Viscoelastic Stress-Strain Relations

Similarly, the differential operator equations for the shear and dilata-tional responses of a viscoelastic material may be written analogous to the one-dimensional case in Chapter 5 (Eq. 5.20 or Eq. 5.26b) as, 

Alternately, the viscoelastic stress-strain relationships in the transform domain may be written as. 

The two Eqs. 9.10 may be recombined in the transform domain to obtain an expression relating the total stress and strain tensors, Oy and By. In doing so, the relationship between Lame s constant, X (s), and bulk and shear moduli, K (s) and G (s), will be recovered. Further manipulations in the transform domain result in the usual relationship between total strain and stress, analogous to Eq. 2.36 

Note that with Eqs. 9.10, 9.12 and 9.14, we again have the viscoelastic constitutive law represented in the transform domain in a form equivalent to elasticity. These relationships will then allow us to utilize the correspondence principle as in Chapter 8 to solve 2D and 3D viscoelastic boundary value problems based on elasticity solutions. 

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The stress-strain relations for viscoelastic materials are reviewed. The simplest case of intrinsic absorption in polymers is a molecular relaxation mechanism with a single relaxation time. However, the relaxation mechanisms which lead to absorption of sound are usually more complicated, and are characterized by a distribution of relaxation times. Under causal linear response conditions the attenuation and dispersion of sound in a... 

Many materials, particularly polymers, exhibit both the capacity to store energy (typical of an elastic material) and the capacity to dissipate energy (typical of a viscous material). When a sudden stress is applied, the response of these materials is an instantaneous elastic deformation followed by a delayed deformation. The delayed deformation is due to various molecular relaxation processes (particularly structural relaxation), which take a finite time to come to equilibrium. Very general stress-strain relations for viscoelastic response were proposed by Boltzmann, who assumed that at low strain amplitudes the effects of prior strains can be superposed linearly. Therefore, the stress at time t at a given point in the material depends both on the strain at time t, and on the previous strain history at that point. The stress-strain relations proposed by Boltzmann are [4,39] ... 

Stress-Strain Relations for Viscoelastic Materials. The viscoelastic behaviour of an elastomer varies with temperature, pressure, and rate of strain. This elastic behaviour varies when stresses are repeatedly reversed. Hence any single mathematical model can only be expected to approximate the elastic behaviour of actual substances under limited conditions 2J. ... 

The viscoelastic stress-strain equation, Equation (4) can be expressed in finite element formulation which relates the stress tensor a.. at time index n and cell centre (ij) to the corresponding strain tensor arising from the movement of the adjoining cell corners. Using backward differences for the time step, at time index n. 

Fibers are linearly elastic up to fracture (ct = Ee), but the stress-strain relations of typical polymer matrices are nonlinear, as a consequence of their viscoelastic behavior (see Fig. 15.10) therefore, ci cannot be replaced by E e . Furthermore, the fibers and the matrix in the laminae are assumed to fail independently, as if they were each tested alone. The behavior of the... 

The failure phenomena of viscoelastic materials consist of two factors. One is the time dependence of the hulk stress-strain relation of the material. For example, higher strain rate increases the stress level of the material and consequently the stress or the strain at break will he different from those of the case of the lower rate. The second factor is the time dependence of the failure mechanism itself, namely, the time dependence of the failure initiation mechanism. 

Biot, M. A., 1954.r/ieory of Stress-strain Relations in Anisotropic Viscoelasticity and Relaxation Phenomena, J. Appl. Phys., 25(11) ppl385-1391. 

For example, the stress-strain relation for shear strain in an isotropic material is <7 = Ge. For a viscoelastic material this relation is modified as follows ... 

The Boltzman Superposition Principle is one starting point for inclusion of structural relaxation losses. An equally valid starting point is to include in equation 9 time derivatives (first-order and higher) of stress and strain. It can he shown that this approach is equivalent to the above integral representation (10). Finally, modified stress-strain relations, to describe viscoelastic response, have also been formulated using fractional derivatives (11). 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

The general factorable single integral constitutive equation is an equation that describes well the viscoelastic properties of a large class of crosslinked rubbers > and the elasticoviscous properties of many polymer melt8 under various types of deformation An appropriate way to compare the stress-strain relation for cured elastomers and polymer melts therefore is to calculate the strain-dependent function contained in this constitutive equation from experimental results and to compare the strain measures so obtained. 

For cross-linked polymers at equilibrium (Chapter 14, Section Cl), an equation of this form provides a good empirical fit to stress-strain relations up to moderate extensions," although it is not a proper constitutive equation and is inconsistent with behavior in other types of deformation. " "" The data of Fig. 13-19 and many other examples "" " " " are quite well described by equation 31 with time-dependent Cl and C2 as illustrated in Fig. 13-20. The sum, 6(C + C2), corresponds to the relaxation modulus E(t) of linear viscoelasticity. The term with Ci (which corresponds to neo-Hookean strain dependence) relaxes first and the term with C2 relaxes about two decades later. 

A full discussion of stress-strain relations at equilibrium for large deformations in rubbery cross-linked polymers is beyond the scope of this chapter there have been many investigations of uniaxial deformations (simple extension, compression,) torsion, and biaxial deformations, which have been critically reviewed elsewhere. A few comments will introduce the discussion of nonlinear viscoelastic behavior. 

The stress-strain relations (6.25) are of the familiar form employed in linear viscoelasticity, except that the retardation spectra incorporate now aging affects, and all instantaneous compliances age with time. Recalling (6.18) for a discrete spectrum of retardation times Xr, expressions (6.25) read... 

Abdel-Tawab K, Weitsman YJ (1998) A Coupled Viscoelasticity/Damage Model with Application to Swirl-Mat Composites. International Journal of Damage Mechanics 7(4) 351-380 Biot MA (1954) Theory of Stress-Strain Relations in Anisotropic Viscoelasticity and Relaxation Phenomena. Journal of Applied Physics 25(11) 1385 Callen HB (1960) Thermodynamics. Wiley, New York... 

Particulate materials, such as clay or particulate gels of the type discussed in Chapter 4, may be plasticy rather than viscoelastic. Two simple types of plastic behavior are illustrated in Fig. 18b a perfectly plastic material is elastic up to t)xt yield stress, Oy > but it deforms without limit if a higher stress is applied in a linearly hardening material there is a finite slope after the yield stress. In real plastic materials, the stress-strain relations are likely to be curved, rather than linear. If the stress is raised above Oy and then released, the elastic strain is recovered but the plastic strain is not. This differs from viscous behavior in its time-dependence if the stress on a linearly hardening plastic material is raised to Oh and held constant, the strain remains at a viscoelastic material would continue to deform at a rate proportional to Ou/ri-... 

II. One-dimensional Constitutive Equations, (a) One-dimensional stress-strain relations for a non-aging linear viscoelastic material take the form... 

In the limit of linear stress-strain relations, the relaxation modulus does not depend on the initial deformation step and the rheological properties are only described by transient functions. Equation 9.11 suggests that the relaxation modulus describes the stress relaxation after the onset of a step function shear strain. In viscoelastic liquids of entangled solutions of rod-shaped micelles, an applied stress is always relaxing to zero after inflnite long periods of time. 

The general formulation of the evolution of linear isotropic viscoelastic material in time is governed by the stress-strain relation, in which the stress can be expressed as a convolution of the strain rate with a relaxation function as in... 

The different forms of the modulus are still widely disputed and different opinions are expressed in the literature. Hsu and Mark " " prepared networks of polybutadiene by endlinking. According to dynamic measurements polybutadiene has a high plateau modulus in viscoelasticity and one can expect a strong contribution to the modulus from entanglements. In this study experiments have been fitted to the stress-strain relations with the Flory constraint-fluctuation model as in ref. 245. The authors concluded that there is no contribution from entanglements to the modulus. The same conclusion was drawn in refs. 245-248. 

Figure 1 Diagram showing the viscoelastic behavior of a mucous gel. The first two panels illustrate stress-strain relations in idealized materials, namely an elastic solid, for which the displacement or strain is proportional to the applied force or stress, and a viscous liquid, for which the rate of strain (displacement/time) is proportional to the stress. Mucus is a viscoelastic semisolid. It responds instantaneously as a solid, with a very rapid displacement in response to an applied force. This is followed by a transition to a liquidlike response, in which the rate of strain is constant with time. Finally a zone of viscous response is reached, in which the rate of displacement is constant with time. After release of the applied force, the mucous gel recoils only partially to its initial position.

The theory relating stress, strain, time and temperature of viscoelastic materials is complex. For many practical purposes it is often better to use an ad hoc system known as the pseudo-elastic design approach. This approach uses classical elastic analysis but employs time- and temperature-dependent data obtained from creep curves and their derivatives. In outline the procedure consists of the following steps ... 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

PP bead foams of a range of densities were compressed using impact and creep loading in an Instron test machine. The stress-strain curves were analysed to determine the effective cell gas pressure as a function of time under load. Creep was controlled by the polymer linear viscoelastic response if the applied stress was low but, at stresses above the foam yield stress, the creep was more rapid until compressed cell gas took the majority of the load. Air was lost from the cells by diffusion through the cell faces, this creep mechanism being more rapid than in extruded foams, because of the small bead size and the open channels at the bead bonndaries. The foam permeability to air conld be related to the PP permeability and the foam density. 15 refs. 

Polymers are viscoelastic materials meaning they can act as liquids, the visco portion, and as solids, the elastic portion. Descriptions of the viscoelastic properties of materials generally falls within the area called rheology. Determination of the viscoelastic behavior of materials generally occurs through stress-strain and related measurements. Whether a material behaves as a viscous or elastic material depends on temperature, the particular polymer and its prior treatment, polymer structure, and the particular measurement or conditions applied to the material. The particular property demonstrated by a material under given conditions allows polymers to act as solid or viscous liquids, as plastics, elastomers, or fibers, etc. This chapter deals with the viscoelastic properties of polymers. 

Dynamic mechanical measurements are performed at very small strains in order to ensure that linear viscoelasticity relations can be applied to the data. Stress-strain data involve large strain behavior and are accumulated in the nonlinear region. In other words, the tensile test itself alters the structure of the test specimen, which usually cannot be cycled back to its initial state. (Similarly, dynamic deformations at large strains test the fatigue resistance of the material.)... 

Rheological properties of food materials over a wide range of phase behavior can be expressed in terms of viscous (viscometric), elastic and viscoelastic functions which relate some components of flie stress tensor to specific components of the strain or shear rate response. In terms of fluid and solid phases, viscometric... 

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