Hookean energy

In most cases the distribution function is Gaussian (or approximately so) and the corresponding free energy fimction can be approximated by a simple parabolic equation (Figure 12.2.15). In this case, we assume a Hookean energy (which is correct at least for the region around the maximum of the distribution curve), so we have ... 

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... 

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. 

A spring with a modulus of G, and a dashpot containing a liquid with a viscosity of rj, have been used as models for Hookean elastic solids and Newtonian liquids, respectively. In these models, the spring stores energy in a reversible process, and the dashpot dissipates energy as heat in an irreversible process. Figure 5.3 is a stress-strain curve for a typical elastomer the straight... 

If no volume or internal energy changes accompany deformation, the network is neo-Hookean (258), obeying the constitutive equation... 

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

It can be easily demonstrated that for a Hookean body a thermomechanical inversion of the internal energy (AU = 0) must occur at the deformation... 

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. 

Retractive forces in animal fibers stretched into the Hookean region of the stress-strain curve are generally attributed to the stretching of chemical bonds and hence to an increase in the internal energy of the fiber (Ast-bury and Haggith, 1953 Peters, 1956). 

In the case of an ideal Hookean body with = 0°, the loss compliance is zero and / relates to the elastic energy, which has been stored in the material. No... 

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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