Viscoelasticity elastic material functions

Elastic behavior of liquids is characterized mainly by the ratio of first differences in normal stress, Ni, to the shear stress, t. This ratio, the Weissenberg number Wi = Nih, is usually represented as a function of the rate of shear y. Figure 7 depicts flow curves of some viscoelastic fluids, and Figure 8 presents a dimensionless standardized material function of these fluids. It again verifies that they behave similarly with respect to viscoelastic behavior under shear stress. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

When a periodic deformation, e = eo-sin cot, is imposed upon a purely elastic material, the stress will also be periodic according to d = do sin cot (the use of a cosine function gives the same results). With viscoelastic materials, however, a phase shift 8 occurs ... 

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).

A study was made of the ability of viscoelastic models to describe the measured material functions of unplasticised PVC during extrusion and to determine whether it was possible to reproduce the elastic properties of the large entrance pressure drop and small extrudate swell during the extrusion of PVC using a capillary rheometer. Models used were the Phan-Thien and Tanner model and the K-BKZ-Wagner model with a single exponential damping... 

The same parameters can also be determined by applying a constant shear stress to the interface and measuring the resulting shear strain as a function of time (see fig. 3.40), so-called interfacial creep tests. At t = 0, a shear stress is suddenly applied, and kept constant thereafter. For ideally viscous monolayers a steady increase of the shear strain with t will be observed, while for an elastic material the observed strain will be instantaneous and constcmt in time. For a viscoelastic material, as in fig. 3.40, there is first am Instantaneous increase AB in the strain, the elastic response followed by a delayed elastic response BC and a viscous... 

FIGURE 5.8 Various kinds of behavior of a material under stress, (a) Stress applied as a function of time, (b) Resulting strain as a function of time for a purely elastic material, (c) Same, for a Newtonian liquid, (d) Same, for a viscoelastic material. 

Stress and Strain Definitions. As remarked, the material functions relate the stress and strain responses of the material through a constitutive equation. For the elastic material, there is no time dependence and the relationships are relatively simple. In the case of linear viscoelasticity, equations that take into accoimt the time history of the stresses or strains are required. First, stress and strain are defined. 

The Viscoelastic Material Functions. In linear viscoelasticity, the moduli discussed for the elastic case can be recast as time- or fi equency-dependent functions. The same is true for the compliance functions that are discussed here. For simplicity, consider the shear modulus G which becomes G(t) or G (a>) in the case of the viscoelastic material. An important point here is that the viscoelastic modulus functions all exhibit time (frequency) dependence. Hence, one will have functions for K(t) and E(t) [or, eg, G t) and v i)] and these are required in the case of a three-dimensional strain or stress field. 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

Finally, the power of finite elasticity theory is that once the material properties [Wi and W2 or are known, the stresses in any deformation field can be calculated. There is an extensive literature on ways to represent the material functions and, in fact, commercial finite element codes use finite elasticity theory in calculations that can be important in applications that range from the stresses in automobile tires (105) to those in earthquake bearings for large buildings (106). One feature of the K-BKZ theory to be discussed next is that it retains the structure of finite elasticity theory and includes time-dependent properties of the viscoelastic materials that were discussed in the earlier sections of this article. 

For linear materials, the stress is proportional to the strain. For linear elastic materials, Young s modulus ( ), equal to the slope of the stress-strain curve, is constant. For viscoelastic materials, E is dependent on the deformation rate. A material is called linear viscoelastic if the stress is proportional to the strain, despite the time-dependence. In this case, the Young s modulus is a function of the deformation rate only. In Fig. 2.13, the stress-strain behavior of a hnear viscoelastic material is shown schematically for different deformation rates, assuming that the material is exposed to a constant strain. 

The notation used to describe the contacts is shown in Figure 1. P t) is the time dependent applied load, S P,t) the deformation, a(P,t) the contact radius, and R and Ri the radii of curvature of the two bodies at the point of contact. We consider only flat substrates so that R R and R2 = >. Each elastic material is described by its Young modulus E, Poisson ratio v, and is assumed to be isotopic so that the shear modulus is G = Ejl + v). Viscoelastic materials are assumed to be linear with stress relaxation functions E t) and creep compliance functions J t), All properties are assumed to be independent of depth. 

For a perfectly elastic material, Hooke s law is obeyed, a = 6. This implies that the width of the hysteresis loop is zero (the dashed line in Fig. 5.17) and evaluation of the integral in Eq. 5.53 results in identically zero. For a viscoelastic material, we can write the stress as a function of the strain via the complex modulus (Eq. 5.38) and then rearrange in terms of the storage and loss moduli... 

Typical examples of tensile (isochronous) linear and nonlinear stress-strain diagrams for elastic and viscoelastic materials are shown in Fig, 10.1. For elastic materials, the response is time independent, so there is a single curve for multiple times and the nonlinearity is apparent as a deviation of the stress-strain response from linear. For linear viscoelastic materials, the isochronous response is linear, but the effective modulus decreases with time so that the stress-strain curves at different times are separated from one another. When a viscoelastic material behaves nonlinearly, the isochronous stress-strain curves begin to deviate from linearity at a certain stress level. Fig. 10.2 shows creep compliance data for an epoxy adhesive as a function of stress level for various time intervals after initial loading. 

A specimen is rapidly strained by an amount sq and then the stress a-(t) required to hold that strain is measured as a function of time. For an elastic material, ER(f) is the Yoimg s modulus and is constant in time. For a purely viscous material, ER(f) quickly goes to zero. To obtain the temperature behavior of a viscoelastic material, (1) is evaluated at some particular time, usually 10s, during the relaxation process. The values of Er(10) are then plotted against temperature as shown in Figure 9.10. 

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10. 

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