Hookean spring element

The simplest mechanical model, the hookean spring element, has an elastic response. The spring is an energy storage element. It releases its energy when it returns to its original form. When subjected to an instantaneous stress Oo, the spring has a response with a strain Eq [10]  

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The last reference system we discuss is the lattice of interacting harmonic oscillators. In this system each atom is connected to its neighbors by a Hookean spring. By diagonalizing the quadratic form of the Hamiltonian, the system may be transformed into a collection of independent harmonic oscillators, for which the free energy is easily obtained. This reference system is the basis for lattice-dynamics treatments of the solid phase [67]. If D is the dynamical matrix for the harmonic system (such that element Dy- describes the force constant for atoms i and j), then the free energy is... 

When this model is subjected to a constant stress, the response includes an instantaneous elastic strain caused by spring 1, retarded elastic strain by the Kelvin component, viscous flow by dashpot 1, instantaneous elastic strain on unloading from spring 1, retarded strain recovery from the Kelvin element and permanent deformation in dashpot 1. The multiparameter model response is shown in Figure 4.13. This model can be described as the combined response of a Hookean elastic element, a Kelvin retarded-elastic solid and a Newtonian viscous fluid. 

Materials, and, in particular, polymers, that show viscoelastic behaviour, can be modelled by a combination of perfectly elastic Hookean springs and Newtonian viscous dashpots. For many polymers the behaviour at temperatures above when strains are small 1 per cent) is approximately represented by the so-called standard linear substance. This consists of a dashpot and spring in series (called a Maxwell element) and this combination in parallel with a second spring of different elastic modulus. In deformation, the strain of the Maxwell element and of the spring will be the same, say e. At a time t let Oi be the stress in the Maxwell element and 02 that in the spring. Then, if Ei is the Young s modulus of the Maxwell element and E2 that of the spring ... 

It is useful as an aid to understanding viscoelastic phenomena to interpret this function in terms of the behavior of a mechanical assembly consisting of a linear (Hookean) spring and a linear dash-pot, connected in series as shown in Fig. 4.5. A dash-pot is an element in which the force is proportional to the rate of displacement and is thus analogous to a Newtonian fluid. This assembly was proposed by Maxwell as a model for the behavior of gases, and it is referred to as Maxwell element. 

A simple way to illustrate the viscoelastic properties of materials subjected to small deformations is to evaluate the stress that results from combining a linear spring that obeys Hooke s law and a simple fluid that obeys Newton s law of viscosity. An example of such combination is the mathematical representation of the Maxwell element. Even though this model is inadequate for quantitative correlation of polymer properties, it illustrates the quahtative nature of real behavior. Furthermore, it can be generahzed by the concept of a distribution of relaxation times so that it becomes adequate for quantitative evaluation. Maxwell s element is a simple one combining one viscous parameter and one elastic parameter. Mechanically, it can be visualized as a Hookean spring and a Newtonian dashpot in series ... 

A Maxwell-type element is constructed of the usual viscous dashpot but with an ideal rubber band in place of the usual Hookean spring. In a creep experiment with a constant load of 1000 Pa, and at a temperature of 27 C, the total deformation in tensile strain units after 3 h is e = 2.00. When the... 

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... 

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). 

The simplest mechanical models for viscoelastic behavior consist of two elements a spring for elastic behavior and a damper for viscous behavior. First it is convenient to introduce the model of a linear spring to represent a Hookean bar under uniaxial tension where the spring constant is the modulus of elasticity. As indicated in Fig. 3.19 the spring constant can be replaced by Young s modulus if the stress replaces P/A and strain replaces 6/L. 

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