Theoretical Study of Finite Deformations of Elastic Materials having Internal Constraints.

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1. Development of the Theory of Finite Deformations of Inextensible(Length-Preserving) Elastic Materials.(1)We established the method of analysis of plane problems and solved some problems of the deflection of beams whose cross-section changes abruptly. (2)For these beams under various loads, we made clear some aspects of deformations, stresses, and singular stresses. Furthermore, we discussed their fracture modes. (3)We determined the structure of singular stress surface; we obtained their equilibrium equations for finite deformations as well as incremental deformations. (4)We made clear the structure of elastostatic shocks and conditions for their appearance. Furthermore, we discussed it in terms of energy dissipation.2. Development of the Theory of Finite Deformations of Inexpansible(Area-Preserving) Elastic Materials.(1)We examined the constitutive relations of bio-membranes and found that some of them were approximately expressible in terms of the internal constraint to inexpansibility. (2)We developed the kinematics of area-preserving materials and obtained the expressions for reaction stresses. (3)We discussed the reaction stresses and their equilibria. (4)We showed that plane problems could be treated in the similar way as those for length-preserving materials. For axisymmetric problems, we proposed the basic governing equations and the method of solution.