Poisson’s constant

When the sample is measured using a differential-type TMA, it is not necessary to make corrections [4]. The volumetric thermal expansion coefficient, a, cannot be measured by TMA. Generally, the expansion coefficient of solids is small rz = 3/j is currently used. However, for samples whose Poisson s constant is not 1.0, a... 

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... 

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. 

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. ...

The constant value of 0.25 for Poisson s ratio versus depth reflects the geology and the rock mechanics of the mature sedimentary basin in the West Texas region. Since mature basins are well cemented, the rock columns of West Texas will act as compressible, brittle, elastic materials. 

Poison s ratio It is the proportion of lateral strain to longitudinal strain under conditions of uniform longitudinal stress within the proportional or elastic limit. When the material s deformation is within the elastic range it results in a lateral to longitudinal strain that will always be constant. In mathematical terms, Poisson s ratio is the diameter of the test specimen before and after elongation divided by the length of the specimen before and after elongation. Poisson s ratio will have more than one value if the material is not isotropic... 

Poisson s ratio always falls within the range of 0 to 0.5. A zero value indicates that the specimen would suffer no reduction in diameter or contraction laterally during elongation but would undergo a reduction in density. A value of 0.5 indicates that the specimen s volume would remain constant during elongation or as the diameter decreases. For most plastics the ratio lies between 0.10 and 0.40 (Tables 2-1 and 2-2). 

The application of Poisson s ratio is frequently required in the design of structures that are markedly 2-D or 3-D, rather than one-dimensional like a beam. For example, it is needed to calculate the so-called plate constant for flat plates that will be subjected to bending loads in use. The higher Poisson s ratio, the greater the plate constant and the more rigid the plate. 

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 ... 

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. 

This means that any solutions of Poisson s equation, for instance U ip) and U2(p), can differ from each other at every point of the volume Fby a constant only, if their normal derivatives coincide on the boundary surface S. Thus, this boundary value problem defines also uniquely the field of attraction, and it can be written as... 

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. 

Tensile stress-strain tests give another elastic constant, called Poisson s ratio, v. Poisson s ratio is defined for very small elongations as the decrease in width of the specimen per unit initial width divided by the increase in. length per unit initial length on the application of a tensile load ... 

In this equation e is the longitudinal strain and er is the strain in the width (transverse) direction or the direction perpendicular to the applied force It can be shown that when Poisson s ratio is 0.50, the volume of the specimen remains constant while being stretched. This condition of constant volume holds for liquids and ideal rubbers. In general, there is an increase in volume, which is given by... 

Another effect o(f orientation shows up as changes in Poisson s ratio, which can be determined as a function of time by combining the results of tension and torsion creep tests. Poisson s ratio of rigid unoriented polymers remains nearly constant or slowly increases with time. Orientation can drastically change Poisson s ratio (254). Such anisotropic materials actually have more than one Poisson s ratio. The Poisson s ratio as determined when a load is applied parallel to the orientation direction is expected to... 

Continuum models are rooted in classical electrostatics, and its applications in the analysis of the dielectric constants of polar liquids. A key relationship is Poisson s equation,... 

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