Shear elastic constant

P Lam6 elastic constant, shear modulus (6.2) Table 6.1... 

A list of the elastic constants of all the heat treated and as received specimens is given in Table 4. The measurements were done in the temperature range 20 - 1250°C by the resonance method [21]. This listing only contains the room temperature and 1200°C values of the elastic constants, shear modulus, Young s modulus and the Poisson s ratio. Almost no change can be seen from these results, if one considers the complexity of the measurements. 

Fig. 7. Relations between elastic constants and ultrasonic wave velocities, (a) Young s modulus (b) shear modulus (c) Poisson s ratio and (d) bulk...

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

In order to use this model it is necessary to estimate the ratio of the longitudinal to shear elastic constants, CjilCbb- In the original model this quantity was... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

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. ...

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

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-... 

For interpreting indentation behavior, a useful parameter is the ratio of the hardness number, H to the shear modulus. For cubic crystals the latter is the elastic constant, C44. This ratio was used by Gilman (1973) and was used more generally by Chin (1975) who showed that it varies systematically with the type of chemical bonding in crystals. It has become known as the Chin-Gilman parameter (H/C44). Some average values for the three main classes of cubic crystals are given in Table 2.1. 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

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. 

If one side of the quartz is coated with material, the spectrum of the resonances is shifted to lower frequencies. It has been observed that the three above mentioned modes have a somewhat differing mass sensitivity and thus experience somewhat different frequency shifts. This difference is utilized to determine the Z value of the material. By using the equations for the individual modes and observing the frequencies for the (100) and the (102) mode, one can calculate the ratio of the two elastic constants Cgg and C55. These two elastic constants are based on the shear motion. The key element in Wajid s theory is the following equation ... 

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

Some fermentation broths are non-Newtonian due to the presence of microbial mycelia or fermentation products, such as polysaccharides. In some cases, a small amount of water-soluble polymer may be added to the broth to reduce stirrer power requirements, or to protect the microbes against excessive shear forces. These additives may develop non-Newtonian viscosity or even viscoelasticity of the broth, which in turn will affect the aeration characteristics of the fermentor. Viscoelastic liquids exhibit elasticity superimposed on viscosity. The elastic constant, an index of elasticity, is defined as the ratio of stress (Pa) to strain (—), while viscosity is shear stress divided by shear rate (Equation 2.4). The relaxation time (s) is viscosity (Pa s) divided by the elastic constant (Pa). 

For anisotropic materials torsion is discussed in the books by Love, Lekhnitskii175 and Hearmon185. The torque M now depends not upon one elastic constant only, as in the isotropic case, but upon two. This makes the determination of shear modulus by a torsion test a difficult task and requires careful experimentation. Early work on this for polymers was done by Raumann195, by Ladizesky and Ward205 and by Arridge and Folkes165. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

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