Limiting function

Theorem 2.14. From the sequence x = of solutions of the problem (2.116) one can choose a subsequence, still denoted byto, such that as 5 0 the convergence (2.119) takes place and, moreover, the limiting function satisfies (2.123). 

To justify the passage to the limit in the relations obtained from (2.127) by a change of variables, we use the convergence (2.128) and the statement analogous to Lemma 2.1. The limiting function % = U,u) is a solution of the variational inequality (2.100) with the function from (2.126), that is X = Xtl>- Finally, it is easy to verify that... 

This precisely means that the limiting function -0 is a solution of the extreme crack shape problem (2.125). 

This obviously means that the limiting function w is a solution of the equilibrium problem for the shell having the crack shape y = x) =0,xG [0,1]. Thus the following statement has been proved. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

Corresponding referential, current spatial, and unrotated spatial inelastic constitutive equations are equivalent, and identical results are obtained from their use, if corresponding moduli and elastic limit functions are used. In the current spatial constitutive equations, if the dependence of the spatial moduli... 

This is the condition (A.85) that the elastic limit function / be an isotropic tensor functions of its arguments. By analogy with the hypoelastic constitutive equation, the name hypoinelasticity suggests itself for this formulation. 

The dependence of the spatial elastic limit function on has been emphasized in writing the elastic limit condition (5.I6O3). This dependence implies that the elastic limit surface is changing as the deformation proceeds, quite apart from its dependence on k. If the referential elastic limit function is assumed to be independent of k, the spatial elastic limit surface would distort... 

This is the condition (A.85) for to be isotropic, as in the hypoinelastic formulation (5.116i). It is therefore clear that, when the moduli c and b, as well as the elastic limit function do not include the special dependence on F indicated in (5.155) and (5.160), then objectivity demands that they be isotropic tensor functions of their arguments, and the spatial formulation reduces to the hypoinelastic formulation. 

The elastic limit function in stress space may be translated into spatial terms by using (5.151) and (A.39) in (5.135)... 

It is the dependence of the spatial constitutive functions on the changing current configuration through F that renders the spatial constitutive equations objective. It is also this dependence that makes their construction relatively more difficult than that of their referential counterparts. If this dependence is omitted, then the spatial moduli and elastic limit functions must be isotropic to satisfy objectivity, and the spatial constitutive equations reduce to those of hypoinelasticity. Of course, there are other possible formulations for the spatial constitutive functions which are objective without requiring isotropy. One of these will be considered in the next section. 

The dependence of the unrotated elastic limit function on U has been emphasized in writing the elastic limit condition (5.1883). This dependence... 

Filter/dry OH-functionality of polyol depends on structure of R Difficult to "cap all 2° OH groups with EO Side reactions (esp. proton abstraction) limit functionality of Urethane-grade polyol product and create unwanted functional groups... 

The calibration-design-dependent LOD approach, - namely the use of the confidence limit function, is endorsed here for reasons of logical consistency, response to optimization endeavors, and easy implementation. Fig. 2.14 gives a (highly schematic) overview ... 

CL is inserted in Eqs. (2.18) and (2.19), with k - oo, and using the + sign. The intercept of the horizontal with the lower confidence limit function of the regression line defines the limit of quantitation, jcloq, any value above which would be quoted as 2f(y ) t s ... 

Slope b is close to zero and/or ires is large, which in effect means the horizontal will not intercept the lower confidence limit function, and... 

The horizontal intercepts the lower confidence limit function twice, i.e., if n is small, ires is large, and all calibration points are close together this can be guarded against by accepting Xloq only if it is smaller than... 

For large h one cannot reduce the error significantly by increasing n. There is obviously a limiting function edoo h) for n — oo, which for large h is given by (C.4). For small h (C.3) is not convenient because it is slowly convergent. 

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