Hookean constant

The rheological behavior of a viscoelastic material can be investigated by applying a small-amplitude sinusoidal deformation. The behavior can be described by a mechanical model, called the Maxwell model [33], consisting of an elastic spring with the Hookean constant, G , and a dashpot with the viscosity, r/<,. The variation of storage modulus (G ) and loss modulus (G") with shear frequency, O), are given by the equations... 

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... 

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] ... 

FIGURE 14.1 Stress-strain plots for a Hookean spring (a) where E (Equation 14.1) is the slope, and a Newtonian dashpot (b) where is a constant (Equation 14.3). 

For most solids, one can neglect the difference between Pp f (ap f/3 for an isotropic body) and the coefficient of thermal expansion at constant P is usually used. Therefore, we may use P and a without subscripts. Assuming that E and p are independent of temperature and ignoring the change in lateral dimensions during defonnation (i.e. we take the Poisson s ratio p = 0, because this simplification gives effects of only the second order of smallness), one can arrive at relations similar to Eqs. (17)—(21). To do this, it is necessary to replace in Eq. (16) the volume deformation e by e, the modulus K by E and a by p (see Fig. 1). For the simple deformation of a Hookean body the characteristic parameter r is also inversely dependent on strain, viz. r = 2PT/e and sinv = —2PT. It is interesting to note that... 

A polymer is said to exhibit Hookean elasticity if the ratio of stress to strain is a constant. In this case, the ratio is called Young s modulus, and is given the symbol G. 

Figure 14.19 Stress-strain plots for (a) a Hookean spring where E is the slope (6) a Newtonian dash pot where s is constant, (c) stresstime plot stress for relaxation in the Maxwell model, and (d) stresstime plot stress for a Voigt-Kelvin model.

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 ... 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

Cantilevers in AFM function as force transducers converting unknown force to measurable deflection. The value of the unknown force can then be expressed by Hookean mechanics following spring constant calibrations. In addition to static point loads, cantilevers can also be vibrated, e.g., by an oscillation piezo to which the fixed end of the beam is attached (or by other approaches). Excitation frequency, oscillation amplitude, and phase relationships are variables that govern dynamic tapping (intermittent contact) imaging. This problem will be discussed in the next section. 

The remarkable insensitivity of (5sxo) to variations of cv can be rationalized as follows. Because of the Hookean regime in the limit as — 0, C44 should be approximately constant and positive in this limit. A typical plot in Fig. 5.20 confirms this notion. However, bc< au.sc of Etp (5.112) one expects C44 to decline from its Hookean value as o-s —> o sxo also in agreement with Fig. 5.20. Furthermore, as Fig. 5.20 show s that the variation of C44 with as-x is not too strong over the range a 0 < a <, it seems sensible to... 

Theories for polymer dynamics of dilute polymer solutions include the elastic (Hookean) spring model (Kuhn, 1934) which considers that the system is mechanically equivalent to a set of beads attached with a spring. The properties are then based on a spring constant between beads and the friction of beads through solvent. The viscosity of a Hookean system is then described by... 

H) Hookean dumbbells The beads are joined by a Hookean spring which has zero length if there are no forces acting on the beads the tension in the connector F(c) = HR, where H is a Hookean spring constant. 

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... 

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