Youngs Modulus and Poissons Ratio

The elasticity of isotropic materials is usually described by Young s modulus E, the Poisson ratio v and by the shear modulus G. These quantities are of ubiquitous use in engineering. It is also quite generally known that these quantities are related by 

We now turn briefly to the question to what extent it might make sense to deflne these quantities also for crystals. Because of the anisotropy of crystals it is evident that this will always be possible only for one or two specified directions. 

Let us choose the direction for which we wish to determine Young s modulus E as the x l axis of a coordinate system rotated with respect to the crystal coordinate system. Let us assume in addition that with respect to this primed coordinate system there is only one component of stress, namely T[, all others being zero. This normal stress T[ causes a strain S[ given by 

In analogy to isotropic bodies we define Young s modulus in the direction considered here by the ratio of 

To visualize the elastic properties of crystals it is common to draw a polar diagram with radius vectors of length equal to E (that is the reciprocal of Yj j) in all directions. The endpoints of these radius vectors define the so called elasticity surface of the given crystal. For quartz it is in the form of an oval. The numerical value of Young s modulus for each specified direction may be calculated from Eq. (3.57) and from the transformation equation for s j j. 

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The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

In this work Young modulus and Poisson ratio were found for sintered copper using circular plate torsial vibrations method [13. Diagram of measuring equipment is displayed in Fig.l. 

Strained set of lattice parameters and calculating the stress from the peak shifts, taking into account the angle of the detected sets of planes relative to the surface (see discussion above). If the assumed unstrained lattice parameters are incorrect not all peaks will give the same values. It should be borne in mind that, because of stoichiometry or impurity effects, modified surface films often have unstrained lattice parameters that are different from the same materials in the bulk form. In addition, thin film mechanical properties (Young s modulus and Poisson ratio) can differ from those of bulk materials. Where pronounced texture and stress are present simultaneously analysis can be particularly difficult. 

Fig. 5. Apparatus used to measure Young s modulus and Poisson ratio of gel films. AM anglemeter, LB bar to apply a longitudinal force, W weight, TB bar to detect thickness change, M mirror, S fulcrum, FB fixed bar, F gel film, TW weight to apply preload on gel film through pins...

Local stress concentration in the matrix at and between particles (depending on particle shape, particle content, Youngs modulus, and Poisson s ratio of matrix polymer and particles)... 

The photodiode-lever separation can be measured sufficiently accurately to ensure that it is a negligible source of error. The value of this distance varies slightly om scan to scan, depending on the beam alignment, but is typically 12.0 mm. Finally one must consider the geometrical and mechanical properties of the cantilever. Because the stoichiometry of the cantilevers can vary from that of bulk amorphous silicon nitride, the values of the Young s modulus and Poisson ratio are not very accurately known. We have used quoted values for bulk silicon nitride, but the accuracy of these values is hard to assess. As mentioned earlier, the quoted thickness of the levers was checked by measuring the lever resonant frequency, and frie width of the levers is accurately known (10 pm). 

In order to predict the permeability change during CBM production, cleat compressibility and adsorption-induced volumetric swelling strain should be studied firstly. According to Fu et al. (2003), the Young modulus and Poisson s ratio of middle rank coal under at the depth of 500-800 m in central and southern part of the Qinshui Basin is 3.05 Mpa and 0.19, respectively. The sample used in this study belongs to middle rank coal and is from about 700 m depth in Changzhi area, south... 

ASTM C-1259 Dynamic Young s modulus, shear modulus, and Poisson ratio for advanced ceramics by impulse excitation of vibration... 

The practical calculations of deformations and displacements of specimens and elements made of concrete-like composites are based on simple assumptions of Young s modulus and Poisson ratio, corrected by experimental observations. In such an approach, the strain-stress behaviour of concretelike composites, as determined after the testing of specimens and elements, is generally considered as non-linear and non-elastic, as shown in Figure 8.3 for a case of bending or axial compression. Therefore, several different values of Young s modulus are distinguished, namely ... 

These formulae were partly verified experimentally. Namely, agreement within a few per cent has been found between calculated and observed values of Young s modulus and Poisson ratio. Also values of were only about 8% higher than the experimental ones. Other proposed formulae derived from the law of mixtures are briefly described below. 

The notation used to describe the contacts is shown in Figure 1. P t) is the time dependent applied load, S P,t) the deformation, a(P,t) the contact radius, and R and Ri the radii of curvature of the two bodies at the point of contact. We consider only flat substrates so that R R and R2 = >. Each elastic material is described by its Young modulus E, Poisson ratio v, and is assumed to be isotopic so that the shear modulus is G = Ejl + v). Viscoelastic materials are assumed to be linear with stress relaxation functions E t) and creep compliance functions J t), All properties are assumed to be independent of depth. 

Tsai conducted experiments to measure the various moduli of glass-fiber-epoxy-resin composite materials [3-1]. The glass fibers and epoxy resin had a Young s modulus and Poisson s ratio of 10.6 x 10 psi (73 GPa) and. 22 and. 5 x 10 psi (3.5 GPa) and. 35, respectively. 

The shear modulus is related to Young s modulus and Poisson s ratio by... 

The designer must be aware that as the degree of anisotropy increases, the number of constants or moduli required to describe the material increases with isotropic construction one could use the usual independent constants to describe the mechanical response of materials, namely, Young s modulus and Poisson s ratio (Chapter 2). With no prior experience or available data for a particular product design, uncertainty of material properties along with questionable applicability of the simple analysis techniques generally used require end use testing of molded products before final approval of its performance is determined. 

Under increasing strain the propint volume increases from the voids created around the unbonded solid particles. Nonlinearities in Young s modulus and Poisson s ratio then occur. Francis (Ref 50) shows this effect for a carboxy-terminated polybutadiene composite propellant with 14% binder as in Figure 12. He concludes that nonlinearities in low-temperature properties reduce the predicted stress and strain values upon cooling a solid motor, and therefore a structural analysis that neglects these effects will be conservative. However, when the predictions are extended to a pressurized fiberglas motor case, the nonlinearities in properties produce greater strains than those predicted with linear analysis... 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

Assuming that a number of NMR data sets (e.g., 2-D or 3-D maps of displacement vectors resulting from an external periodic excitation) from an object are acquired, the remaining difficulty is their reconstruction into viscoelastic parameters. As written in Section 2 the basic physical equation is a partial differential equation (PDE, Eq. (3)) relating the displacement vector to the density, the attenuation, Young s modulus and Poisson s ratio of the medium. The reconstruction problem is indeed two-fold ... 

Young s modulus ( ) and Poisson s ratio (u) for several magnetostrictive films and different substrates... 

Investigations on SmFeB/TbFeB multilayers have shown that strain and stress are transferred effectively at the interface (Shima et al. 1997). In these SmFeBA bFeB multilayers, the thickness of the layers was varied, and it was found that the magnetostriction is sensitively affected by Young s modulus, the Poisson ratio and the thickness of the constituent layers. 

Simple linear FEA programmes, as used for stress analysis of metals, take Young s modulus and Poisson s ratio as input but this is not satisfactory for rubbers because the strains involved cannot be considered as small and the Poisson s ratio is very close to 0.5. Non-linear FEA programmes for use with rubbers take data from a model such as the Mooney-Rivlin equation. More sophisticated programmes will allow a number of models to be used and may also allow direct input of the stress strain data. 

For lower symmetries there is more than one Young Modulus and shear modulus and Poisson s ratio, and their interrelation is not so simple. 

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