Composite criteria

Although application of chemometrics in sample preparation is very uncommon, several optimisation techniques may be used to optimise sample preparation systematically. Those techniques can roughly be divided into simultaneous and sequential methods. The main restrictions of a sequential simplex optimisation [6,7] find their origin in the complexity of the optimisation function needed. This function is a predefined function, often composed of several criteria. Such a composite criterion may lead to ambiguous results [8]. Other important disadvantages of simplex optimisation methods are that not seldom local optima are selected instead of global optima and that the number of experiments needed is not known beforehand. 

This criterion may be used during a sequential optimization process (see chapter 5), leading to an acceptable result and to completion of the optimization process once the threshold value has been reached. Alternatively, it may be used to establish ranges of conditions in the parameter space for which the result will be acceptable. This latter approach has been followed by Glajch et al. [415], by Haddad et al. [424] and by Weyland et al. [425] and was referred to as resolution mapping by the former. Within the permitted area(s) secondary criteria are then required to select the optimum conditions. For example, the conditions at which the k value of the last peak (k is minimal while the minimum value for Rsexceeds 1 may be chosen as the optimum. Such a composite criterion can be described as... 

It is extremely easy to update the old values for the criterion, because all product criteria become zero for chromatograms which contain less than the highest number of peaks, whereas all sum criteria remain unaltered. If a composite criterion is used, in which a time factor occurs, then the previous values for the optimization criterion (C,) may usually only be updated if the values for the individual contributions (the value of the criterion C and a time factor) are stored separately. 

Figure 5.2 (a) Pseudo-isometric three-dimensional response and (b) iso-response contour plot for a two-parameter optimization problem. Parameters (in triangular representation) quaternary mobile phase composition. Criterion normalized resolution product (see section 4.3.2). O, is the location of the optimum. For further details see section 5.5.2. Figure taken from ref. [502]. Reprinted with permission. 

We can also seek to optimize several of these criteria simultaneously. In this case, we can use a composite criterion, defined as a function of the relative weight of each of the initial criteria. 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

The criterion used for "too near the plait point" is that ratio of K s for the two "solvent" components is less than seven with the feed composition in the two-phase region. 

The field-density concept is especially usefiil in recognizing the parallelism of path in different physical situations. The criterion is the number of densities held constant the number of fields is irrelevant. A path to the critical point that holds only fields constant produces a strong divergence a path with one density held constant yields a weak divergence a path with two or more densities held constant is nondivergent. Thus the compressibility Kj,oi a one-component fluid shows a strong divergence, while Cj in the one-component fluid is comparable to (constant pressure and composition) in the two-component fluid and shows a weak... 

The properties of fillers which induence a given end use are many. The overall value of a filler is a complex function of intrinsic material characteristics, eg, tme density, melting point, crystal habit, and chemical composition and of process-dependent factors, eg, particle-si2e distribution, surface chemistry, purity, and bulk density. Fillers impart performance or economic value to the compositions of which they are part. These values, often called functional properties, vary according to the nature of the appHcation. A quantification of the functional properties per unit cost in many cases provides a vaUd criterion for filler comparison and selection. The following are summaries of key filler properties and values. 

Many investigations of relationships between composition and properties take into account only the concentration of the asphaltenes, independendy of any quality criterion. However, a distinction should be made between the asphaltenes which occur in straight mn asphalts and those which occur in blown asphalts. Remembering that asphaltenes are a solubiUty class rather than a distinct chemical class means that vast differences occur in the make-up of this fraction when it is produced by different processes. 

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Convergence was achieved rapidly in five iterations by using Eq. (13-88) as the criterion. Computed compositions for lean gas and rich oil are ... 

The way to obtain the membranes as well their composition has been optimized. The main operational criterion for the membranes is the solubility of applicable ionic pairs. The solubility should be quite low - else the substance will be outwashed from the membrane. At the same time, the ionic pairs which have very low solubility are not suitable too because of the complicated obtaining homogeneous membranes. 

However, it has to be considered that it is neither the content of free formaldehyde itself nor the molar ratio which eventually should be taken as the decisive and the only criterion for the classification of a resin concerning the subsequent formaldehyde emission from the finished board. In reality, the composition of the glue mix as well as the various process parameters during the board production also determine both performance and formaldehyde emission. Depending on the type of board and the manufacturing process, it is sometimes recommended to use a UF-resin with a low molar ratio F/U (e.g. F/U = 1.03), hence low content of free formaldehyde, while sometimes the use of a resin with a higher molar ratio (e.g. F/U = 1.10) and the addition of a formaldehyde catcher/depressant will give better results [17]. Which of these two, or other possible approaches, is the better one in practice can only be decided in each case by trial and error. 

If the fibres are aligned at 15° to the jc-direction, calculate what tensile value of Ox will cause failure according to (i) the Maximum Stress Criterion (ii) the Maximum Strain Criterion and (iii) the Tsai-Hill Criterion. The thickness of the composite is 1 mm. 

A single ply glass/epoxy composite has the properties Usted below. If the fibres are aligned at 30° to the x-direction, determine the value of in-plane stresses, a, which would cause failure according to (a) the Maximum Stress criterion (b) the Maximum Strain criterion and (c) the Tsai-Hill criterion. 

In applications of the maximum stress criterion, the stresses in the body under consideration must be transformed to stresses in the principal material coordinates. For example, Tsai [2-21] considered a unidirec-tionally reinforced composite lamina subjected to uniaxial load at angle 6 to the fibers as shown in Figure 2-35. The biaxial stresses in the principal material coordinates are obtained by transformation of the uniaxial stress, a, as... 

For a unidirectionally reinforced composite material subject to uniaxial load at angle 0 to the fibers (the example problem in Section 2.9.1 on the maximum stress criterion), the allowable stresses can be found from the allowable strains X, Y , etc., in the following manner. 

As with the maximum stress failure criterion, the maximum strain failure criterion can be plotted against available experimental results for uniaxial loading of an off-axis composite material. The discrepancies between experimental results and the prediction in Figure 2-38 are similar to, but even more pronounced than, those for the maximum stress failure criterion in Figure 2-37. Thus, the appropriate failure criterion for this E-glass-epoxy composite material still has not been found. 

The Tsai-Hill failure criterion appears to be much more applicable to failure prediction for this E-glass-epoxy composite material than either the maximum stress criterion or the maximum strain failure criterion. Other less obvious advantages of the Tsai-Hill failure criterion are ... 

For E-glass-epoxy, the Tsai-Hill failure criterion seems the most accurate of the criteria discussed. However, the applicability of a particular failure criterion depends on whether the material being studied is ductile or brittle. Other composite materials might be better treated with the maximum stress or the maximum strain criteria or even some other criterion. 

R. C. Tennyson, D. MacDonald, arrd A. P. Nanyaro, Evaluation of the Tensor Polynomial Failure Criterion for Composite Materials, Journal of Composite Materials, January 1978, pp. 63-75. 

Because of the various characteristics of composite laminates, it is difficult to determine a strength criterion in which all failure modes and their interactions are properly accounted for. Moreover, the verification of a proposed strength criterion is greatly complicated by scatter in measured strengths caused by inconsistent processing techniques (that... 

Two simple invariants, U, and U5, were shown in the previous subsubsection to be the basic indicators of average laminate stiffnesses. For isotropic materials, these invariants reduce to U. =Qi. and U5 = Qqq, the extensional stiffness and shear stiffness. Accordingly, Tsai and Pagano suggested the orthotopic invariants U., and U5 be called the isotropic stiffness and isotropic shear rigidity, respectively [7-16 and 7-17]. They observed that these isotropic properties are a realistic measure of the minimum stiffness capability of composite laminates. These isotropic properties can be compared directly to properties of isotropic materials as well as to properties of other orthotropic laminates. Obviously, the comparison criterion is more complex than for isotropic materials because now we have two measures, and U5, instead of the usual isotropic stiffness or E. Comparison of values of U., alone is not fair because of the degrading influence of the usually low values of U5 for composite materials. 

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