Elastic solids

Johnson K L, Kendall K and Roberts A D 1971 Surface energy and the contact of elastic solids Proc. R. Soc. A 324 301... 

Polymers have found widespread applications because of their mechanical behaviour. They combine the mechanical properties of elastic solids and viscous fluids. Therefore, they are regarded as viscoelastic materials. Viscoelastic... 

Let a solid body occupy the domain fl C with the smooth boundary T. The solid particle coincides with the point x = xi,X2,xs) G fl. An elastic solid is described by the following functions ... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

Fig. 8.1. Stress-strain behaviour for a linear elastic solid. The axes are calibrated for a material such as steel.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

Creep of polymers is a major design problem. The glass temperature Tq, for a polymer, is a criterion of creep-resistance, in much the way that is for a metal or a ceramic. For most polymers, is close to room temperature. Well below Tq, the polymer is a glass (often containing crystalline regions - Chapter 5) and is a brittle, elastic solid -rubber, cooled in liquid nitrogen, is an example. Above Tq the Van der Waals bonds within the polymer melt, and it becomes a rubber (if the polymer chains are cross-linked) or a viscous liquid (if they are not). Thermoplastics, which can be moulded when hot, are a simple example well below Tq they are elastic well above, they are viscous liquids, and flow like treacle. 

Irwin [23] developed an expression for the mode I stress intensity factor around an elliptical crack embedded in an infinite elastic solid subjected to uniform tension. The most general formulation is given by ... 

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. 

Contact mechanics of elastic solids with interfaces in non-equilibrium... 

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... 

Maugis, D., Contact, Adhesion and Rupture of Elastic Solids. Solid State Sciences. Springer, Berlin, 2000. 

Strength and Stiffness. Thermoplastic materials are viscoelastic which means that their mechanical properties reflect the characteristics of both viscous liquids and elastic solids. Thus when a thermoplastic is stressed it responds by exhibiting viscous flow (which dissipates energy) and by elastic displacement (which stores energy). The properties of viscoelastic materials are time, temperature and strain rate dependent. Nevertheless the conventional stress-strain test is frequently used to describe the (short-term) mechanical properties of plastics. It must be remembered, however, that as described in detail in Chapter 2 the information obtained from such tests may only be used for an initial sorting of materials. It is not suitable, or intended, to provide design data which must usually be obtained from long term tests. 

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

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 simple wave produced by impacting vitreous silica has approximately the form of a linear ramp of velocity. When this ramp wave is used to load another elastic solid placed in contact with the vitreous silica, a measurement... 

Elastic modulus Up to the fracture stress, glass behaves, for most practical purposes, as an elastic solid at ordinary temperatures. Most silicate-based commercial glasses display an elastic modulus of about 70GNm", i.e. about 1/3 the value for steel. If stress is applied at temperatures near the annealing range, then delayed elastic effects will be observed and viscous flow may lead to permanent deformation. 

Boa-Teh Chu, Transverse chock waves in incompressible elastic solids, I. Msch. Phys. Solids, 16 (1967), 1—14. 

Rheology is the science that deals with the deformation and flow of matter under various conditions. The rheology of plastics, particularly of TPs, is complex but understandable and manageable. These materials exhibit properties that combine those of an ideal viscous liquid (with pure shear deformations) with those of an ideal elastic solid (with pure elastic deformation). Thus, plastics are said to be viscoelastic. 

Viscoelasticity A combination of viscous and elastic properties in a plastic with the relative contribution of each being dependent on time, temperature, stress, and strain rate. It relates to the mechanical behavior of plastics in which there is a time and temperature dependent relationship between stress and strain. A material having this property is considered to combine the features of a perfectly elastic solid and a perfect fluid. 

Viscoelastic fluids are thus capable of exerting normal stresses. Because most materials, under appropriate circumstances, show simultaneously solid-like and fluid-like behaviours in varying proportions, the notion of an ideal elastic solid or of a purely viscous fluid represents the commonly encountered limiting condition. For instance, the viscosity of ice and the elasticity of water may both pass unnoticed The response of a material may also depend upon the type of deformation to which it is subjected. A material may behave like a highly elastic solid in one flow situation, and like a viscous fluid in another. 

A real material whose behaviour can be modelled in this way initially undergoes irreversible deformation as the stress is applied. This eventually ceases, and the material then behaves effectively as an elastic solid. Release of the stress will cause a rapid return to a less strained state, corresponding to the spring component of the response, but part of the deformation, arising due to viscous flow in the dashpot will not disappear. 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

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. 

---