Poisson negative

The Poisson equation assumes that the solvent is completely homogeneous. However, a solvent can have a significant amount of charge separation. An example of a heterogeneous solution would be a polar solute molecule surrounded by water with NaCl in solution. The positive sodium and negative... 

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... 

Poisson s ratio, is the negative of the ratio of the strain transverse to the fiber direction, 8, and the strain ia the fiber direction, S, when the lamina is loaded ia the fiber direction and can also be expressed ia terms of the properties of the constituents through the rule of mixtures. 

When it strains in this way, the cube usually gets thinner. The amount by which it shrinks inwards is described by Poisson s ratio, v, which is the negative of the ratio of the inward strain to the original tensile strain ... 

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... 

The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

That is, the total strain will be the sum of the tensile strain due to and the negative strain due to the Poisson s ratio effect caused by Oy. 

E, E2, Eg = Young s (extension) moduli in the 1-, 2-, and 3-directions V j = Poisson s ratio (extension-extension coupling coefficient), i.e., the negative of the transverse strain in the j-direction over the strain in the i-direction when stress is applied in the i-direction, I.e.,... 

If the bulk modulus were negative, a hydrostatic pressure would cause expansion of a cube of isotropic material Finally, for isotropic materials, Poisson s ratio is restricted to the range... 

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,... 

When a bar is elongated axially, as in Figure 2-25, it will contract laterally. The negative ratio of the lateral strain to the axial strain is called Poisson s ratio v. For isotropic materials, materials that have the same elastic properties in all directions, Poisson s ratio has a value of about 0.3. 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

There are some other characteristics of the Poisson distribution that differ from the Normal distribution in ways that are of importance to us here. The chief one is that the Poisson distribution does not admit of negative values. This makes intuitive sense since the Poisson distribution is a distribution that results from a counting operation, the smallest number that you can achieve when counting objects is zero. We will be using this fact during the course of our derivations. 

Another characteristic of scintillation noise is that, since it represents the amount of energy in the optical beam, it can never attain a negative value. In this respect it is similar to the Poisson distribution, which also can never attain a negative value. On the other hand, since it is a continuous distribution it will behave the same way as the constant-noise case in regard to achieving an actual zero any given reading can become... 

Fisher B, Costantino J, Redmond C, Poisson R, Bowman D, Couture J, et al. (1989) A randomized clinical trial evaluating tamoxifen in the treatment of patients with node-negative breast cancer who have estrogen-receptor-positive tumors. N Engl J Med 320(8) 479—484... 

The radial (compressive) stress, qo, is caused by the matrix shrinkage and differential thermal contraction of the constituents upon cooling from the processing temperature. It should be noted that q a, z) is compressive (i.e. negative) when the fiber has a lower Poisson ratio than the matrix (vf < Vm) as is the normal case for most fiber composites. It follows that q (a,z) acts in synergy with the compressive radial stress, 0, as opposed to the case of the fiber pull-out test where the two radial stresses counterbalance, to be demonstrated in Section 4.3. Combining Eqs. (4.11), (4.12), (4,18) and (4.29), and for the boundary conditions at the debonded region... 

The minus sign in Eq. (5.9) is to account for the fact that Ad as defined above is usually negative. Thus, Poisson s ratio is normally a positive quantity, though there is nothing that prevents it from having a negative value. Eor constant volume deformations (such as in polymeric elastomers), v = 0.5, but for most metals, Poisson s ratio varies between 0.25 and 0.35. Values of Poisson s ratio for selected materials are presented in Appendix 7. 

The analysis has shown that PAI may only be negative, and PAB ( both positive and negative. Therefore, the thermal effect accompanying a reversible stretching of the model depends on the ratio between p and PA,n and may be a function of strain even at small strains. Besides, Poisson s ratio for such a heterogeneous model may exeed 0.5, Direct measurements of Poisson s ratio for a number of various oriented crystalline polymers are consistent with this suggestion (see Table 5). 

A process with independent increments can be generated by compounding Poisson processes in the following way. Take a random set of dots on the time axis forming shot noise as in (II.3.14) the density fx will now be called p. Define a process Z(t) by stipulating that, at each dot, Z jumps by an amount z (positive or negative), which is random with probability density w(z). Clearly the increment of Z between t and t + T is independent of previous history and its probability distribution has the form (IV.4.7). It is easy to compute. 

The first line shows that the fluctuations of X are the same as in the reaction (1.1), compare (1.13). This could have been predicted, because as far as X is concerned the present reaction scheme is the same. The third line shows that there is a negative correlation between X and Y, an obvious consequence of the fact that each time when two atoms X associate, nx decreases and nY increases. Finally (5.13b) shows that the variance of nY is intermediate between the value belonging to a Poisson distribution and the value belonging to nx. 

Ycganch-Hacri, A Wcidncr, D.J., and J.B. Parise Elasticity of Alpha-Cristobalitc A Silicon Dioxide with a Negative Poisson s Ratio, Science, 650 (July 31, 1992). Yun, W, and R.T. Howe Recent Developments in Silicon Micro-accclcromctcrs, ... 

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