Elasticity Hookean

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

A. Winetnan, Some results for generalized neo-Hookean elastic materials, Int. J. Non-Linear Mech., 40, 271-279, 2005. 

On the other hand, if the top plate moves but its displacement is not directly proportional to the applied force (it may be either more or less than proportional to the force), the material is said to be a nonlinear (i.e., non-Hookean) elastic solid. It can be represented by an equation of the form... 

Viscoelastic behavior can be divided into five subclassifications (Figure 14.4). From 1 to 2, the material behaves as a viscous glass or Hookean elastic or glass, where chain segmental motion is quite restricted and involves mainly only bond bending and bond angle deformation. 

A spring with a modulus of G, and a dashpot containing a liquid with a viscosity of rj, have been used as models for Hookean elastic solids and Newtonian liquids, respectively. In these models, the spring stores energy in a reversible process, and the dashpot dissipates energy as heat in an irreversible process. Figure 5.3 is a stress-strain curve for a typical elastomer the straight... 

The simplest type of elasticity, Hookean elasticity, is defined by the equation... 

A polymer is said to exhibit Hookean elasticity if the ratio of stress to strain is a constant. In this case, the ratio is called Young s modulus, and is given the symbol G. 

The most straightforward rheological behaviour is exhibited on the one hand by Newtonian viscous fluids and on the other by Hookean elastic solids. However, most materials, particularly those of a colloidal nature, exhibit mechanical behaviour which is intermediate between these two extremes, with both viscous and elastic characteristics in evidence. Such materials are termed viscoelastic. 

For Hookean elastic solids the stress and strain are in phase, whereas for purely viscous liquids the strain lags 90° behind the applied stress. 

The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the stress relaxation behaviors of polymers. 

Relevant to this chapter is that the rheological behaviour (property) of any viscoelastic food can be well approximated by an arrangement (structure) of two mechanical elements springs and dashpots. In these models the Hookean elastic contribution is represented by a spring (with modulus E) and the viscous component by a dashpot (operating with a liquid of viscosity /i). 

Let s start by looking at a simple polymer, polyethylene, that has a lot going on in its stress/strain plots (Figure 13-38). Flexible, semi-crystalline polymers such as this (where the T of the amorphous domains is below room temperature) usually display a considerable amount of yielding or cold-drawing, as long as they are not stretched too quickly. For small deformations, Hookean elastic-type behavior (more or less) is observed, but beyond what is called the yield point irreversible deformation occurs. 

Polymeric (and other) solids and liquids are intermediate in behavior between Hookean, elastic solids, and Newtonian, purely viscous fluids. They often exhibit elements of both types of response, depending on the time scale of the experiment. Application of stresses for relatively long times may cause some flow and permanent deformation in solid polymers while rapid shearing will induce elastic behavior in some macromolecular liquids. It is also frequently observed that the value of a measured modulus or viscosity is time dependent and reflects the manner in which the measuring experiment was performed. Tliese phenomena are examples of viscoelastic behavior. 

From Eq. (3-32) for a dilute solution of Hookean elastic dumbbells with relaxation time T and modulus G, calculate polymer contributions to the extensional viscosity as a function of time after start-up of steady extension at extension rate e. 

This equation indicates that an isolated molecular chain behaves like a Hookean elastic element. 

These materials exhibit both viscous and elastic properties. In a purely Hookean elastic solid, the stress corresponding to a given strain is independent of time, whereas for viscoelastic substances the stress will gradually dissipate. In contrast to purely viscous liquids, on the other hand, viscoelastic fluids flow when subjected to stress, but part of their deformation is gradually recovered upon removal of the stress. 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

Problem 2-25. Complex fluids. Consider a suspension consisting of a Newtonian suspending fluid and micrometer-sized particles in which Brownian motion is a factor. The particles are spherical at equilibrium but are made of a Hookean elastic (rubberlike) material. Assume that the suspension is dilute (the motion of each particle is independent of the fact that other particles are present). Discuss whether this material can be described at a continuum level as a Newtonian fluid. Does this depend on the magnitude of the elastic modulus On the shear rate ... 

Nonisothermal Maxwell model extended past the freeze line with the Hookean elastic behavior... 

The response of rubbery materials to mechanical stress is a slight deviation from ideal elastic behavior. They show non-Hookean elastic behavior. This means that although rubbers are elastic, their elasticity is such that stress and strain are not necessarily proportional (Figure 14.3). 

Viscoelastic material such as polymers combine the characteristics of both elastic and viscous materials. They often exhibit elements of both Hookean elastic solid and pure viscous flow depending on the experimental time scale. Application of stresses of relatively long duration may cause some flow and irrecoverable (permanent) deformation, while a rapid shearing will induce elastic response in some polymeric fluids. Other examples of viscoelastic response include creep and stress relaxation, as described previously. 

The large resistance of the membrane to area dilation has been characterized in micromechanical experiments. The changes in surface area that can be produced in the membrane are small, and so they can be characterized in terms of a simple Hookean elastic relationship between the isotropic force resultant N and the fractional change in surface area a = A/Ao — 1 ... 

A practical explanation for the velocity-porosity variations is provided by a simple elastic theory (Wood, 1941) where seawater-saturated, unconsolidated marine sediments are considered to be nonrigid systems, consisting of discrete, noninteracting mineral grains suspended in seawater (Hamilton, 1971,1972). Sound velocity would only depend on the relative proportion of solid and fluid, and their respective compressibilities and densities, expressed through Hookean elastic equations (except for attenuation which must be treated viscoelastically) (Hamilton, 1972,1980). 

Notwithstanding the inherent advantages of the controlled-stress technique in yield studies, it should be borne in mind that an interpretation of the results of creep-compliance measurements in terms of a real yield stress (i.e. a stress below which the sample exhibits Hookean elastic behaviour) is subject to the usual experimental limitations of machine resolution (i.e. of angular displacement) and the role of time-scale in the sample s response to applied stress. 

An interesting three-parameter model (the Burger model has four parameters) was proposed by Hsueh [6] and is shown in Fig. 3b. He demonstrated that for a Hookean elastic element (Ei) in series with a Kelvin solid (E2,ry), the stress-strain rate relations for constant strain rate and constant stress creep tests are,... 

The theory of hydrodynamics similarly describes an ideal liquid behavior making use of the viscosity (see Sect 5.6). The viscosity is the property of a fluid (liquid or gas) by which it resists a change in shape. The word viscous derives from the Latin viscum, the term for the birdlime, the sticky substance made from mistletoe and used to catch birds. One calls the viscosity Newtonian, if the stress is directly proportional to the rate of strain and independent of the strain itself. The proportionality constant is the viscosity, q, as indicated in the center of Fig. 4.157. The definitions and units are listed, and a sketch for the viscous shear-effect between a stationary, lower and an upper, mobile plate is also reproduced in the figure. Schematically, the Newtonian viscosity is represented by the dashpot drawn in the upper left comer, to contrast the Hookean elastic spring in the upper right. 

Similar reasoning applies to the temperature dependence. At low temperatures, t, = rj/G tends toward infinity as the viscosity rj becomes very large only a Hookean elasticity is observed. At high temperatures, on the other hand, the third term (viscous flow) predominates. In between lies a range of temperatures at which the test and orientation times are comparable. A damping will then be observed at these temperatures. 

Elasticity (1664) n. A property that defines the extent to which a material resist small deformations from which a material recovers completely when deforming force is removed. When the deformation is proportional to the applied load, the material is said to exhibit Hookean elasticity or ideal elasticity. Elasticity equals stress divided by strain. Shah V (1998) Handbook of plastics testing technology. John Wiley and Sons, New York. Elias HG (1977) Macromolecules, vols 1-2. Plenum Press, New York. Weast RC (ed) (1978) CRC handbook of chemistry and physics, 59th edn. The Chemical Rubber Co., Boca Raton, EL. 

Hookean elasticity (ideal elasticity) n. Stress-strain behavior in which stress and strain are directly proportional, in accordance with Hooke s law. Serway RA, Faugh JS, Bennett CV (2005) College physics. Thomas, New York. 

Hookean elasticity, where the motion of chain segments is drastically restricted and probably involves only bond stretching and bond angle deformation the material behaves like a glass. 

Vp for Polymer B. This calculation is an approximation since it ignores an anisotropy that can be predicted from the deviation from neo-Hookean elasticity, but the latter is very small as will be seen below. The sum (v + v ) (for which the term reticulation density has been suggesLea ) can now be compared with the corresponding quantity from stress-strain and state of ease measurements, namely, (2/RT)(C.y determinations of... 

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