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... 

The Newtonian viscosity is given by the product of the relaxation time and the Hookean modulus. This result was anticipated in the discussion of Eqs. (2.29) and (2.31). 

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

The mass of the subchain is pictured as concentrated in a bead, connected to adjacent beads by Hookean springs which, individually, obey Eq. (3.45). 

For a component subjected to a uniaxial force, the engineering stress, a, in the material is the applied force (tensile or compressive) divided by the original cross-sectional area. The engineering strain, e, in the material is the extension (or reduction in length) divided by the original length. In a perfectly elastic (Hookean) material the stress, a, is directly proportional to be strain, e, and the relationship may be written, for uniaxial stress and strain, as... 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

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. 

The second major assumption is that the material is elastic, meaning that the strains are directly proportional to the stresses applied and when the load is removed the deformation will disappear. In engineering terms the material is assumed to obey Hooke s Law. This assumption is probably a close approximation of the material s actual behavior in direct stress below its proportional limit, particularly in tension, if the fibers are stiff and elastic in the Hookean sense and carry essentially all the stress. This assumption is probably less valid in shear, where the plastic carries a substantial portion of the stress. The plastic may then undergo plastic flow, leading to creep or relaxation of the stresses, especially when the stresses are high. 

When an engineering plastic is used with the structural foam process, the material produced exhibits behavior that is easily predictable over a large range of temperatures. Its stress-strain curve shows a significantly linearly elastic region like other Hookean materials, up to its proportional limit. However, since thermoplastics are viscoelastic in nature, their properties are dependent on time, temperature, and the strain rate. The ratio of stress and strain is linear at low strain levels of 1 to 2%, and standard elastic design... 

Ea = Arrhenius activation energy Es = excess stress energy AEr = potential barrier for bond rotation Eel = molecular elastic energy F = mean force potential f = average force on the chain fb = bond breaking force H0 = Hookean spring constant kB = Boltzmann constant... 

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

The inclusion of internal viscosity raises considerably the free-energy storage capacity of a rapidly deforming macromolecule as compared to the idealized Hookean spring model and could play a decisive role in mechanochemical reactivity in transient elongational flow. 

For practical purposes, it is convenient to define the relaxation time in terms of macroscopic quantities which can be readily determined. Within the validity limit of Hookean connectors (Eq. 13), the low-shear viscosity of a polymer solution is given by the relation ... 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

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

Rheology deals with the deformation and flow of any material under the influence of an applied stress. In practical apphcations, it is related with flow, transport, and handling any simple and complex fluids [1], It deals with a variety of materials from elastic Hookean solids to viscous Newtonian liquid. In general, rheology is concerned with the deformation of solid materials including metals, plastics, and mbbers, and hquids such as polymer melts, slurries, and polymer solutions. 

Taking the solid to the Hookean, we have a = eY, with Y being Young s modulus. Using Eqs. (3) and (4), we obtain as a scaling law ... 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

First approaches to approximating the relaxation time on the basis of molecular parameters can be traced back to Rouse [33]. The model is based on a number of boundary assumptions (1) the solution is ideally dilute, i.e. intermolecular interactions are negligible (2) hydrodynamic interactions due to disturbance of the medium velocity by segments of the same chain are negligible and (3) the connector tension F(r) obeys an ideal Hookean force law. 

In classic terms, the elastic modulus of a material is the stress divided by the strain (i.e., the slope) of the linear portion of its force versus elongation curve at low strain. In this region, the material is assumed to behave in a Hookean fashion, i.e., stress and strain are linearly proportional, as illustrated in Fig. 8.5 a). Most polymers do not behave in this manner. 

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