Solids, ideal elastic

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. 

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. 

The effect of stress S on an ideal viscous liquid and on an ideal elastic solid may be predicted from Newton s... 

Figure H3.2.1 Deformation pattern of a substance in response to shear. (A) An ideal elastic solid subjected to shear. (B) An ideal viscous fluid subjected to shear, h, height AL, displacement in length.

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

Figure 1-7 Stress Relaxation Response of a Fluid, Ideal Elastic Solid, and a Viscoelastic Material. The experimental response is not shown separately because it is the same as that of the ideal elastic solid are identical.

Figure 5.2 Response of ideal elastic solids (b) and ideal liquids (c) to a constant shear strain (a) in relaxation experiments.

Let us now find the relationship between the tensile and shear moduli in terms of the Poisson ratio. Since for ideal elastic solids = 2(1 + v)G, the correspondence principle establishes that... 

In the rheological structure of most food systems there is a viscous element present, and the deformation curves are often highly influenced by the rate of the imposed strain. This is due to the fact that the material relaxes (or flows) while tested under compression and the resultant deformation of this flow is dependent on the nature of the viscous element (Szczesniak, 1963 Peleg and Bagley, 1983). In the viscoelastic food systems, where during processing it is caused to oscillate sinusoidally, the strain curve may or may not be a sine wave. In cases when a periodic oscillatory strain is applied on a food system like fluid material, oscillating stress can be observed. The ideal elastic solid produces a shear stress wave in phase with... 

In the case of an ideal elastic solid (which obeys Hooke s law), the stress is directly proportional to the strain (y), i.e.,... 

We note that the Voigt model predicts that strain is not a continuous function of stress that is, the element does not deform continuously with the sustained application of a constant stress. The strain approaches an asymptomatic value given by (Oq/E). The strain of the element at equilibrium is simply that of an ideal elastic solid. The only difference is that the element does not assume this strain instantaneously, but approaches it gradually. The element is shown to exhibit retarded elasticity. In creep recovery, the Maxwell element retracts instantaneously but not completely, whereas the Voigt element exhibits retarded elastic recovery, but there is no permanent set. 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

The disturbance in seawater travels rapidly outward, creating a growing spherical cavity. The traveling gas-liquid interface is known as the detonation front. Seawater is not an ideal elastic solid but has a bulk modulus that is a fxmction of pressure. The seawater as it is compressed becomes stiffen Therefore, the velocity of the outward traveling shock wave is a function of the peak pressure of the new wave. As the wave travels outward, the peak pressure decreases as a negative power of radial distance from the source. The value of the negative power decreases from typically -3 to -1.3 with distance from the source (Kramer et al., 1968 Dobrin, 1976). When the initial positive pressure pulse reaches a free water surface, it is reflected as a pulse of opposite polarity. 

In an ideal elastic solid, a one-to-one relationship between stress and strain is expected. In practice, however, there are often small deviations. These are termed anelastic effects and result from internal friction in the material. Part of the strain develops over a period of time. One source of anelasticity is thermoelasticity, in which the volume of a body can be changed by both temperature and applied stress. The interaction will depend on whether a material has time to equilibrate with the surroundings. For example, if a body is rapidly dilated, the sudden... 

Creep recovery experiments inea.sure the capacity of the fluid to recover its original configuration, i.e.. how more elastic than viscous a fluid is. In this sen.se. the portion of the curve after stress removal (see Fig. 14 is analyzed. An ideal elastic solid revert.s the deformation completely (recoil.s), while a Newtonian fluid does not recover at all. For a viscoelastic fluid, the creep recovery function can be defined as... 

We recall that the combination of both liquid and solid behavior is termed viscoelasticity. We have already discussed the basic law for the "simple" liquid, Newton s law. For solids, Hooke s law defines the relationship between stress S and deformation 7, using a material parameter called the modulus of elasticity S = Gy. G represents a shear modulus, while E represents Young s modulus in tension (S = Ey). That is the behavior of so-called "ideal elastic solids." This may occur mostly in metals or rigid materials, while in the case of polymers, Hooke s correlation is foxmd only in the glassy state, below Tg. 

Ideal viscous liquid (3) Ideal elastic solid response response... 

Viscoelasticity, Fig. 1 (a) Creep and recovery tests (7) ideal elastic solid relaxation curve (4) stress relation... 

A Hookian or an ideal elastic solid, whose small reversible deformations are directly proportional... 

Define a Newtonian liquid and a Hookian ideal elastic solid, and an ideal elastomer. 

When a constant stress is imposed (its time derivative cr = 0), this equation describes the ideal Newtonian fluid under steady shear flow. When 77 —> 00, this equation describes the ideal elastic solid. The instantaneous response of the solid to an imposed stress is elastic, and the shear modulus E corresponds to the modulus 00 at high frequency. Consequently, the shear stress will relax down to zero exponentially. Under the condition of y = 0, the exponential function (6.18) can be solved from (6.24), which defines the characteristic relaxation time as... 

When designing with polymers, it is important to keep in mind that many polymers deform over time when they are under a continuous load. This deformation with time of loading is called creep. Ideal elastic solids do not creep since strain (deformation) is proportional to stress, and there is no time dependence. Viscous materials (liquids) deform at a constant rate with a constant applied stress. Equation (11.6) describes the strain in a viscous material under constant load or stress [Pg.268]

Hooke s law Hooke s law describes the relationship between applied stress and strain in an ideal elastic solid body... 

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