Hookean solids

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. 

The relaxation exponent n is restricted to values between 0 and 1. The case of n = 0 corresponds to the limiting behavior of a Hookean solid (the relaxation... 

However, if the top plate moves a distance which is in proportion to the applied force and then stops, the material is called a linear elastic (Hookean) solid (e.g., rubber). The quantitative relation that defines such a material is... 

The characteristic parameter q of the Hookean solid can be found immediately... 

We see from Eq. (21) that the internal energy inversion occurs at compression of the system with a positive thermal expansivity and at extension with the negative one. Occurrence of the thermomechanical internal energy inversion in Hookean solids is a result of a different dependence of the work and heat on strain (Fig. 1). 

Fig. la and b. Mechanical work W (1), elastic heat Q (2), internal energy change AU (3) and heat to work ratio q (4) as a function of strain e (uniform deformation) or s (unidirectional deformation) for quasi-isotropic Hookean solid 8. a — positive a and P b — negative a and p. The arrows indicate inversion points (see text)... 

This result stresses the fact that the heat effects can be detected only at such deformation modes of quasi-isotropic Hookean solids that are accompanied by a change in volume. Thus, the thermal effects are the consequence of the change of the vibrational entropy which in turn is a result of the volume change. It is very important to emphasize now that the internal energy and entropy changes are closely interrelated and their values are of the same order of magnitude. 

Uniaxial compression of a cylindrical shape is a method of measuring the behavior of a Hookean solid. Hooke s law is represented by the following relationship ... 

Figure 11. Stress-strain curves for Hookean solids, rubber-type materials and extensible biological substances (adapted from ref. 57).

In the case of solids it is evident that deformation is either linear elastic - like a Hookean solid (most solids including steel and rubber) - or non-linear elastic or viscoelastic. In the case of liquids, fluids differ between those without yield stress and those with yield stress (so-called plastic materials). Fluids without yield stress will flow if subjected to even slight shear stresses, while fluids with yield stress start to flow only above a material-specific shear stress which is indicated by o0. 

All real materials fall Theologically between two extremes the perfectly elastic Hookean solid, for which stress is directly proportional to strain, and the Newtonian liquid, for which (shear) stress is directly proportional to (shear) strain rate. Strain can be defined as deformation relative to a reference length, area or volume (Barnes et al., 1989) it is dimensionless. Strain... 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

One convenient manner of studying viscoelasticity is by stress relaxation where the time-dependent shear stress is studied for step increase in strain. In Figure 1-7, the stress relaxation of a Hookean solid, and a viscoelastic solid and liquid are shown when subjected to a strain instantaneously and held constant. The relaxation modulus can be calculated as ... 

Because a Hookean solid deforms instantaneously, the imposed strain has a time-dependent profile similar to that of the stress. For relatively small strains, shear normal stresses and the type of deformation (e.g., linear or shear) will not be important. Further, the extensional and the shear relaxation moduli are related as ... 

There have been a number of studies that demonstrate that crystallized AMF and butter exhibit linear (ideal) viscoelastic behavior at low levels of stress or strain (4), where the strain is directly proportional to the applied stress. For most materials, this region occurs when the critical strain (strain where structure breaks down) is less than 1.0%, but for fat networks, the strains typically exceed 0.1% (4, 66). Ideally, within the LVR, mUkfat crystal networks will behave like a Hookean solid where the stress is directly proportional to the strain (i.e., a oc y), as shown in Figure 15 (66, 68). Within the elastic region, stress will increase linearly with strain up to a critical strain. Beyond that critical strain (strain at the limit of linearity), deformation of the network will occur at a point known as the yield point. The elastic limit quickly follows, beyond which permanent deformation and sample fracture occurs. Beyond these points, the structural integrity of the network is compromised and the sample breaks down. 

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... 

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 Maxwell model is a Hookean solid and a Newtonian liquid in series. The... 

Oscillatory strain (solid curve and left axis) and oscillatory stress (dashed curve and right axis) are in-phase for a Hookean solid. 

The term xK replaces the ratio q/G and is referred to as the retardation time. In practice, the retardation time determines the rate at which the sample deforms following application of the stress. In a Hookean solid, the retardation time is zero as the deformation following an applied stress is instantaneous. Thus, the retardation time describes the retarding effects of the viscous properties of the dashpot to sample... 

However, if the material is a linear elastic (Hookean) solid. 

Somewhat beyond the scope of this book is to demonstrate that the theory, when simplified to this point, predicts that a = Gy, even at high strains (see problem 22). Note that the solid defined by the high-strain theory in equation (6-60) reduces to the Hookean solid at low strains, and is referred to occasionally as a neo-Hookean solid (see Section C). 

For a solid material, the typical difference in deformation behavior between a Hookean solid and a viscoelastic solid can be explained in terms of an applied constant load. 

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