Calculation of thin isotropic shells beyond the elastic limit by the method of elastic solutions

The paper focuses on the model of calculation of thin isotropic shells beyond the elastic limit. The determination of the stress-strain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method. In the present work the building of the solution based on the equilibrium equations and geometric relations of linear theory of thin shells in curved coordinate system α and β, and the relations between deformations and forces based on the Hirchhoff-Lave hypothesis and the small elastic-plastic deformations theory are presented. Internal forces tensor is presented in the form of its expansion to the elasticity tensor and the additional terms tensor expressed the physical nonlinearity of the problem. The functions expressed the physical nonlinearity of the material are determined. The relations that allow to determine the range of elastic-plastic deformations on the surface of the present shell and their changing in shell thickness are presented. The examples of the calculation demonstrate the convergence of elastic-plastic deformations method and the range of elastic-plastic deformations in thickness in the spherical shell. Spherical shells with the angle of half-life regarding 90 degree vertical symmetry axis under the action of equally distributed ring loads are observed. 1 State of the problem The constructions in the form of shells take a special place among the spatial structures. The theory of the calculation of stress-stain state of shells is presented in the works [1, 4] and in many other scientific works. The practical necessity of the calculation beyond the elastic material work is caused by the fact that the high tensions exceeded the limit of material proportionality appear in thin-walled constructions in the places of local influence. In the present work the model of the numerical implementation of the calculation of thin isotropic shells the material of which works beyond the elastic limit is observed. The definition of stress-stain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method of A.A. Ilyushin [2]. _______________________________ * Corresponding author: ast965@yandex.ru


State of the problem
The constructions in the form of shells take a special place among the spatial structures.The theory of the calculation of stress-stain state of shells is presented in the works [1,4] and in many other scientific works.The practical necessity of the calculation beyond the elastic material work is caused by the fact that the high tensions exceeded the limit of material proportionality appear in thin-walled constructions in the places of local influence.In the present work the model of the numerical implementation of the calculation of thin isotropic shells the material of which works beyond the elastic limit is observed.The definition of stress-stain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method of A.A. Ilyushin [2].

_______________________________ 2 The method of solution
At the building of the solution, the main ratios are the equations of equilibrium and geometric ratios of the linear theory of thin shells in the curved coordinate system α and β [3,8], Figure 1.Equilibrium equalities are considered in the following form: bending moments, Htorque moments and q 1, q 2 , q 3outer load projections distributed on the surface to the directions of the unit vectors , , 1 2 t t z .They operate at the edges of the infinitely small element the median surface figure 1.
The following equalities represent the ratios between deformations functions and displacements functions (2): where A,B -Lame coefficients, u, υ, wfull vector displacement projections, ε 1 , ε 2 , γ 12relative linear and angular deformation, κ 1 , κ 2bending deformations, κ 12torsion deformations, R 1 , R 2radiuses of the principal curvatures.The complete equation system is received by the connection to the equation system (1) and (2) of the ratios between deformations and forces.According to the Hirchhoff-Lave hypothesis the isotropic shells deformations ε α , ε β γ αβ , in the layer located at the distance z from median surface can be presented in the form of the first two terms of the expansion in powers z (3): In small elastic-plastic deformations theory [6] deformations and tensions are connected with the ratios: where the function ψ is equal to the ratio between the tension intensity σ i and the deformation intensity ε i (5): As a result the relations between tensions and deformations take the forms ( 6), (7): MATEC Web of Conferences 196, 01014 (2018) https://doi.org/10.1051/matecconf/201819601014XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering In equilibrium equalities of thin shells theory (1) the integral characteristics are introduced instead of tensions: longitudinal and shifting forces, bending and twisting moments.The ratios ( 6), (7) allow to express the forces by means of the deformations in the following form: The following values are indicated in the formulas (8): hthickness of the shell, zdistance from the median surface along the normal to the median surface, the functions J 1 , J 2 , J 3the variables along the median surface of the rigidity of the shell.
It should be noticed that the function ψ is equal to the material elasticity module ψ = E, and the rigidities of the shell are J 1 = Eh , J 2 = 0, J 3 = Eh 3 If the relation between tension intensity σ i and deformation intensity ε i is known, according to [6] the deformation intensity is defined by the formulas (12), (13): Meanwhile the rigidities (9) express in the form of nonlinearity functions from the deformations.As a result, elastic and plastic components obtained from the formulas ( 8), (10) and (11) are assigned in elastic solutions method in the internal forces ratios: The functions expressed physical nonlinearity of the material are defined by the formulas (16), (17),(18): Summarizing the ratios ( 16), (17) we get the result where the common internal stress tensor (T) can be presented in the form of its decomposition to elastic (t) and additional tensor (ΔT) expressing physical nonlinearity problems: The right part of the system is obtained from the component in the form of two terms during the substitution of decomposition (19) in equilibrium equalities.Thus, the full equation system is a sum of the following equation systems in the problem solution by elastic solutions method.Substituting internal forces functions in the form (19) in the equilibrium equalities the terms that express physical nonlinearity of thin shell material appear in the right part of the equilibrium equalities system.In each approximation the system (1) where: 1 2 3 , , q q q -the projections of the components of outer load distributed on the the components expressed physical nonlinearity.
The ratios between deformations functions and displacements functions in each approximation are determined by the following formulas (21): Physical ratios take the following form ( 22), (23): Here t 1n , t 2n , s 1n , m 1n , m 2n , h n are internal loads and the moments for elastic shell that has the same deformations as the present elastic-plastic shell ( 22), (23): Using elastic solutions method where the components (26) are supposed to be equal to zero, we get the values of stress-strain state components relevant to the phase of elastic material work in the first approximation with n = 1.The deformation values of the first approximation allow to define the value of the functions (26) for the second approximation, the deformation values of the second approximation serve for the defining the adding functions for the third approximation, etc.The achievement of the demanding the accuracy for the values of the stress-strain state components of thin isotropic shell demands the performing n number of approximations.The functions expressed physical nonlinearity problems (26) are not equal to zero at the points of the formation of plastic deformations.The ratios (26) -(32) allow to determine the area of plastic deformations distribution in the shell thickness.

Results
The study of the convergence of elastic solutions method was conducted by numerical method [5][6][7] where each tenth approximations for all the resolving functions was compared.The results of the study for the spherical shell with the angle of half-life α 0 = 90 0 and with the relation of the shell curve radius R/h =100 are given under the action of the ring load applied to the middle of the Meridian.The diagram Pu z (the loadvertical displacement) depending on the number of approximations is presented in the figure 2a).The section below the point of load application is considered.The area of plastic deformations distribution in the shell thickness (figure 2b)) is presented.The material is steel, the module of the material elasticity E = 2.1•10 5 МПа, the diagram with linear strengthening where λ = 0.95.Under the obtaining the relations, it is occured that elastic solution is useful for the load before P/Eh ≈ 0.15•10 -3 .It takes 10 approximations, about 20-30 approximations for the loads P/Eh ≈ 0.25•10 -3 , and about 100 approximations in the case of high loads P/Eh > 0.35•10 -3 for the achieving the accuracy μ = 0.1 % with the loads before P/Eh ≈ 0.2•10 -3 .

Conclusions
The calculations demonstrate that, with the growing of the load, the necessary number of approximations for the achieving the necessary accuracy is increasing, the plastic deformations area is increasing too.The elastic solutions method converges too slowly for the range of loads, where small elastic-plastic deformations theory becomes unjust.

Fig. 1 .
Fig. 1.Positive directions of outer load projections and of internal forces components.

MATECFig. 2 .
Fig. 2. Rotational shell under the action of the rotation load: a) the diagram P u z (the loadvertical displacement), b) the area of plastic deformations distribution in the shell thickness /12 for the elastic area.Meanwhile MATEC Web of Conferences 196, 01014 (2018) https://doi.org/10.1051/matecconf/201819601014XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering internal forces according to the ratios (8) move to the elasticity ratios with Puasson coefficient equal to ½ that should be indicated by t 1 , t 2 , s 1 , m 1 ,m 2 , h 1 : has the form (20):