Strain viscous

Sridhar and coworkers studied the kinetics of a compressed film on a viscous substrate [30], They performed linear-stability analysis to determine the onset and maximally unstable mode of this mechanical instability as a function of misfit strain, viscous layer thickness, and viscosity. 

A viscous material shows the deformation behavior given in Figure 4.12. While under stress, the strain in a viscous material increases linearly with time. When the stress is removed, the material remains strained. Viscous behavior is described by ... 

Dynamic Mechanical (Low Strain Deformation). When a cyclic strain of small ampUtude is applied to a strip of material, a cyclic stress will be generated in response by the sample. If the material is ideal (Hookian) and stores all the input energy, the cyclic stress is in phase with the applied cyclic strain. Viscous components cause a finite phase lag or phase angle, 8, between the stress and strain. represents the elastic, real, or storage modulus while E" is the viscous, imaginary, or loss modulus. Tan 8 is equal to the ratio E /E" and is related to the molecular relaxations that occur in the sample. Transition temperatures and associated activation energy can be determined (72) by varying the frequency of oscillation at a fixed temperature or the temperature at a fixed frequency. 

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. 

When the load is applied spring 2 extends by 0/M2 and remains stretched. Time dependency is due entirely to the Kelvin unit, in which the total strain = viscous strain, v = elastic strain e-... 

The response exhibits an instantaneous strain, retarded strain, viscous flow, instantaneous recovery strain upon unloading, retarded strain recovery and permanent deformation. Scientifically, the total strain response can be separated into the initial elastic strain and the strain after the initial response, which is the creep strain. For a given material, the creep strain can be written as a fimction of time, temperature and stress as ... 

Gvinf limihng high strain viscous modulus... 

Since the strain is the same in both elements in the Voigt model, the applied stress (subscript 0) must equal the sum of the opposing forces arising from the elastic and viscous response of the model ... 

It is interesting to note that the Voigt model is useless to describe a relaxation experiment. In the latter a constant strain was introduced instantaneously. Only an infinite force could deform the viscous component of the Voigt model instantaneously. By constrast, the Maxwell model can be used to describe a creep experiment. Equation (3.56) is the fundamental differential equation of the Maxwell model. Applied to a creep experiment, da/dt = 0 and the equation becomes... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

A parameter used to characterize ER fluids is the Mason number, Af, which describes the ratio of viscous to electrical forces, and is given by equation 14, where S is the solvent dielectric constant T q, the solvent viscosity 7, the strain or shear rate P, the effective polarizabiUty of the particles and E, the electric field (117). 

Mechanical Behavior of Materials. Different kinds of materials respond differently when they undergo basic mechanical tests. This is illustrated in Eigure 15, which shows stress—strain diagrams for purely viscous and purely elastic materials. With the former, the stress is reheved by viscous flow and is independent of strain. With the latter, there is a direct dependence of stress on strain and the ratio of the two is the modulus E (or G). 

Figure 16 (145). For an elastic material (Fig. 16a), the resulting strain is instantaneous and constant until the stress is removed, at which time the material recovers and the strain immediately drops back to 2ero. In the case of the viscous fluid (Fig. 16b), the strain increases linearly with time. When the load is removed, the strain does not recover but remains constant. Deformation is permanent. The response of the viscoelastic material (Fig. 16c) draws from both kinds of behavior. An initial instantaneous (elastic) strain is followed by a time-dependent strain. When the stress is removed, the initial strain recovery is elastic, but full recovery is delayed to longer times by the viscous component.

Fig. 15. Stress—strain diagrams, (a) Viscous material of viscosity Tj the stress is independent of strain, but dependent on the speed of testing, (b) Elastic material of modulus E the slope is the modulus which is independent of the speed of testing.

Fig. 16. Response (strain) of different idealized materials to an instantaneous appHcation of a stress at time t = tg ( ) elastic, (b) viscous, and (c)...

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

Figure 36 is representative of creep and recovery curves for viscoelastic fluids. Such a curve is obtained when a stress is placed on the specimen and the deformation is monitored as a function of time. During the experiment the stress is removed, and the specimen, if it can, is free to recover. The slope of the linear portion of the creep curve gives the shear rate, and the viscosity is the appHed stress divided by the slope. A steep slope indicates a low viscosity, and a gradual slope a high viscosity. The recovery part of Figure 36 shows that the specimen was viscoelastic because relaxation took place and some of the strain was recovered. A purely viscous material would not have shown any recovery, as shown in Figure 16b. 

Bacterial Cellulose. Development of a new strain of Acetobacter may lead to economical production of another novel ceUulose. CeUulon fiber has a very fine fiber diameter and therefore a much larger surface area, which makes it physicaUy distinct from wood ceUulose. Its physical properties mote closely resemble those of the microcrystalline ceUuloses thus it feels smooth ia the mouth, has a high water-binding capacity, and provides viscous aqueous dispersions at low concentration. It iateracts synergisticaUy with xanthan and CMC for enhanced viscosity and stabUity. 

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

A. Kumar, F.E. Hauser, and J.E. Dorn, Viscous Drag on Dislocations in Aluminum at High Strain Rates, Acta Metall 16, 1189-1197 (1968). 

Fig. 19.7. A rotation viscometer. Rotating the inner cylinder shears the viscous glass. The torque (and thus the shear stress aj is measured for a given rotation rate (and thus shear strain rate y).

One example of this occurs with stress relaxation. If a polymer is deformed to a fixed strain at constant temperature the force required to maintain that strain will decay with time owing to viscous slippage of the molecules. One measure of this rate of decay or stress relaxation is the relaxation time 0, i.e. the time taken for the material to relax to 1/e of its stress on initial application of strain. 

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. 

In a perfectly viscous (Newtonian) fluid the shear stress, t is directly proportional to the rate of strain (dy/dt or y) and the relationship may be written as... 

The dashpot is the viscous component of the response and in this case the stress (72 is proportional to the rate of strain f2> ie... 

Example 2.14 A plastic is subjected to the stress history shown in Fig. 2.45. The behaviour of the material may be assumed to be described by the Maxwell model in which the elastic component = 20 GN/m and the viscous component r) = 1000 GNs/m. Determine the strain in the material (a) after u seconds (b) after 1/2 seconds and (c) after 3 seconds. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

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