ANALYTIC SOLUTION OF THE WAVE EQUATION OF ELASTICITY THEORY IN THE METAL FORMING
DOI:
https://doi.org/10.15587/1729-4061.2012.4519Keywords:
metal forming, elasticity theory, dynamic problem, harmonic function.Abstract
The dynamical problems of the elasticity and plasticity theories occur in the process of metal forming. In the article, an analytical solution of the wave equation of the elasticity was obtained. The differential relations, converting the wave equation into an identity, were obtained, using the product of trigonometric functions with arguments, defined by time and space variables. The conditions of solutions, which are different from the solution, obtained by the method of separation of variables, are described. The arguments of function can be nonlinear dependences from the variables. The restrictions were imposed on the function arguments, suggesting that the latter may have different order, just to satisfy the conditions of solutions. Another peculiarity of the solution is the presence of two simultaneous variables in expressions of arguments. It was shown that between the arguments of trigonometric functions there is a correspondence in differential form. This expands possibilities of the solution, not limiting the argument of function by linear dependence. The suggested solution accommodates well-known solutions, and is considered more general in comparison with the expressions obtained by the method of separation of variables. This allows expanding of the range of dynamic problems for the account of greater variety of boundary and initial conditions.References
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