation points, the singular integral equations are reduced to a system of linear equations for the unknown coefficients. Numerical examples show that the values of the stress intensity factor are influenced by the distance from the cracks to the boundary of the half-plane and the configuration of the cracks.

Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading

In the present work, the static response of symmetric and anti-symmetric laminated composite and sandwich plates is investigated for elevated temperature and moisture conditions. Analysis is carried out using recently developed zigzag model which is based on a trigonometric (secant) shear deformation in the plate. The theory incorporates inter-laminar shear stress continuity along with the traction free boundary conditions at the top and bottom surfaces. The finite element method is employed for analysis, using the eight noded isoparametric serendipity element, accounting for the C0 continuity. Numerous examples covering the various features such as effect of thermal and moisture coefficients, material anisotropy, boundary and loading conditions and span to thickness ratio are solved for symmetric as well as anti-symmetric plates. Results in the form of deflection and stresses are presented and validated with the available results in the existing literature. Further, few new results are also provided in this work, to set a benchmark study for the future research.

2D analysis of laminated composite and sandwich plates using a new fifth-order plate theory

Abstract In the present paper, a new fifth-order shear and normal deformation theory (FOSNDT) is developed for the bi-directional bending analysis of laminated composite and sandwich plates subjected to transverse loads. This theory considered the effects of both transverse shear and normal deformations. In-plane displacements use a polynomial shape function expanded up to fifth-order in terms of the thickness coordinate to properly account the effect of transverse shear deformation. Transverse displacement is the function of x, y and z- coordinates to account the effect of transverse normal deformations i.e. thickness stretching. Hence, the present theory involves nine unknowns in the displacement field. The present theory does not require a problem dependent shear correction factor as it satisfies traction free boundary conditions at top and bottom surfaces of the plate. The governing differential equations and associated boundary conditions are obtained using the principle of virtual work. The plate is analysed for simply supported boundary conditions using Navier’s solution technique. To prove the efficiency of the present theory, the non-dimensional displacements and stresses obtained for laminated composite and sandwich plates are compared with existing exact elasticity solutions and other theories. It is observed from the comparision that the displacements and stresses obtained by the present theory are in excellent agreement with the results obtained by exact elasticity solutions compared to other higher-order plate theories available in the literature.