Lateral contraction

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... 

Fig. 4. Parameters in a coking cycle where (—) represents coke oven wall pressure ( ) coal charge center temperature (—) coke mass lateral contraction ...

Me(A) increases between entanglement points due to the retraction process at constant X. A more detailed treatment of disentanglement would account for the orientation function of the entanglements and lateral contraction, as discussed elsewhere [1]. Eq. 8.4 becomes... 

If the spring is subjected to a 50% overload for 1 day, estimate the percentage increase in the extension over the normal 1 day extension. The shear stress in the material is given by 16 WR/d. Use the creep curves supplied and assume a value of 0.4 for the lateral contraction ratio. 

Querzitronrinde, /. quercitron bark. Querzusammenziehtmg, /. lateral contraction. Quetsche,/. squeezer, wringer, presser, crusher plum. 

Alle the deformation zones contain a finite and equal number of extended chains in their most highly stretched strands. This surprising conformity of the deformation zones may well be the consequence of the imposed plane-strain fracture condition which impedes lateral contraction of the material. However, no quantitative explanation has been presented as yet. A plausible explanation would be to assume that due to the hindered lateral contraction additional tensile stresses are transferred to the most extended strand with each additional chain pulled out of the matrix [112]. 

Fig. 6.1. Model of crack fiber interaction in a simple composite, (a) In the uncracked composite, the fiber is gripped by the matrix, (b) A matrix crack is halted by the fiber. Increasing the load allows the crack to pass around the fiber without breaking the interfacial bond, (c) Interfacial shearing and lateral contraction of the fiber result in debonding and a further increment of crack extension, (d) After considerable debonding the fiber breaks at some weak spot within the matrix and further crack extension occurs, (e) The broken fiber end must be pulled out against the frictional grip of the matrix if total separation of the composite is to occur. After Harris (1980).

Note 5 Poisson s ratio is also sometimes called the lateral contraction ratio and is sometimes used in cases of non-linear deformation. The present definition will not apply in such cases. 

Stress is equal to the force per unit area, and strain or elongation is the extension per unit length. For an isotopic solid, i.e., one having the same properties regardless of direction, the strain is defined by Poisson s ratio, V = y /j which is the change in thickness (lateral contraction) to the change in length. 

Figure 16 illustrates several test specimens which have been used (46) in the multiaxial characterization of solid propellants. The arrows indicate the direction of load application. The strip tension or strip biaxial test has been used extensively in failure studies. It can be seen that the propellant is constrained by the long bonded edge so that lateral contraction is prevented and tension is produced in two axes simultaneously. The sample is free to contract normal to these axes. The ratio of the two principal tensile stresses may be varied from 0 to 0.5 by varying the bonded length of incompressible materials. 

When a material is stretched there is also contraction in the direction perpendicular to the direction of stretching. The ratio of the lateral contraction to the longitudinal extension is Poisson s ratio. For incompressible materials, Poisson s ratio is 0.5 and as rubbers are very nearly incompressible they have values close to this. 

Since the lateral contraction is half the tensile strain and the width of dumbbells is much smaller than the gauge length, a very high performance is needed from the "extensometer" to achieve sensible accuracy. Not surprisingly, accurate measurements have proved very difficult to obtain. 

Polyethylene crystals change shape with time at temperatures between the crystallisation temperature and the final melting point [42]. The process is commonly referred to as crystal thickening because the major effect is that the crystals increase their dimensions along the c axis. These changes may occur without any change in crystal volume, i.e. crystal thickening is associated with a lateral contraction of the crystals (Fig. 9). A second option is that... 

Weir with end contractions. When the length B of the crest of a rectangular weir is less than the width of the channel, there will be a lateral contraction of the nappe so that its width is less than B. It is believed that end contractions are a source of error, and so this type of weir is not considered so accurate as the preceding. Its chief virtue is that the approach channel need not be of uniform cross section or have smooth sides. 

Cipolletti weir. In order to avoid correcting for end contractions, the sides of the Cipolletti weir are given as 0.25 1 batter, which is supposed to add enough to the effective width of the stream to offset the lateral contraction. Thus, if B is the width or length of the crest,... 

It should be pointed out that the material parameter Ge can be determined in principle more precisely by means of equi-biaxial measurements than by uniaxial measurements. This is due to the fact that the first addend of the Ge-term in Eq. (45) increases linearly with X. This behavior results from the high lateral contraction on the equi-biaxial extension X2=X 2). It postulates a close dependency of the equi-biaxial stress on the tube constraint modu-... 

Any Poisson ratio V Change in width per unit width lateral contraction (13.7)... 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

In Eq. 10.18, v is the Poisson s ratio, named after French mathematician Simeon-Denis Poisson (1781-1840). Poisson s ratio is the dimensionless ratio of relative diameter change (lateral contraction per unit breadth) to relative length change (longitudinal... 

An experiment such as that in Fig. 11 -12a can produce changes in the volume as well as the shape of the test specimen. The elastic moduli listed in this figure are related by Poisson s ratio p, which is a measure of the lateral contraction accompanying a longitudinal extension ... 

The Poisson ratio, like the bulk, tensile, and shear creep compliance, is an increasing function of time because the lateral contraction cannot develop instantaneously in uniaxial tension but takes an infinite time to reach its ultimate value. In response to a sinusoidal uniaxial stretch, the complete Poison ratio is obtained ... 

Fig. 1. Tensile creep and recovery of rubber-modified plastics, showing axial strain (x), lateral contraction ( ), and volume strain (o) for ASA polymer and polypropylene copolymer at 20 °C...

Krafft [4] and Krafft and Mulherin [5] later extended the TLI model to describe stress corrosion crack growth. Crack growth was viewed in terms of the instability of tensile ligaments where their lateral contraction was augmented by uniform chemical dissolution of the tensile ligaments. For sustained-load crack growth in an inert environment, on the other hand, the reduction in the cross-sectional area of the ligaments would be associated with the creep rate (Landes and Wei [2], Yin et aL... 

Crazing is an important source of toughness in mbber-modified thermoplastics. A craze can be described as a layer of polymer a nanometer to a few micrometers thick, which has undergone plastic deformation approximately in the direction normal to the craze plane as a response to tension applied in this direction [Kambour, 1986]. Crazing occurs without lateral contraction. As a result, the polymer volume fraction in the craze is proportional to 1/, where is the draw ratio in the craze. The reduction in density occurs on such a small scale that the refractive index is markedly reduced, which accounts for the reflectivity of the craze [Kramer, 1983]. 

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