Acta Mechanica Solida Sinica

, Volume 22, Issue 2, pp 95–108 | Cite as

Analysis of shakedown of FG Bree plate subjected to coupled thermal-mechanical loadings

  • Xianghe Peng
  • Ning Hu
  • Hengwei Zheng
  • Cuirong Fang


The static and kinematic shakedown of a functionally graded (FG) Bree plate is analyzed. The plate is subjected to coupled constant mechanical load and cyclically varying temperature. The material is assumed linearly elastic and nonlinear isotropic hardening with elastic modulus, yield strength and the thermal expansion coefficient varying exponentially through the thickness of the plate. The boundaries between the shakedown area and the areas of elasticity, incremental collapse and reversed plasticity are determined, respectively. The shakedown of the counterpart made of homogeneous material with average material properties is also analyzed. The comparison between the results obtained in the two cases exhibits distinct qualitative and quantitative difference, indicating the importance of shakedown analysis for FG structures. Since FG structures are usually used in the cases where severe coupled cyclic thermal and mechanical loadings are applied, the approach developed and the results obtained are significant for the analysis and design of such kind of structures.

Key words

functionally graded material the Bree plate coupled thermal-mechanical loading shakedown 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ilschner, B., Processing-microstructure-property relationships in graded materials. Journal of Mechanics and Physics of Solids, 1996, 44(5): 647–56.CrossRefGoogle Scholar
  2. [2]
    Lee, Y.D. and Erdogan, F., Residual/thermal stress in FGM and laminated thermal barrier coating. International Journal of Fracture, 1994, 69: 145–165.CrossRefGoogle Scholar
  3. [3]
    Suresh, S. and Mortensen, A., Functionally graded metals and metal-ceramic composites — Part 2: thermomechanical behavior. International Materials Reviews, 1997, 29: 306–312.Google Scholar
  4. [4]
    Wataria, F., Yokoyama, A., Omori, M., Hirai, T., Kondo, H., Uo, M. and Kawasaki, T., Biocompatibility of materials and development to functionally graded implant for bio-medical application. Composites Science and Technology, 2004, 64: 893–908.CrossRefGoogle Scholar
  5. [5]
    Fujihara, K., Teo, K., Gopal, R., Loh, P.L., Ganesh, V.K., Ramakrishna, S., Foong, K.W.C. and Chew, C.L., Fibrous composite materials in dentistry and orthopedics: review and applications. Composites Science and Technology, 2004, 64: 775–788.CrossRefGoogle Scholar
  6. [6]
    Praveen, G.N., Chin, C.D. and Reddy, J.N., Thermoelastic analysis of functionally graded ceramic-metal cylinder. Journal of Engineering Mechanics, 1999, 125: 1259–1267.CrossRefGoogle Scholar
  7. [7]
    Shabana, Y.M. and Noda, N., Numerical evaluation of the thermo mechanical effective properties of a functionally graded material using the homogenization method. International Journal of Solids Structures, 2008, 45: 3494–3506.CrossRefGoogle Scholar
  8. [8]
    Tanigawa, Y., Some basic thermoelastic problems for nonhomogeneous structural materials. Applied Mechanics Reviews, 1995, 48(6): 287–300.CrossRefGoogle Scholar
  9. [9]
    Kim, K.S. and Noda, N., Green’s function approach to solution of transient temperature for thermal stresses of functionally graded material. International Journal Series A — Solids Mechanics and Material Engineering, 2001, 44(1): 31–36.CrossRefGoogle Scholar
  10. [10]
    Loy, C.T., Lam, K.Y. and Reddy, J.N., Vibration of functionally graded cylindrical shells. International Journal of Mechanical Sciences, 1999, 41(3): 309–24.CrossRefGoogle Scholar
  11. [11]
    Yang, J., and Shen, H.S., Dynamic response of initially stressed functionally graded rectangular thin plates. Composite Structures, 2001, 54(4): 497–508.CrossRefGoogle Scholar
  12. [12]
    Guo, L., Wu, L., Sun, Y. and Ma, L., The transient fracture behavior for a functionally graded layered structure subjected to an in-plane impact load. Acta Mechanica Sinica, 2005, 21(3): 257–266.CrossRefGoogle Scholar
  13. [13]
    Liang, J., Wu, S.P. and Du, S.Y., The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves. Acta Mechanica Sinica, 2007, 23(4): 427–435.CrossRefGoogle Scholar
  14. [14]
    Matbuly, M.S., Analysis of mode III crack perpendicular to the interface between two dissimilar strips. Acta Mechanica Sinica, 2008, 24(4): 433–438.CrossRefGoogle Scholar
  15. [15]
    Noda, N., Thermal stresses in functionally graded materials. Journal of Thermal Stresses, 1999, 22: 477–512.CrossRefGoogle Scholar
  16. [16]
    Ootao, Y. and Tanigawa, Y., Three-dimensional solution for transient thermal stresses of an orthotropic functionally graded rectangular plate. Composite Structures, 2007, 80(1): 10–20.CrossRefGoogle Scholar
  17. [17]
    Marin, L. and Lesnic, D., The method of fundamental solutions for nonlinear functionally graded materials. International Journal of Solids Structures, 2007, 44: 6878–6890.CrossRefGoogle Scholar
  18. [18]
    Elishakoff, I. and Gentilini, C., Three-dimensional flexure of rectangular plates made of functionally graded materials. Journal of Applied Mechanics, 2005, 72(5): 788–91.CrossRefGoogle Scholar
  19. [19]
    Kapuria, S., Bhattacharyya, M. and Kumar, A.N., Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation, Composite Structures, 2008, 82: 390–402.CrossRefGoogle Scholar
  20. [20]
    Zhong, Z. and Shang, E.T., Three-dimensional exact analysis of simply supported functionally gradient piezoelectric plates. International Journal of Solids Structures, 2003, 40(20): 5335–5352.CrossRefGoogle Scholar
  21. [21]
    Ying, J., Lu, C.F. and Chen, W.Q., Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 2008, 84: 209–219.CrossRefGoogle Scholar
  22. [22]
    Chen, W.Q. and Ding, H.J., Bending of functionally graded piezoelectric rectangular plates. Acta Mechanica Solida Sinica, 2000, 13(4): 312–319.Google Scholar
  23. [23]
    Dai, H.L., Fu, Y.M. and Yang, J.H., Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere. Acta Mechanica Sinica, 2007, 23(1): 55–63.CrossRefGoogle Scholar
  24. [24]
    Huang, H. and Han, Q., Buckling of imperfect functionally graded cylindrical shells under axial compression. European Journal of Mechanics A/Solids, 2008, 27(6): 1026–1036.CrossRefGoogle Scholar
  25. [25]
    Eslami, M.R., Babaei, M.H. and Poultangari, R., Thermal and mechanical stresses in a functionally graded thick sphere. International Journal of Pressure Vessels and Piping, 2005, 82: 522–527.CrossRefGoogle Scholar
  26. [26]
    Jabbari, M., Sohrabpour, S. and Eslami, M.R., General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads, Journal of Applied Mechanics, 2003, 70: 111–118.CrossRefGoogle Scholar
  27. [27]
    Yang, J., Liew, K.M., Wu, Y.F. and Kitipomchai, S., Thermo-mechanical post-buckling of FGM cylindrical panels with temperature dependent properties. International Materials Reviews of Solids Structures, 2006, 43(2): 307–324.CrossRefGoogle Scholar
  28. [28]
    Na, K.S. and Kim, J.H., Three-dimensional thermomechanical buckling analysis for functionally graded composite plates, Composite Structures, 2006, 73(4): 413–422.CrossRefGoogle Scholar
  29. [29]
    Feldoman, E. and Aboudi, J., Buckling analysis of functionally graded plates subjected to uniaxial loading. Composite Structures, 1997, 38: 29–36.CrossRefGoogle Scholar
  30. [30]
    You, L.H., Ou, H. and Zheng, Z.Y., Creep deformations and stresses in thick-walled cylindrical vessels of functionally graded materials subjected to internal pressure. Composite Structures, 2007, 78: 285–291.CrossRefGoogle Scholar
  31. [31]
    Fischer-Cripps, A.C., Analysis of instrumented indentation test data for functionally graded Materials. Surface and Coatings Technology, 2003, 168: 136–141.CrossRefGoogle Scholar
  32. [32]
    Gu, Y., Nakamura, T., Prchlik, L., Sampath, S. and Wallace, J., Micro-indentation and inverse analysis to characterize elastic-plastic graded materials, Materials Science and Engineering A, 2003, 345: 223–233.CrossRefGoogle Scholar
  33. [33]
    Peng, X., Fan, J. and Zeng, X., Analysis for Plastic buckling of thin-walled cylinders via non-classical constitutive theory of plasticity. International Journal of Solids Structures, 1996, 33(30): 4495–4509.CrossRefGoogle Scholar
  34. [34]
    Peng, X., and Ponter, A.R.S., Extremal properties of Endochronic plasticity — Part II: Extremal path of the Endochronic constitutive equation with a yield surface and application. International Journal of Plasticity, 1993, 9: 567–581.CrossRefGoogle Scholar
  35. [35]
    König, J.A., Shakedown of Elastic-Plastic Structures. PWN-Polish Scientific Publishers, 1987.Google Scholar
  36. [36]
    Bree, J., Elastic plastic behavior of thin tubes subjected to internal pressure and intermittent high heat fluxes with application to fast nuclear reactor fuel elements. Journal of Strain Analysis, 1967, 2: 226–238.CrossRefGoogle Scholar
  37. [37]
    Sankar, B.V., An elasticity solution for functionally graded beams. Composites Science and Technology, 2001, 61(5): 689–696.CrossRefGoogle Scholar
  38. [38]
    Ding, X.Y., Li, H.J. and Chen, W.Q., Three-dimensional analytical solution for functionally graded magneto-electro-elastic circular plates subjected to uniform load. Composite Structures, 2008, 83(4): 381–390.CrossRefGoogle Scholar
  39. [39]
    Nie, G.J. and Zhong, Z., Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Composite Structures, 2008, 84(2): 167–176.CrossRefGoogle Scholar
  40. [40]
    Choi, H.J. and Paulino, G.H., Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation. Journal of the Mechanics and Physics of Solids, 2008, 56(4): 1673–1692.MathSciNetCrossRefGoogle Scholar
  41. [41]
    He, X., Wang, J.S. and Qin, Q.H., Saint-Venant decay analysis of FGPM laminates and dissimilar piezoelectric laminates. Mechanics of Materials, 2007, 39(12): 1053–1065.CrossRefGoogle Scholar
  42. [42]
    Shao, Z.S. and Ma, G.W., Thermo-mechanical stresses in functionally graded circular hollow cylinder with linearly increasing boundary temperature. Composite Structures, 2008, 83(3): 259–265.CrossRefGoogle Scholar
  43. [43]
    Guler, M.A. and Erdogan, F., Contact mechanics of two deformable elastic solids with graded coatings. Mechanics of Materials, 2006, 38(7): 633–647.CrossRefGoogle Scholar
  44. [44]
    Pan, E. and Han, F., Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces. International Journal of Solids Structures, 2005, 42(11–12): 3207–3233.CrossRefGoogle Scholar
  45. [45]
    Guo, L.C., Noda, N. and Wu, L., Thermal fracture model for a functionally graded plate with a crack normal to the surfaces and arbitrary thermomechanical properties. Composites Science and Technology, 2008, 68(3–4): 1034–1041.CrossRefGoogle Scholar
  46. [46]
    Guo, L.C. and Noda, N., Modeling method for a crack problem of functionally graded materials with arbitrary properties-piecewise-exponential model. International Journal of Solids Structures, 2007, 44, (21): 6768–6790.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2009

Authors and Affiliations

  • Xianghe Peng
    • 1
  • Ning Hu
    • 1
    • 2
  • Hengwei Zheng
    • 1
  • Cuirong Fang
    • 1
  1. 1.Department of Engineering MechanicsChongqing UniversityChongqingChina
  2. 2.Department of Mechanical EngineeringChiba UniversityChibaJapan

Personalised recommendations