Effect of structure on elastic behavior

In the last chapter, the formal description of linear elasticity was introduced. It was shown that knowledge of the elastic constants for a particular material allows one to describe the strains produced by any arbitrary state of stress. In materials science one is often interested in controlling a material property and, thus, this chapter is concerned with the influence of structure on the elastic constants. At the most basic level, the elastic constants reflect the ease of deformation of the atomic bonds but it will be shown that other levels of structure can be very important, especially with the use of composite materials. 

If an accurate equation for the interatomic potential is known, the elastic constants can be calculated from first principles. This analysis is straightforward for cubic ionic crystals. The potential for a pair of positive and negative ions is often written in the form 

CARBIDES OXIDES FLUORITES SULFIDES ALKALI HALIDES CESIUM HALIDES 

To determine the energy between nearest neighbors within a particular crystal 

The force constant k for the single bond is obtained by differentiating f twice with respect to r and putting r=a (Eq. (2.2)). Performing these operations one obtains 

---

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

The major or exclusive constituent of yellow brass is P brass which is the intermetallic CuZn phase. It exhibits an A2 structure at high temperatures and a B2 structure at low temperatures, i.e. there is an order-disorder transition at about 460°C (Flinn, 1986 Massalski et al., 1990). Its range of homogeneity - between about 40 and 50 at.% Zn at higher temperatures - depends sensitively on temperature and does not include the stoichiometric 50 at.% composition at intermediate temperatures. This order-disorder transition has been used to study the effect of ordering, e.g. on elastic behavior (Westbrook, 1960 a Quillet and Le Roux, 1967), diffusion (Qirifalco, 1964 Hagel, 1967 Wever et al., 1989 Wever, 1992), recrystallization (Cahn, 1991), and hardness (Westbrook, 1960 a). 

Many industrial products use mixtures of both surfactant and polymer molecules or surfactant and colloid. Although the effects of polymer on the phase behavior and structure of surfactant phases have begun to be investigated in microemulsions, lamellar phases, and vesicle phases, further experimental work in mixed systems is necessary to understand how the polymer or the colloid modifies the elastic properties of the surfactant film. 

Designers of most structures specify material stresses and strains well within the pro-portional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or process-wise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

Different viscoelastic materials may have considerably different creep behavior at the same temperature. A given viscoelastic material may have considerably different creep behavior at different temperatures. Viscoelastic creep data are necessary and extremely important in designing products that must bear long-term loads. It is inappropriate to use an instantaneous (short load) modulus of elasticity to design such structures because they do not reflect the effects of creep. Viscoelastic creep modulus, on the other hand, allows one to estimate the total material strain that will result from a given applied stress acting for a given time at the anticipated use temperature of the structure. 

---