Shear elastic moduli

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. 

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,... 

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. 

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. 

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... 

Fig. 4.4 Yield stress normalized by the elastic shear modulus plotted against a size scale parameter (volume per surface area) illustrating the six orders of magnitude of stress levels and ten orders of magnitude of size related to plastic behavior of single crystal metals [152]...

Figure 11.11. Effect of fat level (half-fat, HF, 17%, w/w full-fat, FF, 32%, w/w) and assay temperature (4 or 40°C) on the elastic shear modulus of Cheddar cheeses measured using low-amplitude strain oscillation at a frequency of 0.1( ), 1 ( ) or 10 (g), Hz.

For crystals of reasonably pure, well-annealed metals at a given temperature, slip begins when the resolved shear stress reaches a certain critical value, which is characteristic of each metal. In the case of aluminum, for example, the observed critical shear stress Uco is usually about 4x10 N/m ( 4 bars = 0.4 MPa). Theoretically, for a perfect crystal, the resolved shear stress is expected to vary periodically as the lattice planes slide over each other and to have a maximum value that is simply related to the elastic shear modulus /t. This was first pointed out in 1926 by Frenkel who, on the basis of a simple model, estimated that the critical resolved shear stress was approximately equal to h/Itt (see Kittel 1968). In the case of aluminum (which is approximately elastically isotropic), = C44 = 2.7x10 N/m, so the theoretical critical resolved shear stress is about lO wco for the slip system <100>(100). 

FIGURE 17.4 Variation in textural properties, (a) Stress (o) versus strain (sh) curves and fracture points (X). (b) Relation between elastic shear modulus G and fracture stress (some types of gels of various concentrations of the networkforming material. (From approximate results by H. Kimura et al. J. Food Sci. 38 (1973) 668.)... 

Using the chemo-elasto-plastic model implemented in the finite elements program LAGAMINE, and the material parameters listed in Table 1, the volumetric strains due to changes in effective stress and concentration are predicted. The value of X was estimated from the volumetric response measured by Fernandez and Quigley (1991) for the application of external vertical effective stress, with the assumption that A o=0.5. The constant k was taken as X/5, which is typical for clays. Values typical for clays were selected for the elastic shear modulus G and the critical state constant M. 

Rock stratum Thickness Bulk modulus of elasticity Elastic shear modulus Gravity density Angle of internal friction Cohesion Uniaxial compressive strength Uniaxial tensile strength... 

---