EXACT ANALYTICAL SOLUTIONS OF A PROBLEM OF THE NONLINEAR THEORY OF ELASTICITY FOR TWO POTENTIALS OF DEFORMATION ENERGY OF AN INCOMPRESSIBLE MATERIAL

Andreeva Yuliya Yur'evna, Senior lecturer, sub-department of applied mathematics, Volgograd State Technical University (28 Lenina avenue, Volgograd, Russia), ajj308@mail.ru
Zhukov Boris Aleksandrovich, Doctor of engineering sciences, professor, sub-department of applied mathematics, Volgograd State Technical University (28 Lenina avenue, Volgograd, Russia); professor, sub-department of algebra, geometry and mathematical analysis, Volgograd State Sociopedagogical University (27 Lenina avenue, Volgograd, Russia), zhukov.b.a@gmail.com

539.3

10.21685/2072-3040-2018-2-7

Background. In the non-linear elastic theory at the present moment there is no uniform equation of state, similar to the Hooke law in the linear theory. Development of new elastomeric materials, research of various biological materials results in need of creation of new models of a response to external influences. Within a hyper elasticity it leads to emergence of new mathematical expressions for strain energy potential. In the commercial packages intended for calculation of an intense strained state of designs from elastoplastics, expressions for the potential of strain energy are rigidly set. For accounting of new models it is necessary to create new packages of applied calculations. Their verification is carried out on the known precise decisions.
Materials and methods. The purpose of work is finding of some precise solutions of one model task with different potentials. As a task for which precise decisions are found the task about terminating longitudinal shift of the circular cylindrical plug between rigid concentric holders is considered. On the one hand it is one of the simplest tasks, on the other hand it has applied value as it is one of shift shockabsorber designs.
Results. The considered problem was solved by many authors for various potentials. In this work precise decisions for two potentials specified in are received for which we did not manage to find solutions in literature. Decisions are received by the semi-inverse method.
Conclusions. The practical significance of precise decisions is not exhausted their application for verification of numerical methods, they allow to investigate the effects which are not described by the linear theory. It is shown that the distinction of potentials brings not only to the quantitative, but also to qualitative distinction of decisions.

finite antiplane deformation, hyperelastic incompressible material, the potential energy of deformation, exact solution

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