Elastic constants shear modulus

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. 

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

Adhesive Layer with Constant Shear Modulus. The assumption is that a lap shear test with small beech specimens connected with this elastic adhesive layer will yield the following linear relationship between the shear stress r and the slip v ... 

Metals are normally homogeneous and isotropic in nature and their reaction to an applied load can be defined by knowing two of the three basic elastic constants (the modulus of elasticity E, the shear modulus G and Poisson s ratio v). A basic uni-... 

Figure 6-43. The modulus of elasticity, the shear modulus, and Poisson s ratio for an isotropic material such as fiber matt-reinforced plastics. Because constants do not vary with the angle of load, the ratio of modulus Ei at any angle to E at any reference direction is unity. The shear modulus is a constant proportion of E, and Poisson s ratio is constant.

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... 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

The plot of the rheological parameters (relaxation time, shear modulus, and stored elastic energy) are shown in Figs. 22-24. The relaxation time increases as the ACM content is increased to attain a maximum at 60 40 = ACM XNBR blend ratio for the preblends. For lower shear rate the rise is sharp and after 60 40 blend ratio, // remains almost constant, whereas for the higher shear rate region the rise is not sharp and after 60 40 blend ratio ty decreases as ACM percent increased in the blend. In the case of the preheated blends the /y increases up to 50 50 blend ratio and then decreases with the addition of ACM in the blend. The preheating increases the ty in both shear rate regions. 

The plot of the rheological parameters (relaxation time, /r shear modulus, G and stored elastic energy, W ) are given in Figs. 28-30. The relaxation time of both preblends and preheated blends remains almost constant up to 50 50 blend ratio and then shoots up drastically at both shear rates. Up to 50 50 blend ratio it is observed that the relaxation time is more at lower shear rate. Preheating of blends lowers the values. 

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... 

For an elastic material the shear stress is related to the shear strain through a constant of proportionality G, railed the shear modulus. The shear strain is dimensionless, and the shear modulus has units of force per unit area. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

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 ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

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. 

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,... 

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. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

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... 

The results obtained by the present mechanical measurements are also consistent with the previous experimental results of the dynamic light scattering studies of the collective diffusion coefficient of gels and the rheological studies of the shear modulus of gels. The studies published by different researchers indicate that the concentration dependence of the collective diffusion constant of the polymer networks of gel and that of the elastic modulus are well represented by the following power law relationships [2, 3, 5]... 

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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