# Analytical and numerical study of temperature stress in the bi-modulus thick cylinder

• Gao, Jinling (School of Aeronautics and Astronautics, Purdue University) ;
• Huang, Peikui (College of Engineering, South China Agricultural University) ;
• Yao, Wenjuan (Department of Civil Engineering, Shanghai University)
• Accepted : 2017.07.12
• Published : 2017.10.10

#### Abstract

Many materials in engineering exhibit different modulus in tension and compression, which are known as bi-modulus materials. Based on the bi-modulus elastic theory, a modified semi-analytical model, by introducing a stress function, is established in this paper to study the mechanical response of a bi-modulus cylinder placed in an axisymmetric temperature field. Meanwhile, a numerical procedure to calculate the temperature stresses in bi-modulus structures is developed. It is proved that the bi-modulus solution can be degenerated to the classical same modulus solution, and is in great accordance with the solutions calculated by the semi-analytical model proposed by Kamiya (1977) and the numerical solutions calculated both by the procedure complied in this paper and by the finite element software ABAQUS, which demonstrates that the semi-analytical model and the numerical procedure are accurate and reliable. The result shows that the modified semi-analytical model simplifies the calculation process and improves the speed of computation. And the numerical procedure simplifies the modeling process and can be extended to study the stress field of bi-modulus structures with complex geometry and boundary conditions. Besides, the necessity to introduce the bi-modulus theory is discussed and some suggestions for the qualitative analysis and the quantitative calculation of such structure are proposed.

#### Acknowledgement

Supported by : National Natural Science Foundation of China, South China Agricultural University

#### References

1. Ambartsumyan, C.A., Wu, R.F. and Zhang, Y.Z. (1986), Different Modulus of Elasticity Theory, China Railway Publishing House, China.
2. Bertoldi, K., Bigoni, D. and Drugan, W. J. (2008), "Nacre: An orthotropic and bimodular elastic material", Compos. Sci. Technol., 68(6), 1363-1375. https://doi.org/10.1016/j.compscitech.2007.11.016
3. Chandrashekhara, K. and Bhimaraddi, A. (1982), "Elasticity solution for a long circular sandwich cylindrical shell subjected to axisymmetric load", Int. J. Solid. Struct., 18(7), 611-618. https://doi.org/10.1016/0020-7683(82)90043-9
4. Fang, X., Yu, S.Y., Wang, H.T. and Li, C.F. (2014), "The mechanical behavior and reliability prediction of the HTR graphite component at various temperature and neutron dose ranges", Nucl. Eng. Des., 276(2), 9-18. https://doi.org/10.1016/j.nucengdes.2014.05.036
5. Geim, A.K. (2009), "Graphene: status and prospects", Sci., 324(5934), 1530-1534. https://doi.org/10.1126/science.1158877
6. Gilbert, G.N.J. (1961), "Stress/strain properties of cast iron and Poisson's ratio in tension and compression", Brit. Cast Res. Assn. J., 9, 347-363.
7. Green, A.E. and Mkrtichian, J.Z. (1977), "Elastic solids with different moduli in tension and compression", J. Elasticity, 7(4), 369-386. https://doi.org/10.1007/BF00041729
8. Guo, Z.H. and Zhang, X.Q. (1987), "Investigation of complete stress-deformation curves for concretes in tension", ACI Mater. J., 84(4), 278-285.
9. He, X.T., Chen, Q., Sun, J.Y. and Chen, S.L. (2010), "Application of Kirchhoff hypotheses to bending thin plates with different moduli in tension and compression", J. Mech. Mater. Struct. 5(5), 755-769. https://doi.org/10.2140/jomms.2010.5.755
10. He, X.T., Chen, Q., Sun, J.Y. and Zheng, Z.L. (2012), "Largedeflection axisymmetric deformation of circular clamped plates with different moduli in tension and compression", Int. J. Mech. Sci., 62(1), 103-110. https://doi.org/10.1016/j.ijmecsci.2012.06.003
11. He, X.T., Chen, S.L. and Sun, J.Y. (2007), "Applying the equivalent section method to solve beam subjected to lateral force and bend-compression column with different moduli", Int. J. Mech. Sci., 49(7), 919-924. https://doi.org/10.1016/j.ijmecsci.2006.11.004
12. He, X.T., Hu, X.J., Sun, J.Y. and Zheng, Z.L. (2010), "An analytical solution of bending thin plates with different moduli in tension and compression", Struct. Eng. Mech., 36(3), 363-380. https://doi.org/10.12989/sem.2010.36.3.363
13. He, X.T., Xu, P., Sun, J.Y. and Zheng, Z.L. (2015), "Analytical solutions for bending curved beams with different moduli in tension and compression", Mech. Adv. Mater. Struct., 22(5), 325-337. https://doi.org/10.1080/15376494.2012.736053
14. He, X.T., Zheng, Z.L., Sun, J.L., Li, Y.M. and Chen, S.L. (2009), "Convergence analysis of a finite element method based on different moduli in tension and compression", Int. J. Solid. Struct., 46(20), 3734-3740. https://doi.org/10.1016/j.ijsolstr.2009.07.003
15. Jones, R.M. (1971), "Buckling of circular cylindrical shells with different moduli in tension and compression", AIAA J., 9(1), 53-61. https://doi.org/10.2514/3.6124
16. Jones, R.M. (1971), "Buckling of stiffened multilayered circular cylindrical shells with different orthotropic moduli in tension and compression", AIAA J., 9(5), 917-923. https://doi.org/10.2514/3.6296
17. Kamiya, N. (1977), "Thermal stress in a bi-modulus thick cylinder", Nucl. Eng. Des., 40, 383-391. https://doi.org/10.1016/0029-5493(77)90047-4
18. Leal, A.A., Deitzel, J.M. and Gillespie, Jr J.W. (2009), "Compressive strength analysis for high performance fibers with different modulus in tension and compression", J. Compos. Mater., 43(6), 661-674. https://doi.org/10.1177/0021998308088589
19. Li, H., Fok, S.L. and Marsden, B. J. (2008), "An analytical study on the irradiation-induced stresses in nuclear graphite moderator bricks", J. Nucl. Mater., 372(2-3), 164-170. https://doi.org/10.1016/j.jnucmat.2007.03.041
20. Liu, X.B. and Meng, Q.C. (2002), "On the convergence of finite element method with different extension-compression elastic modulus", J. Beijing Univ. Aeronaut. Astronaut., 28(2), 232-234.
21. Liu, X.B.and Zhang, Y.Z. (2000), "Modulus of elasticity in shear and accelerate convergence of different extension-compression elastic modulus finite element method", J. Dalian Univ. Technol., 40(5), 527-530.
22. Medri, G. (1982), "A nonlinear elastic model for isotropic materials with different behavior in tension and compression", J. Eng. Mater. Technol., 16(104), 26-28.
23. Patel, B.P., Khan, K. and Nath, Y. (2014), "A new constitutive model for bimodular laminated structures: Application to free vibrations of conical/cylindrical panels", Compos. Struct., 110, 183-191. https://doi.org/10.1016/j.compstruct.2013.11.008
24. Shapiro, G.S. (1971), "Deformation of bodies with different tensile and compressive strengths (stiffnesses)", Mech. Solid., 22(5), 82-89.
25. Shi, J. and Gao, Z.L. (2015), "Ill-loaded layout optimization of bimodulus material", Finite Elem. Anal. Des., 95, 51-61. https://doi.org/10.1016/j.finel.2014.10.005
26. Shi, J., Cai, K. and Qin, Q.H. (2014), "Topology optimization for human proximal femur considering bi-modulus behavior of cortical bones", MBC: Mol. Cell. Biomech., 11(4), 235-248.
27. Timoshenko, S. (1941), Strength of Materials, Part II: Advanced Theory and Problems, 2nd Edition, Van Nostrand , USA.
28. Tsoukleri, G., Parthenios, J., Papagelis, K., Jalil, R., Ferrari, A.C., Geim, A.K., ... and Galiotis, C. (2009), "Subjecting a graphene monolayer to tension and compression", Small, 5(21), 2397-2402. https://doi.org/10.1002/smll.200900802
29. Vijayakumar, K. and Rao, K P. (1987), "Stress-strain relation for composites with different stiffnesses in tension and compression", Comput. Mech., 2(3), 167-175. https://doi.org/10.1007/BF00571022
30. Wu, X., Yang, L.J., Huang, C. and Sun, J. (2010), "Large deflection bending calculation and analysis of bi-modulus rectangular plate", Eng. Mech., 27(1), 17-22.
31. Yang, H.T. and Zhu, Y.L. (2006), "Solving elasticity problems with bi-modulus via a smoothing technique", Chin. J. Comput. Mech., 23(1), 19-23.
32. Yang, H.T., Wu, R.F., Yang, K.J. and Zhang, Y.Z. (1992), "Solution to problem of dual extension-compression elastic modulus with initial stress method", J. Dalian Univ. Technol., 32(1), 35-39.
33. Yao, W.J. and Ma, J.W. (2013), "Semi-analytical buckling solution and experimental study of variable cross-section rod with different moduli", J. Eng. Mech. Div., ASCE, 139(9), 1149-1157. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000473
34. Yao, W.J. and Ye, Z.M. (2004), "Analytical solution for bending beam subject to lateral force with different modulus", J. Appl. Math. Mech. (Engl. Transl.), 25(10), 1107-1117.
35. Yao, W.J. and Ye, Z.M. (2004), "Analytical solution of bendingcompression column using different tension-compression modulus", J. Appl. Math. Mech. (Engl. Transl.), 25(9), 983-993.
36. Yao, W.J. and Ye, Z.M. (2006), "Internal forces for statically indeterminate structures having different moduli", J. Eng. Mech. Div., ASCE, 132(7), 739-746. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:7(739)
37. Yao, W.J., Ma, J.W., Gao, J.L. and Qiu, Y.Z. (2015), "Nonlinear large deflection buckling analysis of compression rod with different moduli", Struct. Eng. Mech., 54(5), 855-875. https://doi.org/10.12989/sem.2015.54.5.855
38. Yao, W.J., Zhang, C.H. and Jiang, X.F. (2006), "Nonlinear mechanical behavior of combined members with different moduli", Int. J. Nonlin. Sci. Numer. Simul., 7(2), 233-238. https://doi.org/10.1515/IJNSNS.2006.7.2.233
39. Ye, Z.M., Yu, H.G. and Yao, W.J. (2001), "A new elasticity and finite element formulation for different Young's modulus when tension and compression loading", J. Shanghai Univ., 5(2), 89-92. https://doi.org/10.1007/s11741-001-0001-0
40. Ye, Z.M., Yu, H.R. and Yao, W.Y. (2001), "A finite element formulation for different Young's modulus when tension and compression loading", Combinational and Computational Mathematics Center Conference on Computational Mathematics, Pohang University of Science and Technology, South Korea.
41. Zhang, Y.Z. and Wang, Z.F. (1989), "The finite element method for elasticity with different moduli in tension and compression", Chin. J. Comput. Mech., 6(1), 236-246.
42. Zhou, J.G. (1981), "Method of calculating the temperature stresses in composite hollow cylinder", J. Build. Struct., 3, 42-55.
43. Zhuang, Z. (2009), Finite element analysis and application based on ABAQUS, Tsinghua University Press, Beijing, China.