The self-consistent method

The self-consistent method is suitable for dealing with systems that contain dilute concentrations of a number of different heterogeneities with different elastic properties, provided that certain conditions are met. The heterogeneities must be of equiaxed shape, must not be clustered in space, and must generally be randomly distributed and present in dilute concentrations, of volume fraction around 0.1 or less, to make elastic interactions between heterogeneities negligible. Plate-like and needle-like heterogeneities are excluded. 

The self-consistent method is based on a classical solution of Eshelby (1957) that spawned a remarkably large number of different apphcations. It states that for an ellipsoidal isotropic elastic inclusion in an infinite elastic medium of different but uniform isotropic elastic properties the state of stress (or strain) inside the inclusion is uniform when the distant body is subjected to a uniform stress (or strain). 

This gives five unknowns, Kc, a, p, and Vc, with five equations. 

Young s modulus of the composite can be determined, if required, from the known connection 

Clearly, obtaining the composite properties and Kc of the heterogeneous composite from eqs. (4.22) 4.26) will require numerical procedures. For cases in which the heterogeneous system contains only one or two distinct types of inclusion, solutions of these equations will be relatively straightforward. As long as the assumptions of the self-consistent method are valid, the results are generally quite good. 

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Laws and McLaughlin30 discuss viscoelastic creep compliances of composite materials using another approach to the problem of the elastic properties of heterogeneous materials - the self-consistent method. 

For aligned transversely isotropic elements the self consistent method gives (Walpole58 ) the relations... 

Laws and McLaughlin30 solve the problem of the viscoelastic ellipsoidal inclusion in anisotropic materials and then use the self consistent method to calculate the overall viscoelastic compliances for a composite. 

In this determinant y>j (ft) indicates that electron /t has been assigned to orbital with spin and (fi) represents the assignment of electron ft to orbital with spin —In accordance with the Pauli exclusion principle, this wave function is antisymmetric under a permutation of any two electrons. The principle of the self-consistent method is then to minimize the integral... 

There are cases for which more than one solution is found, and it is possible that both may possess physical reality under certain conditions [12] (this will arise again in chapter 11). Furthermore, the Hartree-Fock method can be made multiconfigurational, i.e. several configurations can be mixed or superposed. An electron is then shared between different states, which goes beyond the independent particle approximation. The self-consistent method allows the mixing coefficients to be determined, but the configurations to be included must be specified at the outset, and there is no simple prescription as to which ones should be chosen or left out. 

In this connection let us consider the FrShlieh-type Hamiltonian. As it is well known sueh a one-dimensional gas has no stability with Tsspect to the transition to the Peierls dielectric or to the superconductor state. If we use the self-consistent method the gap in the electronic spectrum of Fermi excitations will be of order of g- for Peierls distorted phase and for superconducting one. where ... 

The methods are classified as self-consistent and non-self-consistent. In the self-consistent methods, the effect of electron-electron interactions is explicitly taken into account. In ab initio methods, as soon as the geometry and the basis functions are defined all necessary integrals are explicitly evaluated. It has been proven that long-range effects become highly significant as soon as the unit cell contains permanent dipoles. Multipole expansion must, therefore, be used in order to obtain a satisfactory balance of the... 

In the self-consistent method, the CD spectrum of the protein being analyzed is retained in the matrix C and an initial guess for the secondary structure fractions is included in the matrix F. The guess is refined by an iterative process. Here, the initial guess is the secondary structure content of the protein in the basis set which has a CD spectrum most similar to that of the test protein. 

Until now, much research work has been done on the prediction of composite material coefficient of thermal expansion and elastic modulus by forefathers, and many prediction methods have been developed such as the sparse method (Guanhn Shen, et al. 2006), the Self-Consistent Method (Hill R.A. 1965), the Mori-Tanaka method (Mori T, Tanaka K. 1973) and so on. However, none of these formulas take into account the parameters variation with concrete age, and there is little research on the autogenous shrinkage and creep. In the mesoscopic simulation of concrete, thermal and mechanical parameters of mortar and aggregate (coefficient of thermal expansion, autogenous shrinkage, elasticity modulus, creep, strength) are important input parameters. In fact, there is abundant of test data on concrete, but much less data on mortar while it is one of the important components. Also parameter inversion is an essential method to obtain the data, but there are few studies on this so far. 

The Fock matrix represents the average effects of the field of all the electrons and nuclei in each orbital as the orbitals depend on the molecular orbital expansion coefficients, thus the self-consistent method (SCF) is used to solve... 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

In fact, another problem exists in chain statistics, where the self-consistent method does not benefit from the sanK cancellations. This is the case of a charged chain (polyelectrolyte) for which a self-consistent approach was attempted very earlyHere the neglect of correlations is not too serious because most of the repulsion comes from very distant monomers. Thus point (i) is improved, but pomt (ii) remains weak the net result is a formula for v in charged systems which gives incorrect values for 3 < rf < 6. We return to this problem in Chapter XI. 

A program for the self-consistent method is available from the authors upon request. A software package CDPro for the analysis of protein CD spectra which contains the SELCON and a modified CONTIN/LL program is available from http //lamar.colostate.edu/ sreeram/CDPro/. 

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