Elastic behavior

Various theories of the elastic extension of oriented fibers have been developed. In a series model, uniform stress throughout the fiber is assumed. Two kinds of series models have been proposed. The classical series model, first applied by Ward to polymer fibers, is based on a series arrangement of cube-shaped elements, whereas in the modified 

Recently Allen and Roche presented equations for the modulus and the tensile curve of a fiber that had already been derived earlier by Northolt. However, in both papers the distinction with respect to the 

Other possible models for fibers are the uniform strain and the elastic unwrinkling model. The first does not describe the observed linear relation between the dynamic compliance and the orientation distribution parameter of the chains in the polymer fibers the second is not essentially different from the classical series model. 

For the relatively narrow orientation distributions commonly found in polymer fibers the second moment defined according to eqn (6.16) hardly differs from the one calculated with the standard averaging procedure. 

The equation for the elastic tensile curve of the fiber with well-oriented chains is 

Fully reversible elastic behavior is characterized by a direct proportionality between stresses and deformations and is described by Hook s law  

Physical-Chemical Mechanics of Disperse Systems and Materials 

FIGURE 3.1 Deformation (strain), y, resulting from the applied shear stress t. 

In the rheology of condensed phases, the elastic modulus G is often used as the sole characteristic of elastic behavior. In isotropic media mechanics, it has been established that for solid-like bodies, the modulus G 2/5 of the Young s modulus [10]. 

Elastic behavior is typical of solid bodies. The nature of elasticity is related to the reversibility of small deformations of interatomic bonds. Within the limits of small deformations, the potential energy curve can be approximated with a quadratic parabola, which conforms to a linear relationship between y and t. The magnitudes of the elastic modulus G depend on the nature of the interactions in the solid body. For molecular crystals, G is 10 N/m, while for metals and covalent crystals, the value of G is on the order of 10 N/m. The elastic modulus G reveals either a weak dependence from temperature or no temperature dependence at all. 

As noted in the introduction, atomic or molecular volumes indicate bonding strength. Small atomic volume (or small atom separation) indicates strong interatomic cohesion. Thus, the radii of 

The above transition metal carbides and nitrides are often referred to as interstitial phases because the small C atoms occupy interstitial positions within the metallic sublattice, leading to close-packed phases. Thus, these phases exhibit small atomic or molecular volumes. In fact, phases (both metallic and nonmetallic) containing the small radius atoms or ions B, C, N, O, Be, Al, and Si generally are the strongest (Table 10). 

Hardness tests, which result in plastic as well as an elastic deformation, are also qualitative indicators of the strength of materials. Nowotny has suggested the following expression for the Vickers hardness H of the monocarbides of the transition elements  

Young s Moduli and Melting Points for a Number of Materials  

Mitra and Marshall showed that the bulk moduli of 16 alkali halides with the NaCl structure are also equal to a constant/u , where a is the lattice constant and n is closer to three than to four. Whereas Keyes s constant is given by q, no such simple constant could be given for the alkali halides, which are essentially ionically bonded phases. Kittel derived the following expression for the bulk modulus K, applicable to the above alkali halides  

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This test attempts to characterize the brittleness of bitumen at low temperatures. It consists of measuring the temperature at which fissures appear on a bitumen film spread on a blade as it is repeatedly flexed. This test is delicate and of questionable reliability, but it is currently the only one that allows the elastic behavior of bitumen on decreasing temperature to be characterized. It is standardized in France (T 66-026). 

In this chapter we examine the elastic behavior of polymers. We shall see that this behavior is quite different from the elasticity displayed by metals and substances composed of small molecules. This is a direct consequence of the chain structure of the polymer molecules. In many polymers elasticity does not occur alone, but coupled with viscous phenomena. The combination of these effects is called viscoelasticity. We shall examine this behavior as well. 

Minimization of the elastic behavior of the fluid at high deformation rates that are present when high molecular weight water-soluble polymers are used to obtain cost-efficient viscosities at low shear rates. 

Much more information can be obtained by examining the mechanical properties of a viscoelastic material over an extensive temperature range. A convenient nondestmctive method is the measurement of torsional modulus. A number of instmments are available (13—18). More details on use and interpretation of these measurements may be found in references 8 and 19—25. An increase in modulus value means an increase in polymer hardness or stiffness. The various regions of elastic behavior are shown in Figure 1. Curve A of Figure 1 is that of a soft polymer, curve B of a hard polymer. To a close approximation both are transpositions of each other on the temperature scale. A copolymer curve would fall between those of the homopolymers, with the displacement depending on the amount of hard monomer in the copolymer (26—28). 

Elastic Behavior. In the following discussion of the equations relevant to the design of thick-walled hoUow cylinders, it should be assumed that the material of which the cylinder is made is isotropic and that the cylinder is long and initially free from stress. It may be shown (1,2) that if a cylinder of inner radius, and outer radius, is subjected to a uniform internal pressure, the principal stresses in the radial and tangential directions, and <7, at any radius r, such that > r > are given by... 

Film. Blown film manufactured from PB has a high tensile strength and exhibits good resistance to tear, impact, and puncture (47). Such film also exhibits hard-elastic behavior that is, it can recover its original length even after extensive stretching. Some properties of PB film are given in Table 5. 

Detailed treatments of the rheology of various dispersed systems are available (71—73), as are reviews of the viscous and elastic behavior of dispersions (74,75), of the flow properties of concentrated suspensions (75—82), and of viscoelastic properties (83—85). References are also available that deal with blood red ceU suspensions (69,70,86). 

The Weissenberg Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characteri2ation of viscoelastic materials. Its capabiUties include measurement of steady-state rotational shear within a viscosity range of 10 — mPa-s at shear rates of, of normal forces (elastic... 

Elastic Behavior. Elastic deformation is defined as the reversible deformation that occurs when a load is appHed. Most ceramics deform in a linear elastic fashion, ie, the amount of reversible deformation is a linear function of the appHed stress up to a certain stress level. If the appHed stress is increased any further the ceramic fractures catastrophically. This is in contrast to most metals which initially deform elastically and then begin to deform plastically. Plastic deformation allows stresses to be dissipated rather than building to the point where bonds break irreversibly. 

Elastic behavior is commonly quantified by the Young s modulus E, the proportionality constant between the appHed tensile stress O, and the tensile strain (A length/original length). 

Coating solutions often exhibit a mixture of viscous and elastic behavior, with the response of a particular system depending on the stmcture of the material and the extent of deformation. Eor example, polymer melts can be highly elastic if a polymer chain can stretch when subjected to deformation. 

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

Displacement Strains The concepts of strain imposed by restraint of thermal expansion or contraction and by external movement described for metallic piping apply in principle to nonmetals. Nevertheless, the assumption that stresses throughout the piping system can be predic ted from these strains because of fully elastic behavior of the piping materials is not generally valid for nonmetals. 

Elastic Behavior The assumption that displacement strains will produce proportional stress over a sufficiently wide range to justify an elastic-stress analysis often is not valid for nonmetals. In brittle nonmetallic piping, strains initially will produce relatively large elastic stresses. The total displacement strain must be kept small, however, since overstrain results in failure rather than plastic deformation. In plastic and resin nonmetallic piping strains generally will produce stresses of the overstrained (plasfic) type even at relatively low values of total displacement strain. 

It is usual in the classical theory to assume that the stress rate is independent of the hardening parameters, since the elastic behavior is expected to be unaffected by plastic deformation. Consequently, the stress rate relation (5.23) reduces to... 

Since elastic behavior is important for inelastic constitutive equations, it is instructive to examine the behavior of the hypoelastic constitutive equation (5.112) in some detail. This has been addressed variously by Truesdell and Noll [20], Eringen [16], Atluri [17], and others. 

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

The second- and higher-order elastic constant studies in single crystals with large Hugoniot limits have provided an examination of elastic behavior... 

The properties of the lamina constituents, the fibers and the matrix, have been only briefly discussed so far. Their stress-strain behavior is typified as one of the four classes depicted in Figure 1-8. Fibers generally exhibit linear elastic behavior, although reinforcing steel bars in concrete are more nearly elastic-pertectly plastic. Aluminum, as well as... 

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

Finally, if the usual restriction to linear elastic behavior to failure is made. 

Hong T. Hahn and Stephen W. Tsai, Nonlinear Elastic Behavior of Unidirectional Composite Laminae. Journal of Composite Materials, January 1973, pp. 102-118. 

The analysis of stresses in the laminae of a laminate is a straight-fonvard, but sometimes tedious, task. The reader is presumed to be familiar with the basic lamination principles that were discussed earlier in this chapter. There, the stresses were seen to be a linear function of the applied loads if the laminae exhibit linear elastic behavior. Thus, a single stress analysis suffices to determine the stress field that causes failure of an individual lamina. That is, if all laminae stresses are known, then the stresses in each lamina can be compared with the lamina failure criterion and uniformly scaled upward to determine the load at which failure occurs. 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

Linear elastic behavior to failure occurs for individual laminae. 

N. J. Pagano and Sharon J. Hatfield, Elastic Behavior of Multilayered Bidirectional Composites, AIAA Journal, July 1972, pp. 931-933. 

Elastic-plastic transition It is the changes from recoverable elastic behavior to non-recoverable plastic strain which occurs on stressing a material beyond its yield point. 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

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