NONLINEAR DEFORMATION OF CYLINDERS FROM MATERIALS WITH DIFFERENT BEHAVIOR IN TENSION AND COMPRESSION 

A new numerical-analytical method for solving physically nonlinear deformation problems of axisymmetrically loaded cylinders made of materials with different behavior in tension and compression has been developed. To linearize the problem, the uninterrupted parameter continuation method was used. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible displacement rates, is constructed. To find the main unknowns of the problem of physically nonlinear cylinder deformation, the Cauchy problem for the system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta-Merson method with automatic step selection. The initial conditions were established by solving the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson’s scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method. The test problem for the nonlinear elastic deformation of a thin cylindrical shell is solved. Coincidence of the spatial solution with the shell solution was obtained. Physically non-linear deformation of a thick-walled cylinder was studied. It is shown that failure to take into account the different behavior of the material under tension and compression leads to significant errors in the calculations of stress-strain state parameters.


Introduction
Axisymmetrically loaded cylinders are widely used in modern technology, for example, as pressure vessels (hydraulic cylinders, gun barrels, nozzles, boilers, fuel tanks), battery housings, cylinders for the aerospace industry, nuclear reactor pipelines, etc.
Many materials (light alloys, superalloys, gray cast iron, polymers, composites, etc.) are characterized by different resistance to tension and compression beyond linear elasticity [1][2][3].The problem of deformation of bodies made of such materials becomes physically nonlinear.However, when studying the physically nonlinear deformation of cylinders, scientists face certain mathematical difficulties associated with modeling the nonlinear behavior of the material, developing methods of linearization, and solving nonlinear boundary value problems.
The study of physically nonlinear deformation (nonlinear-elastic, elastic-plastic problems, creep problems) of cylinders and cylindrical shells is studied in, for example, papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].Only a few papers [11][12][13][14][15][16][17][18][19] study the nonlinear deformation of cylinders and shells made from materials with different behavior in tension and compression.Thus, in paper [11], the solution of the problem of nonlinear elastic bending of a cylindrical shell made from material with different behavior in tension and compression was obtained by integrating the Cauchy problem by the Runge-Kutta-Merson's method with a simultaneous five-fold solution of the boundary problems at each step for the original equations by the method of discrete orthogonalization.In the monograph [12, P. 1], the problems of elastic-plastic and nonlinear-elastic deformation of thick-walled cylinders were reduced to the solution of initial boundary value problems.The methods of discrete orthogonalization of S. K. Godunov and movement along the load in combination with iterative refinement of the solution were used to integrate the equilibrium equations.In the paper [13], a comparative analysis of the spatial and shell solutions of the axisymmetric problem of creep and damage of a cylinder under the action of external pressure is performed.Both in the spatial and in the shell formulation, the problem was reduced to an initial boundary problem.Integration by time was performed using the Runge-Kutta-Merson's method, and boundary value problems at each step were solved using the R-function method and the S. K. Godunov discrete orthogonalization method.
The theory and methods of calculating the nonlinear deformation of cylinders made of nontraditional materials are currently being developed.The aim of the paper is the development of a numerical and analytical method for solving the problems of physically nonlinear deformation of cylinders from materials with different behavior in tension and compression, based on the use of the uninterrupted parameter continuation, Ritz and Runge-Kutta-Merson's methods.

Formulation of the problem. Solution method
An axisymmetrically loaded isotropic hollow cylinder of finite length in a cylindrical coordinate system Orφz is considered.Axis Oz coincides with the axis of symmetry.
To formulate and linearize the problem of physically nonlinear deformation of cylinders, we will use the uninterrupted parameter solution continuation method [20].Let's introduce a parameter , which is related to the external load acting on the cylinder.In this case t 0 is the value of the parameter at which the deformation problem is linear, and t * corresponds to the specified loading level.Let's mark the derivative with respect to the parameter with a dot above the symbol t.Further on derivatives by parameter t will be called rates.
Let's assume that the components kl   of strain rate tensor consist of linear kl e  , which are subject to Hooke's law, and non-linear kl   components, i.e After differentiating the Cauchy dependencies in cylindrical coordinates by the load parameter t [21], we obtain the relation between the strain rates and the displacement rates where z r u u   , are displacement rates along the axes Or and Oz.Similarly, by differentiating Hooke's law [21] with respect to t and taking into account (1), we obtain the relation between stress rates and strain rates where ; E, G, ν are Young's modulus, shear modulus and Poisson's ratio of the material.
To describe the nonlinear behavior of the material, we use tensor-linear constitutive equations that describe the different behavior of the material under tension and compression [22] where is the equivalent stress; and quadratic invariants of the stress tensor; A, B, C -material parameters determined from experimental data.
If elastic-plastic deformation is studied, then relations (4) should be supplemented by the condition of plasticity.
Consider the method of determining the parameters of the material A, B, C in relations (4).For this, it is necessary to have experimental data for material samples, for example, under uniaxial stress state and under pure torsion.
Let's assume that as a result of experiments on uniaxial tension (σ 11 >0) it is established that in the direction of the applied load is (6) Similarly for pure torsion (σ 12 ≠0) it is where In the case of a simple load, equations ( 4) can be integrated and written in the form From relations (8) for uniaxial tension, we will have Next, matching ( 5) and ( 9), ( 6) and ( 10), ( 7) and ( 11) in pairs, we get the following system of equations , from which it is easy to find the material parameters For the variational statement of the problem, we will use a functional in the Lagrange form, given on the kinematically possible displacement rates, which for a rotation body has the form [12] where ∂Ω p is the part of the border ∂Ω, to which surface loads are applied; P n 0 , P τ 0 are normal and tangent components of external loads; n, τ are external normal and tangent to the contour ∂Ω p ; are direction cosines of the normal n.In formula (12) the rates of the nonlinear components are considered given for each fixed value of the parameter t and do not vary.Substituting (2), ( 3) into (12) and integrating along the angular coordinate, we obtain the following functional: where are "fictitious" forces caused by nonlinear components of deformation.Solution of the variational equation δΛ=0 gives the distribution of the displacement rate fields for fixed values of the parameter t>t 0 , at any point of the cylinder.The main unknown problems of nonlinear deformation can be found by integrating the corresponding rate fields from the solution of the Cauchy problem with the parameter t for a system of ordinary differential equations of the form The nonlinearity of the system ( 14) is due to the nonlinearity of the constitutive equations ( 4).The initial conditions for the sought functions are found from the solution of the problem of linear elastic deformation.For this purpose, we can use a functional of the form (13), in which we replace the rates of the functions with the functions themselves and assume that the "fictitious" forces are 0 We will solve the initial problem for the system of equations ( 14) using the Runge-Kutta-Merson's method with automatic step selection [23].To calculate the right-hand sides of equations ( 14) at fixed values t>t 0 , corresponding to the Runge-Kutta-Merson's scheme, it is necessary to solve the variational problems for the functional ( 13) five times at each step.Variational problems were solved using the Ritz method.

Numerical results
As a test example, the non-linear elastic deformation of a thin cylindrical shell made of gray cast iron СЧ 15-32, which is loaded with an internal pressure of P inn =4 MPa, is considered.The shell is rigidly fixed on one edge and free from fastening and effort on the other.The geometric dimensions are as follows: length l=0.2 m, the radius of the inner surface r 1 =0.195 m, the radius of the outer surface r 2 =0.205 м.
For gray cast iron, the equality of the modulus of elasticity under tension and compression on the initial linear sections of the deformation diagrams was experimentally established.At a higher load, the nonlinear character of the deformation, in which the diagrams of deformation under tension and compression differ significantly, is manifested [1].
A linear law for the load is assumed as where     t t , 0 .The origin of the coordinates is placed on the fixed edge.Then the kinematic boundary conditions will have the form 0 Sequences of coordinate functions, which satisfy the conditions (16), can be written as: where coefficients, which are found at each step by the Ritz method; t -some fixed value of the load parameter; are systems of linearly independent functions.In this paper, bicubic Schoenberg splines were . Spline systems were built on a regular grid N r ×N z , where N r , N z is the number of discretization segments along the axes Or and Oz, respectively.
Rates of equivalent stresses e   in the constitutive equations were assumed to be constant at each step by t and were calculated from the stress values and stress rates in the previous step.
Figs. 1, 2 show graphs of changes of radial w=u r (r 0 , z) and axial u z0 =u z (r 0 , z) displacements of the middle surface of the shell 2 along the axis Oz.
Dashed lines show the results obtained within the framework of the theory of shells [11], using relations (4), and solid lines show the results obtained using the method developed in the paper.When using it for the loading in formula (15), the following parameters were accepted: P 1 =0.02MPa, P 2 =10 -1 MPa, and the initial step and the given calculation error in the Runge-Kutta-Merson's method were equal to: Δt 0 =10 -2 , ε=10 -5 .
From Figs. 1, 2, it can be seen that the results of the shell displacements calculation, obtained by the shell and spatial models, almost completely coincided.The ends of the cylinder are free of loading and fixed in such a way that the radial displacements are equal to zero.The origin of the coordinates is placed in the center of the cylinder.Then the kinematic boundary condition will be written as 0 In this case, sequences of coordinate functions have the form In formulas (18), the second expression is not related to the satisfaction of boundary conditions, but only ensures the fulfillment of symmetry conditions for axial displacements.
Fig. 3 shows graphs of changes along the axis of the cylinder of the radial displacements of the middle surface, and Fig. 4 -graphs of the distribution of axial stresses σ zz =σ zz (r 1 , z) on the inner surface of the cylinder.The solid lines show the results obtained on the basis of the constitutive equations (4), and the dashed lines show the results based on a simplified model built only on experimental data obtained during uniaxial tension [12].For loading, the following parameters were taken in formula ( 15): P 1 =1.0 MPa, P 2 =0.1 MPa, and the initial step and the given calculation error in the Runge-Kutta-Merson's method were equal to: Δt 0 =10 -2 , ε=10 -5 .
The graphs given in Figs. 3, 4 show that failure to take into account the different behavior of the material under tension and compression leads to significant errors in determining the components of the stressstrain state.
Spline systems in both solved problems were built on a regular dimensional grid N r ×N z =5×10.At the same time, the total number of coordinate functions was equal to 208.

Conclusions
A numerical-analytical method for solving the problems of physically nonlinear deformation of cylinders from materials with different behavior in tension and compression has been developed.The method is based on the joint use of uninterrupted parameter continuation, Ritz and Runge-Kutta-Merson's methods.The test problem of the deformation of the cylindrical shell was solved, and a coincidence with the shell solution was obtained.The calculation of nonlinear elastic deformation of a thick-walled hollow cylinder under the action of internal pressure was performed.It is shown that failure to take into account the different behavior of the material under tension and compression leads to significant errors in the results of the stressstrain state parameters calculation.

Fig. 1 .
Fig. 1.Radial displacements along the shell axis Fig. 2. Axial displacements along the shell axisNext, the nonlinear deformation of a thick-walled hollow cylinder made of the material СЧ 15-32, loaded with internal pressure of P inn =22.0 MPa was considered.Geometric dimensions were: l=0.2 m, r 1 =0.18 m, r 2 =0.22 m.The ends of the cylinder are free of loading and fixed in such a way that the radial displacements are equal to zero.The origin of the coordinates is placed in the center of the cylinder.Then the kinematic boundary condition will be written as 0 