Partial Differential Equations Of Equilibrium Research Articles

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Thermomechanical modeling of dissimilar-material interfaces in composite structures

Due to the heterogeneities in material properties of strongly-bonded multi-material composite structures, there exist singularities of the stress fields at the material interface as well as the interface corners on the bounding surfaces. The finite-element method is probably the only plausible option to analyze the stress around such interfaces between dissimilar materials subjected to combined thermo-mechanical loading. In this article, we propose a novel mathematical model for the computational analysis of coupled thermo-elastic fields with perfectly bonded interfaces of dissimilar isotropic/orthotropic composite-material structures. A scalar function of the spatial variables, namely, the displacement function, is used to integrate the temperature field and the associated displacement-function field to obtain two nonhomogeneous partial-differential equations of equilibrium that govern two thermo-mechanical sub-problems defined in terms of the appropriate conditions of orthogonal temperature gradients. For a specific temperature field, obtained as a steady-state solution of the given thermal problem, the two thermo-mechanical sub-problems are solved independently to obtain the full-field solution of the thermo-elastic problem using the method of superposition. As an alternative to the conventional computational approach, we employed a single variable semi-analytical method of solution to determine the thermo-mechanical response of the multi-material composite structures. The continuity of the temperature and the heat flux in the thermal field problem and the continuity of the displacement and the traction vectors are utilized in the elastic field problem to determine the appropriate algebraic equations suitable for nodal application at the interfaces as well as interface corners on the boundaries of the respective problems. The quantitative comparison of the present displacement-function solutions to several mixed-boundary-value problems with those obtained by the conventional finite-element method verifies the high accuracy and effectiveness of the present method. This computational model has potential applications in identifying near-interface high-stress locations as well as possible critical regions of the physical boundary of composite structures subjected to mixed modes of thermal and mechanical loadings.

Parametric study for modified couple stress theory on postbuckling of size-dependent FG saturated porous mindlin microplates

Using modified couple stress and Mindlin plate theory, the present study is a parametric study for revealing the effect of length scale parameter on postbuckling of saturated poroelastic rectangular microplates with pore distribution being functionally graded across thickness. The microplate postbuckling is a nonlinear eigenvalue problem with coupled partial differential equations of equilibrium. In order to find the mode shapes, the eigenfunctions have been approximated by some complete polynomial series expansions in accordance with the conventional Ritz approximation technique. After discretizing the governing equations throughout the microplate grid points, the assembled difference equations construct a nonlinear eigenvalue problem with coupled algebraic equations. The nontrivial coefficients of the expansions were calculated such that the eigenfunctions extremize the microplate potential energy for satisfying the equilibrium homogeneous equations. Variation of the potential energy have been derived with respect to the expansions coefficients from the vantage of using Maple software. By gradual reduction in the initial end shortening of the movable supports and solving the assembled difference equations via Newton-Raphson iterations, the critical postbuckling displacement imposed to the movable supports are found along with the nontrivial coefficients regarding eigenfunctions. The results of the parametric study have revealed how geometric parameters, pore distribution, the Skempton’s coefficient, porosity, and different Supports modify the influence of length scale parameter on the microplate postbuckling.

НЕКОТО РЫЕ ВОПРОСЫ СТАТ ИКИ ГЕОМЕТРИЧЕСКИ И ФИЗИЧЕСКИ НЕЛИНЕЙНОЙ СПЛОШНОЙ СРЕДЫ

We consider the generalized plane deformation of physically and geometrically nonlinear continuous medium. The mathematical model of medium is described by physically nonlinear relations in the form of arbitrary cross dependences between the first invariants of tensors and the second invariants of stress and strain deviators, taking into account geometrical nonlinearity (according to V.V. Novozhilov). The system of resolving partial differential equations of equilibrium, done in displacements, is quasilinear. This type is being investigated. Despite the fact that the type of the system of quasilinear partial differential equations in a certain part of space can be determined only for a specific solution, it is stated that in the general case the system of differential equations of equilibrium of generalized plane deformation is a system of mixed type. The type of the system of partial differential equations in a given area of space is completely determined by the values of its coefficients, and therefore depends both on the value of physical constants of continuous medium material and also on the value of derivatives of displacements in spatial coordinates.

Displacement-function modeling of thermo-mechanical behavior of fiber-reinforced composite structures

A new modeling scheme has been developed for the analysis of thermo-mechanical stresses in structural problems of fiber-reinforced composite materials with mixed mode of physical conditions. In contrast with conventional approaches, the coupled thermal and mechanical problems of structural mechanics are integrated here using a single scalar function of space variables, which is defined in terms of the displacement components of plane elasticity. The displacement-function is governed by a single partial-differential equation of equilibrium that takes into account the two orthogonal gradients of the resulting non-uniform temperature field through two different versions of the formulation. The temperature-dependent elastic field is determined using a two-step solution scheme that utilizes the two versions of displacement-function formulations in two sequential steps. The scheme is developed in such a flexible fashion that even the solution of the corresponding isotropic problems can readily be obtained with thermal and/or mechanical loading as a limiting case of the formulation. The application of the proposed modeling scheme is demonstrated through the solution of a number of problems of structural mechanics, namely plates and beams subjected to thermal as well as thermo-mechanical loadings with both uniform and mixed mode of physical conditions. An attempt is also made to verify the soundness and accuracy of the resulting thermoelastic fields by comparing them with the corresponding analytical as well as computational solutions available in the literature. Moreover, the correspondence between the versions of the displacement-function formulation is also verified by solving identical problems using different versions or different combinations of the versions of the formulation.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

In this paper, a geometrically nonlinear size-dependent Elastic Kirchhoff nanoplate model is developed based on the second-order negative strain gradient nonlocal theory useful for capturing the size effects. Taking the mid-plane stretching into consideration as the source of the nonlinear behaviour, weak form of the governing partial differential equations of equilibrium and the relevant classical/non-classical boundary conditions are derived using the variational method. For finite element analysis, the weak form requires C1 continuity of the in-plane displacements and C2 continuity of the transverse displacement. In the present work, a new computationally efficient subparametric nonconforming 4-noded finite element of arbitrary quadrilateral shape is presented for the first time for the modelling of nanoplates using the second-order negative strain gradient theory. The performance of the developed finite element is investigated for static bending of rectangular nanoplates with all edges simply supported and all edges clamped boundary conditions. The proposed element is found to be accurate and depicts good convergence characteristics for rectangular and non-rectangular meshes. The strain gradient model with negative nonlocal coefficient predicts results matching with those from the lattice model available in the literature.

Thermal buckling and postbuckling of edge-cracked functionally graded multilayer graphene nanocomposite beams on an elastic foundation

This paper investigates thermal buckling and postbuckling behaviors of functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer beams containing an open edge crack and resting on a Pasternak-type elastic foundation based on the first-order shear deformation beam theory including von Kármán geometric nonlinearity. The material properties of functionally graded GPL-reinforced composites (GPLRCs), which exhibit piece-wise variation along the thickness direction, are evaluated using micromechanics based models. The bending stiffness of the cracked section is estimated by the rotational spring model. The obtained nonlinear partial differential equations of equilibrium are discretized by the differential quadrature method, and then an iterative method is used to obtain the thermal buckling loads and postbuckling load-deflection curves. Detailed parametric studies are conducted to investigate the effects of crack length, GPL distribution pattern, GPL weight fraction, GPL length-to-width and length-to-thickness ratios, boundary conditions, and foundation stiffnesses on the thermal buckling loads and postbuckling response of the cracked GPLRC beams.

Elastic deformation behavior of freestanding MoS2 films using a continuum approach

In this paper, an approximate solution based on the continuum mechanics is presented to obtain the mechanical response of freestanding single and two layers molybdenum disulfide (MoS2) films. Both circular and rectangular geometries as well as point load and uniform pressure are investigated. The partial differential equilibrium equations are established based on the non-linear Föppl membrane theory for both circular and rectangular films. The Galerkin method is then employed to solve the non-linear partial differential equilibrium equations in case of circular films. Also, the principle of virtual work affords another means of obtaining an approximate solution for rectangular film. Theoretical predictions for the width of membrane regions enable us to estimate the analytical deflection and stress distributions. Finally, the obtained results are compared to the available experimental and theoretical results and a good agreement is seen. It can be concluded that the continuum approach can be used to study the behavior of the MoS2 films.

Analytical investigation of fiber-orientation dependent stresses in a thick stiffened fiber-reinforced composite beam on simple supports

Fiber orientation dependent stresses in a thick, stiffened composite beam on simple supports are investigated analytically using a scalar potential field expressed in terms of displacement components of plane elasticity. The potential field of the present mixed-boundary-value problem is obtained by solving a single partial differential equation of equilibrium, which is then used to derive the expressions of relevant displacement and stress components in terms of Fourier series. Investigations are carried out for both the cases of lateral and axial stiffeners at the opposing ends of the beam. Two limiting cases of fiber orientation are considered for a number of fiber reinforced composite materials with a wide range of beam aspect ratio. The present analysis verifies the fiber orientation to be one of the important design variables not only for thick beams but also for moderately long beams, especially when the ends of the beam are stiffened.

Buckling and post-buckling analysis of geometrically non-linear composite plates exhibiting large initial imperfections

This paper discusses the buckling and post-buckling of thick composite plates having large initial geometrical imperfections. Within the theory of large transformations, the proposed formulation uses generalised forces and deformations derived from the energy conjugacy of the second Piola-Kirchhoff stress and Green-Lagrange strain. In addition, it provides partial differential equations of equilibrium that are easy to solve using common techniques such as the Galerkin-Ritz method. Unlike existing contributions, which are (i) limited to imperfections that are proportional to current deformations, (ii) address small to moderate initial imperfections only, and/or (iii) require advanced and time consuming numerical methods, the present approach is based on analytical solution and takes into account large initial imperfections. Despite the complexity of the energy balance formulation, the resulting governing equations are simple and applicable to design practical composite structures. The proposed model was compared to numerical results obtained using the finite element method. Overall, good agreements have been obtained when the assumptions fall within the range of validity of the commercial software used for simulation.

A Comparison of Fourier Spectral Iterative Perturbation Method and Finite Element Method in Solving Phase-Field Equilibrium Equations

AbstractThis paper systematically compares the numerical implementation and computational cost between the Fourier spectral iterative perturbation method (FSIPM) and the finite element method (FEM) in solving partial differential equilibrium equations with inhomogeneous material coefficients and eigen-fields (e.g., stress-free strain and spontaneous electric polarization) involved in phase-field models. Four benchmark numerical examples, including inhomogeneous elastic, electrostatic, and steady-state heat conduction problems demonstrate that (1) the FSIPM rigorously requires uniform hexahedral (3D) and quadrilateral (2D) mesh and periodic boundary conditions for numerical implementation while the FEM permits arbitrary mesh and boundary conditions; (2) the FSIPM solutions are comparable to their FEM counterparts, and both of them agree with the analytic solutions, (3) the FSIPM is much faster in solving equilibrium equations than the FEM to achieve the accurate solutions, thus exhibiting a greater potential for large-scale 3D computations.

Axisymmetric solutions for the multi-layered transversely isotropic piezoelectric medium

By introducing the precise integration algorithm (PIA) and the technique of the mixed variable formalism accounting for any number of layers, variations of components of the axisymmetric piezoelectric field including elastic displacement, vertical normal stress, electric potential and electric displacement along the axis of symmetry in the cylindrical coordinate system are explored. The piezoelectric field is induced by the vertical loadings of mechanical or electric types uniformly distributed over a circular region. With the aid of the mixed variable formulations, along with the Hankel integral transform and the corresponding matrix algebraic operations, the governing partial differential equations of equilibrium expressed in terms of displacements and electric potential are reduced to the first order ordinary differential matrix equations. As a highly accurate algorithm, the PIA is provided to evaluate the obtained ordinary differential matrix equations in the transformed domain. Both mechanical and electrical quantities in the physical domain can be acquired by taking the inversion of the Hankel integral transform. An example as benchmark is illustrated to examine the applicability and performance of the proposed technique compared with results in the literature. Numerical examples are used to demonstrate the influence of the degree of the material anisotropy and stratified parameters.

Computational analysis of elastic field of deep curved beams/rings using displacement-function equilibrium method

A new computational approach is proposed for determining the elastic field of deep curved beams/rings, which is based on a displacement-function equilibrium method. The displacement function is defined in terms of the radial and circumferential components of displacements, which is governed by a single partial differential equation of equilibrium. Solution of the elastic field is obtained in terms of the displacement function, in which both the uniform and the mixed mode of boundary conditions are treated with equal sophistication. The application of the single function computational method is demonstrated through the numerical solutions of a number of problems of curved beams and rings with mixed boundary conditions. The soundness and accuracy of the investigation are verified through the comparison of results with those obtained by conventional computational and analytical approaches available.

A Novel Method for Investigating the Casimir Effect on Pull-In Instability of Electrostatically Actuated Fully Clamped Rectangular Nano/Microplates

The objective of the present paper is to represent a novel method to investigate the stable and unstable behaviors of fully clamped rectangular nano/microplates under the effects of electrostatic and Casimir pressures. To this end, the governing partial differential equation of equilibrium is considered and reduced to an algebraic equation using a simple and computationally efficient single degree of freedom (SDOF) model through the Galerkin weighted residual method. The linear and undamped mode-shapes of the plate are used in the Galerkin procedure as the weight function which is obtained by the extended Kantorovich method (EKM). The present findings are compared and validated by available empirical and theoretical results in the literature as well as those obtained by finite element (FE) simulation carried out using COMSOL Multiphysics commercial software and excellent agreements between them are observed.

Strength and buckling of a sandwich beam with thin binding layers between faces and a metal foam core

The strength and buckling problem of a five layer sandwich beam under axial compression or bending is presented. Two faces of the beam are thin aluminium sheets and the core is made of aluminium foam. Between the faces and the core there are two thin binding glue layers. In the paper a mathematical model of the field of displacements, which includes a share effect and a bending moment, is presented. The system of partial differential equations of equilibrium for the five layer sandwich beam is derived on the basis of the principle of stationary total potential energy. The equations are analytically solved and the critical load is obtained. For comparison reasons a finite element model of the beam is formulated. For the case of bended beam the static analysis has been performed to obtain the stress distribution across the height of the beam. For the axially compressed beam the buckling analysis was carried out to determine the buckling load and buckling shape. Moreover, experimental investigations are carried out for two beams. The comparison of the results obtained in the analytical and numerical (FEM) analysis is shown in graphs and figures. The main aim of the paper is to present an analytical model of the five layer beam and to compare the results of the theoretical, numerical and experimental analyses.

The Influence of the Aspect Ratio on the Composite Material Bridge Deck Structures

Theories for composite material structures are too difficult for such design engineers for construction and some simple but accurate enough methods are necessary. The author has reported that some laminate orientations have decreasing values of <TEX>$D_{16}$</TEX>, <TEX>$B_{16}$</TEX>, <TEX>$D_{26}$</TEX> and <TEX>$B_{26}$</TEX> stiffnesses as the ply number increases. For such plates, the fiber orientations given above behave as specially orthotropic plates and simple formulas developed by the author. Most of the bridge deck structures on girders have large aspect ratios. For such cases further simplification is possible by neglecting the effect of the longitudinal moment terms <TEX>$M_x$</TEX> on the relevant partial differential equations of equilibrium. In this paper, the result of the study on the subject problem is presented.

面外せん断変形を考慮した異方性円筒殻の軸圧縮・ねじりおよび複合荷重に対する閉じた解

Simple, closed-form solutions, which can be easily calculated using spreadsheet software, for thin and moderately thick anisotropic cylinders under torsion, axial compression and these combined loads that includes the effect of transverse shear deformation are derived. Previously derived partial differential equations of equilibrium, which are including both layup anisotropy and transverse shear deformation, are separated two equation groups by neglecting coupling stiffnesses (tension-shear, bending-torsion and tension-torsion couplings). One is tension-bending equation group and the other is torsional equation group, and these equation groups are solved independently and the closed-form solutions are obtained. One of the two tension-bending solutions satisfies a simply supported condition, another a fully fixed condition and both of these can be applied to cylinders that are shorter and longer than the length of the bending boundary layer. In addition, it is shown that previously derived bending-boundary solution can be applied as the same form when the length of the cylinder is longer than the length of its bending boundary layer. Moreover, Donnell-type shallow-shell approximation is inappropriate for solving torsional deformation of moderately thick cylinder is also shown. Comparison between closed-form solution and precise solution including both layup anisotropy and transverse shear deformation shows that the difference can be negligible when cylinder is honeycomb-sandwich with symmetric facesheet but cannot be negligible when the symmetric cylinder has non-negligible tension-torsion coupling stiffnesses. Therefore, the estimation formula for the effect of tension-torsion coupling stiffnesses is also derived.

Stress Distribution at the Stiffened Edges of a Thick Curved Flat Bar

The present paper focuses on the determination of stress distribution at the stiffened edges of a thick curved flat bar using a new computational elasticity approach. The bar is rigidly fixed at its one end and the two opposing curved edges are subjected to circumferential stiffeners. The corresponding stress field of the curved bar is obtained from the finite-difference solution of an elliptic partial differential equation of equilibrium in polar coordinate system. The analysis is carried out for two different conditions of loading at the bar end. The effect of bar aspect ratio on the state of stresses at the stiffened edges is also included in the present analysis.

Analysis of stresses in a simply-supported composite beam with stiffened lateral ends using displacement-potential field

Abstract The elastic field of a stiffened simply-supported fiber-reinforced composite beam is investigated using an analytical scheme based on the displacement-potential field. A single fourth-order partial differential equation of equilibrium is solved to determine the potential field of the present mixed-boundary-value elastic problem, and the expressions of relevant displacement and stress components are derived in terms of the potential function using Fourier series. Solutions are obtained for two different types of stiffeners at the opposing lateral ends of the beam, where the fibers are assumed to be directed along the beam axis. The effect of beam aspect ratio is also investigated on the state of stresses at different critical sections of the beam. The soundness and accuracy of the present solutions are verified through a comparison made with the corresponding solutions of classical beam theory and standard computational method.