Ideal Stress-Strain Behavior

Fig. 11.22 A schematic representation of the domain-cavitation model of craze growth in spherical-morphology PS/PB diblock copol5mers (a) the idealized stress-strain behavior of a cubical cell containing a cavitated PB sphere and (b) the model of the craze-tip process including the likely craze-tip traction distribution (after Schwier et al. (1985a) courtesy of Taylor and Francis).

The relationship between the tensile stress and tensile strain that a fiber displays is known as that fiber s stress-strain behavior. If a fiber obeys the Hooke s law, its tensile stress is directly proportional to the tensile strain, up to the point of failure, where the fiber breaks without yielding or plastic deformation (Figure 15.5). In this case, the fiber exhibits ideal stress-strain behavior. 

A simple model eanbe used to describe the ideal stress-strain behavior. Figure 15.7 shows a one-dimensional array of atoms held together by chemical bonds. This array eanbe used to represent part of an ideal material that has identical rows of atoms placed adjacent to one another. When a tensile force is applied to this material, the atoms are displaced from their equilibrinm positions, creating a restoring force that is equal, but opposite to the applied tensile force. The mechanical response of the material is governed by the internal eneigy, i.e., the sum of the... 

Therefore, this simple model indicates the mechanical resporrse of the material follows the Hooke s law and exhibits the ideal stress-strain behavior if the tensile strain is small. However, deviations occur under larger strains. 

Deviations FROM Ideal Stress-Strain Behavior... 

In general, polymer fibers do not exhibit ideal stress-strain behavior. Why ... 

Quested et al. [16] have conducted an extensive experimental program on the stress-strain behavior of the elastomer solithane while subjected to an ambient at high pressure. Some of their experimental results are reproduced in Fig. 13. (Note that the reported stress is the deviatoric, not the total, stress as observed from the fact that the reported stress is zero for X = 1 for the various imposed ambient pressures). For the classic ideal affine network model (all stress caused by ideal Nc Gaussian chains in a volume v with no nonbonded interactions)... 

If a material exhibits linear-elastic stress-strain behavior prior to rupture (an ideal behavior approximated by many thermosets), then a simple relationship exists between the material s fracture toughness and its fracture surface energy, J (or G),... 

One of the most important rheological properties of lipids is their plastic behavior. A plastic material is defined as one that does not undergo a permanent deformation until a certain yield-stress has been exceeded. The example of an ideal stress-strain curve is depicted in Figure 4.11. Under influence of a small stress from zero to Xq, no deformation occurs. At the point Xq, stress increases, and the flow will appear from zero to El with stress Xq constant. The area (A) in Figure 4.11 is the value of total... 

The ideal elastie response is typified by the stress-strain behavior of a spring. A spring has a constant modulus that is independent of the strain rate or the speed of testing stress is a funetion of strain only. For the pure Hookean spring the inertial effects are neglected. For the ideal elastic material, the mechanical response is deseribed by Hooke s law ... 

Polymeric materials exhibit stress-strain behavior which falls somewhere between these two ideal cases hence, they are termed viscoelastic. In a viscoelastic material the stress is a function of both strain and time and so may be described by an equation of the form... 

Hookean elasticity (ideal elasticity) n. Stress-strain behavior in which stress and strain are directly proportional, in accordance with Hooke s law. Serway RA, Faugh JS, Bennett CV (2005) College physics. Thomas, New York. 

It is always easy to calculate idealized scattering curves for perfect networks. The experimental systems vary from the ideal to a greater or lesser degree. Accordingly, any estimate of the correctness of a theoretical analysis which is based on an interpretation of experiment must be put forth with caution since defects in the network may play a role in the physical properties being measured. This caveat applies to the SANS measurement of chain dimensions as well as to the more common determinations of stress-strain and swelling behavior. 

Swollen tensile and compression techniques avoid both of these problems since equilibrium swelling is not required, and the method is based on interfacial bond release and plasticization rather than solution thermodynamics. The technique relies upon the approach to ideal rubberlike behavior which results when lightly crosslinked polymers are swelled. At small to moderate elongations, the stress-strain properties of rubbers... 

Many examples have been given (Baird, 1982 Hemqvist, 1983 Princen, 1983 Hermansson, 1994 Morrison, 1994 Breitschuh and Windhab, 1997) that illustrate the complexity of the rheological behavior of lipids-based foods. The rheology of lipids, as in other food systems, is the science of deformation and flow of real materials, in terms of stress, strain, and time effects, not merely those of ideal... 

Fig. 8.20 The elastomeric stress-strain curve of PET at 353 K, re-plotted against the Gaussian strain function g X), showing near-ideal rubbery behavior. The slope suggests an entanglement molecular weight of Me = 2342 g/mole.

A sinusoidal stress applied to an ideal elastic material produces a sinusoidal strain proportional to the stress amplitude and in phase with it. For ideal viscous materials the stress and strain are out of phase by 90°. Figure 15 gives an example of a stress-strain diagram for a sinusoidal stress applied to a real material. The amplitude of the deformation (strain) in response to the stress is proportional to that of the stress, but lags behind the strain curve by some angle 5 between 0 and 90°, depending on the elastic/viscous characteristic of the material. This behavior is usually analyzed by the use of complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, x and y > respectively, are... 

Figure 1.6 Idealized modulus-temperature behavior of an amorphous polymer. Young s modulus, stress/strain, Is a measure of stiffness.

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