One-dimensional mathematical models of tension and compression of solids

Authors

DOI:

https://doi.org/10.15587/1729-4061.2014.23308

Keywords:

tension and compression processes, limiting transition to the continuum, Hooke’s law

Abstract

Mathematical models of tension and compression processes for one-dimensional solid systems based on behavior analysis at an applied constant load of chains of particles with finite mass are considered. One-, two-, and N-particle discrete models are built. For the latter, limiting transitions to the continuum (continuous models) are performed. The phenomenon of failure of the Hooke’s law for all models without considering the mechanical energy dissipation and its validity when taking into account the dissipation is revealed.

Relevance of the conducted researches is caused by previously unnoticed paradox: the Hooke’s law, which describes the steady state of a macroscopic body with constant load applied to it, is used as the basis for the elasticity theory, equations of which are dynamic. In addition, in the Hooke’s law there are no mechanical energy dissipation “traces”, which is a very rough idealization.

Telegraph tension-compression equation, alternative to the Lame’s equation in the elasticity theory is obtained. Unlike the latter, the equation found describes the evolving displacement field and can serve as a basis for generalizations: plasticity and creep equations in homogeneous and heterogeneous media. Relation, corresponding to the Hooke’s law and describing the steady state of a deformable rod taking into account the mechanical energy dissipation is obtained.

The obtained results are of theoretical and practical importance since determining relations between the Hooke’s law and discrete models and deriving telegraph viscoelasticity equations are prerequisites for improving the theory of deformation of solids and for the mathematical modeling of movements of real continuous media (rocks in particular). 

Author Biography

Игорь Романович Венгеров, Institute of Mining Processes NAS Str. Rosa Luxemburg, 72, Donetsk, Ukraine, 83114

Candidate of Physical and Mathematical Sciences, Senior Researcher 

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Published

2014-04-08

How to Cite

Венгеров, И. Р. (2014). One-dimensional mathematical models of tension and compression of solids. Eastern-European Journal of Enterprise Technologies, 2(7(68), 23–27. https://doi.org/10.15587/1729-4061.2014.23308

Issue

Vol. 2 No. 7(68) (2014): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2014 Игорь Романович Венгеров

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