The paper presents the following four kinds of shape analysis based on the basic equations in the structural analysis. The common feature of these four analyses is in the use of the generalized inverse matrix. (1) Shape analysis of unstable structures In the first part, the definition and the characteristics of unstable structure are explained. Then, an analytical method for shape finding of unstable structure under the prescribed load is presented with some illustrative examples. (2) Analysis of self-equilibrated stress system In order to stabilize unstable structure by introducing self-equilibrated stress system, an analytical method for finding such stress system is presented based on the equilibrium equation with zero load term. (3) Shape analysis of structures subjected to the constraint on the prescribed stress The shape analyses of minimum surface, free form shell, membrane structure and surfaces with no wrinkling are introduced. (4) Shape analysis of structures subjected to the constraint on the prescribed displacement The load-displacement relation whose coefficient marix is a function of an unknown position vector representing the structural shape, and the subsidiary condition for the prescribed displacement are introduced. Then, these two equations are theoretically analyzed by using the projection matrix depending upon the subsidiary condition in order to get the nonlinear equation for the shape anaysis.

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Theories of diffusion-controlled migration of interphase boundaries in metals and alloys are presented with an emphasis on the mathematical treatment of a diffusion equation. The methods of treating various complications which are inherent in metallic materials are described. They include the interfacial reaction (friction due to the lattice change at interfaces) , the diffusion interaction among solute species, the concentration dependence of diffusivity, non-isothermal heat treatment processes etc.. In view of the fact that several thermodynamic databanks of calculating phase relations in a multicomponent system are available, it is of considerable interest to develop kinetic models based on diffusion and phase transformation theories to improve our capability of alloy design and process simulation.

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Though it is adequate to describe the macroscopic behavior of materials from the viewpoint of the continuum theory, the existence of microstructures of materials has been ignored on these theories. The concept of polar materials has been introduced in order to modify the classical continuum approach. In this paper, the entropy inequality is described by the integral form in order to express the noninstantaneity and nonlocality. Second order terms of the fluctuation from the average value in a small region (mesodomain) are considered. Theses terms may express the additional entropy production rates caused by the noninstantaneity and nonlocality. Materials with these effects can be called chronopolar materials and thermopolar materials. It is shown that the temperature gradient, curvature and velocity should not be omitted from the arguments of the thermodynamic potential or stress constitutive equations. The hyperbolic differential equation of heat conduction is derived, which gives finite speeds of temperature propagation.

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