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.

本文致力于在非平稳外部源存在下弹性杆纵向变形的非线性模型的研究。非平稳外部源的存在大大简化了该模型的研究。通过对称性分析的方法，可以找到具有不同对称性的通用模型的所有基本模型。与其他基本模型相比，一个可以接受最多Lie变换的基本模型描述了在初始时刻存在非常强的奇异外部源时的变形。对于该模型，获得了所有不变的子模型。可以明确找到描述这些子模型的解，或者将其搜索简化为非线性积分方程的解。对于明显获得的解决方案所描述的子模型，当达到应力的临界值（材料的极限强度）时，我们发现了杆破坏的时间和地点。应用所获得的公式生成新的解将给出包含其他任意常数的解族。描述其余子模型的积分方程中存在任意常数，使我们能够研究具有物理意义的边值问题。对于这些不变的子模型，我们研究了弹性杆的非线性纵向变形，对于该弹性杆，其纵向位移和变化速度在固定点的初始时刻指定。我们确定了在某些附加条件下这些边值问题的解的存在性和唯一性。他们的解简化为积分方程的解，对于这些参数中收件箱的某些值，我们已经用数值方法对其进行了求解。