Linear viscoelasticity elastic material functions

Figure 2.34 Schematic of Newtonian, elastic, linear, and non-linear viscoelastic regimes as a function of deformation and Deborah number during deformation of polymeric materials.

These linear viscoelastic dynamic moduli are functions of frequency. For a suspension or an emulsitm material at low frequency, elastic stresses relax and viscous stresses dominate with the result that the loss modulus, G", is higher than the storage modulus, G. For a dilute solution, G" is larger than G over the entire frequency range, but they approach each other at higher frequencies as shown in Fig. 3. 

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. 

For small stresses and strains, if both the elastic deformation at equilibrium and the rate of viscous flow are simple functions of the stress, the polymer is said to exhibit linear viscoelasticity. While most of the material in this chapter refers to linear viscoelasticity, much of the material in Chapters 9 and 10 concerns nonlinear behavior. 

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. 

Elastomers are typically viscoelastic and exhibit both viscous as well as elastic characteristics when rmdergoing deformations. The viscous component (o = t], where, a is the stress, t] is the coefficient of viscosity and is the change of strain as a function of time) takes care of the energy dissipated as heat after a strain/stress is applied and followed by its removal while the elastic component (o = Ee) brings back the material towards original dimension, or, in other words, strain depends on stress applied to the materials and time. The overall strain is then governed by the equation where the two terms are separable, as in Eq. (2) is called as linear viscoelasticity and usually it is applicable only for small deformations, where, t is... 

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. 

This chapter deals with fundamental definitions, constitutive equations of a viscoelastic medium subject to infinitesimal strain, and the nature and properties of the associated viscoelastic functions. General dynamical equations are written down. Also, the boundary value problems that will be discussed in later chapters are stated in general terms. Familiar concepts from the Theory of Linear Elasticity are introduced in a summary manner. For a fuller discussion of these, we refer to standard references (Love (1934), Sokolnikoff (1956), Green and Zerna (1968), Gurtin (1972)). Coleman and Noll (1961) have shown that the theory described here may be considered to be a limit, for infinitesimal deformations, of the general (non-linear) theory of materials with memory. 

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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