Hookian

The SFA, originally developed by Tabor and Winterton [56], and later modified by Israelachvili and coworkers [57,58], is ideally suited for measuring molecular level adhesion and deformations. The SFA, shown schematically in Fig. 8i,ii, has been used extensively to measure forces between a variety of surfaces. The SFA combines a Hookian mechanism for measuring force with an interferometer to measure the distance between surfaces. The experimental surfaces are in the form of thin transparent films, and are mounted on cylindrical glass lenses in the SFA using an appropriate adhesive. SFA has been traditionally employed to measure forces between modified mica surfaces. (For a summary of these measurements, see refs. [59,60].) In recent years, several researchers have developed techniques to measure forces between glassy and semicrystalline polymer films, [61-63] silica [64], and silver surfaees [65,66]. The details on the SFA experimental procedure, and the summary of the SFA measurements may be obtained elsewhere (see refs. [57,58], for example.). 

For a Hookian material, the concept of minimum strain energy states that a material fails, for example cell wall disruption occurs, when the total strain energy per unit volume attains a critical value. Such an approach has been used in the past to describe a number of experimental observations on the breakage of filamentous micro-organisms [78,79]. Unfortunately, little direct experimental data are available on the Young s modulus of elasticity, E, or shear modulus of elasticity G representing the wall properties of biomaterial. Few (natural) materials behave in an ideal Hookian manner and in the absence of any other information, it is not unreasonable to assume that the mechanical properties of the external walls of biomaterials will be anisotropic and anelastic. 

Stark, R. W. and Heckl, W. M. (2000). Fourier transformed atomic force microscopy tapping mode atomic force microscopy beyond the Hookian approximation. Surface Science 457,219-28. [319]... 

Linear amorphous polymers can behave as either Hookian elastic (glassy) materials, or highly elastic (rubbery) substances or as viscous melts according to prevailing temperature and time scale of experiments. The different transitions as shown schematically in Figure 5.1 are manifestations of viscoelastic deformations, which are time dependent [1]. 

Oefor -motion Hookian Glassy Secondary transition Primary transition Highly elastic rubbery Flow... 

Dynamic Mechanical (Low Strain Deformation). When a cyclic strain of small ampUtude is applied to a strip of material, a cyclic stress will be generated in response by the sample. If the material is ideal (Hookian) and stores all the input energy, the cyclic stress is in phase with the applied cyclic strain. Viscous components cause a finite phase lag or phase angle, 8, between the stress and strain. represents the elastic, real, or storage modulus while E" is the viscous, imaginary, or loss modulus. Tan 8 is equal to the ratio E /E" and is related to the molecular relaxations that occur in the sample. Transition temperatures and associated activation energy can be determined (72) by varying the frequency of oscillation at a fixed temperature or the temperature at a fixed frequency. 

The constitutive equation of elasticity is represented by the Hookian spring (Fig. 16). Hook s law states that the stress is proportional to the strain... 

Rheological and elastic properties under flow and deformations are highly characteristic for many soft materials like complex fluids, pastes, sands, and gels, viz. soft (often metastable) solids of dissolved macromolecular constituents [1]. Shear deformations, which conserve volume but stretch material elements, often provide the simplest experimental route to investigate the materials. Moreover, solids and fluids respond in a characteristically different way to shear, the former elastically, the latter by flow. The former are characterized by a shear modulus Go, corresponding to a Hookian spring constant, the latter by a Newtonian viscosity r]o, which quantifies the dissipation. 

Above the transition, the quiescent system forms an (idealized) glass [2, 38], whose density correlators arrest at the glass form factors fq from Fig. 10, and which exhibits a flnite elastic constant G , that describes the (zero-frequency) Hookian response of the amorphous solid to a small applied shear strain y cr = Geo/ for y 0 the plateau can be seen in Fig. 3 and for intermediate times in Fig. 12. If steady flow is imposed on the system, however, the glass melts for any arbitrarily small shear rate. Particles are freed from their cages and diffusion perpendicular to the shear plane also becomes possible. Any finite shear rate, however small, sets a finite longest relaxation time, beyond which ergodicity is restored see Figs. 1 l(b,c) and 12. 

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

This equation has the correct limiting behavior it reduces to an equation for a simple Newtonian fluid when dx/dt approaches to 0 for steady shear flow. When the stress changes rapidly with time, and X is negligible compared with dx/dt, it reduces to the constitutive equation of a Hookian solid. 

For the first two terms, the bond stretching and angle bending, the energy is calculated according to a Hookian function, assuming harmonic behaviour. In each case the energy is with respect to deviations from what can be considered as a "natural or equilibrium value. 

As classic brittle materials, glasses exhibit nearly perfect Hookian behavior on application of a stress. The ratio of the strain, e, resulting from application of a stress, a, is a constant which is known as the elastic modulus, or Young s modulus, E, which is defined by the expression ... 

The extension from the three atom, one-dimensional case to that of an isolated polymer chain of finite length is straightforward. As before, the linear ehain is composed only of equal masses, m, separated by a constant distance, d (the repeat distance). Hookian springs with force constants, f, join the masses. Again, the chain is assumed to vibrate in only one dimension, say the x-axis. The equation of motion for the n mass, when it is displaced from its equilibrium position, is (analogous to equation 6.9)... 

Consider a one dimensional approximation of a triatomic linear molecule represented by identical mass points with mass, m, linked by weightless Hookian springs with force constants, (Figure 6.2). The equations of motion for this system is given (equation 6.9) by... 

Where ""u is the Hookian potential energy term which is quadratic in the symmetric strain tensor e, contracted with the elastic constant tensor C. The Greek indices (i.e.,... 

A Hookian or an ideal elastic solid, whose small reversible deformations are directly proportional... 

In the elastic response for a Hookian liquid the stress-strain relationship is a(t) = Gyif). For the Newtonian liquid it is shear strain and shear rate at time t, while a t) is the shear stress. 

Define a Newtonian liquid and a Hookian ideal elastic solid, and an ideal elastomer. 

Rheology is the study of flow of matter and deformation and these techniques are based on their stress and strain relationship and show behavior intermediate between that of solids and liquids. The rheological measurements of foodstuffs can be based on either empirical or fundamental methods. In the empirical test, the properties of a material are related to a simple system such as Newtonian fluids or Hookian solids. The Warner-Bratzler technique is an empirical test for evaluating the texture of food materials. Empirical tests are easy to perform as any convenient geometry of the sample can be used. The relationship measures the way in which rheological properties (viscosity, elastic modulus) vary under a... 

Fig. 32. Simple relationship between stress (a) and strain (e) in dynamic mechanical tests illustrating the role of the phase angle, showing (a) Hookian elastic response, (b) viscoelastic response, and (c) vectorial representations of complex oscillatory stress/strain response (74).

Rheology is the study of flow and deformahon of matter (Barnes et al., 1989). It encompasses a wide range of mechanical behaviour from Hookian elashc... 

If the output data is recorded on a X-Y recorder or an oscilloscope, a hysteresis loop as shown in Fig. lOB is obtained. The area of the loop is proportional to the damping or energy dissipation of the system. The loop is a circle for the Newtonian fluid, a line for a Hookian body, and an ellipse for a viscoelastic body. The values for , ", and sin 6 for an elastic body can be calculated from the hysteresis ellipse using... 

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