Purely elastic solid

The function iKt-t ) may be interpreted as a memory function having a form as shown in Figure 3.14. For an elastic solid, iff has the value unity at all times, while for a purely viscous liquid iff has the value unity at thfe current time but zero at all other times. Thus, a solid behaves as if it remembers the whole of its deformation history, while a purely viscous liquid responds only to its instantaneous deformation rate and is uninfluenced by its history. The viscoelastic fluid is intermediate, behaving as if it had a memory that fades exponentially with time. The purely elastic solid and the purely viscous fluid are just extreme cases of viscoelastic behaviour. 

Let us remind ourselves that for a purely elastic solid a = Es, where E is independent of s, whereas for a purely viscous (Newtonian) liquid s = qy, where q (viscosity) is independent of y (shear rate). 

Unfortunately the theo is derived for purely elastic solids and cannot handle the inelastic deformations (and thus energy losses) whidi normally predominate in... 

Relaxation Time. Figure 5.10 illustrates what may happen in stress relaxation experiments. A material is somehow deformed until a given strain is obtained and then kept at that strain (a). In (b) the response of the stress is given. For a Newtonian liquid, the stress will instantaneously go to zero. For a purely elastic solid, the stress will remain constant. For a viscoelastic material, the stress will gradually relax. The figure illustrates the simplest case, where... 

Polymers are also unique in their viscoelastic nature, a behavior that is situated between that of a pure elastic solid and that of a pure viscous liquid-like material their mechanical properties present a strong dependence on time and temperature. Given all the factors that have to be taken into account to determine the mechanical properties of polymers, their measurement would appear to be very complex. However, there is a series of general principles that determine the different mechanical properties and that give a general idea of the expected results in different mechanical tests. These principles can be organized in a systematic manner to determine the interrelation of polymer structure and the observed mechanical properties, using equations and characteristic parameters of polymeric materials. 

In fact purely elastic solids do not exist. Solids like elastomers are viscoelastic they lose energy when subjected to a cycle of deformation, and particularly at a crack tip where stresses and strain rates are high. G - w is the crack extension force applied to the crack tip under this force the crack takes a limiting speed v, instead of continuously accelerate as for elastic solids, and one can write (6)... 

At low temperatures, the material can best be described as a glassy sohd. It has a high modulus, and behavior in this state is characterized ideally as a purely elastic solid. In this... 

For Newtonian behaviour r = ri where r is the shear stress, 77 is the viscosity and y is the strain rate. Kaolinite dispersions are cohesive sediments and show shear thinning behaviour ie rj decreases as y increases. The extrapolated value of r at zero shear rate is called the Bingham yield stress, Tg. For a Hookean solid r = G y, where G is the modulus of rigidity. Most substances are neither purely elastic (solid-like) nor purely viscous (liquidlike) but are viscoelastic. Kaolinite is in this category. For viscoelastic behaviour... 

Here, 5 is the solid network stress and i)) the porosity, and a negative hquid pressure is considered as compressive equivalent equations hold for the y- and z-directions. For a small cube of gel or very low evaporation rates m, liquid can easily flow to keep pressure gradients negligible, and uniform can be assumed. Then, the liquid pressure is entirely balanced by the solid network stress a = lyPw (negative stress indicates compression), and the wet gel shrinks with no total stress, 0 = 0. If the gel has a purely elastic solid network, then volumetric strain, that is, relative volume change, is given by... 

In the case of purely viscous materials, the initial stress would be very large but would relax almost instantaneously. In the case of a pure elastic solid, the stress would be proportional to the strain with no relaxation. In the case of viscoelastic fluid, it would exhibit a complete relaxation, although at a finite rate, whereas a viscoelastic solid would relax to some extent at a finite rate, but not completely. 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

---