Elastic constants values 232

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

Table 3. Measured Values of Elastic Constants at Small Extensions and 25°C...

The are the elastic constants the bulk modulus of the material is computed as -B = + 2c 2 )/3- Values in parentheses are estimates. 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

In Figure 9.7 stresses are imposed on a body showing ordinary elastic deformation only, a second body showing high elastic deformation only and a third body showing viscous deformation only. The stress is imposed at time to and held at a constant value until time t, when it is removed. Deformation... 

The stiffness of a plastic is expressed in terms of a modulus of elasticity. Most values of elastic modulus quoted in technical literature represent the slope of a tangent to the stress-strain curve at the origin (see Fig. 1.6). This is often referred to as Youngs modulus, E, but it should be remembered that for a plastic this will not be a constant and, as mentioned earlier, is only useful for quality... 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

Fig. 3.24 shows the variation of these elastic constants for all values of 6 between 0 and 90°. 

The restrictions on engineering constants can also be used in the solution of practical engineering analysis problems. For example, consider a differential equation that has several solutions depending on the relative values of the coefficients in the differential equation. Those coefficients in a physical problem of deformation of a body involve the elastic constants. The restrictions on elastic constants can then be used to determine which solution to the differential equation is applicable. 

From the value for the elastic constant, C, where 1 denotes zz, can... 

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

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... 

Table I. Experimental and calculated lattice constants a (in A), elastic constants, bulk and shear moduli (in units of 10 ) for the M3X (X = Mn, Al, Ga, Ge, Si) intermetallic series. Also listed are values of the anisotropy factor A and Poisson s ratio V. The experimental data for a are from Ref. . The experimental data for B, the elastic constants, A and v are taken from Ref. . The theoretical values for NiaSi are from Ref.. Also listed in the table are values of the polycrystalline elastic quantities-shear moduli G, Yoimg moduli (in units of and the ratio The experimental data for these quantities are from Ref. ...

The constant value of 0.25 for Poisson s ratio versus depth reflects the geology and the rock mechanics of the mature sedimentary basin in the West Texas region. Since mature basins are well cemented, the rock columns of West Texas will act as compressible, brittle, elastic materials. 

In the following discussion of design theory the values of a number of elastic constants... 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

To examine this peculiar behavior, we have converted the elastic compressibility modulus, per unit area, Y (Fig. 12a), to the modulus per chain, Y = F/10 F (Fig. 12b). The elastic compressibility modulus per chain is practically constant, 0.6 0.1 pN/chain, at high densities and jumps to another constant value, 4.4 0.7 pN/chain, when the density decreases below the critical value. The ionization degree, a, of the carboxylic acid determined by FTIR spectroscopy gradually decreases with increasing chain density due to the charge regulation mechanism (also plotted in Fig. 12b). This shows that a does not account for the abrupt change in the elastic compressibihty modulus. 

A tube-plate is essentially a perforated plate with an unperforated rim, supported at its periphery. The tube holes weaken the plate and reduce its flexual rigidity. The equations developed for the stress analysis of unperforated plates (Section 13.3.5) can be used for perforated plates by substituting virtual (effective) values for the elastic constants E and v, in place of the normal values for the plate material. The virtual elastic constants E and v are functions of the plate ligament efficiency, Figure 13.38 see O Donnell and Langer (1962). The ligament efficiency of a perforated plate is defined as ... 

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