Three-dimensional Elasticity Equations Research Articles

Numerical models of earthquake mechanism

The tectonic creep and its variation after particular earthquakes are studied by the Stokes equation. The stress state of the region is modelled according to a hypothesis of plate tectonics in which the lithosphere of the region is laterally compressed across the Eurasian, Indian, and Arabian plates. The 1966 Tashkent (Uzbekistan) earthquake and the 1976 Gazli (Uzbekistan) earthquake are selected as examples to study different models of earthquake focal mechanisms. Based on the specifics of the geodynamic formulation, the three-dimensional equations of moment elasticity and hydromechanics are reduced to two-dimensional equations for averaged stresses, displacements, and displacement velocities. The two-dimensional equations are solved by boundary integral equations. The stresses can be useful in zoning maps. The vertical velocities obtained from the creep model of the earth's crust can serve as additional data to Central Asia's horizontal velocities from GPS measurements.

Static and dynamic responses of simply supported sandwich plates using non-polynomial zigzag theory

The static, free vibration and transient responses of simply-supported laminated composites and sandwich plates are analytically derived using a non-polynomial zigzag theory. This mathematical model is displacement-based with five field variables and uses a non-polynomial mathematical function to introduce the non-linearity of transverse shear stresses through-thickness, thus eliminating the use of a large number of higher-order terms in the kinematic field. The zigzag kinematics is implemented by the inclusion of auxiliary variables defined at the interfaces of the plates with a piecewise linear function of the thickness coordinate. The dynamic governing equations of equilibrium are derived with Hamilton’s principle that generates five coupled partial differential equations. The Navier’s solution technique with the Newmark’s time integration scheme is adopted to find the solutions of the coupled equations. The responses in the form of displacements, stresses, natural frequencies, and displacement–time responses are obtained by solving several standard problems in the literature. The transverse shear stresses are accurately obtained by integrating the three-dimensional equilibrium equations of elasticity. The forced-vibration analysis of the laminated composites and sandwich plates are presented in detail by considering various forms of time-dependent loads that also includes the various forms of blast loads. The present results agree closely with the published results in the literature for both laminated composites and sandwich plates.

Flexural Behavior of Functionally Graded Plates with Piezoelectric Materials

In the perspective of significant industrial applications and research works being conducted on functionally graded materials (FGMs), the present article seeks to demonstrate the flexural behavior of smart functionally graded plate with the use of recently developed inverse hyperbolic shear deformation theory. The kinematic field of this theory consists of five independent primary variables and inherently satisfies the traction-free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate. The piezoelectric fiber-reinforced composite (PFRC) lamina is considered as the actuator of the smart structure. The Navier’s solution technique is used to convert the system of partial differential equations derived from the virtual work principle to a set of five algebraic equations. The response of the smart plate is determined in the form of deflection and stresses subjected to electromechanical load. The three-dimensional equilibrium equations of elasticity are used to accurately determine the transverse shear stresses. The responses obtained from the numerical examples of FGM plate integrated with PFRC layer are compared with the elasticity solutions and with the various kinematic models available in the existing literature. The actuation in the response of the smart FGM plate due to the PFRC layer is also presented in the form of graphs for verifying the controlling capacity.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Application of Carrera unified formulation in conjunction with finite strip method in static and stability analysis of functionally graded plates

This paper presents the static and mechanical buckling analyses of thick functionally graded (FG) plates. For this purpose, Carrera’s unified formulation (CUF) and the principle of virtual displacement are employed in the numerical finite strip method (FSM). CUF transforms the governing three-dimensional (3D) elasticity equations to quasi-3D ones by employing a set of thickness functions. Since this formulation is capable of considering the effects of shear deformations in a realistic manner, it is suitable for analyzing the structures in which these deformations play a major role and cannot be ignored. Another major advantage of CUF is that the governing equations are expressed in terms of a few fundamental nuclei, which are independent of the order of thickness functions used in the transverse direction and the description of the equivalent single layer or layer-wise variables. The accuracy of the proposed method is evaluated by comparing the obtained results with those available in the literature. The effect of various parameters such as boundary conditions, aspect ratio, plate thickness, and material distribution across the thickness of the plate are investigated.

Virtual element method (VEM)-based topology optimization: an integrated framework

We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for convenient handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method, the VEM does not require numerical integration (when linear elements are used) and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using the VEM basis functions. When compared to the standard element-wise constant approximation, the continuous approximation enriches the geometrical representation of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, is available in the (electronic) Supplementary Material accompanying this publication.

Three-dimensional dynamics of functionally graded and laminated doubly-curved composite structures having arbitrary geometries and boundary conditions

This paper focuses on the dynamics of doubly-curved functionally graded and laminated composite structures with arbitrary geometries and boundary conditions. Integral boundary value problem is obtained following an energy-based approach where the strain energy of the structure is expressed using three-dimensional elasticity equations. The effective properties of functionally graded materials can be described based on Mori–Tanaka or theory of mixtures methods. To simplify the domain of the problem, coordinate transformations are applied to map the curved structure into a straight one; and furthermore, a one-to-one mapping technique is applied to map the (complex) curved geometry to a master geometry in the case of composites with arbitrary geometries. Then, the integral boundary value problem is discretized by means of Gauss–Lobatto sampling and solved using the three-dimensional spectral-Tchebychev approach. In this method, the system matrices are calculated through the exact evaluation of differentiation and integration operations using the derived Tchebychev matrix operators. Finally, if necessary, to impose the essential boundary conditions on the boundary value problem and to assemble multiple layers, the projection matrices approach is used. Various case studies including (i) doubly-curved structures, (ii) doubly-curved laminated composites and (iii) doubly-curved laminated composite structures with arbitrary geometries are analyzed. In each case study, to present the accuracy/precision of the developed solution technique, the predicted (non-dimensional) natural frequencies and mode shapes are compared to those obtained using either a commercial finite element software and/or to those found in literature. It is shown that the developed three-dimensional spectral-Tchebychev solution technique enables accurately and efficiently capturing the vibration behavior of doubly-curved laminated composite structures having arbitrary geometries under different boundary conditions.

A computational approach to obtain nonlinearly elastic constitutive relations of special Cosserat rods

We present a computational framework to obtain nonlinearly elastic constitutive relations of one-dimensional continua modeled as special Cosserat rods. The kinematics of the recently proposed Helical Cauchy–Born rule is used to construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations which are subjected to uniform strain field along their arc-length. This uniformity along the rod’s arc-length results in the reduction of three-dimensional equations of elasticity to just the rod’s cross-section which further allows us to obtain the induced force, moment and stiffnesses of the rod by solving a nonlinear cross-sectional warping problem for every state of strain. The formulation is general in that the rod’s material could obey arbitrary three-dimensional hyperelastic constitutive relations. A nonlinear finite element formulation is presented to solve the cross-sectional warping problem and further obtain the induced force, moment and stiffnesses numerically. Several numerical examples are presented illustrating warping due to bending, shearing and torsion in rectangular as well as circular rods and how the warping affects stiffnesses. We also obtain all the stiffnesses of helically reinforced tubes and show the variation in their stiffnesses with the tube’s fiber angle.

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

A higher order theory for functionally graded shells

New theory of higher order for functionally graded (FG) shells, which is based on the expansion of the three-dimensional (3D) equations of elasticity for functionally graded materials (FGMs) into Legendre’s polynomials series is developed here. The stress and strain tensors, the displacement, traction and body force vectors of the 3D equations of elasticity, are expanded into Legendre’s polynomials series in terms of in the thickness coordinate. The mechanical parameters that describe the functionally graded material properties are also represented in the form of Legendre’s polynomials series expansion. As result the equations of the 3D elasticity are turned into the infinite number of two-dimensional (2D) equations for the Legendre’s polynomials series expansion coefficients. Considering finite number of the Legendre’s polynomials series coefficients and substituting kinematic relations into generalized Hooke’s law and the obtained result into the equations of motion the differential equations of motion the equations of motion in form of displacements have been obtained. The first order equations for the FG axisymmetric cylindrical plate and spherical shell are considered in more details. Corresponding boundary-value problems are solved using the finite element method (FEM) implemented in the MATEMATICA software. The numerical results are presented and discussed.

An asymptotic membrane model for wrinkling of very thin films

In this work, a formal deduction of a two-dimensional membrane theory, similar to Landau–Lifshitz model, is performed via an asymptotic development of the weak formulation of the three-dimensional equations of elasticity. Some interesting aspects of the deduced model are investigated, in particular the property of obtaining a hyperbolic equation for the out-of-plane displacement under a certain class of boundary conditions and loads. Some simple cases are analyzed to show the relevant aspects of the model and the phenomenology that can be addressed. In particular, it is shown how this mathematical formulation is capable to describe instabilities well known as wrinkling, often observed for the buckling of very thin membranes.

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material properties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England’s method, the problem can be solved by determining the expressions of four analytic functions. Expanding the transverse load in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.

Three-dimensional vibration behavior of bi-directional functionally graded curved parallelepipeds using spectral Tchebychev approach

This paper presents the application of the three-dimensional spectral-Tchebychev technique to accurately predict the vibration behavior of bi-directional functionally graded material curved parallelepipeds including geometries such as beams, thin/thick plates, and solids. In this study, the material distribution within the domain of the structure is obtained using bi-directional Mori-Tanaka method. To derive the boundary value problem governing the dynamics of functionally graded curved parallelepipeds, three-dimensional elasticity equations are used together with extended Hamilton’s principle. Numerical solution of the integral boundary value problem is performed using the three-dimensional spectral Tchebychev approach. To validate and assess the performance of the presented solution approach, a number of case studies are conducted. In each case study, the non-dimensional natural frequencies and mode shapes are calculated and compared to those found using a finite element solution approach. Furthermore, computational time of the simulation is measured in each case. It is shown that the presented solution technique enables accurate prediction of vibration behavior of bi-directional functionally graded curved parallelepipeds as precise as a finite elements method, but for a fraction of the computational cost.

THE REFINED THEORY OF ELASTIC PLATES AS ASYMPTOTIC APPROACH OF 3D PROBLEM

The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.

Dispersion Properties of Lamb Waves in an Elastic Layer–Ideal Liquid Half-Space System

The problem of the propagation of normal waves in an elastic layer interacting with a compressible ideal liquid half-space is stated and solved using the three-dimensional linear equations of classical elasticity for the solid and the three-dimensional linearized Euler equations for the compressible ideal fluid. The dispersion curves are plotted for quasi-Lamb modes over a wide frequency range. The effect of the ideal compressible fluid and the elastic layer thickness on the dispersion of the phase velocities of quasi-Lamb modes in hydroelastic waveguides is analyzed. The localization properties of these modes in hydroelastic waveguides are studied. The numerical results are presented in the form of plots and analyzed.

Shear Stiffness of Homogeneous, Orthotropic, Prismatic Beams

Exact analytical expressions for shear stiffness constants are provided herein for homogeneous, orthotropic, prismatic beams with arbitrary cross-sectional shapes. The analytical expressions are extracted, without invoking any approximations, from exact solutions of the linear equations of three-dimensional elasticity for flexure. Owing to the elasticity solutions being in terms of a stress function for cross sections with arbitrary shapes, the shear stiffness constants are also provided in terms of the same stress function. For elliptical and rectangular cross sections, this stress function is known so that shear stiffness constants are provided in closed form. These closed-form expressions constitute a standard with which the accuracy of two-dimensional beam cross-sectional analyses to calculate shear stiffness in the presence of orthotropic materials may be assessed. The calculated shear stiffness constants, from two such cross-sectional analyses, are successfully validated in this manner.

A non-local asymptotic theory for thin elastic plates

The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.

Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models

In this paper, we provide alternative Uflyand–Mindlin's plate equations taking into account rotary inertia and shear deformation, based on both asymptotic expansion and variational arguments. The aim is to derive truncated versions of Uflyand–Mindlin's equations, specifically without the fourth order derivative term with respect to time. The truncated version of Uflyand–Mindlin's plate model may be derived starting from three-dimensional elasticity equations, by using asymptotic arguments based on expansion of displacements with respect to a small geometrical parameter. This expansion method also leads to a proper identification of the shear correction factor. It is shown that suitably modified variational derivation leads to an additional term which is shown to be negligible for determination of the fundamental natural frequency of the all-round simply supported plates, but may contribute significantly in estimation of higher natural frequencies. It is argued that the proposed version of Uflyand–Mindlin's plate equations is simpler and more consistent than the original Uflyand–Mindlin equations. Likewise, it is advantageous over the equation that stems from neglecting the fourth order time derivative in original Uflyand–Mindlin equations. The two alternative truncated models serve as intermediate theories between the classical plate theory and the original Uflyand–Mindlin theory their usefulness depending on the problem at hand.

Wave propagation across double-walled laminated composite cylindrical shells along with air-gap using three-dimensional theory

This paper presents a complete model to calculate the sound transmission loss across laminated composite cylindrical shells based on three-dimensional equations of anisotropic elasticity. This model is included a double-walled laminated composite cylindrical shell with infinite length along with an air-gap impinged upon incidence oblique plane wave and immersed in a fluid. Therefore, the equations of motion are derived for each monoclinic anisotropic layer of both walls of double-walled laminated composite cylindrical shell; then, these equations are rewritten using the state space method. Hence, the state space governing formulation of each layer of both walls has been solved using the approximate laminate model along with the local transfer matrix approach. Finally, using the global transfer matrix method and considering the appropriate boundary conditions the transmission loss (TL) of the double-walled laminated composite cylindrical shell is calculated. Comparison of the presented results with those of other researchers for both single-walled and double-walled cylindrical shells indicates the accuracy and validity of the present study. Moreover, the effects of each parameter on TL are investigated. The results show the TLs of three-dimensional theory is of higher accuracy rather than other theories, because it doesn’t consider any assumptions in simplifying the governing equations.

3-D exact solution of two-layer plate bonded by a viscoelastic interlayer with memory effect

An exact solution for simply supported two-layer plate sandwiching a thin viscoelastic interlayer under transverse load is proposed. The deformation of each plate layer is described by the exact three-dimensional linear elasticity equations. The viscoelastic characteristic of interlayer is simulated by the standard linear solid model, considering the memory effect of the material. The shear and transverse normal strains in the interlayer are simultaneously considered in the analysis. The analytical approach is presented to obtain the solution of stresses and displacements in the plate. The present solution can be used as a benchmark to access other simplified solutions for the problem aforementioned. The comparison analysis shows that the finite element solution has a close agreement with the present solution. However, the solution based on Kirchhoff-Love hypothesis is greatly different from the present solution for thick plates. It is indicated that the effect of the transverse normal strain in the interlayer increases with the increase of the interlayer thickness, especially for thick plates. Finally, the influences of geometric and material parameters on the plate deflections are investigated in detail.