Modulus lattice

Figure 52. Lattice modulus with negative Poisson coefficient (a) lattice with coordination number N = 3 (b) a chain of bonds.

Theoretical moduli of elasticity calculated in this way are largely confirmed experimentally by what are known as lattice moduli, or crystal moduli. The lattice moduli are determined by following the Bragg angle of selected reflexes by X-ray crystallography as a function of applied stress. The apparent lattice modulus so obtained ... 

Matsuo M and Sawatari C (1990) Morphological and mechanical properties of poly(ethyleue terephthalate) gel and melt films in terms of the crystal lattice modulus, molecular orientation, and small angle X-ray scattering intensity distribution, Polym J 22 518-538. 

For the same lattice strains, the larger the valency difference between solute and solvent, the greater the hardening. The strengthening influence of alloying elements persists to temperatures at least as high as 815°C. Valency effects may be explained by modulus differences between the various alloys... 

For nickel, cobalt, and hon-base alloys the amount of solute, particularly tungsten or molybdenum, intentionally added for strengthening by lattice or modulus misfit is generally limited by the instability of the alloy to unwanted CJ-phase formation. However, the Group 5(VB) bcc metals rely on additions of the Group 6(VIB) metals Mo and W for sohd-solution strengthening. 

So ceramics, at room temperature, generally have a very large lattice resistance. The stress required to make dislocations move is a large fraction of Young s modulus typically, around E/30, compared with E/10 or less for the soft metals like copper or... 

Strained set of lattice parameters and calculating the stress from the peak shifts, taking into account the angle of the detected sets of planes relative to the surface (see discussion above). If the assumed unstrained lattice parameters are incorrect not all peaks will give the same values. It should be borne in mind that, because of stoichiometry or impurity effects, modified surface films often have unstrained lattice parameters that are different from the same materials in the bulk form. In addition, thin film mechanical properties (Young s modulus and Poisson ratio) can differ from those of bulk materials. Where pronounced texture and stress are present simultaneously analysis can be particularly difficult. 

The transmission of forces through a lattice as a function of the fraction p, of bonds in the lattice has been analyzed by Kantor and Webman [63], Feng and colleagues [64-66], Thorpe et al. [68-72] and others [73]. The normalized elastic modulus E/Eo, of the lattice as a function of p was found to obey relations similar to scalar percolation... 

Crystals are sohds. Sohds, on the other hand can be crystalhne, quasi-crystal-hne, or amorphous. Sohds differ from liquids by a shear modulus different from zero so that solids can support shearing forces. Microscopically this means that there exists some long-range orientational order in the sohd. The orientation between a pair of atoms at some point in the solid and a second (arbitrary) pair of atoms at a distant point must on average remain fixed if a shear modulus should exist. Crystals have this orientational order and in addition a translational order their atoms are arranged in regular lattices. 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

The geometry and structure of a bone consist of a mineralised tissue populated with cells. This bone tissue has two distinct structural forms dense cortical and lattice-like cancellous bone, see Figure 7.2(a). Cortical bone is a nearly transversely isotropic material, made up of osteons, longitudinal cylinders of bone centred around blood vessels. Cancellous bone is an orthotropic material, with a porous architecture formed by individual struts or trabeculae. This high surface area structure represents only 20 per cent of the skeletal mass but has 50 per cent of the metabolic activity. The density of cancellous bone varies significantly, and its mechanical behaviour is influenced by density and architecture. The elastic modulus and strength of both tissue structures are functions of the apparent density. 

Table4.6 Lattice constants a, volume V, cohesive energy and bulk modulus 6 for fee gold from nonrelativistic and relativistic pseudopotential DFT calculations (from Ref [402]).

Figure 5. Variation in apparent lattice-site Young s Modulus ) with molar anorthite content of the host crystal for the plagioclase partitioning experiments of Blundy and Wood (1994)., Ej and...

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

In cellulose II with a chain modulus of 88 GPa the likely shear planes are the 110 and 020 lattice planes, both with a spacing of dc=0.41 nm [26]. The periodic spacing of the force centres in the shear direction along the chain axis is the distance between the interchain hydrogen bonds p=c/2=0.51 nm (c chain axis). There are four monomers in the unit cell with a volume Vcen=68-10-30 m3. The activation energy for creep of rayon yarns has been determined by Halsey et al. [37]. They found at a relative humidity (RH) of 57% that Wa=86.6 kj mole-1, at an RH of 4% Wa =97.5 kj mole 1 and at an RH of <0.5% Wa= 102.5 kj mole-1. Extrapolation to an RH of 65% gives Wa=86 kj mole-1 (the molar volume of cellulose taken by Halsey in his model for creep is equal to the volume of the unit cell instead of one fourth thereof). 

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