Nonlinear longitudinal deformations of an elastic rod under the action of a non-stationary singular source

• Published in: International journal of non-linear mechanics 2020-09, Vol.124, p.103514, Article 103514
• Main Author: Chirkunov, Yu.A.
• Format: Article
• Language: English
• Subjects: Boundary value problems; Constants; Elastic deformation; Integral equations; Invariant solutions; Invariant submodels; Invariants; Mathematical models; Non-stationary source; Nonlinear equations; Nonlinear longitudinal deformation of an elastic rod; Symmetry; Symmetry analysis; Time and place of destruction of the rod; Ultimate tensile strength
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2020.103514

This paper is devoted to the study of the nonlinear model of the longitudinal deformations of an elastic rod in the presence of the non-stationary external source. The presence of the non-stationary external source greatly complicates the study of this model. By methods of the symmetry analysis, all basic models of the general model, having different symmetry properties are found. A basic model that admits the widest group of Lie transformations, in comparison with other basic models, describes a deformation in the presence of a very strong singular external source at the initial moment of the time. For this model all invariant submodels are obtained. The solutions describing these submodels were found either explicitly, or their search is reduced to solving of the nonlinear integral equations. For the submodels which are described by obviously obtained solutions, we found the time and place of the destruction of the rod when a critical value of the stress (an ultimate strength of the material) was reached. Applying of the obtained formula generating the new solutions gives the families of the solutions containing additional arbitrary constants. The presence of arbitrary constants in the integral equations describing the remaining submodels allows us to study having physical meaning boundary value problems. For these invariant submodels we researched nonlinear longitudinal deformations of the elastic rod for which a longitudinal displacement and speed of its change are specified at the initial moment of the time at a fixed point. We are established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Their solution is reduced to the solution of the integral equations, which we have solved numerically for some values of the inbox in them parameters. •All basic models, having different symmetry properties, are found.•For the model, admiting the widest group, all invariant submodels are obtained.•The solutions describing these submodels are found either explicitly, or their search is reduced to solving of the integral equations.•The existence and uniqueness of the solutions of some boundary value problems are established.•These boundary value problems were numerically solved for some values of the parameters.