Strain invariant

Over a long time period it may well not be possible to duplicate library cell culture conditions. What happens when the lot of media used in the final culture step prior to pyrolysis has been consumed Can culture media suppliers assure nutritional identity between batches Media types for growth of fastidious strains invariably include natural products such as brewer s yeast, tryptic soy, serum, egg, chocolate, and/or sheep blood. Trace components in natural products cannot be controlled to assure an infinite, invariable supply. The microtiter plate wells used here do not hold much media. Even so, the day will come when all media supplies are consumed and a change in batch is unavoidable. When that happens, if there were no effective way to compensate spectra for the resulting distortions, it would be necessary to re-culture and re-analyze replicates for every strain in the reference library. Until recently the potential for obsolescence was a major disincentive for developing PyMS spectral libraries of bacteria. Why this is no longer an insurmountable problem is discussed in the next section. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

The non-linear response of elastomers to stress can also be handled by abandoning molecular theories and using continuum mechanics. In this approach, the restrictions imposed by Hooke s law are eliminated and the derivation proceeds through the strain energy using something called strain invariants (you don t want to know ). The result, called the Mooney-Rivlin equation, can be written (for uniaxial extension)—Equation 13-60 ... 

For viscoelastic fluids, both strain energy and stress can be assumed to depend on the strain history through the strain invariants ... 

The active site in the enzyme is easily recognizable in the crystal structure as a cavity, approximately 8 A deep and 15 A across, formed by loops on the outer-most surface of the protein. About 20 amino acids in and around the active site are strain-invariant. Many of the side chains that directly bind substrate are charged, and their net charge is approximately balanced. On the tetramer the active sites are located at a roughly 45° angle to the four-fold axis and are separated by approximately 45 A. 

From experiments in our laboratory on biaxial deformations of thin sheets, it is found that in some materials cracks are formed without any evidence of necking, while at the same levels of strain in uniaxial extension necking had already occurred. This is not surprising since the potential function w depends on the strain Invariants and for biaxial experiments, the solution given in section III has to be modified because the strain potential now has to be differentiated with respect to the first and second strain invariants. More work in biaxial deformations will lead to a better description of the failure mechanism In general. 

Note that there are alternative, but equivalent, definitions of these in the literature, comprising sums or products of the ones shown here for example, I2=J [2+ A22 + A32, which is quotient of the Ix and I2 in equation (6-75).] The third strain invariant is obviously... 

A blood vessel generally exhibits anisotropic behavior when subjected to large variations in internal pressure and distending force. When the degree of anisotropy is small, the blood vessel may be treated as isotropic. For isotropic materials it is convenient to introduce the strain invariants ... 

An alternative isotropic strain energy density function which can predict the appropriate type of stress stiffening for blood vessels is an exponential where the arguments is a polynomial of the strain invariants. 

Symmetry considerations suggest that appropriate measures of strain are given by three strain invariants, defined as... 

Strain invariants are independent of the axes used to define the geometry, enabling calculations for inhomogeneous deformations without explicit consideration of the principal directions. During a homogenous deformation,... 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

The Phantom Model. In this model polymer chains are allowed to move freely through one another and the network junctions fluctuate around their mean positions [3,91-93], The conformation of each chain depends only on the position of its ends and is independent of the conformations of the surrounding chains with which they share the same region of space. The junctions in the network are free to fluctuate around their mean positions and the magnitude of the fluctuations is strain invariant. The positions of the junctions and of the domains of fluctuations deform affinely with macroscopic strain. The result is that the deformation of the mean positions of the end-to-end vectors is not affine in the strain. This is because it is the convolution of the distribution of the mean positions (which is affine) with the distribution of the fluctuations (which is strain invariant, i.e., nonaffine). The elastic free energy of deformation is given by... 

The symbol I represents the strain invariants analogous to the stress invariants given as J in Eqs. (1.22e) and (1.23). The coefficients in Eq. (1.98c) are the results of the engineering shear strain being ... 

A further simple strain invariant is / = X X Xj, which for an incompressible rubber is always unity, because A1A2A3 = 1. Therefore / can also be written as... 

In addition to these theoretical considerations, which suggest that we need not be restricted to squares of extension ratios in formulating the strain energy function, it has been found by experimentalists that there is high sensitivity to experimental error when small values of the strain invariants I and f are involved. It is therefore natural to postulate that the only constraint on the form U is that imposed by the invariance of U with respect to the axis lables, which implies that U i, X-i, A3) should be a symmetric function of the extension ratios, i.e. invariant to any permutation of the indices 1,2, 3. 

The third strain invariant is equal to the square of the volume change. 

Figure 1. The remanent strain saturation curve dividing remanent strain space into regions that are attainable and unattainable by a polycrystal assembled from randomly oriented tetragonal single crystals. Only remanent strain states below the curve are attainable by such a material. The dots are numerical results from Landis (2003a) obtained using a micromechanical self consistent model, and the line is one divided by the function/given in Eqs. (2.7) and (2.8). The remanent strain invariants 4 and are defined in Eq. (2.5) and the results are normalized by the saturation strain in axisymmetric compression s. ...

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