Viscoelasticity material constants

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.

Penetration—Indentation. Penetration and indentation tests have long been used to characterize viscoelastic materials such as asphalt, mbber, plastics, and coatings. The basic test consists of pressing an indentor of prescribed geometry against the test surface. Most instmments have an indenting tip, eg, cone, needle, or hemisphere, attached to a short rod that is held vertically. The load is controlled at some constant value, and the time of indentation is specified the size or depth of the indentation is measured. Instmments have been built which allow loads as low as 10 N with penetration depths less than mm. The entire experiment is carried out in the vacuum chamber of a scanning electron microscope with which the penetration is monitored (248). 

The most characteristic features of viscoelastic materials are that they exhibit a time dependent strain response to a constant stress (creep) and a time dependent stress response to a constant strain (relaxation). In addition when the... 

When a linear viscoelastic material is subjected to a constant stress, a, at time, /i, then the creep strain, e(t), at any subsequent time, t, may be expressed as... 

When a viscoelastic material is subjected to a constant stress, it undergoes a time-dependent increase in strain. This behavior is called creep. The viscoelastic creep behavior typical of many TPs is illustrated in Figs. 2-22 and 2-23. At time to the material is suddenly subjected to a constant stress that is main-... 

When a viscoelastic material is subjected to a constant strain, the stress initially induced within it decays in a time-dependent manner. This behavior is called stress relaxation. The viscoelastic stress relaxation behavior is typical of many TPs. The material specimen is a system to which a strain-versus-time profile is applied as input and from which a stress-versus-time profile is obtained as an output. Initially the material is subjected to a constant strain that is maintained for a long period of time. An immediate initial stress gradually approaches zero as time passes. The material responds with an immediate initial stress that decreases with time. When the applied strain is removed, the material responds with an immediate decrease in stress that may result in a change from tensile to compressive stress. The residual stress then gradually approaches zero. 

TTie strain readings of a creep test can be more accessible to a designer if they are presented as a creep modulus. In a viscoelastic material, namely plastic, the strain continues to increase with time while the stress level remains constant. Since the creep modulus equals stress divided by strain, we thus have the appearance of a changing modulus. 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

Time characterizing the response of a viscoelastic material to the instantaneous application of a constant stress. 

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

The reduced expressions of Table 4 form a set of universal viscoelastic functions. Given the polymer molecular weight, material constants [Jg, Je, etc.), and one extremal relaxation/retardation time, one should be able to predict, roughly, the nature of the system response (within the framework of the linear models) from Eqs. (T 1)—(T 6) and Fig. 2—5. 

Hooke s Law, which states that a proportional relationship exists between stress and strain, usually holds for a viscoelastic material at a small strain. This phenomenon is called linear viscoelasticity (LVE). Within the LVE region, the viscoelastic parameters G and G" remain constant when the amplitude of the applied deformation is changed. Consequently, parameters measured within the LVE region are considered material characteristics at the observation time (frequency). 

The sample must have reached steady state before cessation of the test or the application of a second step. Steady state in a creep test is seen as a constant slope in the strain curve. A constant slope in the stress curve may also be seen in a stress relaxation test, but often the signal is lost in the noise. A material that is liquid-like in real time will need a test period of 5 to 10 min. A stress relaxation test is likely to be somewhat shorter than a creep test since the signal inevitably decays into the noise at some point. A creep test will last indefinitely but will probably reach steady state within an hour. For a material that is a solid in real time, all experiments should be longer as molecular motion is, by definition, slower. Viscoelastic materials will lie in between these extremes. Polymer melts can take 1 hr or more to respond in a creep test, but somewhat less time in a stress relaxation test. 

Real (viscoelastic) materials give an intermediate response that is an exponential curve. The exponential time constants associated with the curve are used to approximate the relaxation times of the material itself. Thus, the shape of the output curve is analyzed to give viscoelastic information, although this model fitting is only strictly legitimate in the linear viscoelastic region. Workers have shown that the mechanical parts of the models (springs and dashpots) can be associated with specific parts of a food s makeup. 

Thus, we may give a good description of a linear viscoelastic material in terms of relaxed, and unrelaxed elastic constants and a distribution of relaxation times (- this is not necessarily the same distribution for each elastic constant ). These all have to be found from experiments. In general it is possible to find some of the relaxed and unrelaxed elastic constants and to estimate the distribution of relaxation times. 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

The number of PPE particles dispersed in the SAN matrix, i.e., the potential nucleation density for foam cells, is a result of the competing mechanisms of dispersion and coalescence. Dispersion dominates only at rather small contents of the dispersed blend phase, up to the so-called percolation limit which again depends on the particular blend system. The size of the dispersed phase is controlled by the processing history and physical characteristics of the two blend phases, such as the viscosity ratio, the interfacial tension and the viscoelastic behavior. While a continuous increase in nucleation density with PPE content is found below the percolation limit, the phase size and in turn the nucleation density reduces again at elevated contents. Experimentally, it was found that the particle size of immiscible blends, d, follows the relation d --6 I Cdispersed phase and C is a material constant depending on the blend system. Subsequently, the theoretical nucleation density, N , is given by... 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

A viscoelastic material is characterized by at least three phenomena the presence of hysteresis, which is observed on stress-strain curves, stress relaxation which take place where step constant strain causes decreasing stress and creep occurs where step constant stress causes increasing strain. 

When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep. 

At a time to, a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period. The material responds to the stress with a strain that increases until the material ultimately fails. When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time ti, after which the strain immediately decreases (discontinuity) then gradually decreases at times t > ti to a residual strain [2,23-26],... 

For viscoelastic materials combinations of these two models can be used, e.g. a spring and a dashpot in series or parallel. The first combination is called the Maxwell element, its response under constant stress is the sum of that of its two components ... 

Figure 7.2. Definitions of stress, strain, and modulus. Stress is defined as force per unit area, and strain is the change in length divided by the original length. When stress is plotted versus strain, then the slope is the modulus (A). When the load is removed, any strain remaining is called permanent or plastic deformation (B). When elastic materials are loaded, they are characterized by a constant strain as a function of time, whereas viscoelastic materials have strains that increase with time (C).

Another approach that has physical merit is to model the behavior of viscoelastic materials as a series of springs (elastic elements) and dashpots (viscous elements) either in series or parallel (see Figure 8.1). If the spring and dashpot are in series, which is described as a Maxwell mechanical element, the stress in the element is constant and independent of the time and the strain increases with time. 

For common liquids, the viscosity is a material constant which is only dependent on temperature and pressure but not on rate of deformation and time. For polymeric liquids, the situation is much more complicated viscosities and normal stress coefficients differ with deformation conditions. Because polymer melts are viscoelastic their flow is accompanied by elastic effects, due to which part of the energy exerted on the system is stored in the form of recoverable energy. For this reason the viscosities are time and rate dependent polymer melts are viscoelastic. 

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