Three-dimensional Elasticity Equations Research Articles

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.

On the general homogenization of von Kármán plate equations from three-dimensional nonlinear elasticity

Starting from three-dimensional elasticity equations we derive the model of the homogenized von Kármán plate by means of [Formula: see text]-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic.

Three-dimensional elasticity solution of layered plates with viscoelastic interlayers

An analytical solution for simply supported layered plates with viscoelastic interlayers under a transverse load is proposed. The deformation of each plate layer is described by the exact three-dimensional elasticity equations. The viscoelastic property of interlayer is simulated by the generalized Maxwell model. The constitutive relation of the interlayer is simplified by the quasi-elastic approximation, which significantly simplifies the analytical process. The solution of stress and displacement fields with undetermined coefficients is derived by solving a group of ordinary differential equations. The undetermined coefficients can be efficiently deduced by using the recursive matrix technique for the plate with any number of layers. The practical convergence is observed during numerical tests. The comparison analysis indicates that the present solution has a close agreement with the finite element solution. However, the solution based on the Mindlin–Reissner hypothesis is significantly different from the present solution for thick plates. Finally, the effect of interlayer thickness on stress and displacement distributions of a five-layer plate is discussed in detail.

Accurate analytical solutions for shear-deformable web-core sandwich plates

An accurate discrete model and analytical solutions thereof are presented for shear-deformable web-core sandwich plates. The face-plates are analyzed using the equations of three-dimensional elasticity, while the webs are accurately modelled using the classical plate theory with a plane stress solution for transverse bending and a Levy-type methodology for lateral bending. It is shown that this obviates the need for a complete three-dimensional analysis of the sandwich plate. Results obtained by this approach are used to highlight the effect of shear deformation of the face-plates.

Polynomial approach for modeling a piezoelectric disc resonator partially covered with electrodes

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach.

Higher-order structural theories for the static analysis of doubly-curved laminated composite panels reinforced by curvilinear fibers

The main aim of this paper is the evaluation of the through-the-thickness profiles of strain, stress and displacement components of several doubly-curved panels reinforced by curvilinear fibers. The placement of the reinforcing phase along curved paths allows to obtain mechanical properties which change point by point and affects the static behavior of shell structures. Some numerical applications based on both higher-order Equivalent Single Layer (ESL) and Layer-Wise (LW) theories are shown in order to underline the curvilinear fiber influence on the static analysis. The structural model, which is based on the so-called Carrera Unified Formulation (CUF), is completely general and can deal easily with variable stiffness shells. An appropriate recovery procedure based on the three-dimensional elasticity equations in principal curvilinear coordinates is presented to compute strains and stresses. The equation system which governs the static problem under consideration is solved numerically through the Generalized Differential Quadrature (GDQ) method. The same numerical technique is employed to evaluate the geometrical parameters needed for the characterization of the shell reference surface, according to the differential geometry.

The effect of surface stresses on the stress–strain state of shells

The effect of surface stresses on the stress–strain state (SSS) of an elastic shell is investigated. The surface stresses are represented as a static pre-loading localized in ultra-thin layers of the shell close to its surface. It is assumed that the mechanical properties of the surface layers differ from the properties of the material far from the surface. The three-dimensional elasticity equations are analysed by an asymptotic method using several asymptotic parameters. Equations are obtained by the method of reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of shell theory in which surface stresses have to be taken into account for a sufficiently small shell thickness. Asymptotic estimates of the effect of surface stresses on the SSS are obtained as a function of the ratio of the elastic moduli of the shell material and of the layer close to the surface, the ratio of the shell thickness to the thickness of the surface layer and the type of SSS and its variability with respect to the coordinates.

A Highly Scalable Multilevel Schwarz Method with Boundary Geometry Preserving Coarse Spaces for 3D Elasticity Problems on Domains with Complex Geometry

We consider overlapping Schwarz algorithms for solving linear and nonlinear systems of equations arising from the finite element discretization of elasticity problems on unstructured meshes in three dimensions. The parallel scalability of Schwarz methods is determined almost completely by how the coarse space is constructed. In this paper, we introduce a low cost, boundary geometry preserving coarse mesh that shares the same boundary geometry with the fine mesh but has a better scalability than the coarse mesh obtained by uniformly coarsening the fine mesh. A new coarsening algorithm and a partitioning strategy are developed. We numerically show that a multilevel Schwarz method with the new coarse spaces is highly scalable, in terms of the total compute time, for solving some three-dimensional linear and nonlinear elasticity equations discretized on unstructured meshes with several hundreds of millions of unknowns and on a supercomputer with over 10,000 processor cores.

High-accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems

We investigate the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the three-dimensional axisymmetric elasticity equations with Dirichlet’s problems. The convergence of the MQMs is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion of the MQMs shows that the approximation order is of O(h3). Numerical examples are tested and the results verify the theoretical analysis.

Refined Zigzag Theory for laminated composite and sandwich plates derived from Reissner’s Mixed Variational Theorem

A mixed-field Refined Zigzag Theory (RZT(m)) for laminated plates is presented. The theory is developed using Reissner’s Mixed Variational Theorem (RMVT) and employs the kinematic assumptions of the displacement-based Refined Zigzag Theory (RZT). In addition, a robust set of assumed transverse-shear stresses is implemented. The stresses, initially derived by integration of the three-dimensional elasticity equations, satisfy a priori the continuity conditions along the layer interfaces and on the bounding surfaces. With the aid of the strain-compatibility variational statement of RMVT, the transverse-shear stresses are expressed in terms of first-order derivatives of the kinematic variables. The RZT(m) retains a fixed number of kinematic variables (seven) regardless of the number of material layers. To ascertain the importance of transverse-shear stress assumptions, the layer-wise polynomial approximation scheme is also implemented. Numerical results concerning the elasto-static and vibration problems of simply supported and clamped plates, demonstrate that RZT(m) is more accurate than RZT, both in terms of local and global responses. These results also reveal that the transverse-shear stresses achieved by a layer-wise polynomial scheme are considerably less accurate, particularly for highly heterogeneous laminates. Furthermore, the RZT(m) is well suited for developing C0-continuous finite elements, thus resulting attractive for large-scale analysis of laminated structures.

Resonances of a submerged fluid-filled spherically isotropic microsphere with partial-slip interface condition

Motivated by the numerous applications of spherical shell models in micro and nano scales (such as microbubbles, bacterial cells, and viral capsids), we have considered the axisymmetric free vibrations of a spherically isotropic fluid-filled thick microspherical shell suspended in another unbounded fluid. A partial-slip condition is considered at the solid-fluid interface(s). Three-dimensional linear elasticity equations for the spherically isotropic shell dynamics and linearized Navier-Stokes equations for the two compressible viscous fluids are used in the analysis. The eigenvalue problem is discretized and solved to find the resonances and quality factors. A perfectly matched layer technique is used to separate the solid driven spectrum from the boundary reflecting spectrum. An example of air filled polymer shell suspended in water is presented. The added mass effect and partial-slip condition from water (air) on the frequencies and quality factors are found to be significant (negligible). Spherical isotropy is found to have major influence on the low frequency and large meridional wave number region of the resonance spectrum. High quality eigenmodes are observed due to very small viscous penetration depth compared to the shell size. In the thin-shell limit, the eigenvalue problem can have only two modes of vibration for any meridional wave number greater than or equal to two. This explains the reason for the second resonance frequency found for the quadrupole shape oscillations of various bacterium cells in the earlier work. The partial-slip condition is found to have very small influence on the first few modes of vibration. Surface tension is found to have significant influence only on the lowest frequency trend of the eigenspectrum. Perfectly matched layer technique used in the present analysis is found to be very effective in handling the boundary truncated problems.

Continuum-based stability of anisotropic and auxetic beams

The critical buckling loads of pinned-pinned and cantilever beams are computed using the equations of three-dimensional elasticity rather than typical beam theories. These loads are influenced both by the nature of the assumed displacement field over the beam cross-section and by the inclusion of the terms from the full constitutive tensor. Of special interest are beams that are either anisotropic or auxetic. For anisotropic beams, an increased ratio of longitudinal to shear modulus for cantilevered beams increases the generation of shear buckling rather than flexural buckling. For isotropic auxetic beams, the values of Poisson ratio that define the limit between buckling loads that approach the classical buckling load from above or below are discussed.

Extensional Waves in a Sandwich Plate With Interface Slip

The two-dimensional (2D) equations for thin elastic plates are used to study extensional motions of a sandwich plate with weak interfaces. The interfaces are governed by the shear-slip model that possesses interface elasticity and allows for a discontinuity of the tangential displacements at the interfaces. Equations for the individual layers of the sandwich plate are coupled by the interface conditions. Through a procedure initiated by Mindlin, the layer equations can be written into equations for the collective motion of the layers representing the extensional motion of the sandwich plate, and equations for the relative motions of the layers with respect to each other representing the symmetric thickness-shear motion of the sandwich plate. The use of plate equations results in relatively simpler models compared to the equations of three-dimensional (3D) elasticity. Solutions to a few useful problems are presented. These include the propagation of straight-crested waves in an unbounded plate with weak interfaces, the reflection of extensional waves at the joint between a perfectly bonded sandwich plate and a sandwich plate with weak interfaces, and the vibration of a finite sandwich plate with weak interfaces.

Three-dimensional wave propagation on orthotropic cylindrical shells with arbitrary thickness considering state space method

This work presents an analytical solution based on the three-dimensional exact equations of anisotropic elasticity for sound transmission through orthotropic cylindrical shells with arbitrary thickness using a laminate approximate model and a state space method along with the transfer matrix approach. The shell is assumed to be infinitely long and is subjected to an oblique plane wave with uniform external airflow in the external fluid medium. The developed equations are applied to the calculation of the diffuse field transmission of a cylinder. Predictions with the presented models are compared with those of previous models for thin shells. Similar results are observed due to restricted effects of shear and rotation on transmission loss (TL). However, the presented model demonstrates more accurate results for thick shells as the shear and rotation effects become more significant in lower R/h ratio. Additionally, the effects of related parameters on transmission loss such as material and geometrical properties are discussed.

On the occurrence of lumped forces at corners in classical plate theories: a physically based interpretation

The paradigmatic example of a twisted square plate is here considered. An equivalent partition of the plate in a grid of beams a la Grashof is found such that, as the number of beams tends to infinity, the grid exhibits the same deflection of the plate. This scheme is used to interpret, through the distinction between Euler‐Bernoulli and Timoshenko beam theories, the different types of natural boundary conditions that can arise in the Kirchhoff‐Love and Mindlin‐Reissner theories of plates. A physically based interpretation for the occurrence of lumped forces at the plate corners through the formation of a boundary layer is provided. It is well known that the solution of the biharmonic equation governing the bending of plates in Kirchhoff‐ Love theory is compatible with only two distinct conditions at each boundary point, whereas in general three boundary data can be independently assigned on an unconstrained border. This contradiction for the order of the equation is a two-hundred-year-old problem. The paradox arose when the three-boundarydata statement by Poisson [1829] was criticized by Kirchhoff [1850], who obtained only two natural conditions at the border within a variational framework, using a static equivalence sometimes referred to as the “Kirchhoff transformation” [Vasil’ev 2012]. This result arose from the first variation of the energy functional, but it was not corroborated by any physically based interpretation. A long discussion ensued among the most eminent scientists of the period with the purpose of reconciling the Poisson and Kirchhoff theories. The dispute culminated with the elementary interpretation by Thomson and Tait [1883], who showed how to reduce the torque per unit length on the contour to a shear transverse force. Friedrichs and Dressler [1961] and Gol’Denveiser and Kolos [1965] have independently shown that the plate theory is the leading term of the expansion solution (in a small thickness parameter) for the linear elastostatics of thin, flat, isotropic bodies. As expected, this leading term alone is unable to satisfy arbitrarily prescribed edge conditions. There has been a renewed interest during the last years in the fundamental problem of understanding the relationship between the three-dimensional elasticity theory and theories for lower-dimensional objects (plates, shells, rods). Due to the availability of sophisticated methods of variational convergence [Ciarlet 1997], important achievements have been obtained by showing that various theories of plates arise as a rigorous variational limit (or 0-limit) of the equations of three-dimensional elasticity as the

Acoustic scattering from submerged laminated composite cylindrical shells

The scattering of an oblique plane progressive monochromatic acoustic field upon a laminated composite cylindrical shell is studied based on the three-dimensional exact equations of anisotropic elasticity. Each layer of laminated structure is made of helically filament wound (fiber reinforced) homogeneous material whose degree of anisotropy is considered as monoclinic type. An approximate laminate model along with the local transfer matrix solution is used to solve the state space governing formulation within each layer. Considering the perfect bonding between the adjacent layers, the global transfer matrix is constructed as the product of the local transfer matrices employed to solve for the unknown scattering coefficients. The so-called back-scattered form function amplitude is evaluated for two cases of (strong anisotropy) Graphite/Epoxy and (weak anisotropy) Glass/Epoxy multilayered cylindrical shells, for different number of layers and different stacking sequence. Furthermore, the axisymmetric and non-axisymmetric dynamic behavior of the composite shells are studied by evaluating the directivity patterns of the far-field form function amplitude at the selected resonance frequency corresponding to the first overtone of n=2 vibration mode. Some limiting cases are considered and good agreements with the solutions available in the literature are obtained.

Stiffness Constants of Homogeneous, Anisotropic, Prismatic Beams

This paper presents a complete set of analytical expressions for the stiffness constants of a generalized Euler–Bernoulli beam theory for homogeneous, anisotropic, prismatic beams with arbitrary cross-sectional shapes. These expressions are extracted from exact solutions of the linear equations of three-dimensional elasticity for the cases of loading by axial forces, torques, and bending moments about two orthogonal directions. Closed-form expressions are derived for the extensional stiffness and the extension-related coupling terms. Expressions for the remaining stiffness constants are derived in terms of the torsional stiffness: the expression of which is in terms of a function that needs to be obtained. The resulting expressions reveal both similarities and differences from its isotropic and orthotropic counterparts. For elliptical, anisotropic cross sections and rectangular, orthotropic cross sections, all stiffness constants are known in closed form. These closed-form expressions constitute a standard with which the ability of two-dimensional beam cross-sectional analyses to model material anisotropy may be assessed. The calculated stiffness constants, from one such cross-sectional analysis, are successfully validated in this manner.

Gradient index lenses for flexural waves based on thickness variations

This work presents a method for the realization of gradient index devices for flexural waves in thin plates. Unlike recent approaches based on phononic crystals, the present approach is based on the thickness-dependence of the dispersion relation of flexural waves, which is used to create gradient index devices by means of local variations of the plate's thickness. Numerical simulations of known circularly symmetrical gradient index lenses have been performed. These simulations have been done using the multilayer multiple scattering method and the results prove their broadband efficiency and omnidirectional properties. Finally, finite element simulations employing the full three-dimensional elasticity equations also support the validity of the designed approach.

Two multilayered plate models with transverse shear warping functions issued from three dimensional elasticity equations

A multilayered plate theory which uses transverse shear warping functions is presented. Two methods to obtain the transverse shear warping functions from three-dimensional elasticity equations are proposed. The warping functions are issued from the variations of transverse shear stresses computed at specific points of a simply supported plate. The first method considers an exact 3D solution of the problem. The second method uses the solution provided by the model itself: the transverse shear stresses are computed integrating equilibrium equations. Hence, an iterative process is applied, the model is updated with the new warping functions, and so on. Once the sets of warping functions are obtained, the stiffness and mass matrices of the models are computed. These two models are compared to other models and to analytical solutions for the bending of simply supported plates. Four different laminates and a sandwich plate are considered. Their length-to-thickness ratios vary from 2 to 100. An additional analytical solution that simulates the behavior of laminates under the plane stress hypothesis – shared by all the considered models – is computed. Both presented models give results very close to this exact solution, for all laminates and all length-to-thickness ratios.

Interaction of a plane progressive sound wave with anisotropic cylindrical shells

An exact analysis based on the wave function expansion is carried out to study the scattering of a plane harmonic acoustic wave incident at an arbitrary angle upon an arbitrarily thick helically filament-wound (anisotropic) cylindrical shell submerged in and filled with compressible ideal fluids. Using the laminated approximation method, a modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their cylindrical harmonics to present an analytical solution based on the three-dimensional exact equations of anisotropic elasticity. Taylor’s expansion theorem is then employed to obtain the solution to the modal state equation, ultimately leading to calculation of a transfer matrix. Following the classic acoustic resonance scattering theory (RST), the scattered field and response to surface waves are determined by constructing the partial waves and obtaining the background (non-resonance) and resonance components from it. The solution is particularly used for the isolation and identification of excited resonances of an air-filled and water submerged Graphite/Epoxy cylindrical shell as the circumnavigating helically propagating waves. In addition, the sensitivity of resonances associated with various modes of wave propagation appearing in the backscattered amplitude to the perturbation in the material’s elastic constants is examined. Furthermore, non-axisymmetric dynamic behavior of the anisotropic shell is illustrated by analyzing the directivity pattern associated to the angular distribution of the far-field form function amplitude. The effects of winding angle of filaments and the shell wall-thickness on the frequency response of the shell are also investigated. For verification, the wave propagation characteristics of the anisotropic shell (which have been extracted from the main body of the solution) and the far-field form function amplitude of a limiting case are considered and fair agreement with the solutions available in the literature are established.