Siam journal on control and optimization siam society for. The main purpose of the present paper is to find homogenized equations in explicit form of the theory of linear elasticity in a twodimensional domain with an interface rapidly oscillating between two concentric circles. The book is distinct from the traditional text aimed at graduate students in solid mechanics by introducing the subject at a level appropriate for. The corresponding feedthrough operator is found to be the identity operator on the input space. Seismology and the earths deep interior elasticity and seismic waves linearlinear and nonlinear stress and strainand nonlinear stress and strain stress vs. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. Rivlin, exact solutions in incompressible nonlinear elasticity rubber. These are examples of relations in which stress not only depends on deforma tion, but. Lectures on non linear wave equations download ebook pdf.
Secondorder hyperbolic partial differential equations wave equation linear wave equation 2. The governing equations for the nonlinear case can then be linearized to obtain the simpler theory of linear elasticity. Since odd elasticity cannot be obtained from a free energy, the presence of a and ko entails the ability to take an odd elastic medium through a closed cy. They involve pulses of arbitrary shape that travel at speeds determined by the elastic properties and density of the medium, and these pulses are often decomposed into harmonic wave solutions involving sine and cosine functions. Chapter 1 covers linear elasticity through the displacement equations of. Modified equation and amplification factor are the same as original laxwendroff method. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In this section we summarize and derive the solutions to various elementary problems in dynamic linear elasticity.
Jun 04, 2016 the classic linear wave is discussed in section the linear wave equation with some further examples given in section linear wave equation examples. The elasticity of the material provides the restoring force of the wave. The most general solution has two unknown constants, which. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials. Perfectly matched layer for secondorder timedomain. When applied to linear wave equation, twostep laxwendroff method. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. Greens medium, a transverse isotropic medium, and an orthotropic medium were found to satisfy the scalar wave equation form. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. Numerical results are presented and discussed for both isotropic and anisotropic media. Because the wave equation is linear, any linear combination of waves is also. Theoretical study of nonlinear elastic wave propagation. The wave equation is often encountered in elasticity, aerodynamics, acoustics, and. Theoretical study of nonlinear elastic wave propagation k.
This site is like a library, use search box in the widget to get ebook that you want. Dec 30, 2010 the main purpose of the present paper is to find homogenized equations in explicit form of the theory of linear elasticity in a twodimensional domain with an interface rapidly oscillating between two concentric circles. Elastodynamics is the study of elastic waves and involves linear elasticity with variation in time. The generalized scalar wave equation and linear differential. An ordinary di erential equation ode is an equation for a function which depends on one independent variable which involves the. We now derive several auxiliary field equations which follow as a consequence of the field equations listed in the preceding section. With the material linear elastic, the only nonzero stress is xy 2 xy. Homogenized equations of the linear elasticity theory in two. First, we note that the thirdorder nonlinear contributions to the elastic energy do not affect a transverse wave of the form, say, u 2 x 1ux, which obeys the. Dalemberts solution see the appendix to this section for its derivation. On the singular neumann problem in linear elasticity kuchta. Together with the heat conduction equation, they are sometimes referred to as the evolution equations because their solutions evolve, or change, with passing time. The acoustic equations, while based on linear elasticity, are explicitly set up to result in propagating wave solutions assuming small stresses and strains. The dirac equation is a relativistic wave equation detailing electromagnetic interactions.
Exact solutions linear partial differential equations secondorder hyperbolic partial differential equations wave equation linear wave equation 2. Homogenized equations of the linear elasticity theory in. Pdf a dispersive wave equation using nonlocal elasticity. Introduction to linear elasticity, 3rd edition, provides an applicationsoriented grounding in the tensorbased theory of elasticity for students in mechanical, civil, aeronautical, and biomedical engineering, as well as materials and earth science.
Derivation of the wave equation in these notes we apply newtons law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the wave equation. The media which admit this differential equation form are investigated. Stressequations of motion and boundary conditions 12 5. By means of a partial fourier transform, it is shown that the system is wellposed and regular in the sense of d. Surface subjected to time varying normal pressure an isotropic, linear elastic half space with shear modulus and poissons ratio and mass density occupies the region.
Throughout, we use the term \ wave equations to describe a broad class of pdes, both linear and nonlinear, whose principal part consists of the wave operator. Module 4 boundary value problems in linear elasticity. A dispersive wave equation using nonlocal elasticity article pdf available in comptes rendus mecanique 3378. Theoretical and numerical solutions of linear and nonlinear elastic. Elastic waves in infinite homogeneous isotropic media. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. We coupled the rotational elasticity to linear elasticity describing displacements and searched for plane wave solutions of the coupled model in the fully nonlinear setting. Linear partial differential equations for scientists and engineers. What is the difference between the equation for linear. An ordinary di erential equation ode is an equation for a function which depends on one independent variable which involves the independent variable. On a nonlinear wave equation in elasticity sciencedirect. And there is one other equation relating e 1111, e 1122. In order to do that, we use the equations of linear elasticity in polar coordinates, and write them and the continuity conditions on the interface in matrix form.
The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. The model of waves as a nearly linear superposition of harmonic components, at random phase, is con. Abstract pdf 280 kb 2010 energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics the fundamental linearizing assumptions of linear elasticity are. Second order linear partial differential equations part iv. This is not so informative so lets break it down a bit. Linear elasticity theory is thus the best known and most widely used branch of solid mechanics. We were able to find various types of solutions by assuming that the rotational and displacement waves were either transversal or longitudinal, respectively.
The second type of second order linear partial differential equations in 2 independent variables is the onedimensional wave equation. The 1d wave equation for light waves 22 22 0 ee xt where. On the singular neumann problem in linear elasticity. Tyn myintu lokenath debnath linear partial differential. Differential equations for scientists and engineers. Schematic of generic problem in linear elasticity or alternatively the equations of strain compatibility 6 equations, 6 unknowns, see. Throughout, we use the term \wave equations to describe a broad class of pdes, both linear and nonlinear, whose principal part consists of the wave operator. Pdf 280 kb 2010 energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation.
From them we shall derive a linear wave equation for the displacement vector. The main difference between the methods of this report and those of 1 arises from the fact that the stress function here has to satisfy a generalized biharmonic equation which is of the same type as that occurring in the. W ij c ijkl e kl c o stiffnessor elastic coefficient. The stress strain relationship for 2d state of stress are given by. Seismology and the earths deep interior elasticity and seismic waves linear elasticity deformation tensorlinear elasticity deformation tensor the symmetric part is called the deformation tensor p 0 q qq 0. Our derivation provides us with a simple almost trivial example of the general procedure dis. We simplify it to the standard form by modeling the material as series of homogeneous layers.
Click download or read online button to get lectures on non linear wave equations book now. Seismology and the earths deep interior elasticity and seismic waves linearlinear and non linear stress and strainand non linear stress and strain stress vs. For wave propagation problems assuming linear elasticity is usually sufficient. Feedthrough operator for linear elasticity system with boundary control and observation. The most general linear relationship between stress and strain is. An elastic wave is a type of mechanical wave that propagates in elastic or viscoelastic materials. The problem is solved numerically and tested against the solution of wave propagation in a homogeneous viscoelastic medium, obtained by using the correspondence principle. The purpose of this short paper is not to provide a recipe for. Rotational elasticity and couplings to linear elasticity.
Greens medium, a transverse isotropic medium, and an orthotropic medium were found to. The seismic wave equation using the stress and strain theory developed in the previous chapter, we now con. Scott school of mathematics, university of east anglia, norwich, nr4 7tj. Stressstrain relations and the elasticity tensor in unified, you saw particular examples of this, but we now want to. Pdf nonlinear wave modulation in nanorods using nonlocal. The threedimensional hookes law for isotropic linear elastic solids part i, eqns. This is often used for analysis of static or slowly changing problems. In this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory.
A theoretical study of the propagation of a plane wave in a material with nonlinear response is presented. Seismology and the earths deep interior the elastic wave equation solutions to the wave equation solutions to the wave equation ggeneraleneral let us consider a region without sources. Nov 24, 2003 first, we note that the thirdorder non linear contributions to the elastic energy do not affect a transverse wave of the form, say, u 2 x 1ux, which obeys the same equation of motion as for linear elasticity. Periodically layered medium our first three examples illustrate wave behavior. In this paper we analyze a multidimensional controlled wave equation on a bounded domain, subject to partial dirichlet control and colocated observation. We discussed two types of waves pwavescompressional swavesshear finally, if we assume no shearing then we reduced it to an acoustic wave equation.
Vekuas theory to the linear elasticity setting, but it has not been accomplished yet see the comments at the end of section 3. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. Linear waves are modelled by pdes that are linear in the dependent variable, \u\,\ and its first and higher derivatives, if they exist. Chapter 2 introduces the onedimensional wave equation and the. Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. The formulae governing wave propagation are recast as a firstorder differential equation in time, in the vector represented by the displacements and memory variables.
Ex,t is the electric field is the magnetic permeability is the dielectric permittivity this is a linear, secondorder, homogeneous differential equation. If a photoelastic model exhibits a perfectly linear elastic behavior, the difference in the principal strain can be measured by established the fringe order n. The wave equation is solved subject to the initial conditions and boundary conditions. The generalized scalar wave equation with the generalized linear differential invariant is defined for linear elastic analysis. The seismic wave equation x 1 x 2 x 3 t x 1 t x 1 dx 1 dx 2 dx 3 figure 3. The goal of this talk is to describe the lame operator which drives the propagation of linear. Eringens nonlocal elasticity theory is employed to derive nonlinear equations for the. Pdf introduction to elastic wave propagation researchgate. And there is one other equation relating e 1111, e 1122 and e 2323 2 independent components of e.
The book is distinct from the traditional text aimed at graduate students in solid mechanics by. The wave equation is often encountered in elasticity, aerodynamics, acoustics, and electrodynamics. D alemberts solution see the appendix to this section for its derivation. A partial di erential equation pde is an equation involving partial derivatives. This assumption turns out to be an excellent predictor of the response of components which undergo small deformations.
453 1548 1125 760 1482 214 1138 1068 827 547 1298 680 856 1339 365 1299 743 623 1545 1429 612 392 1502 1158 29 150 1093 1344 137 432 369 646 807 1326 1378 1385 244