Viscoelasticity time dependence

One of the fundamental methods used to characterize the viscoelastic time-dependent behavior of a polymer is the relaxation test. In a relaxation test, a constant strain is applied quasi-statically to a uniaxial tensile (or compression or torsion) bar at zero time. That is, the bar is suddenly stretched to a new position and rigidly fixed such that the strain remains constant for the duration of the test. The sudden strain must not induce any dynamic or inertia effects (which explains the term quasi-static, i.e., the loading motion is sufficiently slow that inertia effects can be ignored). 

The Larson-Miller and other similar methods have been widely used for metals but here it is important to note that difficulties arise for fiber reinforced composite laminates because the constants are only valid for one configuration of the plies and a more general approach is needed. Dillard (1981) developed an incremental viscoelastic time dependent lamination theory approach that included the Tsai-Hill failure law modified to account for delayed failures using the Zhurkov time dependent failure model that will be discussed in the next section. The advantage of the Dillard approach is that information on the viscoelastic behavior as well as the delayed failure behavior of 0°, 10° and 90° plies can be used to predict the behavior of general laminate configurations. 

The second term on the right side of (6.29) expresses the effects of viscoelastic, time-dependent motions of the polymeric network on the moisture absorption process, while the third term represents the role of free volume on diffusion. 

In Sect. 2.5, we discussed briefly the case where the ratio of viscoelastic (time-dependent terms in the functions G(t), J(t)) to purely elastic effects (constant terms in these functions) is small. The possibility of expressing solutions as power series in this ratio was noted. In the present section, we shall consider solutions to first order in this parameter. It turns out that the problem of the moving load greatly simplifies, to the extent that explicit expressions for all quantities of interest may be written down, in contrast to the highly implicit equations which emerged from the exact analysis in the previous section. 

The relaxation modulus is directly used in viscoelastic time-dependent stress analysis to predict deformations and is therefore a useful relation to illustrate the association between basic molecular characteristics and engineering design parameters. 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

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 elongation of a stretched fiber is best described as a combination of instantaneous extension and a time-dependent extension or creep. This viscoelastic behavior is common to many textile fibers, including acetate. Conversely, recovery of viscoelastic fibers is typically described as a combination of immediate elastic recovery, delayed recovery, and permanent set or secondary creep. The permanent set is the residual extension that is not recoverable. These three components of recovery for acetate are given in Table 1 (4). The elastic recovery of acetate fibers alone and in blends has also been reported (5). In textile processing strains of more than 10% are avoided in order to produce a fabric of acceptable dimensional or shape stabiUty. 

Eracture mechanics concepts can also be appHed to fatigue crack growth under a constant static load, but in this case the material behavior is nonlinear and time-dependent (29,30). Slow, stable crack growth data can be presented in terms of the crack growth rate per unit of time against the appHed R or J, if the nonlinearity is not too great. Eor extensive nonlinearity a viscoelastic analysis can become very complex (11) and a number of schemes based on the time rate of change of/have been proposed (31,32). 

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

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.

Steady state, fuUy developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate-pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. 

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

The premise of the above analysis is the fact that it has treated the interfacial and bulk viscoelasticity equally (linearly viscoelastic experiencing similar time scales of relaxation). Falsafi et al. make an assumption that the adhesion energy G is constant in the course of loading experiments and its value corresponds to the thermodynamic work of adhesion W. By incorporating the time-dependent part of K t) into the left-hand side (LHS) of Eq. 61 and convoluting it with the evolution of the cube of the contact radius in the entire course of the contact, one can generate a set of [LHS(t), P(0J data. By applying the same procedure described for the elastic case, now the set of [LHS(t), / (Ol points can be fitted to the Eq. 61 for the best values of A"(I) and W. 

Fig. 22. Nomialized pull-off energy measured for polyethylene-polyethylene contact measured using the SFA. (a) P versus rate of crack propagation for PE-PE contact. Change in the rate of separation does not seem to affect the measured pull-off force, (b) Normalized pull-off energy, Pn as a function of contact time for PE-PE contact. At shorter contact times, P does not significantly depend on contact time. However, as the surfaces remain in contact for long times, the pull-off energy increases with time. In seinicrystalline PE, the crystalline domains act as physical crosslinks for the relatively mobile amorphous domains. These amorphous domains can interdiffuse across the interface and thereby increase the adhesion of the interface. This time dependence of the adhesion strength is different from viscoelastic behavior in the sense that it is independent of rate of crack propagation.

Another example of time-dependent adhesion has been reported by Rimai et al. [110], In this study, adhesion-induced stresses were found to cause an alternating block polyester/PDMS copolymer was found to flow over stacks of particles during a 2-week-long interval, suggestive of the occurrence of viscoelastic flow. 

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... 

For a linear viscoelastic material in which the strain recovery may be regarded as the reversal of creep then the material behaviour may be represented by Fig. 2.49. Thus the time-dependent residual strain, Sr(t), may be expressed as... 

The viscoelastic nature of the matrix in many fibre reinforced plastics causes their properties to be time and temperature dependent. Under a constant stress they exhibit creep which will be more pronounced as the temperature increases. However, since fibres exhibit negligible creep, the time dependence of the properties of fibre reinforced plastics is very much less than that for the unreinforced matrix. 

Mechanical properties of plastics are invariably time-dependent. Rheology of plastics involves plastics in all possible states from the molten state to the glassy or crystalline state (Chapter 6). The rheology of solid plastics within a range of small strains, within the range of linear viscoelasticity, has shown that mechanical behavior has often been successfully related to molecular structure. Studies in this area can have two objectives (1) mechanical characterization of... 

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. 

Time dependence Viscoelastic deformation is a transition type behavior that is characterized by the occurrence of both elastic strain and time-dependent flow. It is the time dependence of the mechanical properties of plastics that makes the behavior of these materials difficult to analyze by mathematical theory. 

The mechanical behavior of plastics on time-dependent applied loading can cause different important effects on materials viscoelasticity. Loads applied for short times and at normal rates (Chapter 2) causes material response that is essentially elastic in character. However, under sustained load plastics, particularly TPs, tend to creep, a factor that is included in the design analysis. 

Stress relaxation tests are alternative ways of measuring the same basic phenomenon in viscoelastic polymers as creep tests, Le. the time-dependent nature of their response to an applied stress. As such, they have also been of value in understanding the behaviour of these materials. The essence of stress relaxation tests is that strain increases with time for a given stress, so that if stress is decreased with time in a controlled manner ( relaxed ), a state... 

The general mode of operation in dynamic tests is to vary the stress sinusoidally with time. A viscoelastic solid in which the viscous element is that of a Newtonian liquid (as defined earlier) responds with a sinusoidal strain of identical oscillation frequency. However, because of the time-dependent relaxation processes taking place within the material, the strain lags behind the stress, as illustrated in Figure 7.9. 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

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