Poisson constant

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

It follows that the constant CQ is equal to ViE and also equal to the shear modulus G, because for rubbers the Poisson constant v V2. 

In the equations above a is the true tensile stress, i.e. F/A. In practice in general use is made of engineering stress, which is equal to F/Aa, where F is the tensile load and A and A0 are the cross-sectional surface areas of the sample in the deformed and non-deformed state, respectively. Because the Poisson constant Vi for rubbers A = A0/A, so that the equations for the tensile stress become ... 

An extensive study by Koppelmann (1958) of viscoelastic functions through one of the secondary mechanisms in Poly(methyl methacrylate) is shown in Fig. 13.15. Dynamic storage moduli and tan <5 s near 25 °C are plotted vs. angular frequency. It shows first, that a secondary mechanism is present, secondly that E and G are not completely parallel, because the Poisson constant is not a real constant, but also dependent on frequency (the numbers in between both moduli are the actual, calculated Poisson constants) and thirdly that tan <5E is practically equal to tan <5G. 

A sharp central crack of length 30 mm in a wide, thick plate of a glassy plastic commences to propagate at [Pg.475]

Hgure 4. Shear (G) and Young (Y) moduli, and Poisson constant p of a poly (vinyl acetate) glasn as functions of temperature at atmospheric pressure. Reprc uc frcxn ref. 16. Copyright 1968 American Chemical Society. 

Part (c) The Poisson equation is solved with a Poisson constant determined from Eq. (6.49) ... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The Poisson equation relates spatial variation of the potential 4> at position r to the density of the charge distribution, p, in a medium with a dielectric constant e... 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

The charge density is simply the distribution of charge throughout the system and has 1 units of Cm . The Poisson equation is thus a second-order differential equation (V the usual abbreviation for (d /dr ) + (f /dx/) + (d /dz )). For a set of point charges in constant dielectric the Poisson equation reduces to Coulomb s law. However, if the dielectr... 

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... 

One final point. We earlier defined Poisson s ratio as the negative of the lateral shrinkage strain to the tensile strain. This quantity, Poisson s ratio, is also an elastic constant, so we have four elastic constants E, G, K and v. In a moment when we give data for the elastic constants we list data only for . For many materials it is useful to know that... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

The modulus term in this equation can be obtained in the same way as in the previous example. However, the difference in this case is the term V. For elastic materials this is called Poissons Ratio and is the ratio of the transverse strain to the axial strain (See Appendix C). For any particular metal this is a constant, generally in the range 0.28 to 0.35. For plastics V is not a constant. It is dependent on time, temperature, stress, etc and so it is often given the alternative names of Creep Contraction Ratio or Lateral Strain Ratio. There is very little published information on the creep contraction ratio for plastics but generally it varies from about 0.33 for hard plastics (such as acrylic) to almost 0.5 for elastomers. Some typical values are given in Table 2.1 but do remember that these may change in specific loading situations. 

Engineering constants (sometimes known as technical constants) are generalized Young s moduli, Poisson s ratios, and shear moduli as well as some other behavioral constants that will be discussed in Section 2.6. These constants are measured in simple tests such as uniaxial tension or pure shear tests. Thus, these constants with their obvious physical interpretation have more direct meaning than the components... 

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 preceding restrictions on engineering constants for orthotropic materials are used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model. For boron-epoxy composite materials, Dickerson and DiMartino [2-3] measured Poisson s ratios as high as 1.97 for the negative of the strain in the 2-direction over the strain in the 1-direction due to loading in the 1-direction (v 2)- The reported values of the Young s moduli for the two directions are E = 11.86 x 10 psi (81.77 GPa) and E2 = 1.33x10 psi (9.17 GPa). Thus,... 

A simplified performance index for stiffness is readily obtained from the essentials of micromechanics theory (see, for example. Chapter 3). The fundamental engineering constants for a unidirectionally reinforced lamina, ., 2, v.,2, and G.,2, are easily analyzed with simple back-of-the-envelope calculations that reveal which engineering constants are dominated by the fiber properties, which by the matrix properties, and which are not dominated by either fiber or matrix properties. Recall that the fiber-direction modulus, is fiber-dominated. Moreover, both the modulus transverse to the fibers, 2, and the shear modulus, G12. are matrix-dominated. Finally, the Poisson s ratio, v.,2, is neither fiber-dominated nor matrix-dominated. Accordingly, if for design purposes the matrix has been selected but the value of 1 is insufficient, then another more-capable fiber system is necessary. Flowever, if 2 and/or G12 are insufficient, then selection of a different fiber system will do no practical good. The actual problem is the matrix systemi The same arguments apply to variations in the relative percentages of fiber and matrix for a fixed material system. 

The curves for 3M XP251S fiberglass-epoxy are shown in Figures C-1 through C-5 [C-1]. Curves are given for both tensile and compressive behavior of the direct stresses. Note that the behavior in the fiber direction is essentially linear in both tension and compression. Transverse to the fiber direction, the behavior is nearly linear in tension, but very nonlinear in compression. The shear stress-strain curve is highly nonlinear. The Poisson s ratios (not shown) are essentially constant with values v.,2 =. 25 and V21 =. 09. 

Tlie pdf of the Poisson distribution can be derived by taking the limit of tlie binomial pdf as n qo, p 0, and np = remains constant. The Poisson pdf is given by... 

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