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Das Variationsproblem der Elastostatik
Citation Link: https://doi.org/10.15480/882.907
Publikationstyp
Technical Report
Date Issued
1989
Sprache
German
Author(s)
TORE-DOI
This work is based upon the differential equations of the linear-elastic continuum and - as a result thereof - NAVIER's equations. General mixed boundary conditions are formulated. First of all the theorem of energy of the theory of elasticity, the strain energy and the total potential energy are derived.
The simple statement of the variation of displacernents, being identical with the principle of virtual displacements, is defined definitely. The application leads to the principle of virtual works. The identity of the principle of virtual works and the principle of stationary total potential TI is proven for a general elastic stress-strain relation. Through the introduction of a linear-elastic constitutive relation results the minimal quality of the total potential energy and therefore the minimal principle of displacements.
Eventually the three-dimensional variational problem of elasticity is formulated. EULER-LAGRANGE's differential equations, which belong to this variational problem of elasticity, are deduced. They turn out to be identical with NAVIER's equations. Consequently the result of the variational problem filters out exactly the set of the field of displacement, which in turn fulfills
the requirements of NAVIER's equations, and therefore also the differential equations of the system.
The simple statement of the variation of displacernents, being identical with the principle of virtual displacements, is defined definitely. The application leads to the principle of virtual works. The identity of the principle of virtual works and the principle of stationary total potential TI is proven for a general elastic stress-strain relation. Through the introduction of a linear-elastic constitutive relation results the minimal quality of the total potential energy and therefore the minimal principle of displacements.
Eventually the three-dimensional variational problem of elasticity is formulated. EULER-LAGRANGE's differential equations, which belong to this variational problem of elasticity, are deduced. They turn out to be identical with NAVIER's equations. Consequently the result of the variational problem filters out exactly the set of the field of displacement, which in turn fulfills
the requirements of NAVIER's equations, and therefore also the differential equations of the system.
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