Solid polymers nonlinear viscoelasticity

D. W. Hadley and I. M. Ward, Anisotropic and nonlinear viscoelastic behaviour in solid polymers. Rep. Prog. Phys. 38(10) 1143 (1975). 

Nonlinear Viscoelastic Response of Solid-like Polymers. The study of the nonlinear viscoelastic response of solid or solid-like polymers is one that has... 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

Viscoelasticity is a phenomenon observed in most of the polymers since they possess elastic and viscous characteristics when deformed. The properties such as creep, stress relaxation, mechanical damping, vibration absorption and hysteresis are included in viscoelasticity. If a material shows linear variation of strain upon the application of stress on it, its behavior is said to be linear viscoelastic. Elastomers and soft biological tissues undergo large deformations and exhibit time dependent stress strain behavior and are nonlinear viscoelastic materials. The non-linear viscoelastic properties of solid polymers are often based on creep and stress-... 

Material functions must however be considered with respect to the mode of deformation and whether the applied strain is constant or not in time. Two simple modes of deformation can be considered simple shear and uniaxial extension. When the applied strain (or strain rate) is constant, then one considers steady material functions, e.g. q(y,T) or ri (e,T), respectively the shear and extensional viscosity functions. When the strain (purposely) varies with time, the only material functions that can realistically be considered from an experimental point of view are the so-called dynamic functions, e.g. G ((D,y,T) and ri (a), y,T) or E (o),y,T) and qg(o),y, T) where the complex modulus G (and its associated complex viscosity T] ) specifically refers to shear deformation, whilst E and stand for tensile deformation. It is worth noting here that shear and tensile dynamic deformations can be applied to solid systems with currently available instruments, whUst in the case of molten or fluid systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear viscoelasticity, relevant to processing. 

The large strain response in the glassy or semicrystalline state is that of a nonlinear viscoelastic solid. However, both engineering and theoretical approaches to plasticity in polymers have largely developed as an independent discipline, in which (Ty plays a central role, in spite of its somewhat arbitrary definition (indeed it is not always possible to associate cty with a maximum in the force-deformation curve [5]). This is because in practice the yield point, rather than the ultimate strength, is usually considered to be the failure criterion for ductile materials. 

Rheology is a branch of physics concerned with the time-dependent deformation of solids and the viscous flow of liquids. Rheological models can be used to illustrate the nonlinear viscoelastic response of rPET polymer concrete. These models are mechanical comparisons that demonstrate the interrelationship between the elastic and viscous response of polymers. Simple and complex models can be proposed to... 

Brereton, M.S., Croll, S.G., Duckett, R.A. etal. (1974) Nonlinear viscoelastic behavior of polymers - an implicit equation approach. J. Mech. Phys. Solids, 22, 97. 

Nonlinear Viscoelastic Response of Solid-like Polymers. The study of the nonlinear viscoelastic response of solid or solid-like polymers is one that has been relatively neglected. One reason is that there is no real molecular framework for the description of these materials, particularly when they are amorphous. The other reason is that many workers in the field have adopted the framework of metal plasticity and then made modifications to try to adapt it to, for example, the fact that amorphous polymers do not readily admit to treatment with the physics of dislocations. In the case of semicrystalline polymers, the... 

R. M. Shay and J. M. Caruthers, A New Nonlinear Viscoelastic Constitutive Equation for Predicting Yield in Amorphous Solid Polymers , J. Rheol. 30, 871-827 (1986). 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

Non-Newtonian fluids have both viscous and elastic properties, and they are called viscoelastic fluids. An example is so-called "silly putty," which is made from poly-dimethyl-siloxane (silicone). It flows like a liquid out of the container, but when it forms a ball, it behaves as elastic, i.e., it bounces back. The crucial factor determining the viscous and elastic behavior is the time period of the force applied short force pulse leads to elastic response, whereas long-lasting force causes flow. The viscoelasticity in polymers is due to shear-induced entanglements and nonlinear behavior of tire chains, coils. A well-known natural viscoelastic material is for, example, the egg white, which springs back when a shear force is released. A polymer resembles both liquid and solids. 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

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