Viscoelastic deformation

Some basic lamina and laminate behavioral characteristics were deliberately overlooked in the preceding discussion. Among them are plastic or nonlinear deformations, viscoelastic behavior, and wave propagation. 

The mechanisms of fatigue crack closure have been identified for metals. These are illustrated by Anderson [16], who has also expanded upon the effects of crack closure on the fatigue properties of metals. The phenomenon can also manifest itself in polymer-based materials because of the plastic deformation, viscoelasticity, residual stresses, and environmental conditions. 

The word vixt oeUislic encompasses many fluids that exhibit both elasticity (solidlike behavior) and flow (liquid-like behavior) when sheared. Most concentrated pastes, emulsions, and gels are viscoelastic. Under small deformations, viscoelastic fluids literally behave as elastic solids under higher deformations they flow as liquids. 

Definition of terms relating to the non-ultimate mechanical properties of polymers. " This document defines mechanical terms of significance prior to failure of polymers. It deals in particular with bulk polymers and concentrated polymer solutions and their respective viscoelastic properties. It contains the basic definitions of experimentally observed stress, strain, deformations, viscoelastic properties, and the corresponding quantities that are commonly met in conventional mechanical characterization of polymeric materials. Definitions from ISO and ASTM publications are adapted. Only isotropic polymeric materials are considered. 

If the stressed polymer has deformed viscoelastically by relaxation, the deformed configuration has a higher energy than the initial one. Upon unloading, the molecules return to their initial positions. This process again requires thermal activation and is therefore time-dependent as well. 

Polymers deform viscoelastically. Under cyclic loads, the stress-strain curve upon unloading is not the same as upon loading. Therefore, there is a hysteresis between stress and strain, causing energy dissipation during the deformation, thus producing heat. This hysteresis is discussed in detail in exercise 26. 

However, amorphous water-soluble materials, such as food materials, deform viscoelastically. The deformation and relaxation behavior of such materials can be described by means of various viscoelastic models. Depending on the nature of the stress/strain applied, either the storage and loss modulus or the elasticity and the viscosity are included as material parameters in these models. These rheological material parameters depend on the temperature and the water content as well as on the applied strain rate. The viscoelastic deformation enlarges the contact area and decreases the distance between the particles (see Fig. 7.3). If the stress decreases once again, the achieved deformation is partially reversed (structural relaxation). 

For a viscoelastic liquid in shear, the ratio oz t)/y in equations 56 to 58 is sometimes treated as a time-dependent viscosity J (r) which increases monoton-ically (provided the viscoelasticity is linear) to approach the steady-state viscosity Meissner has pointed out that, in an experiment involving deformation at constant strain rate followed by stress relaxation at constant deformation, viscoelastic information can be obtained without imposing the restriction in equation 11 of Chapter 1 that the loading interval is small compared with the time elapsed at the first experimental stress measurement. ... 

At small deformations, viscoelastic information can in principle be obtained from stress-strain measurements at a constant strain rate, as shown for shear deformations in equations S6 to S9 of Chapter 3. Such experiments are often made in simple extension, but the deformations can become rather large so there are marked deviations from linear viscoelastic behavior. The most commonly used instrument is the Instron tester other carefully designed devices have been described. - The sample is usually a dumbbell or a ring. In the former case, the strain in the narrow section as checked by separations of several fiducial marks can be calculated from the separation between the clamps by a suitable multiplication factor. In... 

Huber N, Tsakmakis C (2000) Finite deformation viscoelasticity laws. Mech Mater 32 1-18... 

The deformation of the polymers consists of elastic deformation, viscoelastic deformation and irreversible deformation (Anon, 2003). The total deformation e of the segmented polyurethanes with higher HSC can be given by ... 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The viscosity given by Eq. (3.98) not only follows from a different model than the Debye viscosity equation, but it also describes a totally different experimental situation. Viscoelastic studies are done on solid samples for which flow is not measurable. A viscous deformation is present, however, and this result shows that it is equivalent to what would be measured directly, if such a measurement were possible. 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Deformation is the relative displacement of points of a body. It can be divided into two types flow and elasticity. Flow is irreversible deformation when the stress is removed, the material does not revert to its original form. This means that work is converted to heat. Elasticity is reversible deformation the deformed body recovers its original shape, and the appHed work is largely recoverable. Viscoelastic materials show both flow and elasticity. A good example is SiEy Putty, which bounces like a mbber ball when dropped, but slowly flows when allowed to stand. Viscoelastic materials provide special challenges in terms of modeling behavior and devising measurement techniques. 

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.

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. 

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