Viscoelastic response functions compliance

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 viscoelastic response of equilibrium rubber networks can be obtained by measuring the shear and tensile moduli or compliances as a function of time, or the corresponding dynamic moduli and compliances as a function of frequency. As discussed in Section 5.2, the measurements of any viscoelastic function can be converted to another viscoelastic function. 

A further set of tests was conducted in order to evaluate the accuracy of the finite-element code for the case where creep is followed by creep recovery. A qualitative depiction of the loading and the resulting creep strain is given in Figure 11. Rochefort and Brinson O) presented experimental data and analytical predictions on the creep and creep recovery characteristics of FM-73 adhesive at constant temperature. The Schapery parameters necessary to characterize the viscoelastic response of FM-73 at a fixed temperature of 30 °C are obtained by applying a least-squares curve fit to the data presented in Reference 50. The resulting analytical expressions for the creep compliance function D(i ), the shift function and the nonlinear parameters go> 82 presented in Table 4. 

Equation (2.32) indicates that the strains arising from various molecular mechanisms add simply in the compliances and in principle can be separated. On the other hand, the stresses do not add and the various mechanisms cannot be easily resolved in modulus functions. To understand the individual contributions to the viscoelastic response, this additivity property of the creep complicance J(t) is helpful. The effect in low molecular weight polymers discussed in Section 2.6.1 has helped to determine the maximum compliance contributed by the local segmental (a) modes... 

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

These two mathematical Equations (4.59) and (4.60) illustrate an important feature about linear viscoelastic measurements, i.e. the central role played by the relaxation function and the compliance. These terms can be used to describe the response of a material to any deformation history. If these can be modelled in terms of the chemistry of the system the complete linear rheological response of our material can be obtained. 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Dynamic mechanical tests have been widely applied in the viscoelastic analysis of polymers and other materials. The reason for this has been the technical simplicity of the method and the low tensions and deformations used. The response of materials to dynamic perturbation fields provides information concerning the moduli and the compliances for storage and loss. Dynamic properties are of considerable interest when they are analyzed as a function of both frequency and temperature. They permit the evaluation of the energy dissipated per cycle and also provide information concerning the structure of the material, phase transitions, chemical reactions, and other technical properties, such as fatigue or the resistance to impact. Of particular relevance are the applications in the field of the isolation of vibrations in mechanical engineering. The dynamic measurements are a... 

Viscoelastic behavior is a time-dependent mechanical response and usually is characterized with creep compliance, stress-relaxation, or dynamic mechanical measurements. Since time is an additional variable to deformation and force, to obtain unique characterizing functions in these measurements one of the usual variables is held constant. 

The viscoelastic behavior is evaluated by means of two types of methods static tests and dynamic tests. In the first calegtuy a step change of stress or strain is applied and the stress or strain response is recorded as a function of time. Stress relaxation, creep compliance, and creep recovery are static methods. The dynamic tests involve the imposition of an oscillatory strain or stress. Every technique is described in the following sections. 

Although the Cole-Cole plot was first introduced in the context of a dielectric relaxation spectrum, it helped discover that the molecular mechanism underlying both dielectric relaxation and stress relaxation are substantially identical (44). Figure 8.13 provides an illustration, with temperature instead of frequency. Specifically, the same molecular motions that generate a frequency dependence for the dielectric spectrum are also responsible for the relaxation of orientation in polymers above Tg. Subsequently the Cole-Cole type of plot has been applied to the linear viscoelastic mechanical properties of polymers, especially in the vicinity of the glass transition, including the dynamic compliance and dynamic viscosity functions. 

For a viscoelastic liquid in the creep phase, the strain begins at some small value, then builds up rapidly but at a decreasing rate until finally reaching a steady state at which strain simply increases linearly with time. In the recovery phase, a viscoelastic liquid recoils back toward zero and finally reaches equilibrium at some smaller total strain than that at the time of unloading. The response over the entire range of time is linear if the shear stress is chosen small enough. In the linear range, the ratio of shear strain to shear stress in the creep phase is a material function of time alone, the shear creep compliance J t) ... 

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. 

In the above unsteady tests, if one keeps the level of imposed stress and strain low enough, the measured material functions show an independence from these applied stimuli levels, exhibiting only a dependence on time (or frequency). This type of response indicates linear viscoelastic behavior. The primary modes of deformation employed in these tests are either shear or extension. If there is no volume change accompanying the deformation, a single modulus or compliance, whether real or complex, but a function of time (or frequency) and temperature only, suffices to characterize the material behavior. We will define moduli and compliances further below. Let us now start examining these and other key topics in linear viscoelasticity. 

We will now consider another type of transient response. Let a shear stress a be applied at the viscoelastic solution at = 0. In a general case, a time-dependent shear strain is developed that can be measured to get the creep comphance J(t). The creep compliance has the dimensions of a reciprocal modulus, and it is therefore an increasing function of time. From theories of viscoelasticity, it is possible to calcrflate the creep compliance from the relaxation modrflus and inverse ... 

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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