Longitudinal strain, volume

Volume Strain Vs. Longitudinal Strain. Volume strains (AV/V0) were calculated from longitudinal strain ci and lateral strain c2 by use of the following expression ... 

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... 

Elasticity of solids determines their strain response to stress. Small elastic changes produce proportional, recoverable strains. The coefficient of proportionality is the modulus of elasticity, which varies with the mode of deformation. In axial tension, E is Young s modulus for changes in shape, G is the shear modulus for changes in volume, B is the bulk modulus. For isotropic solids, the three moduli are interrelated by Poisson s ratio, the ratio of traverse to longitudinal strain under axial load. 

Longitudinal strain es was measured in the central 20 mm of the specimen, and lateral strain e1 was measured simultaneously at the center of the gage portion is usually negative in a tensile test. The lateral strain e2 was not measured. In the calculations all specimens, including those cut from a drawn sheet, were assumed to be transversely isotropic— i.e., ei = e2. On the basis of this assumption the volume strain AV/V was calculated from the expression ... 

Earlier work (4,6) demonstrated that a high gradient in the volume strain-longitudinal strain curve is associated with a large drop in the modulus of the material. This is to be expected, as the crazes have much lower moduli than the material from which they were formed (7). As a result of the creep tests described above, the 100 sec tensile modulus of Luran S 757R at a strain of 0.5% fell from 1.99 to 1.30 GN/m2, and the modulus of Luran S 776S at a similar strain fell from 1.65 to 0.98 GN/m2. 

Figure 7. Volume strain vs. longitudinal strain curves for three specimens of ABS 1 at 10,000 in./min. Theoretical cavitation and shear curves are also included.

The (1 -f- c2)2 term arises from the assumption that specimens deform isotropically in width and thickness. Volume strain vs. longitudinal strain curves are shown for three specimens of ABS 1 in Figure 7. For comparison, theoretical curves for total cavitation, i.e., c2 = 0, and pure shear, i.e., AV = 0, are also shown. Volume strain vs. longitudinal strain curves, together with nominal stress vs. nominal strain curves, for ABS 1, ABS 2, and ABS 3 are shown in Figure 8. Because of the size of the strains involved, it is not possible to approximate Equation 3 with an expression which contains only first-order powers of strain when calculating volume strains. 

Figure 3.20. Relationship between volume strain A F/Fand longitudinal strain AL/L for creep of HiPS and blends of HiPS with PPO (from 12.5 to 50% ABS) (Bucknall et u/., 1972a). Note that the fraction of deformation due to crazing as opposed to shear drops from about 100 % in the case of HiPS alone to about 60% for a 50-50 blend with PPO. It is interesting that the HiPS, itself a craze-prone polymer, promotes shear yielding in the somewhat ductile PPO.

Figure 3.21. Relationship between volume strain AF/F and longitudinal strain for creep of a high-impact ABS resin, showing mechanism of creep as a function of strain at five different stresses. (Bucknall and Drink water, 1973.)...

Where a is the longitudinal stress, e is corresponding strain, and E is called Young s modulus (or the modulus of elasticity). Similarly, in shear deformation, the modulus is called the shear modulus or the modulus of rigidity (G). When a hydrostatic force is applied, a third elastic modulus is used the modulus of compressibility or bulk modulus (K). It is defined as the ratio of hydrostatic pressure to volume strain. A deformation (elongation or compression) caused by an axial force is always associated with an opposite deformation (contraction or expansion) in the lateral direction. The ratio of the lateral strain to the longitudinal strain is the fourth elastic constant called Poisson s ratio (v). For a small deformation, elastic parameters can be correlated in the following way ... 

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... 

Figure 17. Longitudinal vs. transverse stress-strain behavior for 50-volume rayon composite...

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 ... 

The first quantitative study of deformation mechanisms in ABS polymers was made by Bucknall and Drinkwater, who used accurate exten-someters to make simultaneous measurements of longitudinal and lateral strains during tensile creep tests (4). Volume strains calculated from these data were used to determine the extent of craze formation, and lateral strains were used to follow shear processes. Thus the tensile deformation was analyzed in terms of the two mechanisms, and the kinetics of each mechanism were studied separately. Bucknall and Drinkwater showed that both crazing and shear processes contribute significantly to the creep of Cycolac T—an ABS emulsion polymer—at room temperature and at relatively low stresses and strain rates. 

III. The third step of direct longitudinal transmission of strain onto connected crystalline blocks leads to a perfect stretching of these fibrils. Because of the alignment of the molecules the fibers in this condition should possess a strength about 1 to 2 orders of magnitude higher than the yield stress of randomly distributed folded polycrystals. As the fibrils are able to stabilize the enhanced micro-void volume between them, a lateral coalescence of these voids finally provides a local deformation zone in the shape of a craze as known from amorphous polymers. 

The applications of Hooke s law [equations (2-14) and (2-18)] discussed above have assumed that the volume of the material is invariant with strain during a tensile deformation. However, because the pressure is not zero, this may not be the case, and the strains in each direction must be known to account for this. By measuring the actual transverse (yyy) and longitudinal (y ) strains, one can define the ratio of these two strains as a material property. This is called Poisson s ratio ju, and is defined as ... 

---