Ideal elastic response

Polymeric materials often tend to behave as springs (that is, they have a tendency to retract on stretching), thus displaying some degree of elasticity. An ideal elastic material responds instantaneously to application or removal of stress, with the strain (y) being proportional to the stress (Hookean), independently of the strain rate [Eq. (33)]. Here, the constant G is the modulus of the elastic material. 

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II. SIMPLE RHEOLOGICAL RESPONSES A. THE IDEAL ELASTIC RESPONSE... 

The creep response of a viscoelastic polymer under a fixed stress can be separated into three different contributions a fast (ideal) elastic response, a slow elastic response, and a viscous response (or flow). For purely elastic materials, the creep compliance is a constant. In a viscous material, the creep increases indefinitely as the material flows irreversibly. You can think about creep as the gradual deformation of a normally elastic material. A new foam seat cushion will spring back after each time someone sits on it, although over years of use the cushion will gradually deform. 

For a given amplitude of the quasi-elastic release wave, the more the release wave approaches the ideal elastic-plastic response the greater the strength at pressure of the material. The lack of an ideally elastic-plastic release wave in copper appears to suggest a limited reversal component, however, this is much less than in the silicon bronze. Collectively, the differences in wave profiles between these two materials are consistent with a micro-structurally controlled Bauschinger component as supported by the shock-recovery results. Further study is required to quantify these findings and... 

Fig. 2.5. The idealized elastic/perfectly plastic behavior results in a well defined, two-step wave form propagating in response to a loading within the elastic-plastic regime. Such behavior is seldom, if ever, observed.

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 transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

Voigt-Kelvin model or element Model consisting of an ideal spring and dashpot in parallel in which the elastic response is retarded by viscous resistance of the fluid in the dashpot. 

The overall response of the crystal to such a stress cycle is shown in Fig. 8.16. When the stress a0 is applied suddenly, the crystal instantaneously undergoes an ideally elastic strain following Eq. 8.62. As the stress is maintained, the crystal undergoes further time-dependent strain due to the re-population of the interstitials. When the stress is released, the ideally elastic strain is recovered instantaneously and the remaining anelastic strain will be recovered in a time-dependent fashion as the interstitials regain their random distribution. 

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.

Figure H3.2.2 Responses of an ideal elastic, viscous, and viscoelastic material to a sinusoidal deformation. 8, phase angle y, shear strain co, angular frequency o, shear stress.

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. 

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.

The elastic and viscous mechanisms involved in the viscoelastic responses have traditionally been modeled by combining ideal elastic elements, represented by springs, and ideal viscous elements, represented by dashpots. We shall refer to this treatment in Chapter 10 here we use another approach,... 

As is well known, springs and dashpots represent, respectively, ideal elastic and viscous responses to step stress perturbations. In a similar way, a combination of the two can be used to describe the viscoelastic behavior of materials. The Maxwell model, a spring in series with a dashpot, is the more immediate idealization of this behavior (Fig. 10.1). 

The same parameters can also be determined by applying a constant shear stress to the interface and measuring the resulting shear strain as a function of time (see fig. 3.40), so-called interfacial creep tests. At t = 0, a shear stress is suddenly applied, and kept constant thereafter. For ideally viscous monolayers a steady increase of the shear strain with t will be observed, while for an elastic material the observed strain will be instantaneous and constcmt in time. For a viscoelastic material, as in fig. 3.40, there is first am Instantaneous increase AB in the strain, the elastic response followed by a delayed elastic response BC and a viscous... 

The ideally elastie material exhibits no time effects and negligible inertial effects. The material responds instantaneously to applied stress. When this stress is removed, the sample recovers its original dimensions completely and instantaneously. In addition, the induced strain, e, is always proportional to the applied stress and is independent of the rate at which the body is deformed (Hookean behavior). Figure 14.1 shows the response of an ideally elastic material. 

The ideal elastie response is typified by the stress-strain behavior of a spring. A spring has a constant modulus that is independent of the strain rate or the speed of testing stress is a funetion of strain only. For the pure Hookean spring the inertial effects are neglected. For the ideal elastic material, the mechanical response is deseribed by Hooke s law ... 

However, no real material shows either ideal elastic behavior or pure viscous flow. Some materials, for example, steel, obey Hooke s law over a wide range of stress and strain, but no material responds without inertial effects. Similarly, the behavior of some fluids, like water, approximate Newtonian response. Typical deviations from linear elastic response are shown by rubber elasticity and viscoelasticity. 

The response of rubbery materials to mechanical stress is a slight deviation from ideal elastic behavior. They show non-Hookean elastic behavior. This means that although rubbers are elastic, their elasticity is such that stress and strain are not necessarily proportional (Figure 14.3). 

As Equation 14.7 shows, the Maxwell element is merely a linear combination of the behavior of an ideally elastic material and pure viscous flow. Now let us examine the response of the Maxwell element to two typical experiments used to monitor the viscoelastic behavior of polymer. 

The creep and recovery of plastics can be simulated by an appropriate combination of elementary mechanical models for ideal elastic and ideal viscous deformations. Although there are no discrete molecular structures which behave like individual elements of the models, they nevertheless aid in understanding the response of plastic materials. 

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

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