TOTAL POTENTIAL ENERGY FUNCTIONAL

The generalized decoration-iteration transformation is adapted for the exact study of a coupled spin-electron model on 2D lattices in which localized Ising spins reside on nodal lattice sites and mobile electrons are delocalized over pairs of decorating sites. The model takes into account a hopping term for mobile electrons, the Ising coupling between mobile electrons and localized spins as well as the Ising coupling between localized spins (J ′). The ground state, spontaneous magnetization and specific heat are examined for both ferromagnetic (J ′ > 0) as well as antiferromagnetic (J ′ < 0) interaction between the localized spins. Several kinds of reentrant transitions between the paramagnetic (P), antiferromagnetic (AF) and ferromagnetic (F) phases have been found either with a single critical point, or with two consecutive critical points (P − AF/F − P) and three successive critical points AF/F − P − F/AF − P. Striking thermal variations of the spontaneous magnetization depict a strong reduction due to the interplay between annealed disorder and quantum fluctuations in addition to the aforementioned reentrance. It is shown that the specific heat displays diverse thermal dependencies including finite cusps at the critical temperatures.

This paper presents an split-deflection method of classical rectangular plate analysis. In this method, the deflection was split into x and y components of deflection. It was assumed that the deflection of the rectangular plate is the product of these two components. With this assumption, total potential energy functional was derived from principles of theory of elasticity. By direct variation of the potential energy functional, direct governing equation was obtained. Polynomial deflection and trigonometric deflection were respectively used for the x and y components of the deflection for ssss plate. This deflection components were substituted into this direct governing equation and the coefficient of the deflection was obtained to be 0.0132629qa4/D. With this coefficient, the maximum central deflection of the plate was obtained to be 0.0041446qa4/D. This was compared with the maximum central deflection of Navier’s (0.00416qa4/D) and Levi’s(0.00406qa4/D) plates. It was observed that...

Structurally disordered s-d model of magnetism in metals is investigated. The simplified model of binary alloy structure is offered, the structural correlation functions are calculated. The self-consistent equations for calculation of a spectrum, magnetization, electronic spin polarization and temperature of phase transition are obtained.

Starting from an anisotropic super-exchange Hamiltonian as is found for compounds with strongly correlated electrons in multi-orbital bands and subject to strong spin-orbit interaction we calculate the contribution of thermal and quantum spin fluctuations to the free energy. While the mean field solution of ordered states for such systems usually has full rotational symmetry, we show here that the fluctuations lead to a pinning of the spontaneous magnetization along some preferred direction of the lattice. The fluctuations choose preferred directions for the magnetic order parameter amongst the classically degenerate manifold of states.

Vibration analysis of line continuum with new matrices of elastic and inertia stiffness is introduced in this research. The matrices were developed using Ritz method and assumed six term Taylor's series shape function. Two deformable nodes were introduced at the centre and at the ends of line continuum which brings the number of deformable node to six. The six term Taylor's series shape function assumed was substituted into strain energy equation and into inertia work (Kinetic energy) equation. Their resulting functional were minimized, resulting in 6 x 6 elastic stiffness matrix and 6 x 6 inertia stiffness matrix respectively, for vibration analysis. The two matrices were employed, as well as the traditional 4 x 4 matrices in classical free vibration analysis of four line continua with different boundary condition. The results from the new 6x6 matrices of elastic and inertia stiffness were very close to exact results, with average percentage difference of 0.212425421% from exact solution. Whereas those from the traditional 4 x 4 matrices and 5 x 5 matrices differed from exact results with average percentage difference of 14.72352281% and 0.275% respectively. Thus the newly developed 6 x 6 matrices of elastic and inertia stiffness are suitable for classical free vibration analysis of line continua

We consider the case of electromagnetic field inside a rectangular cavity with conducting walls as a form of a system described by classical mechanics equations. We pass these equations through the Lagrangian formalism to obtain the Hamiltonian formulation. Finally we apply canonical quantization to end up with a quantum theory of the electromagnetic field. Since classical electrodynamics can be interpreted as the quantum theory of a one photon system, then the above quantization is taken as the “quantization of the quantum theory of the electromagnetic field” or simply second quantization.

This paper presents an split-deflection method of classical rectangular plate analysis. In this method, the deflection was split into x and y components of deflection. It was assumed that the deflection of the rectangular plate is the product of these two components. With this assumption, total potential energy functional was derived from principles of theory of elasticity. By direct variation of the potential energy functional, direct governing equation was obtained. Polynomial deflection and trigonometric deflection were respectively used for the x and y components of the deflection for ssss plate. This deflection components were substituted into this direct governing equation and the coefficient of the deflection was obtained to be 0.0132629qa4/D. With this coefficient, the maximum central deflection of the plate was obtained to be 0.0041446qa4/D. This was compared with the maximum central deflection of Navier’s (0.00416qa4/D) and Levi’s(0.00406qa4/D) plates. It was observed that...

This work studied the buckling analysis of biaxially compressed all-round simply supported (SSSS) thin rectangular isotropic plates using the Galerkin's method. The study was limited to thin rectangular isotropic plates having aspect ratios ranging from 1 to 2. The general equation for the critical buckling load of a biaxially loaded plate, was formulated from the overall governing differential equation for plates, using the Galerkin's method. The derived general equation, was expressed as the load in the x-axis in terms of that in the y-axis. This was done by means of a linear relationship which was obtained for the buckling load on the y-axis in terms of that on the x-axis. The SSSS plate deflection equation, was obtained using the polynomial series, and was substituted into the general equation of the critical buckling load of a biaxially loaded plate. This yielded the unique expression for the critical buckling load of a biaxially loaded SSSS plate. Different values (0 to 2) of aspect ratios and "k" (relationship constant between forces on the Y-axis and forces on the X-axis) values (0.1 to 1) were substituted into the critical buckling equation for an SSSS plate, and the critical buckling load coefficients were obtained. The critical buckling load coefficient of a square plate (i.e. at k equal to 1), was obtained as 19.754. When compared with the exact value (19.744) obtained by other researchers who used the trigonometric series, a percentage difference of 0.047 was discovered. At k equal to zero, and for different aspect ratios, the results of the present study showed a maximum percentage difference of 0.069 with those given in the literature. It was therefore concluded that this paper has provided the critical buckling load equation, and results for a biaxially loaded SSSS plates having different k values and aspect ratios, whose result has not been found in the literature.

The behavior of a buckled isotropic rectangular plate of Clamed-Simply-Free-Simple (CSFS) plate was critically examined in this paper; this was accomplished by truncating the polynomial series at the fifth orthogonal terms, which satisfied the boundary condition leading to a particular shape function that was applied in the Ritz method. This shape function was substituted into the total potential energy functional, which was subsequently minimized and the critical buckling load was obtained. The resulting values of non dimensional parameters of the buckling load were compared with the previous research works of Iyengar, Timoshenko and Ibearugbulem within the range of aspect ratios from 0.1 to 2.0. A graph of critical buckling load against aspect ratio was plotted. The behavior of the graph shows that as the aspect ratios increases from 0.1 to 2.0, the critical buckling load decreases. For aspect ratios of 0.2; 0.4; 0.8; 1.0, the values of non dimensional parameters of the buckling load were 25.35; 6.61; 2.08; 1.63 respectively. It was discovered that for the aspect ratio of 0.2; 0.4; 0.8, the percentage difference between the critical buckling load of the present study and those of Iyengar and Ibearugbulem are-0.00394%; 4.3217%;-3.9352% and 37.2442%, 8.6043%,-14.6256% respectively. For aspect ratio of 1.0, the percentage difference between the present study and Timoshenko is-4.1765%.

This paper presents new approach to pure bending analysis of platforms with one clamped and three simply supported edges (SCSS and CSSS). Taylor-Mclaurin's polynomial shape function was derived and substituted into the Galerkin's functional that were minimized to obtain the coefficient of the maximum deflection of the platform. The aspect ratios from 0.1 to 2.0 with 0.1 increments were considered. The coefficient of the deflection and the deflection function profile were used to determine the centre of platform deflections for different aspect ratios. These centre of platform deflection were compared with those from previous studies. For aspect ratio of 1.0, the centre of platform deflection parameter is 0.0028 qb 4 /D.Comparison of the centre of platform deflections from this study and previous studies showed that significant difference does not exist.

This paper presents buckling analysis of rectangular plate by split-deflection method. Here the deflection was taken as the product of these two components in x and y directions. The study formulated the total potential energy functional from principles of theory of elasticity based on work-error approach using this assumption. By direct variation, the energy functional was minimized by and equation for critical buckling load was obtained. Two examples, one withall edges simply supported and the other with all edges clamped were used to test this method. The one used polynomial function for x component of deflection and trigonometric function for y component of. For the second example polynomial function for both x and y components of deflection.Critical buckling load (in non dimensional forms) of the two examples for aspect ratios ranging from 1.0 to 2.0 (at increment of 0.1) were determined and compared with the values from previous study. From the comparison, it was observed that the maximum percentage difference of 0.69 was recorded. Thesmall valeus of percentage difference from this studyshow that this present method is sufficient and reliable for classical plate theory (CPT) buckling analysis of rectangular plates.

This paper provides an efficient solution for determining very good a approximate solutions to the CSCS and SCSC platform problems i.e. platforms with two opposite edges clamped and two opposite edges simply supported. Taylor-Mclaurin's polynomial shape function derived was substituted into the Galerkin functional which was minimized to obtain the maximum deflection of the platform. This research focused on aspect ratios from 0.1 to 2.0 with 0.1 increments. The values of maximum deflection of the present solution were compared with those of previous research works. The maximum deflection of the platforms at aspect ratio of 0.1 is 0.00199 qb4/D. The comparisons of the present work and the previous ones indicate no significant difference. Hence, the Taylor-Mclaurin polynomials shape function obtained for the CSCS and SCSC platforms are very good approximation for the platforms. The results of this study show reasonable agreement with the other available results.

All the previous studies on buckling and postbuckling loads of plate having all four edges simply supported (SSSS) have been limited to the use of assumed double trigonometric functions of displacement and stress. This has constrained further studies on practical plates' problems, as the buckling and postbuckling load equation derived thereafter by this approach lacked practical interpretation and application because major associated parameters such as the displacement parameter, W uv , stress coefficient, W uv 2 and load factor, K cx were not determined. Hence, this paper obtained the displacement and stress functions of buckling and postbuckling loads of SSSS plate by applying the direct integration theory to the Kirchhoff's linear buckling governing differential equation and Von Karman's non-linear governing differential compatibility equation consecutively. Work principle was applied to the Von Karman's non-linear governing differential equilibrium equation to obtain the buckling and postbuckling loads of the plate. Yield/maximum stress of the plate was also obtained by imposing the bending stress influences of the in-plane loads on their direct stress influences. W uv , W uv 2 and K cx were also defined. Therefore, the buckling/postbuckling load and critical yield stress characteristics of SSSS plate under uniformly distributed uniaxial loads could be analyzed completely.

All the previous studies on buckling and postbuckling loads of plate having all four edges simply supported (SSSS) have been limited to the use of assumed double trigonometric functions of displacement and stress. This has constrained further studies on practical plates' problems, as the buckling and postbuckling load equation derived thereafter by this approach lacked practical interpretation and application because major associated parameters such as the displacement parameter, W uv , stress coefficient, W uv 2 and load factor, K cx were not determined. Hence, this paper obtained the displacement and stress functions of buckling and postbuckling loads of SSSS plate by applying the direct integration theory to the Kirchhoff's linear buckling governing differential equation and Von Karman's non-linear governing differential compatibility equation consecutively. Work principle was applied to the Von Karman's non-linear governing differential equilibrium equation to obtain the buckling and postbuckling loads of the plate. Yield/maximum stress of the plate was also obtained by imposing the bending stress influences of the in-plane loads on their direct stress influences. W uv , W uv 2 and K cx were also defined. Therefore, the buckling/postbuckling load and critical yield stress characteristics of SSSS plate under uniformly distributed uniaxial loads could be analyzed completely.

This paper presents deflection function for plate analysis in the form product of two mutually perpendicular truncated polynomial series. The aim herein is to adopt this function as a very good approximate deflection function for first mode analysis (pure bending, stability, vibration and thermal bending) of plate continuum. Energy methods (Galerkin, Ritz, work principle etc) are veritable tools that employ this function for first mode analysis.. When the polynomials are truncated at the fifth term, the kinematic and the kinetic boundary conditions are satisfied. In the same way the obtained data from using this truncated series function in energy methods are very close to the date obtained from numerical and other methods.

This paper presents use of variational calculus to evolve third order functionals for continuum analysis. The governing equilibrium
equation of forces of a line continuum was integrated in the open domain with respect to deflection to obtain three different valid
forms of total energy functional for the continuum. They are second order (Ritz energy functional), fourth order (work error
functional) and any functional hereinafter called the third order energy functional. Third order energy functional for a rectangular
plate was also formulated. These third order energy functionals were subjected to direct variation (differentiating with respect to
the coefficient of deflection) to obtain the weak form equilibrium of forces of continuums. Line continuum of four different boundary
conditions and a plate with one edge clamped and the other three edges simply supported were used to test this new third order
energy functional. In this numerical study, pure bending, buckling and free vibration analysis were performed. The results obtained
indicated that the values obtained using this new method are exactly the same as the values obtained using either Ritz or work error
energy functionals. Thus, one can comfortably and confidently use the third order energy functionals in continuum analysis 



Keywords:  variational calculus, continuum, deflection, energy functional, third order energy functional, equilibrium of forces,
energy functional

This work deals with buckling analysis of a three dimensional isotropic thick plate clamped in all the edges (CCCC) subjected to a uniaxial compressive load, using the variational Energy method. Total potential energy equation of a thick plate was formulated from the three-dimensional constitutive relations, thereafter the compatibility equations was established to obtain the relations between the deflection and shear deformation rotation along the direction of x and y coordinates. By minimizing the potential energy equation with respect to the coefficient deflection and shear deformation rotation, the formulae for calculating the critical buckling load is obtained. Using a mathematical modelling technique based on polynomial displacement function obtained from the compatibility governing equation, a buckling solution was obtained by applying the boundary conditions of the plate and substituted on buckling equation derived. From the numerical analysis obtained, it is found that the value of the critical buckling load increase as the span-thickness ratio increases. This means that an increase in plate thickness improves structural the safety of the plate. The proposed solution were validated and compared with the solution of the trigonometric function using the same model as developed. The critical buckling load was comparable for both functions at varying aspect ratio and the total average percentage difference obtained is 4.3%. The difference being close proved high convergence and accuracy of the approach in the thick plate analysis.

This work deals with buckling analysis of a three dimensional isotropic thick plate clamped in all the edges (CCCC) subjected to a uniaxial compressive load, using the variational Energy method. Total potential energy equation of a thick plate was formulated from the three-dimensional constitutive relations, thereafter the compatibility equations was established to obtain the relations between the deflection and shear deformation rotation along the direction of x and y coordinates. By minimizing the potential energy equation with respect to the coefficient deflection and shear deformation rotation, the formulae for calculating the critical buckling load is obtained. Using a mathematical modelling technique based on polynomial displacement function obtained from the compatibility governing equation, a buckling solution was obtained by applying the boundary conditions of the plate and substituted on buckling equation derived. From the numerical analysis obtained, it is found that the value of the critical buckling load increase as the span-thickness ratio increases. This means that an increase in plate thickness improves structural the safety of the plate. The proposed solution were validated and compared with the solution of the trigonometric function using the same model as developed. The critical buckling load was comparable for both functions at varying aspect ratio and the total average percentage difference obtained is 4.3%. The difference being close proved high convergence and accuracy of the approach in the thick plate analysis.

The broadening of the density of magnetic-impurity-spin states, in a dilute magnetic alloy, is discussed. It is shown that the Ruderman-Kittel-Kasuya-Yosida (RKKY} interaction between the impurities can account for such a broadening. The existance of this broadening can explain the appearance of a maximum in the Kondo resistivity for the weakly interacting systems. The broadening is found to be temperature independent, and dependent on the magnetic-impurity concentration as etc ". It has been recently shown that, assuming the spin of a magnetic impurity in a dilute magnetic alloy to have a broadened density of states, it is pos- sible to explain the maximum observed in the Kondo resistivity for weakly interacting systems, such as Cu-Mn. Mattuck et al. have shown that such a maximum is expected to appear also in the strong- ly interacting systems, such as Cu-Fe. However, a recent calculation by Larsen indicates that this may not be the case if the broadening, y, is smaller than the Kondo temperature T~, but the maximum will appear if y & T~. It seems, now, to be weQ established that for those systems where a maximum in the Kondo resistivity is found, the maximum can be explained to be due to a broadening of the density of the im- purity-spin states with y & T~. In these cases there is no need to assume the existence of magnetic ordering in the form of internal magnetic fields in order to explain the maximum in the resistivity. The latter assumption seems not to be supported by experimental evidence, since the existence of such fields was not found in the relevant impurity concentrations and temperature regions.

This paper presents exact approach to buckling analysis of SSSS and CCCC thin rectangular plates under vibration using split-deflection method. In this method, deflection function, is split into . Total potential energy functional for a thin rectangular plate subjected to both buckling load and vibration was formulated from principles of theory of elasticity. By direct variation, the total potential energy functional was minimized with respect to deflection coefficient to obtain the equation for critical buckling load under vibration. The deflection function is of trigonometric-polynomial family. Numerical analyses for plate with all edges simply supported (SSSS) and a plate with all edges clamped (CCCC) were performed. The non-dimensional critical buckling load results from the present work under no vibration at aspect ratios (0.1 were compared with ones from previous scholars. The average percentage differences from the previous works and the present study for SSSS and CCCC plates when compared stood at 0.013%; 0.066% and 1.654%; 3.538% respectively. This shows that this study gave very close values to the exact solution. Also, the non-dimensional critical buckling loads for SSSS and CCCC thin rectangular plates under vibration at aspect ratios (0. and the corresponding resonating frequency ratios (0 were determined.

This paper presents exact approach to buckling analysis of SSSS and CCCC thin rectangular plates under vibration using split-deflection method. In this method, deflection function, is split into . Total potential energy functional for a thin rectangular plate subjected to both buckling load and vibration was formulated from principles of theory of elasticity. By direct variation, the total potential energy functional was minimized with respect to deflection coefficient to obtain the equation for critical buckling load under vibration. The deflection function is of trigonometric-polynomial family. Numerical analyses for plate with all edges simply supported (SSSS) and a plate with all edges clamped (CCCC) were performed. The non-dimensional critical buckling load results from the present work under no vibration at aspect ratios (0.1 were compared with ones from previous scholars. The average percentage differences from the previous works and the present study for SSSS and CCCC plates when compared stood at 0.013%; 0.066% and 1.654%; 3.538% respectively. This shows that this study gave very close values to the exact solution. Also, the non-dimensional critical buckling loads for SSSS and CCCC thin rectangular plates under vibration at aspect ratios (0. and the corresponding resonating frequency ratios (0 were determined.

This paper presents exact approach to buckling analysis of SSSS and CCCC thin rectangular plates under vibration using split-deflection method. In this method, deflection function, is split into. Total potential energy functional for a thin rectangular plate subjected to both buckling load and vibration was formulated from principles of theory of elasticity. By direct variation, the total potential energy functional was minimized with respect to deflection coefficient to obtain the equation for critical buckling load under vibration. The deflection function is of trigonometric-polynomial family. Numerical analyses for plate with all edges simply supported (SSSS) and a plate with all edges clamped (CCCC) were performed. The non-dimensional critical buckling load results from the present work under no vibration at aspect ratios (0.1 were compared with ones from previous scholars. The average percentage differences from the previous works and the present study for SSSS and CCCC plates when compared stood at 0.013%; 0.066% and 1.654%; 3.538% respectively. This shows that this study gave very close values to the exact solution. Also, the non-dimensional critical buckling loads for SSSS and CCCC thin rectangular plates under vibration at aspect ratios (0. and the corresponding resonating frequency ratios (0 were determined.

This paper presents a new, simple and exact approach to post-buckling analysis of thin rectangular plates. In the study, the Airy's stress functions are not incorporated as the middle surface axial displacement equations are determined as direct functions of middle surface deflection. With this the bending and membrane stresses and strains, which are direct functions of middle surface deflection are obtained. These stresses and strains are used to obtain the total potential energy functional. The minimization of the total potential energy gives the governing equation and compatibility equation for rectangular thin plates buckling with large deflection. The compatibility equations and the governing equation are solved to obtain the deflection function for the problem. Direct variation is applied on the total potential energy function to get the formula to calculate the buckling loads. A numeric analysis is performed for a plate with all the four edges simply supported (SSS plate). It is observed that when deflection to thickness ratio (w/t) is zero the buckling load obtained coincides with the critical buckling from small deflection (linear) analysis. Another observation is that the values of buckling load for given values of w/t obtained in the present study do not vary significant with those obtained by Samuel Levy. The recorded average percentage difference is 12.65%. It is also observed that the maximum w/t to be considered when small deflection analysis is to be used is 0.225. When w/t is more than 0.225, using small deflection analysis will give erroneous results. Thus, large deflection analysis is recommended when w/t is above 0.25. We conclude and recommend that this new equation for analysis of thin plates is a better alternative to the popular von Karman equation.

Key words: Post-buckling buckling load, membrane strains, total potential energy, minimization, direct variation

One of the major problems of rectangular platebuckling under in-plane load is the rigorous approach use in its analysis. In this study, the problem of buckling was addressed by developing a Matlab based computer program for ease of analysis of rectangular plates for critical buckling load, which is needed for safe design.The plates were assumed to be loaded axially along the x-axis, and polynomial shape functions used in Ritz energy equation to formulate a general solutionwhich is computer user-friendly. The critical buckling load coefficients’ 'n' values obtained from this program were compared with thoseavailablein scholarly literaturesso as to demonstrate theirvalidity. These values were found to be very close to existing values in literature. It therefore implies that, this general computer program for buckling analysis of rectangular plates is a better and quicker means of obtaining the critical buckling load of rectangular thin isotropic plates.

Two thin rectangular plates, one simply supported on all sides (SSSS), and the other plate simply supported and fixed on
alternate opposite sides(CSCS), were analyzed for buckling or stability. Polynomial series were used to formulate shape
functions for plates loaded axially along the x- axis. The shape functions were used in Ritz energy equation to obtain the
critical buckling loads 'Nx' for the plates studied. Computer programs based on the formulated critical buckling load
equation were developed in MATLAB language for easier and faster stability analysis of aforementioned plates. In order to
validate the results obtained from the computer program, comparison was made with other results from literature. For
simply supported plate with aspect ratios of 1.0, 1.2, 1.5, 1.6 and 2.0, the maximum percentage difference is 0.02%. And for
the plate with alternate opposite simply supported and fixed edge, the maximum percentage difference from the computed
values of 'n' and those in literature is -0.19%. In both cases, the percentage differences are less than 1% which is consider
very insignificant. Therefore, the developed MATLAB programs can be used to analyze accurately the critical buckling load
of SSSS and CSCS plates.

This paper presents use of variational calculus to evolve third order functionals for continuum analysis. The governing equilibrium equation of forces of a line continuum was integrated in the open domain with respect to deflection to obtain three different valid forms of total energy functional for the continuum. They are second order (Ritz energy functional), fourth order (work error functional) and any functional hereinafter called the third order energy functional. Third order energy functional for a rectangular plate was also formulated. These third order energy functionals were subjected to direct variation (differentiating with respect to the coefficient of deflection) to obtain the weak form equilibrium of forces of continuums. Line continuum of four different boundary conditions and a plate with one edge clamped and the other three edges simply supported were used to test this new third order energy functional. In this numerical study, pure bending, buckling and free vibration analysis were performed. The results obtained indicated that the values obtained using this new method are exactly the same as the values obtained using either Ritz or work error energy functionals. Thus, one can comfortably and confidently use the third order energy functionals in continuum analysis

This paper presents an split-deflection method of classical rectangular plate analysis. In this method, the deflection was split into x and y components of deflection. It was assumed that the deflection of the rectangular plate is the product of these two components. With this assumption, total potential energy functional was derived from principles of theory of elasticity. By direct variation of the potential energy functional, direct governing equation was obtained. Polynomial deflection and trigonometric deflection were respectively used for the x and y components of the deflection for ssss plate. This deflection components were substituted into this direct governing equation and the coefficient of the deflection was obtained to be 0.0132629qa4/D. With this coefficient, the maximum central deflection of the plate was obtained to be 0.0041446qa4/D. This was compared with the maximum central deflection of Navier's (0.00416qa4/D) and Levi's(0.00406qa4/D) plates. It was observed that the value from the present study makes lower bound and upper bound differences of 0.37% and 2.08% with the values from Navier and Levi respectively. This shows that this present method is reliable.

Spin-electron exchange model is generalized and used for description of magnetic states of amorphous substitutional alloys with the structural disorder of the liquid type. A scheme of consistently accounting for the contributions of structural fluctuations to the thermodynamic functions and observable quantities is considered. Using the perturbation theory, the functional of thermodynamic potential is constructed as a functional power series. In the random phase approximation (RPA), the grand thermodynamic potential of the model is calculated. Self-consistency conditions are given, from which equations for magnetizations and critical temperature of the paramagnetic-ferromagnetic transition are obtained.

A general method for obtaining the oscillation periods of the interlayer exchange coupling is presented. It is shown that it is possible for the coupling to oscillate with additional periods beyond the ones predicted by the RKKY theory. The relation between the oscillation periods and the spacer Fermi surface is clarified, showing that non-RKKY periods do not bear a direct correspondence with the Fermi surface. The interesting case of a FCC structure is investigated, unmistakably proving the existence and relevance of non-RKKY oscillations. The general conditions for the occurrence of non-RKKY oscillations are also presented.

A new approach to the analysis of S-S short cylindrical shells subject to internal hydrostatic pressure is presented. Short cylindrical shells with both ends simply supported (S-S) loaded with axisymmetric internal hydrostatic pressure was considered. By satisfying the boundary conditions of S-S short cylindrical shell in the general polynomial series shape function, a particular shape function for the shell was obtained. This shape function was substituted into the total potential energy functional of conservation of work principle, and by minimizing the functional; the unknown coefficient of the particular polynomial shape function was obtained. Bending moments, shear forces and deflections of the shell were determined, and used in plotting graphs for a range of aspect ratios, 1 ≤ L/r ≤ 4. Stresses and deflections at various points of the shell were determined for different cases. Considering case one, with aspect ratio 1, maximum values of deflection, rotation, bending moment and shear force were m,-3.29878 radians, KNm and KN respectively. Thus we observed that the stresses and strains along the S-S short cylindrical reservoir vary inversely as the aspect ratio.

The use of polynomial series function in the buckling analysis of a CCFC is presented. The polynomial series shape function was truncated at the fifth orthogonal terms, which satisfied all the boundary conditions of the plate to obtain a peculiar shape function, which was applied in Ritz method. The peculiar shape function is substituted into the total potential energy functional, which was minimized, and the critical buckling load of the plate was obtained. The critical buckling load is a function of a coefficient, " K ". The values of K from earlier and the present studies were compared within the range of aspect ratios from 0.1 to 2.0. A graph of critical buckling load against aspect ratio was plotted. It was discovered that for aspect ratios of 0.4, 0.5 and 1.0, the critical buckling loads coefficients were 26.94, 17.39 and 4.83. It was also observed from the behavior of the graph that as aspect ratio increases from 0.1 to 2.0, the critical buckling load decreases.

The traditional approach in the analysis of axisymmetrically loaded short cylindrical shells has been to solve the fourth order differential equation using the Krylov's equation. This involved a transition from exponential functions to krylov's functions using Euler's expressions. This approach is grossly limited by the difficulty in the transition from exponential functions to Krylov's functions. A new approach to static analysis of C-C short cylindrical shell subject to internal liquid pressure is presented in this paper. This involves substituting a polynomial series shape function into the Pasternak's differential equation, by satisfying the boundary conditions for C-C short cylindrical shell, a particular shape function was obtained. This shape function was substituted into the total potential energy functional of the Ritz method and minimized to obtain the unknown coefficient. Stresses and deflections at various points of the shell were determined for different cases of aspect ratio with range 1 ≤ L/r ≥ 4. For case 1, maximum values of deflection, rotation, bending moment and shear force were 9.856*10 -3 metres, -3.23*10 -3 radians, -1366.64KNm and -9566.4639KN respectively. It was observed that as the aspect ratio increases from 1 to 4, the deflections and stresses decreases, and the shell tends to behave like long cylindrical shell.

The use of polynomial series function in the buckling analysis of a CCFC is presented. The polynomial series shape function was truncated at the fifth orthogonal terms, which satisfied all the boundary conditions of the plate to obtain a peculiar shape function, which was applied in Ritz method. The peculiar shape function is substituted into the total potential energy functional, which was minimized, and the critical buckling load of the plate was obtained. The critical buckling load is a function of a coefficient, "K". The values of K from earlier and the present studies were compared within the range of aspect ratios from 0.1 to 2.0. A graph of critical buckling load against aspect ratio was plotted. It was discovered that for aspect ratios of 0.4, 0.5 and 1.0, the critical buckling loads coefficients were 26.94, 17.39 and 4.83. It was also observed from the behavior of the graph that as aspect ratio increases from 0.1 to 2.0, the critical buckling load decreases.

The work is to use the energy approach in the form of indirect variational principle (Galerkin's method) for buckling analysis elastic of thin rectangular plates with all edges clamped. The Galerkin method has been used to solve problems in structural engineering such as structural mechanics, dynamics, fluid flow, heat and mass transfer, acoustics, neutron transport and others. The Galerkin method can be used to approximate the solution to ordinary differential equations, partial differential equations and integral equations with a polynomial involving a set of parameters called characteristic orthogonal polynomials. The buckling loads from this study were compared with those of previous researches. The results showed that the average percentage differences recorded for CCCC plates are 0.735 to 4.702. These differences showed that the shape functions formulated by COP has rapid convergence and is very good approximation of the exact displacement functions of the deformed thin rectangular plate under in-plane loading when applied to Galerkin's buckling load for isotropic plates. An indirect variation principle (based on Galerkin's method) could be used in confidence to satisfactorily analyze real time rectangular thin plates of various boundary conditions under in-plane loadings. The results obtained herein are very close to the results obtained by previous research works that used different methods of analysis.

This study considers the application of characteristic orthogonal polonomial to Galerkin indirect variational method for buckling analysis elastic of thin rectangular plates with all edges simply supported. The Galerkin method has been used to solve problems in structural engineering such as structural mechanics, dynamics, fluid flow, heat and mass transfer, acoustics, neutron transport and others. The Galerkin method can be used to approximate the solution to ordinary differential equations, partial differential equations and integral equations with a polynomial involving a set of parameters called characteristic orthogonal polynomials. The buckling loads from this study were compared with those of previous researches. The results showed that the average percentage differences recorded for SSSS plates are 0.014% to 0.055%. These differences showed that the shape functions formulated by COP has rapid convergence and is very good approximation of the exact displacement functions of the deformed thin rectangular plate under in-plane loading when applied to Galerkin's buckling load for isotropic plates. An indirect variation principle (based on Galerkin's method) could be used in confidence to satisfactorily analyze real time rectangular thin plates of various boundary conditions under in-plane loadings. The results obtained herein are very close to the results obtained by previous research works that used different methods of analysis.

This paper studied the stability analysis of orthotropic reinforced concrete shear wall panel with all edges simply supported by the application of Ritz method. The study was carried out through a theoretical formulation based on applying polynomial series deflection function in Ritz method. The polynomial series as used here was truncated at the fifth term, which satisfied all the boundary conditions of the panel and resulted to a particular shape function for simply supported edges (SSSS) panel. Orthotropic parameter ratios of flexural plate rigidities in x, y and x-y directions to flexural plate rigidity in x direction are 1.0, 0.5 and 0.1 respectively. The critical buckling load obtained herein based on this orthotropy for different aspect ratios were compared with those obtained by Iyengar. This comparison showed that the maximum percentage difference recorded is 0.01382%. For aspect ratios of 0.5 and 1.0, the critical buckling loads are 42.74Dx/b2 and 16.799 Dx/b2.

Using first-principles density-functional-theory-based calculations, we analyze the structural stability of small clusters of 3d late transition metals. We consider the relative stability of the two structures: layer-like structures with hexagonal closed packed stacking and more compact structures of icosahedral symmetry. We find that the Co clusters show an unusual stability in hexagonal symmetry compared to the small clusters of other members, which are found to stabilize in icosahedral-symmetry-based structure. Our study reveals that this is driven by the interplay between the magnetic-energy gain and the gain in covalency through the s-d hybridization effect. Although we have focused our study primarily on clusters with 19 atoms, we find this behavior to be general for clusters with between 15 and 20 atoms.

A self-consistent approach is applied for the calculations within the two-time temperature Green functions formalism in the random phase approximation. The effective mass of 4 He atom is computed as m * = 1.58 m. The excitation spectrum is found to be in a satisfactory agreement with the experiment. The sound velocity is calculated as 230 m/s. The temperature of the Bose-condensation with the effective mass taken into consideration is estimated as 1.99 K.