Hookean behavior

Figure 5.3 Stress-strain curve for a typical elastomer. The dashed line indicates Hookean behavior.

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

During this same period, the equilibrium stress-strain properties of well characterized cross-linked networks were being studied intensively. More complex responses than the neo-Hookean behavior predicted by kinetic theory were observed. Among other possibilities it was speculated that, in some unspecified way, chain entanglements might be a contributing factor. 

If the crosslink is removed, the system reverts to two independent chains. To preserve the symmetry of the system one needs to consider the average for three tetrahedra operating together, one for each of the three possible pairings of the strands. The result is again neo-Hookean behavior but with a lower modulus contribution. 

This result differs somewhat from the expression obtained using the Doi-Edwards model (Eq.40), and it gives a larger departure from neo-Hookean behavior for uniaxial extension (A q>endix II and Fig. 9). In the limit of small deformations the entire contribution to stress comes from the first term in Eq. 62. The entanglement contribution to the infinitesimal shear modulus is predsely the same as the Doi-Edwards expression for the plateau modulus (Eq. 37)... 

An ideal elastic body (also called Hooke s body) is defined as a material that deforms reversibly and for which the strain is proportional to the stress, with recovery to the original volume and shape occurring immediately upon release of the stress. In a Hooke body, stress is directly proportional to strain, as illustrated in Fig. 3. The relationship is known as Hooke s law, and the behavior is referred to as Hookean behavior. 

The magnitude of the strain is directly or linearly proportional to the magnitude of the stress for material that exhibits Hookean behavior. This relation between stress and strain is known as Hooke s law and may be written as... 

It is known that up to a certain limiting load, a solid will recover its original dimensions on the removal of the applied loads. This ability of deformed bodies to recover their original dimensions is known as elastic behavior. Beyond the limit of elastic behavior (elastic limit), a material will experience a permanent set or deformation even when the load is removed. Such a material is said to have undergone plastic deformation. For most materials, Hooke s law is obeyed within the elastic limit, that is, stress proportional to strain. However, proportionality between stress and strain does not always hold when a material exhibits elastic behavior. A typical example is rubber, which is elastic but does not show Hookean behavior over the entire elastic region. 

Our discussion thus far has focused in a rather superficial way on the general evolution of the important area of fracture mechanics. The basic objective of fracture mechanics is to provide a useful parameter that is characteristic of the given material and independent of test specimen geometry. We wUl now consider how such a parameter, such as G (, is derived for polymers. In doing so we confine our discussion to the concepts of linear elastic fracture mechanics (LEFM). As the name suggests, LEFM apphes to materials that exhibit Hookean behavior. 

The ideally elastie material exhibits no time effects and negligible inertial effects. The material responds instantaneously to applied stress. When this stress is removed, the sample recovers its original dimensions completely and instantaneously. In addition, the induced strain, e, is always proportional to the applied stress and is independent of the rate at which the body is deformed (Hookean behavior). Figure 14.1 shows the response of an ideally elastic material. 

A point worth noting here is that several of the molecular models that will be described in the subsequent sections are Neo-Hookean in form. Normally, dry rubbers do not exhibit Neo-Hookean behavior. As for the Mooney-Rivlin form of strain energy density function, rubbers may follow such behavior in extension, yet they do not behave as Mooney-Rivlin materials in compression. In Fig. 29.2, we depict typical experimental data for a polydimethylsiloxane network [39] and compare the response to Mooney-Rivlin and Neo-Hookean behaviors. The horizontal lines represent the affine and the phantom limits (see Network Models in Section 29.2.2). The straight line in the range A <1 shows the fit of the Mooney-Rivlin equation to the experimental data points. 

Figure 17.16 A generalized stress-strain curve for an acrylic fiber. Region A - follows Hookean behavior.

Stiffness n. Refers to the deformation behavior in the elastic region. Elastic or Hookean behavior implies that deformation effects due to load are completely recoverable -i.e., no permanent dimensional change occurs. In a theoretical sense this is seldom observed. Practical approximations are usually adequate. When plotted, the horizontal axis is strain, the vertical axis is stress. 

In Fig. 1.1b, a glass-like ceramic, which is usually amorphous, is also shown indicating the same Hookean behavior without plastic deformation, but at a much lower fracture stress than in a crystalline alumina, for example. 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

True stress-strain curves of tension and compression have been experimentally observed to coincide. Yet, this observation is not tme for brittle materials (that behave like glass), in which fracture and yield stress coincide. Thus, in brittle materials, e.g., ceramics, the strain seldom exceeds 1 %, depending to a large degree on the type of deformation involved. Classical Hookean behavior is observed in such brittle materials. The presence of imperfections of various kinds, including porosities, has a profound effect on the mechanical behavior of ceramics. 

The described initial linear (Hookean) behavior is typical for a very large number of solidlike objects. Conversely, objects like soap thin films or modeling clays do not exhibit this type of behavior. A single parameter characterizing the elastic properties of an object is Young s modulus, that is, the dimensional proportionality constant, E = a/e = Ff/AlS. In Chapter 4, we listed some typical values of E. In continuum mechanics, the elasticity moduli are typically described by the letter c and the inverse quantity, compliance, by the letter s, that is, 5 oc 1/c. 

Since the film reveals Hookean behavior over its entire area, one can carry out integration with respect to dc ), that is,... 

Because of the limited usefulness of the Mooney-Rivlin equation, it is probably not worthwhile to seek a molecular interpretation of the coefficients C and C2 the deviations from neo-Hookean behavior should be examined in some other framework. However, if the deviations are expressed in terms of the ratio = C2/(C] + C2), this quantity can be correlated rather successfully with the relative numbers of trapped entanglements and cross-links in the network. It may be inferred from this and other studies that both trapped entanglements and cross-links contribute to C, but that C2 is associated with trapped entanglements only. 

A wide, short, and thin sheet of rubber, Lo is clamped along its wide edges and pulled from its original length Lo to L with a force f. This deformation is called planar extension, plain strain, or sometimes pure shear. Assume neo-Hookean behavior, eq. 1.5.2. 

A simple model for plastic material is Hookean behavior at stresses below yield and Newtonian behavior above. For onedimensional deformations... 

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