Retarded elasticity

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

The total deformation in the four-element model consists of an instantaneous elastic deformation, delayed or retarded elastic deformation, and viscous flow. The first two deformations are recoverable upon removal of the load, and the third results in a permanent deformation in the material. Instantaneous elastic deformation is little affected by temperature as compared to retarded elastic deformation and viscous deformation, which are highly temperature-dependent. In Figure 5.62b, the total viscoelastic deformation is given by the curve OABDC. Upon unloading (dashed curve DFFG),... 

Numerous attempts have been made to fit simplified mechanical models to the two behavior patterns described by Eq. (6). One can picture the elastic element as a spring-anayed network parallel with the viscous element to give essentially a (Kelvin) solid with retarded elastic behavior, wherein ... 

When viscoelastic materials are stressed, some of the energy involved is stored elastically, various parts of the system being deformed into new non-equilibrium positions relative to one another. The remainder is dissipated as heat, various parts of the system flowing into new equilibrium positions relative to one another. If the relative motion of the segments into non-equilibrium positions is hampered, the elastic deformation and recovery of the material is time-dependent (retarded elasticity). 

The Voigt-Kelvin element (retarded elastic response), represented by a spring and a dashpot in parallel. The elastic response is not instantaneous but retarded by a viscous resistance. The two contributions to the stress are additive in this model whereas the strains are equal ... 

In bodies showing retarded elasticity, the deformation is a function of time as well as stress. Such a stress-strain curve is shown in Figure 8-10. The upward part of the curve represents increasing values of stress when the stress is reduced, the corresponding strains are greater on the downward part of the curve. When the stress reaches 0, the strain has a finite value, which will slowly return to zero. There is no permanent deformation. The corresponding relaxation (stress-time) and creep (strain-time) curves... 

Figure 8-10 Stress-Strain Curve of a Retarded Elastic Body...

Tsardaka, K. D., and Rees, J. E. (1989), Plastic deformation and retarded elastic deformation of particulate solids using creep experiments, /. Pharm. Pharmacol., 41, 28P. 

The retarded elastic response displayed by the O-rings is an example of viscoelastic behavior, something everybody has come across in everyday life, usually without real-... 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

FIGURE 13-93 Schematic diagram of a retarded elastic response. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

This does display the three elements of real behavior, an instantaneous elastic response, primary creep (retarded elastic response) and secondary creep (permanent deformation). However, the fit to real data is not good and again it is because real materials have behavior that is characterized by a spectrum of relaxation times. 

Which of these models does not do a good job of describing the retarded elastic response that is characteristic of creep Identify the model (Voigt Maxwell ) and explain why. 

Region B-C corresponds to a time dependent retarded elastic region with a compliance Jr. In this region the bonds break and reform, but all of them do not break and reform at the same rate. The equation for this part using mean values for the parameters is ... 

In Equation (3.85), Jm is the mean compliance of all the bonds and Tm is the mean retardation time Tm equals Jmt m where ijm is the mean viscosity associated with elasticity. One can replace the mean quantities with a spectrum of retarded elastic moduli (Gj) and the viscosities (iji), where, J-, = l/G,. Typically, one or two Kelvin-Voigt elements can be used to describe the retarded elastic region. 

When the stress is suddenly removed at D, the pattern consists of an elastic recovery (D-E) followed by a retarded elastic recovery (E-F). Because bonds between structural units are broken in region C-D, a part of the structure is not recovered. For values of time after the applied stress is removed, the recovered strain approaches a maximum value, 5r, called recoverable shear (Dealy, 1982) ... 

The terms Jo, J, and J2, are the instantaneous compliance, and the compliances associated with retarded elastic behavior, respectively ti = r]]/G and 12 = m/Gi are the retardation times associated with retarded elasticity r], t]2, and are the viscosities associated with Newtonian flow. 

After the stress has been removed (point D in Fig. 13A), the recovery phase follows a pattern mirroring the creep compliance curve to some degree First, there is some instantaneous elastic recovery (D-E return of spring 1 into its original shape Fig. 13A, B). Second, there is a retarded elastic recovery phase (E-F slow movement of the Kelvin unit into its original state Fig. 13A, B). However, during the Newtonian phase, links between the individual structural elements had been destroyed, and viscous deformation is non-recoverable. Hence, some deformation of the sample will remain this is in the mechanical model reflected in dash-pot 2, which remains extended (Fig. 13B). 

If shear continued, more links between the structural units would break and re-form, but as weaker links do so at smaller time points, there is some retardation of this process. In Fig. 13A, this phase, which is called the retarded elastic region, is presented by the curved compliance-time profile between the points B and C. In the mechanical model (Fig. 13B), this region corresponds to a slow movement of spring 2 and dashpot 1, i.e., the Kelvin unit. The value of the retarded compliance can be obtained from ... 

In an un-cross-linked amorphous polymer, above its glass temperature, the molecular chains are continuously wriggling from one conformation to another. If a mechanical stress is imposed on such a system of wriggling chains, it can respond in three distinct ways instantaneous elastic response retarded (conformational) elastic response or viscous flow. Actually, in order to fit experimental data adequately, the retarded elastic element must be expanded into a whole series of such elements, some with shorter and some with longer response times. The local "kinkiness" of the chains can be straightened out (by stress) more rapidly than the... 

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. 

Under creep, the total strain will be due to the instantaneous elastic deformation of the spring of modulus E, and irrecoverable viscous flow due to the dashpot of viscosity 1)2, and the recoverable retarded elastic deformation due to the Voigt element with a spring of modulus E3 and dashpot of viscosity rij. Thus, the total strain is the sum of these three elements. That is. 

In creep recovery, say, the load is removed at time t, the deformation, Oo i> due to the spring of modulus El is recovered instantaneously. This will be followed by the retarded elastic creep reeovery due to the Voigt element given by 83 or... 

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

---