A method of stress-strain analysis of elastoplastic bodies with large displacements, rotations, and finite strains is developed. The incremental loading technique is used within the framework of the arbitrary Lagrangian-Eulerian formulation. Constitutive equations are derived which relate the Jaumann derivative of the Cauchy-Euler stress tensor and the strain rate. The spatial discretization is based on the FEM and multilinear three-dimensional isoparametric approximation. An algorithm of stress-strain analysis of elastic, hyperelastic, and perfectly plastic bodies is given. Numerical examples demonstrate the capabilities of the method and its software implementation Keywords: three-dimensional body, large elastoplastic strains, stress-strain state, incremental loading technique, FEM Introduction. Classical Lagrangian formulations are widely used to solve nonlinear problems of solid mechanics (considering physical and geometrical nonlinearity). According to these formulations, the state of an elementary volume is described in terms of the components of the displacement vector (from the undeformed state to a deformed state) and the second Piola-Kirchhoff stress tensor, also referred to the undeformed volume. In this case, boundary-value problems can be formulated well in differential or variational form [1,4,6,8,12] and then solved using various numerical methods. However, this approach has an essential fault when strains are large. In that case, it becomes very difficult to derive constitutive relations, especially when they have differential (rate) form.Step-by-step methods (incremental loading) developed from modern numerical methods (FEM, FDM, etc.) [19, 21] represent the deformation process as a sequence of equilibrium states and describe the transition from one state to another as a load increment (changed boundary conditions, domain of definition, etc.). These methods [20] can be divided [9] into three groups:(i) methods of the first group are based on the virtual-displacement principle, in which all quantities are referred to the initial undeformed state;(ii) methods of the second group are based on the same variational equation, but the current metric is used as a basis one [1,6,14,15,18];(iii) methods of the third group are based on the Lagrangian-Eulerian formulation, where the behavior of a particle (elementary volume) is described using the Lagrange method, and for the current state a plastic-flow problem is formulated according to the Euler approach [2, 3, 15-17].We will use the third approach to analyze large strains of a medium with elastic and plastic properties. In particular, we will formulate a problem of the corresponding class in the current metric for strain rates, which in combination with incremental loading and continuous geometry update will make it possible use employ the FEM.1. Kinematics of Medium. Following the Lagrange approach, we introduce the position vector r r r x e
