Density of elastic energy

If now a solid of volume V0 with no elastic stresses within it initially is subjected to an amount of work Wo prior to fracture of the solid, we may by dimensional analysis reckon the average density of elastic energy. Let Wo/ Vo represent this average then, assuming that this energy is a function of and Eu above, we have... 

To have the free energy density in a more general form including all the nine elementary distortions tzi, 2... agWQ should add the terms dnjdz, dnjdx and dnj dy and rewrite the Eq. (8.15b) in the vector notations for arbitrary distortion of n with respect to the Cartesian frame. Then we obtain Frank formula for the density of elastic energy in the general vector form ... 

To calculate Ui, Griffith specified that the introduction of a crack into a two-dimensional plate-type specimen entails in its immediate vicinity i.e., in a volume ita l around this crack, a density of elastic energy close to zero I represents the sample thickness assumed equal to 1. Under these conditions, Ui, which corresponds to the elastic energy lowered by the introduction of a crack, is in first approximation... 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

According to Hooke s law, the elastic deformation of solid leads to the accumulation of elastic energy in it. The density of this energy is given by... 

However, one important energy loss which was explained was the effect of the viscoelastic behavior of the polymer. This was studied by varying the crosslink density of the rubber, to alter the loss of elastic energy as the material relaxed. As the viscoelastic loss increased, so did the adhesive hysteresis, as shown in Fig. 8.13. 

The density of elastic distortion energy (a scalar quantity) is quadratic in strain ... 

TTie elastic energy contribution Uei describes the rubber elasticity of the cross-linked polymer chains, and is proportional to the cross-link density which is the number density of elastic strands in the undeformed polymer network. We use the Flory model [35] to specify U ... 

The density of dislocations is usually stated in terms of the number of dislocation lines intersecting unit area in the crystal it ranges from 10 cm for good crystals to 10 cm" in cold-worked metals. Thus, dislocations are separated by 10 -10 A, or every crystal grain larger than about 100 A will have dislocations on its surface one surface atom in a thousand is apt to be near a dislocation. By elastic theory, the increased potential energy of the lattice near... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

This behavior is similar to the cut growth and fatigue behavior of rubber compounds. The rate of the growth of a cut is a function of the tearing energy [38,39] which itself is proportional to the stored elastic energy density in the test piece. The exact value depends on the shape of the test piece. 

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