Strain or stress

Several functions are used to characterize tire response of a material to an applied strain or stress [4T]. The tensile relaxation modulus E(t) describes tire response to tire application of a constant tensile strain l/e -. 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

This stress is to be accounted for when computing the total stress in the sub wall. The calculation can be done easily in real time with a computer however, it is easier and probably more accurate to measure a difference in strain (or stress) in the sub between the off-bottom position and while drilling. This value should be related closely to the true weight-on-bit. 

The strain or stress will either lead to narrower or broader d bands that are shifted up or down in energy, respectively. An upward shift leads to a stronger interaction with the 2jt orbital of adsorbed CO and thus to a stronger chemisorption bond. Stress has the opposite effect. 

In driven dynamic testing an oscillating strain (or stress) is applied to a specimen. This is almost always sinusoidal for ease of analysis. In this case... 

Systematic measurements of stress and strain can be made and the results plotted as a rheogram. If our material behaves in a simple manner - and it is surprising how many materials do, especially if the strains (or stresses) are not too large - we find a linear dependence of stress on strain and we say our material obeys Hooke s law, i. e. our material is Hookean. This statement implies that the material is isotropic and that the pressure in the material is uniform. This latter point will not worry us if our material is incompressible but can be important if this is not the case. 

Firstly, it helps to provide a cross-check on whether the response of the material is linear or can be treated as such. Sometimes a material is so fragile that it is not possible to apply a low enough strain or stress to obtain a linear response. However, it is also possible to find non-linear responses with a stress/strain relationship that will allow satisfactory application of some of the basic features of linear viscoelasticity. Comparison between the transformed data and the experiment will indicate the validity of the application of linear models. 

Ultimate stress and strain, or stress and strain at break, are the values corresponding to the breaking of the samples. 

The fatigue life is the number of cycles a specimen can remain at a specified strain or stress before specimen failure occurs. As stress is decreased, there is a point beyond which failure does not occur regardless of the number of cycles the specimen experiences. This stress value is called the endurance limit or endurance strength. 

In the above thermomechanical equations of state we neglect the variation of K, E, oc and P with temperature and strain (or stress). As we will see later, this is a good... 

Conclusions about the nature of strain Strain or stress ... 

ISO 4666 Part 1 remarks on the care necessary in measuring temperature rise and the fact that the result depends on where the temperature is measured and on the test piece geometry44. It recommends testing at a series of strain or stress levels because a comparison of rubbers at one level only can be misleading. The standard also mentions the measurement of creep and set in the test piece after periods of dynamic cycling. 

Figure H3.1.1 Responses from purely elastic or purely viscous foods under oscillatory testing. The controlled parameters of the oscillatory applied stress or strain are amplitude, frequency, temperature, and duration of measurement. The measured parameters are amplitude and phase shift (8) of the strain or stress response.

A strain or stress sweep is used to establish the LVE region (Figure H3.2.4). The LVE region is a characteristic of a material. While the strain value at the limit of LVE rarely exceeds 0.1 for colloidal gels, a larger LVE region with a strain of up to 1 or more is usually observed for biopolymer gels (Clark and Ross-Murphy, 1987). 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

The fundamental rheological characterization of a material requires the experimental determination of a constitutive equation (a rheological equation of state) that mathematically relates stress and strain, or stress and strain rate. The constants in the constitutive equation are the rheological properties of the material. 

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

An increase in M is also reported to increase the temperature at which a shear to craze transition occurs in both PC and poly(ether sulfone) (PES) and, above the transition temperature, the craze strain, or stress, is greater for the higher molecular weight samples These changes are a result of a greater resistance to disentanglement in the higher M polymers. 

Rate Theory for Unfolding and Refolding of a Helix-like Polymer Exposed to Strain or Stress Pili Elongated in Region II... 

Rate Theory for Elongation and Contraction of a Linear Polymer Exposed to Strain or Stress -Pili Elongated in Region III... 

Similar expressions can be written for any strain or stress component in any reference system. In Equation (79) a,- is in fact a placeholder for a,-, e,-, cr,-, Si and also for 8h that gives the diffraction peak shift caused by the strain in a crystallite. To calculate the peak shift for a polycrystalline sample, 8h (R + r, g) given by Equation (79) must be averaged over r, k and g, where g represents those crystallite orientations for which h is parallel to y, the direction in the sample of the scattering vector. Taking account that the sample could be textured this multiple average is the following ... 

A more sensitive rheological techniques for following the stability of multiple emulsions is to use oscillatory techniques. In this case, a sinusoidal strain or stress is applied to the sample, which is placed in the gap of the concentric cylinder or cone-and-plate geometry the resulting stress or strain sine wave is followed at the same time. For a viscoelastic system, as is the case with multiple emulsions, the stress and strain sine waves oscillate with the same frequency, but out of phase. 

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