Shear elasticity

The dislocations in a tangle can lower their potential energy by aligning themselves to form dipoles and higher multipoles. The stress needed to push subsequent dislocations through a tangle (dipoles and multipoles) is proportional to the elastic shear modulus so it may be expected that the hardnesses of simple metals are proportional to their shear moduli. Figure 2.7 confirms this. 

Figure 2.7 Plastic flow stresses of from Brinell spherical indentation-hardnesses versus elastic shear moduli. Nominally pure fee metals at 200K (Gilman, 1960).

The glide planes on which dislocations move in the diamond and zincblende crystals are the octahedral (111) planes. The covalent bonds lie perpendicular to these planes. Therefore, the elastic shear stiffnesses of the covalent bonds... 

Figure 5.6 Correlation of octahedral shear stiffnesses with bond moduli for Group IV crystals. The octahedral stiffnesses measure the elastic shear resistances of the covalent bonds across the (111) planes.

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

U0 = Gb3 where G = elastic shear modulus = 26.2 GPa. for aluminum, so the elastic part of the line energy is about 2.5 eV/atom length, or roughly ten times the core energy. 

Stresses can can be concentrated by various mechanisms. Perhaps the most simple of these is the one used by Zener (1946) to explain the grain size dependence of the yield stresses of polycrystals. This is the case of the shear crack which was studied by Inglis (1913). Consider a penny-shaped plane region in an elastic material of diameter, D, on which slip occurs freely and which has a radius of curvature, p at its edge. Then the shear stress concentration factor at its edge will be = (D/p)1/2.The shear stress needed to cause plastic shear is given by a proportionality constant, a times the elastic shear modulus,... 

Figure 14.6 Hardnesses of metallic glasses vs. their elastic shear moduli. Data from Davis et al., 1994. The glass compositions are Cu68Zr32, Fe4oNi38Mo4Bi8, FeB (various). The line in the graph has a slope of G/2jl...

Chemical hardness is an energy parameter that measures the stabilities of molecules—atoms (Pearson, 1997).This is fine for measuring molecular stability, but energy alone is inadequate for solids because they have two types of stability size and shape. The elastic bulk modulus measures the size stability, while the elastic shear modulus measures the shape stability. The less symmetric solids require the full set of elastic tensor coefficients to describe their stabilities. Therefore, solid structures of high symmetry require at least two parameters to describe their stability. 

Another property that is related to chemical hardness is polarizability (Pearson, 1997). Polarizability, a, has the dimensions of volume polarizability (Brinck, Murray, and Politzer, 1993). It requires that an electron be excited from the valence to the conduction band (i.e., across the band gap) in order to change the symmetry of the wave function(s) from spherical to uniaxial. An approximate expression for the polarizability is a = p (N/A2) where p is a constant, N is the number of participating electrons, and A is the excitation gap (Atkins, 1983). The constant, p = (qh)/(2n 2m) with q = electron charge, m = electron mass, and h = Planck s constant. Then, if N = 1, (1/a) is proportional to A2, and elastic shear stiffness is proportional to (1/a). 

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

Therefore, the rate at which chemical bonds break increases with elastic shear stressing of the material. The rupture of chemical bonds, hence fracture of material, leads to its fragmentation into particles. This reduces the average particle size in powder as fractured particles multiply into even smaller particles. Equation (1.24) points to the importance of elastic shear strains in mechanical activation of chemical bonds for particle size refinement and production of nanoparticles. 

High levels of elastic shear and other stresses induced by milling... 

Silvery-white lustrous metal face-centered cubic crystal structure ductile ferromagnetic density 8.908 g/cm at 20°C hardness 3.8 Mohs melts at 1,455°C vaporizes at 2,730°C electrical resistivity 6.97 microhm-cm at 20°C total emissivity 0.045, 0.060 and 0.190 erg/s.cm2 at 25, 100 and 1,000°C, respectively modulus of elasticity (tension) 206.0x10 MPa, modulus of elasticity (shear) 73.6x10 MPa Poisson s ratio 0.30 thermal neutron cross section (for neutron velocity of 2,200 m/s) absorption 4.5 barns, reaction cross section 17.5 barns insoluble in water dissolves in dilute nitric acid shghtly soluble in dilute HCl and H2SO4 insoluble in ammonia solution. Thermochemical Properties... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

It is now well established that formation of hard or stiff gels is the result of association of micelles into cubic phases. The notation hard gel follows Hvidt and co-workers (Almgren et al 1995 Hvidt et al. 1994) and refers to a micellar solution with a dynamic elastic shear modulus G > 103Pa. The correlation between the formation of a cubic phase and the onset of plastic flow (i.e. formation of a gel with a finite yield stress) was first made for PS-PI solutions in... 

Fig. 4.2 Illustrating regions of hard and soft gel for aqueous solutions of PE04lPH0s with indicated copolymer concentrations (in wt%). The dynamic elastic shear modulus is plotted as a function of temperature at a frequency of 1 Hz (Booth et al. 1997 Li et al. 1997).

A comparative study of the readout options for the SAW sensor with additional film has shown that for a single SAW sensor the highest signal-to-noise ratio is obtained from the amplitude measurement (Wohltjen and Dessy, 1979). Voltage output related to the phase-shift as discussed above works well for dual delay lines. There are also inherent advantages in measurement of the change of the resonant frequency. The frequency shift due to deposited film of low elastic shear modulus p is... 

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