Elastic solid theories

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The JKR theory is essentially an equilibrium balance of energy released due to interfacial bond formation and the stored elastic energy. For simple elastic solids the deformation as a function of load, according to the JKR theory is given by... 

The continuum theory of deformation of elastic solids is old and well developed [65T01, 74T01], and, in its linear version, is widely applied. Nonlinear theory is of much more recent origin. Most application of nonlinear theory has been to the behavior of highly deformable materials such as rubber or to the explanation of subtle effects observed by precise ultrasonic... 

The rheological characteristics of AB cements are complex. Mostly, the unset cement paste behaves as a plastic or plastoelastic body, rather than as a Newtonian or viscoelastic substance. In other words, it does not flow unless the applied stress exceeds a certain value known as the yield point. Below the yield point a plastoelastic body behaves as an elastic solid and above the yield point it behaves as a viscoelastic one (Andrade, 1947). This makes a mathematical treatment complicated, and although the theories of viscoelasticity are well developed, as are those of an ideal plastic (Bingham body), plastoelasticity has received much less attention. In many AB cements, yield stress appears to be more important than viscosity in determining the stiffness of a paste. 

Crystals have played a dominant role in the development of experimental knowledge about dislocations. Thus, it is often forgotten that the concept of dislocations was developed within the theory of continuous elastic solids. The very name was coined by A. E. H. Love, an elastician (Love, 1944). Therefore, dislocations need not have fixed displacement vectors. 

J.W. Cahn s early contributions to elastic coherency theory were motivated by his work on spinodal decomposition. His subsequent work with F. Larche created a rigorous thermodynamic foundation for coherency theory and stressed solids in general. A single volume, The Selected Works of John W. Cohn [15], contains papers that provide background and advanced reading for many topics in this textbook. This derivation follows from one in a publication included in that collection [16]. 

The macroscopic behavior of a saturated porous material undergoing a dissolution of its linear elastic solid matrix is therefore described by the classical Biot s theory, where the poroelastic properties now depend on the morphological parameter . Formally, plays the role of a damage parameter accounting for the dissolution. 

During a collision, the colliding solids undergo both elastic and inelastic (or plastic) deformations. These deformations are caused by the changes of stresses and strains, which depend on the material properties of the solids and the applied external forces. Theories on the elastic deformations of two elastic bodies in contact are introduced in the literature utilizing Hertzian theory for frictionless contact and Mindlin s approach for frictional contact. As for inelastic deformations, few theories have been developed and the available ones are usually based on elastic contact theories. Hence, an introduction to the theories on elastic contact of solids is essential. 

The book contains two parts each part comprises six chapters. Part I deals with basic relationships and phenomena of gas-solid flows while Part II is concerned with the characteristics of selected gas-solid flow systems. Specifically, the geometric features (size and size distributions) and material properties of particles are presented in Chapter 1. Basic particle sizing techniques associated with various definitions of equivalent diameters of particles are also included in the chapter. In Chapter 2, the collisional mechanics of solids, based primarily on elastic deformation theories, is introduced. The contact time, area, and... 

In principle, then, the surface work which determines the fracture stress of the body can be calculated from the physical properties of the material. In practice this is not easy, since the energy density distribution can only be calculated exactly for linear elastic solids, for which 1 and Eq. (5) reverts to the Griffith theory. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

Stokes GG (1845) On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids. Trans Camb Phil Soc 8 287-305... 

The theory was based on a simple physical model which treats the quartz as a lossless elastic solid, and the liquid as purely viscous fluid. The frequency shifts arise from coupling the oscillation of the crystal, a standing shear wave with a damped propagating shear wave in the liquid. The accuracy of this model was demonstrated using aqueous solutions of glucose and ethanol at various concentrations. 

The assumption that the contraction process is ideally adiabatic, while perhaps not entirely permissible practically, seems indicated by modern theory of the behavior of molecular chains, which pictures these as undergoing, when freed of restraints, a sort of segmental diffusion, much like the adiabatic expansion of an ideal gas into a vacuum (155). In the case of the molecular chain, it diffuses to the most probable, randomly coiled configuration, which is much less asymmetric, hence shorter, than an initially extended chain. Because rubber most nearly presents this ideal behavior, those fibers which develop increased tension (a measure of the tendency toward assumption of the contracted form) when held isometrically under conditions of increasing temperature (favoring the diffusion ) are said to be rubber-like. Most normal elastic solids upon stress are strained from some stable structure and relax as the temperature is raised. 

When fluids can seep through pores, interacting mechanically with the solid skeleton, the material is composed of more than one constituent thus we need to use a mixture theory in which we could clearly make out each part filled by different constituents on a scale which is rather large in comparison with molecular dimensions so we put forward a new continuum theory of an immiscible mixture consisting both of a continuum with ellipsoidal microstructure (the porous elastic solid) and of two classical media (see, also, the conservative case examined by Giovine (2000)). In accordance with Biot (1956), we consider virtual mass effects due to diffusion we also introduce the microinertia associated with the rates of change of the constituents local densities, as well as the one due to the deformation of the pores close to their boundaries. 

Giovine, P. 1999. A linear theory of porous elastic solids. Trans. Porous Media 34 pp.305-318. 

The mechanical behavior of metals in service can often be assumed to be that of a linear, isotropic, and elastic solid. Thus, design analysis can be based on classical strength of materials theory extensively reviewed in textbooks and literature. Practically, results may be used in the form of standard formulae, or design charts for a selected class of applications. Such uses are most appropriate to components of simple geometric shapes for which standard solutions exist, or for more shapes that are complex where they can possibly be used for initial approximate design calculations. 

Adhesive contact of two elastic solids can be studied from two points of view thermodynamics and theory of elasticity. They give the same result, but they enlighten the problem differently. 

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