Stuart S. Antman  Nonlinear Problems of Elasticity

Название: 
Nonlinear Problems of Elasticity 
Автор: 
Stuart S. Antman 
Категория: 
Математика

Тип: 
Книга 
Дата: 
31.12.2008 00:20:49 
Скачано: 
105 
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Описание: 
The scientists of the seventeenth and eighteenth centuries, led by Jas. Bernoulli and Euler, created a coherent theory of the mechanics of strings and rods undergoing planar deformations. They introduced the basic concepts of strain, both extensional and flexural, of contact force with its components of tension and shear force, and of contact couple. They extended Newton's Law of Motion for a mass point to a law valid for any deformable body. Euler formulated its independent and much subtler complement, the Angular Momentum Principle. (Euler also gave effective variational characterizations of the governing equations.) These scientists breathed life into the theory by proposing, formulating, and solving the problems of the suspension bridge, the catenary, the velaria, the elastica, and the small transverse vibrations of an elastic string. (The level of difficulty of some of these problems is such that even today their descriptions are seldom vouchsafed to undergraduates. The realization that such profound and beautiful results could be deduced by mathematical reasoning from fundamental physical principles furnished a significant contribution to the intellectual climate of the Age of Reason.) At first, those who solved these problems did not distinguish between linear and nonlinear equations, and so were not intimidated by the latter.
By the middle of the nineteenth century, Cauchy had constructed the basic framework of threedimensional continuum mechanics on the foundations built by his eighteenthcentury predecessors. The dominant influence on the direction of further work on elasticity (and on every other field of classical physics) up through the middle of the twentieth century was the development of effective practical tools for solving linear partial differential equations on suitably shaped domains. So thoroughly did the concept of linearity pervade scientific thought during this period that mathematical physics was virtually identified with the study of differential equations containing the Laplacian. In this environment, the respect of the scientists of the eighteenth century for a (typically nonlinear) model of a physical process based upon fundamental physical and geometrical principles was lost.
The return to a serious consideration of nonlinear problems (other than those admitting closedform solutions in terms of elliptic functions) was led by Poincare and Lyapunov in their development of qualitative methods for the study of ordinary differential equations (of discrete mechanics) at the end of the nineteenth century and at the beginning of the twentieth century. Methods for handling nonlinear boundaryvalue problems were 
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