Viscoelastic behavior stress relaxation

Maxwell model A mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus E) in series with a dashpot of coefficient of viscosity (ri). It is an isostress model (with stress 8), the strain (e) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as stress relaxation and creep with Newtonian flow analysis. Also called Maxwell fluid model. See stress relaxation viscoelasticity. Maxwell-Wagner efifect See dielectric, Maxwell-Wagner effect. 

While the stresses O12 and O22 consist of an elastic term plus a dissipative integral as the standard one-dimensional model of linear viscoelasticity, On is a purely elastic force distribution, i.e., there is no stress-relaxation. This behavior seems in contrast to the assumed isotropy of the material. 

Polymer properties exhibit time-dependent behavior, which is dependent on the test conditions and polymer type. Figure 1.7 shows a typical viscoelastic response of a polymer to changes in testing rate or temperature. Increases in testing rate or decreases in temperature cause the material to appear more rigid, while an increase in temperature or decrease in rate will cause the material to appear softer. This time-dependent behavior can also result in long-term effects such as stress relaxation or creep. These two time-dependent behaviors are shown in Fig. 1.8. Under a fixed displacement, the stress on the material will decrease over time, and this is called stress relaxation. This behavior can be modeled nsing a... 

It is known that bone is a semi-brittle material and will experience fracture with forces greater than its ultimate strength. Based on the maximum amount of stress, the viscoelastic behavior of tibia showed a tendency to propagate the fracture lines and also higher stresses were reached compared to the isotropic and transversely isotropic property cases of tibia during the impact cycle. This indicates the dependency of a viscoelastic material to the time and also the strain rate, as expected. Also, the observed reduction in the maximum amount of stress after removing the impact load can be seen as a consequence of stress relaxation in tibia. The minimum stress was found to be when a transversely isotropic material property was considered for tibia. 

Long time dynamic load involves behaviors such as creep, fatigue, and impact. T vo of the most important types of long-term material behavior are more specifically viscoelastic creep and stress relaxation. Whereas stress-strain behavior usually occurs in less than one or two hours, creep and stress relaxation may continue over the entire life of the structure such as 100,000 hours or more. 

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. 

Viscoelastic stress-relaxation data are usually presented in one of two ways. In the first, the stress manifested as a function of time. Families of such curves may be presented at each temperature of interest. Each curve representing the stress-relaxation behavior of the material at a different level of... 

Creep and stress relation Creep and stress relaxation behavior for plastics are closely related to each other and one can be predicted from knowledge of the other. Therefore, such deformations in plastics can be predicted by the use of standard elastic stress analysis formulas where the elastic constants E and y can be replaced by their viscoelastic equivalents given in Eqs. 2-19 and 2-20. 

Most pigmented systems are considered viscoelastic. At low shear rates and slow deformation, these systems are largely viscous. As the rate of deformation or shear rate increases, however, the viscous response cannot keep up, and the elasticity of the material increases. There is a certain amount of emphasis on viscoelastic behavior in connection with pigment dispersion as well as ink transportation and transformation processes in high-speed printing machines (see below). Under periodic strain, a viscoelastic material will behave as an elastic solid if the time scale of the experiment approaches the time required for the system to respond, i.e., the relaxation time. Elastic response can be visualized as a failure of the material to flow quickly enough to keep up with extremely short and fast stress/strain periods. 

The effects attributed to entangling interactions, e.g., the plateau region in stress relaxation, appear most prominently at high concentrations and in melts. It is important, however, to distinguish this interaction from other types which are present at lower polymer concentrations. To make the separation properly, it is necessary to examine viscoelastic behavior at all levels of concentration, beginning at infinite dilution. 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

The creep of a viscoelastic body or the stress relaxation of an elasacoviscous one is employed in the evaluation of T] and G. In such studies, the long-time behavior of a material at low temperatures resembles the short-time response at high temperatures. A means of superimposing data over a wide range of temperatures has resulted which permits the mechanical behavior of viscoelastic materials to be expressed as a master curve over a reduced time scale covering as much as twenty decades (powers of ten). 

In a stress relaxation test, a polymer test specimen is deformed by a fixed amount, eo, and the stress required to hold that amount of deformation is recorded over time. This test is very cumbersome to perform, so the design engineer and the material scientist have tended to ignore it. In fact, several years ago, the standard relaxation test ASTM D2991 was dropped by ASTM. Rheologists and scientists, however, have been consistently using the stress relaxation test to interpret the viscoelastic behavior of polymers. 

Summary In this chapter, a discussion of the viscoelastic properties of selected polymeric materials is performed. The basic concepts of viscoelasticity, dealing with the fact that polymers above glass-transition temperature exhibit high entropic elasticity, are described at beginner level. The analysis of stress-strain for some polymeric materials is shortly described. Dielectric and dynamic mechanical behavior of aliphatic, cyclic saturated and aromatic substituted poly(methacrylate)s is well explained. An interesting approach of the relaxational processes is presented under the experience of the authors in these polymeric systems. The viscoelastic behavior of poly(itaconate)s with mono- and disubstitutions and the effect of the substituents and the functional groups is extensively discussed. The behavior of viscoelastic behavior of different poly(thiocarbonate)s is also analyzed. 

Because all tissues are viscoelastic this means that their mechanical properties are time dependent and their behavior is characterized both by properties of elastic solids and those of viscous liquids. The classic method to characterize a viscoelastic material is to observe the decay of the stress required to hold a sample at a fixed strain (stress relaxation) or by the increasing strain required to hold a sample at a fixed stress (creep) as diagrammed in Figure 7.1 and explained further in Figure 7.2. Viscoelastic materials undergo processes that both store (elastic) and dissipate (viscous)... 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

This behavior results from stress relaxation and other viscoelastic phenomena that are typical of TPs. In addition to using heat TPs such as polyolefins, neoprenes, silicones, and other cross-linkable TPs are example of plastics that can be given memory either by radiation or by chemically curing. Fluoroplastics need no such curing. When this phenomenon of memory is applied to fluoroplastics such as TFE, FEP, ETFE, ECTFE, CTFE, and PVF, interesting and useful high-temperature or wear-resistant applications become possible. 

In this section we are going to examine such viscoelastic properties in some detail and we will start by examining in turn three important mechanical methods of measurement creep, stress relaxation, and dynamic mechanical analysis. This will lead us to interesting things like time-temperature equivalence and a discussion of the molecular basis of what we have referred to as relaxation behavior. 

Further aspects of the viscoelastic behavior of ESIs which have been reported to date include linear stress relaxation behavior of amorphous ESI [40] and the creep behavior of amorphous ESI in the glass transition region [41]. Chen et al. [42]... 

Three types of experiments are used in the study of viscoelasticity. These involve creep, stress relaxation, and dynamic techniques. In creep studies a body is subjected to a constant stress and the sample dimensions are monitored as a function of lime. When the polymer is lirst loaded an immediate deformation occurs, followed by progressively slower dimensional changes as the sample creeps towards a limiting shape. Figure 1-3 shows examples of the different behaviors observed in such experiments. 

Limitations to the effectiveness of mechanical models occur because actual polymers are characterized by many relaxation times instead of single values and because use of the models mentioned assumes linear viscoelastic behavior which is observed only at small levels of stress and strain. The linear elements are nevertheless useful in constructing appropriate mathematical expressions for viscoelastic behavior and for understanding such phenomena. 

Figure 4.13 shows the deformation behavior of a viscoelastic material. In response to a stress at time tj, the material deforms exponentially with time. When the stress is removed (or lowered, as is the case when the pad passes from a high region to a low region), the viscoelastic material rebounds with an exponential time dependence. However, the strain in the viscoelastic material does not necessarily completely relax upon unloading. Viscoelastic behavior is described mathematically by ... 

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