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- 4 N.M. Sergeeva, S.P. Bogdanov, A.V. Redkov.pdf
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- 6 V.V. Skripnyak, A.A Kozulyn, V.A. Skripnyak.pdf
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- 8 V.G. Zubchaninov, E.G. Alekseeva, A.A. Alekseev, V.I. Gultiaev.pdf
- In a linear combined space E6 of stresses and strains with an orthonormal basis for stress and strain tensors
- vectors are assigned
- where vectors coordinates , related to tensor , and deviator components , noted by unequivocal transformation [1,2]. The volume strain in E6 is assumed to be elastic according to the law , where K is modulus of volume elasticity (Bulk modulus). Acc...
- where , – functionals, depending on the following parameters of complex loading: s – the arc length of the trajectories of strain and its break angle . The angle of delay characterizes the direction of the vector towards the shearing to strain tr...
- In the mathematical model of materials plastic deformation process theory in addition to the equations (3) uses universal approximations of functionals proposed by V.G. Zubchaninov 
- where – universal function of Odkvist-Il'yushin for simple (proportional) loading; , – elastic and secant shear modulus; ; – arc length at the break point K of trajectory;
- – complex loading function; – observed approximation parameters . The index «zero» refers to the quantities at the break point of trajectory. The numerical solution of the defining relations (3) for the acquainted functionals (4) is implemente...
- 3. Comparison of numerical results with experimental ones
- Using the foregoing mathematical model, experimental studies and numerical calculations of the processes of deformation of thin-walled tubular specimens of steel 45 on the SN-EVM testing complex along two-link broken strain trajectory in the plane , p...
- Figures 1-4 show the results of calculations and experimental data for two-link strain trajectory with the break angle 45º. The experimental data are marked with circles. Figure 1 shows the implemented strain trajectory in the plane , and on Fig. 2 th...
- Fig. 1. Strain trajectorywith the break angle 45º
- Fig. 2. The response
- Fig. 3. Chart of deformation
- Fig. 4. Chart
- Figures 5-8 present the results of calculation and experimental data for two-link strain trajectory with the break angle 90º and Fig. 9-12 – with the break angle 135º.
- Fig. 5. Strain trajectory with the break angle 90º
- Fig. 6. The response
- Fig. 8. Chart
- Fig. 7. Chart of deformation
- Fig. 9. Strain trajectorywith the break angle 135º
- Fig. 10. The response
- Fig. 11. Chart of deformation
- Fig. 12. Chart
- As we can see, the numerical calculations according to the mathematical model are in good agreement with the experimental data on scalar and vector properties, both qualitatively and quantitatively. This shows the integrity of the calculated data, suf...
- 4. The property of delay vector and scalar characteristics of the materials
- In  A.A. Ilyushin noted that the delay of vector properties is a common property of all plastic materials. The experiments show that the direction of the stress vector at some point K relative to the strain trajectory does not depend on the entire ...
- By hypothesis of R.A. Vasin, it is considered  that the trace of delay is exhausted when the angle of delay 6º÷7º is achieved, which is equivalent to the Hencky-Il'yushin theory of small elastic-plastic deformations. The magnitude estimation of th...
- Figure 13 shows the combined dependency graphs for the implemented two-link strain trajectories, and on Fig. 14 the calculated dependence of the trace of delay of the vector characteristics of the material on the break angle of the trajectory is giv...
- Fig. 13. Chart for different break angles of trajectories
- Fig. 14. Chart
- If we divide the current value of the angle of delay by the value of the angle of delay at the break point of trajectory , we get a relative angle of delay . However, it is impossible to postpone the value on the axis with a dimensionless quantity. ...
- Figure 15 shows the dependence of the relative angle of delay on the increment of the arc length of the strain trajectory after a break. It can be seen that for all the break angles, the calculated charts practically coincide with each other. Thus, i...
- V.S. Lensky in his work  introduced the concept of trace of delay of the scalar properties of materials. Fig. 16 shows the dependence on the break angle of the trajectory for the scalar trace of delay of material . For the value on the diagram w...
- This dependence (Fig. 16) shows that the characteristic is also unstable, and when compared with Fig. 14, it is seen that , while the trace of delay of vector characteristics was several times larger than the trace of delay of the scalar characteris...
- Fig. 16. Chart
- Fig. 15. Chart for different break angles of trajectories
- To determine the effect of the length of the first link of the strain trajectory on the scalar and vector properties of the material, numerical and experimental studies were conducted on the strain programs of testing are shown on Fig. 17. The specim...
- Fig. 17. Strain trajectories with different lengths of the first link
- Fig. 18. Charts
- It is seen that the experimental points practically coincide with each other and are consistent with the numerical calculations of the proposed mathematical model. This means that the length of the first link of the strain trajectory, in the case of a...
- 5. Conclusion
- The theoretical positions of the mathematical model of the theory of processes during deformation along two-link strain trajectories are verified in comparison with experimental data. Verification results indicate the correct modeling of elastoplastic...
- It is established that the trace of delay of vector properties and the trace of delay of the scalar characteristics of the material significantly depend on the break angle of the trajectories. This indicates the instability of the value of the trace o...
- 9 O.K. Garishin, V.V. Shadrin, Yu.V. Kornev.pdf
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-  Garishin OK, Shadrin VV, Svistkov AL, Sokolov AK. Experimental studies of rubbers filled by layered clay nanoparticles. In: Proceedings of the XLVI Summer School-Conference "Advanced problems in mechanics". St.Petersburg; 2017. p. 168-174.
- 12 E.A. Ivanova, D.V. Matyas, M.D. Stepanov.pdf
- EMPLOYMENT OF EULERIAN, LAGRANGIAN, AND ARBITRARY LAGRANGIAN-EULERIAN DESCRIPTION FOR CRACK OPENING PROBLEM
- 13 V.I. Ivlev, A.F. Sigachyov, V.A. Yudin.pdf
-  Chauhan A, Chauhan P. Natural Fibers Reinforced Advanced Materials. J. Chem. Eng. Process Technol. 2013;S6: 003. Available from: doi:10.4172/2157-7048.S6-003.
-  Pickering KL, Aruan Efendy MG, Le TM. A review of recent developments in natural fibre composites and their mechanical performance. Composites Part A: Applied Science and Manufacturing. 2016;83: 98-112.
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