Hookean elements

Both Equations 8 and 9 assume linear elements. The calculations have also been performed (by computer) for neo-Hookean elements. This leads to a slight dependence of K[Pg.219]

Finally Keller has developed a theory to explain the deformation behaviour of SBS copolymers with an hexr onal structure formed by polystyrene cylinders in a polybutadiene matrix. His theory, called Random-break treatment is based on a model consisting of a system of broken cylinders of the same initial length with the breaks distributed randomly along the cylinders. The author r ards the cylinders unstrained compared with the matrix material. He assumes the matrix to consist of small Hookean elements joining each rod to the six nearest rods and he calculates the stress in the cylinders due to the tensile strain of the matrix. The theoretical predictions are in a good agreement with the birefringence and electron microscopy results. 

Additional results have been obtained from the two dimensional theory [8] for which it is found that the non-affine character of x varies in the order uniaxial tension > tension with constrained transverse strain > equal biaxial tension. For the last case, no reorientation of elements occurs (x - 0) a.nd the t extremal for Hookean elements is identically affine. (It is emphasized that biaxial tension in a three-dimensional network does result in element reorientation.)... 

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... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

With further loading of the fiber, the curve (BC) becomes linear. In this region, stress is proportional to strain with the ratio referred to as the Hookean slope (i.e. follows Hook s law). The point C at which the curve becomes nonlinear is referred to as the yield point where the loaded elements begin to deform in a nonelastic or irreversible fashion and redistribute 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). 

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. 

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

For a material that behaves as a Hookean solid in hydrostatic compression but as a Maxwell element in shear, the corresponding values of the operators are... 

Equations 40 and 41 involve different averages of the elemental stretches and give slightly different shear moduli for small deformations, 4/5 vNkT and vNkT respectively. However, the shapes of the stress-strain curves would be difficult to distinguish experimentally (see Sect. IIC.). Thus, although this rather natural application of the Doi-Edwards model to permanent networks leads directly to a simple additive contribution of trapped entanj ements to the initial modulus, as observed experimentally (G = it does not account for the quite substantial departures from the neo-Hookean form (Eq.41) which are commonly observed at even moderate deformations ... 

Polymers are viscoelastic, meaning that they have intermediate properties between Newtonian liquids and Hookean solids. The simplest model of viscoelasticity is the Maxwell model, which combines a perfectly elastic element with a perfectly viscous element in series, as shown in Fig. 7.21. -Since the elements are in series, the total shear strain 7 is the sum of the... 

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... 

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. 

We now give the solution to Eq. (13.17) for the simplest possible model, namely Hookean dumbbells, for which there is but one element in the D-matnx and it is identically equal to zero. For this equation we can postulate a Gaus-sian-form solution ... 

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 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] ... 

The total force acting on each node contains contributions from the elastic and osmotic properties of the system. In other words, we have shown [3] that the total force acting on node n of the element m consists of two contributions F (m) = Fi, (m) -H F2, (m). The first term, Fi, (m), describes the neo-Hookean elasticity contribution to the energy of the system, and can be expressed as a... 

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. 

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