Deformation of cylindrical shells of steel 45 under complex loading

Reviewing the numerical simulation of the process of nonproportional elastic-plastic deformation of steel 45 by a knot of constant curvature, taking into account the complex nature of deformation under complex subcritical loading by axial compressive force and torque for a thin-walled circular cylindrical shell. The theory of Quas and simple processes of A. A. Ilyushin and the mathematical model of V. Zubchaninov were applied taking into consider the parameters of the complex loading for plane trajectories To assess the accuracy of accepted theories, the simulation results are compared with experimental results, received on the automated complex СNcomputer in the laboratory of the faculty of «Strength of materials and theory of elasticity and plasticity» of the Tver state technical University. Was introduced the scheme of calculations disproportionate plastic deformation of steel 45 using the proposed mathematical model showed a satisfactory result and recommended for further use. Remarks, that in the described processes the lack of some parameters complex loading in approximations reduces the accuracy of the final calculated values, differences significantly compared to the experimental data.


Introduction
The fundamental system-forming of the theory of processes elastic-plastic deformation of materials and basic equations are accepted by formulas [1-8, 10, 11]: where 00 11 , , , 33 -modules of ball tensors and deviator tensors respectively (first and second invariants); ** 00 , , , -the components of the diverters and their guide's tensors; ij  -the symbol of Kronecker.
With proportional (simple) loading of guide tensors stresses and strains matches, that , and taking into account the elasticity of the volumetrically deformation have a place the relation of the theory of small elastic-plastic deformations of Ilyushin (4)     00 3 , 2 , ( , 1, 2,3) and the modules of the vectors  and Э relatively equal ,.
In space E . It is obvious that the direction and length of vectors  and Э , it is obvious that the direction and length of vectors, will depend on invariants of tensors, parameters of curvature and torsion of the trajectory m k (m=1,2,3,4), temperatures Т and parameters  .
Based on the particular postulate of isotropy A. A. Ilyushin [1], V. G. Zubchaninov received [3] defining relations between stress and strain vectors in 5 E , the local form of which for threedimensional problems has the form In the case of flat trajectories at 22 0, 0 k   from (8)

Mathematical model of the theory of processes in flat tasks.
The main equations of the mathematical model of the theory of processes in flat tasks are the defining relations (13) and universal approximations of functional V. G. Zubchaninov [3,9]  function of complex loading, that taking into account the orientation of the stress vector in the deformation process and its value at the break point of the trajectory; , , , , ,

A B b p q
 material parameters for each construction material, experimentally determined from basic experiments. Generalized to complex loading, the effect of baushinger regarded as a manifestation of General properties of delay scalar properties of materials [3].
Under given initial conditions with specified functional (15)-(16), constitutive relation (13) leads to Cauchy problem, which was solved using the fourth-order numerical Runge-Kutta method, The solutions obtained by comparing the calculated and experimental data allow the verification of different versions of the model, including when some parameters of the complexity of the process are not taken into account in the approximation of the functionals.

Results of the performed experiments and numerical simulation
The experimental results were obtained on the automated complex SN-computer in the laboratory of the faculty of «strength of materials, theory of elasticity and plasticity» Tver state technical University [4]. The experiments were carried out on thin-walled circular cylindrical shells made of steel 45. The studies were carried out at elastic-plastic deformation (hard loading) thin-walled tubular sample along a flat trajectory, containing two straight sections and a section of a circle of constant curvature.  Comparison of deformation diagrams allowed us to conclude that, the material of specimens is conventionally isotropic, as in the developed plastic deformation ranges of values of the module of the vector of stresses does not exceed 10%.
Test programs of complex loading under normal temperature conditions under disproportionate influence of axial force and torque were carried out at a constant speed 6 10    sec -1 in a plane 13 ЭЭ  deviator space of deformations 5 Е . In a series of tests 5 carried out, there are multiple trajectories of deformation with sections of different constant curvature 1 к const  [12,13]. One of these three-link trajectories is shown in the figure 3.

Fig. 3. Stress-strain diagram of the material under complex loading processes
On the first straight section torsion to the value was realized  Figure 4 shows the response to the implemented deformation trajectory in the plane deviator stress space    ,, s к  for flat trajectories and a generalized effect of baushinger. As you can see, numerical calculations on the presented mathematical model of the theory of processes using functional approximations (15) and (16) quite well correspond to the experimental data for this type of trajectory from scalar data.