Elastic Equation Research Articles

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

A new lattice Boltzmann scheme for linear elastic solids: periodic problems

We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The novel scheme is systematically derived using the asymptotic expansion technique and a detailed analysis of the leading-order error behavior is provided. As demonstrated by a linear stability analysis, the method is stable for a very large range of Poisson’s ratios. We consider periodic problems to focus on the governing equations and rule out the influence of boundary conditions. The analytical derivations are verified by numerical experiments and convergence studies.

The effect of axial boundary conditions on breakout noise from finite cylindrical ducts

Ductborne noise within HVAC and exhaust systems may transfer into the surrounding environment through the walls of the duct. This breakout noise normally needs to be lowered in order to meet health and safety guidelines. This means that the ability of the duct walls to lower breakout noise needs to be predicted during the design stage of the duct systems. Significant work has been conducted into the prediction of breakout noise for infinite length ducts, however there is little to be found on finite length ducts. In particular there are very few studies on the difference between finite and infinite length duct breakout, especially where the noise source lies within the internal fluid. The aim of this work is therefore to investigate the difference in breakout noise between finite length and equivalent infinite length ducts when excited by an internal noise source. This is performed through numerical experiments using the semi analytical finite element method which enables the equations of elasticity for the duct wall, as well as a surrounding fluid, to be accommodated. Axial continuity equations at each end of a finite length elastic duct are then enforced through the point collocation method. It is observed that high sound power levels are emitted at axial resonances of the finite length duct. In the low frequency region these resonances have a narrow bandwidth and are expected to be of little practical significance. However, above the critical frequency the resonance bandwidth increases and this is observed to significantly lower the transverse transmission loss of the duct wall when compared to an infinite length duct. This phenomenon is observed for both clamped and simply supported ducts, as well as for two different internal sound sources. It is concluded that breakout noise from axial resonances in finite length ducts should be examined in design calculations in order to avoid excessive breakout noise.

Determination of three-dimensional thermo-mechanical behavior of bidirectional functionally graded rectangular plates with the theory of elasticity

In this study, the thermo-mechanical behavior of rectangular plates functionally graded (FG) in two directions along the plane is investigated with three-dimensional (3D) elasticity equations. In numerical analysis, the finite difference method (FDM) is used and the directional variation of the derivatives of the material properties is taken into account. The thermo-mechanical behavior of the plates is investigated for different compositional gradient exponents and it is emphasized that the variation of the compositional gradient exponent in both directions significantly changed the thermal stress distribution and the results are confirmed.

Thermomagnetic behavior of a semiconductor material heated by pulsed excitation based on the fourth-order MGT photothermal model

This article proposes a photothermal model to reveal the thermo-magneto-mechanical properties of semiconductor materials, including coupled diffusion equations for thermal conductivity, elasticity, and excess carrier density. The proposed model is developed to account for the optical heating that occurs through the semiconductor medium. The Moore–Gibson–Thompson (MGT) equation of the fourth-order serves as the theoretical framework to establish the photothermal model. It is well-known that the optical and heat transfer properties of such materials behave as random functions of photoexcited-carrier density; therefore, the current model is remarkably more reliable compared to the earlier closed-form theories which are limited to a single form. The constructed theoretical framework is able to investigate the magneto-photo-thermoelastic problems in a semiconductor medium due to laser pulse excitation as a case study. Some parametric studies are used to exhibit the impact of thermal parameters, electromagnetic fields, laser pulses and thermoelectric coupling factors on the thermomagnetic behavior of physical variables. Finally, several numerical examples have been presented to draw the distributions of the examined field variables.

An elasticity solution of laminated cylindrical panel

In this paper, an exact elasticity solution of cylindrical panel under cylindrical bending is presented. The formulation is based on the displacement approach in which 3D elasticity equations are reduced to 2-D equations using plane-strain conditions. In the present formulation, an exact elasticity solution is feasible due to the consideration of simply supported boundary conditions with applied loads expressed in harmonic forms. Single layered and multi-layered orthotropic composite panels subjected to realistic transverse normal loads are analysed. Solutions have been presented for uni-directional, bi-directional, and symmetric three-layered cross-ply laminated composite cylindrical panels. Variation of displacements and stresses through the thickness has been investigated for the panel subjected to single sinusoidal load, uniformly distributed load, hydrostatic load, and strip load. Validation of results is presented using reference results in the literature. The extensive results presented in this paper can be served as benchmark solutions for the assessment of improved panel theories.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

Global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law

AbstractIn this paper, we consider the global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law. First, we will give the representation of the traveling wave solution to the recast hyperbolic system. Then, by using the weighted function method, we obtain the global stability of longitudinal wave solution for nonlinear elastic string equation.

On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation

A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.

Study on the Thermohydrodynamic Friction Characteristics of Surface-Textured Valve Plate of Axial Piston Pumps.

The purpose of this paper is to study the oil film and friction characteristics of valve plates with a micro-textured surface and to explore the influence of textures of different shapes and sizes on the valve plates. Firstly, on the basis of thermohydrodynamic theory, this paper established the lubrication model of the oil film on the valve plate pair of swashplate axial piston pumps, according to the Reynolds equation. Secondly, the micro-texture was added to the mathematical model of the valve plate pair's oil film. A combination of the energy equation, oil-film-thickness equation, elastic deformation equation, viscosity-pressure and viscosity-temperature equation, the finite difference method, as well as the relaxation iteration method, was used to solve the problem, and the textured and non-textured valve plate surfaces were simulated. The nephogram of the oil-film-thickness distribution, elastic deformation distribution, oil-film-pressure distribution and oil-film-temperature distribution were generated. Then, the control variable method was used to change the cylinder rotational speed, tilt angle, oil viscosity, initial oil film thickness and other parameters to analyze their effects on oil film characteristics. In addition, the friction characteristics of non-textured surfaces, square textured surfaces, triangular textured surfaces and circular textured surfaces were compared and analyzed. It was found that the textured surface of valve plates can obviously improve friction efficiency under the same operating conditions. The square texture, especially, is the preferable shape, rather than the triangular texture and the circular texture, and the friction performance is at its best when the texture depths are between 20 μm and 50 μm. The results provide a theoretical basis for the design and improvement of the valve plate.

Athermal resistance to phase interface motion due to precipitates: A phase field study

Athermal resistance to the motion of a phase interface due to a precipitate is investigated. The coupled phase field and elasticity equations are solved for the phase transformation (PT). The volumetric misfit strain within the precipitate is included using the error and rectangular functions. Due to the presence of precipitates, the critical thermal driving forces (athermal friction) remarkably differ between the direct and reverse PTs, resulting in a hysteresis behavior. For many cases, the critical thermal driving force increases like cx, x=0.5–0.6, vs. the precipitate concentration c for both the direct and reverse PTs. This is similar to c0.5 for the known effect of solute atoms on the athermal friction, which are also dilation centers, but without surface energy. Change (40% reduction) in the precipitate surface energy during PT significantly changes the PT morphology and the critical thermal driving forces. For the precipitate radius small compared to the interface width, the misfit strain does not practically show any effect on the critical thermal driving force. In the opposite case, for both the constant surface energy (CSE) and variable surface energy (VSE) boundary conditions (BCs) at the precipitate surface, the critical thermal driving force linearly increases vs. the misfit strain for the direct PT while it is almost independent of it for the reverse PT. For any concentration, the VSE BCs result in higher thermal critical driving forces, but a smaller hysteresis range, and a larger transformation rate. The obtained critical microstructure and thermal driving forces are validated using the thermodynamic phase equilibrium condition for stationary interfaces. Increase in the interface width reduces the interphase friction. After neglecting misfit and transformation strain and change in surface energy, our simulations describe well the Zener pinning pressure for the grain boundary. The obtained results give an important generic understanding of athermal friction mechanism for phase interfaces for various PTs at the nanoscale.

On the natural frequency investigation of the thick hollow cylinder with 2D-FGM Mori-Tanaka scheme

ABSTRACT In this paper, we conduct the natural frequency analysis for a 2D-FGM (functionally graded material) finite-length, thick-walled cylinder by the graded finite element method. The material properties gradations are considered along with the thickness and the longitudinal axis of the cylinder according to the Mori-Tanaka scheme and power-law volume fraction distributions. Therefore, the graded properties are implemented by user subroutine UMAT in ABAQUS/Standard™. The eigenvalue problem is solved using three-dimensional elasticity equations, and the current technique has been confirmed by earlier research. The major purpose of this study is to see how FGM configurations affect the vibration behaviour of the cylinder. The variation of the cylinder's first natural frequencies for various thickness-to-outer-radius and length-to-outer-radius ratios is presented, taking into account varied material distributions. According to the findings, raising the power exponents, particularly the radial one, increases the natural frequencies.

SASSIER22: Full‐Waveform Tomography of the Eastern Indonesian Region That Includes Topography, Bathymetry, and the Fluid Ocean

AbstractWe present a new 3‐D seismic structural model of the eastern Indonesian region and its surroundings from full‐waveform inversion (FWI) that exploits seismic data filtered at periods between 15–150 s. SASSY21—a recent 3‐D FWI tomographic model of Southeast Asia—is used as a starting model, and our study region is characterized by particularly good data coverage, which facilitates a more refined image. We use the spectral‐element solver Salvus to determine the full 3‐D wavefield, accounting for the fluid ocean explicitly by solving a coupled system of acoustic and elastic wave equations. This is computationally more expensive but allows seismic waves within the water layer to be simulated, which becomes important for periods ≤20 s. We investigate path‐dependent effects of surface elevation (topography and bathymetry) and the fluid ocean on synthetic waveforms, and compare our final model to the tomographic result obtained with the frequently used ocean loading approximation. Furthermore, we highlight some of the key features of our final model—SASSIER22—after 34 L‐BFGS iterations, which reveals detailed anomalies down to the mantle transition zone, including a convergent double‐subduction zone along the southern segment of the Philippine Trench, which was not evident in the starting model. A more detailed illumination of the slab beneath the North Sulawesi Trench reveals a pronounced positive wavespeed anomaly down to 200 km depth, consistent with the maximum depth of seismicity, and a more diffuse but aseismic positive wavespeed anomaly that continues to the 410 km discontinuity.

Decay estimates of solutions to nonlinear elastic wave equations with viscoelastic terms in the framework of Lp-Sobolev spaces

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in the Lp framework. Sharp estimates of the global solutions in t for the higher order derivatives are established. The proof is based on the Gronwall type argument for 1<p<∞, the use of the difference of the propagation speed of the fundamental solutions for p=1 and the bootstrap argument for p=∞, combined with the sharp estimates for the linearized solutions.

Local multiscale model reduction using discontinuous Galerkin coupling for elasticity problems

In this paper, we consider the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with discontinuous Galerkin (DG) coupling for the linear elasticity equations in highly heterogeneous and high contrast media. We will introduce the construction of a DG version of the CEM-GMsFEM, such as auxiliary basis functions and offline basis functions. The DG version of the method offers some advantages such as flexibility in coarse grid construction and sparsity of resulting discrete systems. Moreover, to our best knowledge, this is the first time where the proof of the convergence of the CEM-GMsFEM in the DG form is given. Some numerical examples will be presented to illustrate the performance of the method.

Elastic wave dispersion equation considering material and geometric nonlinearities

In this letter, we describe the propagation of longitudinal waves in one-dimensional nonlinear elastic thin rods with material nonlinearity and geometric nonlinearity. Mathematical analysis is used to derive the analytical dispersion relationship of longitudinal waves; subsequently, the numerical Fourier spectrum method is used to solve the nonlinear wave equation directly and the results are used to verify the correctness of the derived nonlinear dispersion relation.

Buoyancy-driven mixed convection flow of FENE-P fluids over a flat plate

The primary purpose of this study is to investigate the buoyancy mixed convection flow of non-Newtonian fluid over a flat plate. The addition of a small amount of polymers into a Newtonian solvent raises the viscosity and generates elastic properties in the resulting solution. To study the behavior of these viscoelastic fluids, finite extensible nonlinear elastic constitutive equations along with Peterlin’s closure (FENE-P model) are used. Along with mass, momentum and energy equations, viscoelastic constitutive equations are also used to examine the rheology of the resulting polymer solution. Similarity transformations are introduced to convert the governing equations into nondimensional forms. The nondimensional equations are solved using the fourth-order boundary value solver in MATLAB. The distribution of the velocity and temperature fields is displayed graphically under the impact of various involved parameters like Eckert number (Ec), Richardson number (Ri), Prandtl number (Pr). The addition of polymers increases the friction among the different fluid layers, leading to viscous dissipation in the fluid. The presented model’s validation is done with the Newtonian fluid to verify the results. The Nusselt number is also computed and analyzed to study the heat transfer rate. The effects of viscoelastic parameters like Weissenberg number (W[Formula: see text]), polymer viscosity ratio ([Formula: see text]) and polymer extensibility parameter ([Formula: see text]) on heat transfer rate are also shown graphically. Buoyancy parameter (Richardson number, [Formula: see text]) represents the dominance of natural convection relative to that of forced convection. The temperature of the resulting fluid falls with the increase in the value of Ri. The Nusselt number tends to decrease with increasing Richardson number when viscous dissipation effects are active.

A 2D Model Which Accounts for Transverse Strains in a Linear Elastic Thick Shell

In this paper we present a 2D four-parameter model that accounts for the variation of transverse deformation through-the-thickness in thick linear elastic shells. This model is deduced directly from the 3D elasticity equations of the traction-displacement boundary value problem. Transverse shear strains and thickness variation are accountable through additional terms which appear in the final equations besides well-known terms in the classical Kirchhoff-Love models for thin shells. A unique solution of the variational equation is established, and numerical and analytical results are compared with satisfaction.

Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

Periods of periodic travelling wave solutions for an elastic beam equation

Periods of periodic travelling wave solutions for an elastic beam equation