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Mathematical Problems in Engineering
Volume 2015, Article ID 292809, 16 pages
http://dx.doi.org/10.1155/2015/292809
Research Article

Application of Base Force Element Method on Complementary Energy Principle to Rock Mechanics Problems

The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China

Received 24 August 2014; Accepted 5 September 2014

Academic Editor: Song Cen

Copyright © 2015 Yijiang Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. H. H. Pian, “Derivation of element stiffness matrices by assumed stress distributions,” AIAA Journal, vol. 2, no. 7, pp. 1333–1336, 1964. View at Publisher · View at Google Scholar
  2. T. H. Pian and D. P. Chen, “Alternative ways for formulation of hybrid stress elements,” International Journal for Numerical Methods in Engineering, vol. 18, no. 11, pp. 1679–1684, 1982. View at Google Scholar · View at Scopus
  3. T. H. H. Pian and K. Sumihara, “Rational approach for assumed stress finite elements,” International Journal for Numerical Methods in Engineering, vol. 20, no. 9, pp. 1685–1695, 1984. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Zhang, D. Wang, J. Zhang, W. Feng, and Q. Huang, “On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation,” Finite Elements in Analysis and Design, vol. 43, no. 4, pp. 321–332, 2007. View at Publisher · View at Google Scholar · View at Scopus
  5. B. Fraeijs de Veubeke, “Displacement and equilibrium models in the finite element method,” in Stress Analysis, O. C. Zienkiewicz and G. S. Holister, Eds., pp. 145–197, John Wiley & Sons, New York, NY, USA, 1965. View at Google Scholar
  6. B. F. de Veubeke, “A new variational principle for finite elastic displacements,” International Journal of Engineering Science, vol. 10, no. 9, pp. 745–763, 1972. View at Publisher · View at Google Scholar · View at Scopus
  7. R. L. Taylor and O. C. Zienkiewicz, “Complementary energy with penalty functions in finite element analysis,” in Energy Methods in Finite Element Analysis, R. Glowinski, Ed., pp. 153–174, John Wiley & Sons, New York, NY, USA, 1979. View at Google Scholar
  8. S. N. Patniak, “An integrated force method for discrete analysis,” International Journal for Numerical Methods in Engineering, vol. 6, no. 2, pp. 237–251, 1973. View at Google Scholar · View at Scopus
  9. S. N. Patnaik, “The integrated force method versus the standard force method,” Computers and Structures, vol. 22, no. 2, pp. 151–163, 1986. View at Publisher · View at Google Scholar · View at Scopus
  10. S. N. Patnaik, “The variational energy formulation for the integrated force method,” AIAA Journal, vol. 24, no. 1, pp. 129–137, 1986. View at Publisher · View at Google Scholar · View at Scopus
  11. S. N. Patnaik, L. Berke, and R. H. Gallagher, “Integrated force method versus displacement method for finite element analysis,” Computers and Structures, vol. 38, no. 4, pp. 377–407, 1991. View at Publisher · View at Google Scholar · View at Scopus
  12. E. L. Wilson, R. L. Tayler, W. P. Doherty, and J. Ghaboussi, “Incompatible displacement models,” in Numerical and Computational Methods in Structural Mechanics, S. J. Fenves, N. Perrone, A. R. Robinson, and W. C. Schnobrich, Eds., pp. 43–57, Academic Press, New York, NY, USA, 1973. View at Google Scholar
  13. R. L. Taylor, P. J. Beresford, and E. L. Wilson, “A non-conforming element for stress analysis,” International Journal for Numerical Methods in Engineering, vol. 10, no. 6, pp. 1211–1219, 1976. View at Publisher · View at Google Scholar
  14. J. C. Simo and T. J. R. Hughes, “On the variational foundations of assumed strain methods,” Journal of Applied Mechanics, vol. 53, no. 1, pp. 51–54, 1986. View at Publisher · View at Google Scholar · View at Scopus
  15. J. C. Simo and M. S. Rifai, “Class of mixed assumed strain methods and the method of incompatible modes,” International Journal for Numerical Methods in Engineering, vol. 29, no. 8, pp. 1595–1638, 1990. View at Publisher · View at Google Scholar · View at Scopus
  16. R. H. Macneal, “Derivation of element stiffness matrices by assumed strain distributions,” Nuclear Engineering and Design, vol. 70, no. 1, pp. 3–12, 1982. View at Publisher · View at Google Scholar · View at Scopus
  17. T. Belytschko and L. P. Bindeman, “Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems,” Computer Methods in Applied Mechanics and Engineering, vol. 88, no. 3, pp. 311–340, 1991. View at Publisher · View at Google Scholar · View at Scopus
  18. R. Piltner and R. L. Taylor, “A quadrilateral mixed finite element with two enhanced strain modes,” International Journal for Numerical Methods in Engineering, vol. 38, no. 11, pp. 1783–1808, 1995. View at Google Scholar
  19. R. Piltner and R. L. Taylor, “A systematic constructions of B-bar functions for linear and nonlinear mixed-enhanced finite elements for plane elasticity problems,” International Journal for Numerical Methods in Engineering, vol. 44, no. 5, pp. 615–639, 1997. View at Google Scholar
  20. T. J. R. Hughes, “Generalization of selective integration procedures to anisotropic and nonlinear media,” International Journal for Numerical Methods in Engineering, vol. 15, no. 9, pp. 1413–1418, 1980. View at Publisher · View at Google Scholar · View at Scopus
  21. T. Limin, C. Wanji, and L. Yingxi, “Formulation of quasi-conforming element and Hu-Washizu principle,” Computers and Structures, vol. 19, no. 1-2, pp. 247–250, 1984. View at Publisher · View at Google Scholar · View at Scopus
  22. L. Yu-qiu and H. Min-feng, “A generalized conforming isoparametric element,” Applied Mathematics and Mechanics, vol. 9, no. 10, pp. 929–936, 1988. View at Publisher · View at Google Scholar · View at Scopus
  23. G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam, “A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 45–48, pp. 3883–3897, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. J. Chen, C.-J. Li, and W.-J. Chen, “A 17-node quadrilateral spline finite element using the triangular area coordinates,” Applied Mathematics and Mechanics (English Edition), vol. 31, no. 1, pp. 125–134, 2010. View at Publisher · View at Google Scholar · View at Scopus
  25. J. Chen, C.-J. Li, and W.-J. Chen, “A family of spline finite elements,” Computers and Structures, vol. 88, no. 11-12, pp. 718–727, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. S. Rajendran and K. M. Liew, “A novel unsymmetric 8-node plane element immune to mesh distortion under a quadratic displacement field,” International Journal for Numerical Methods in Engineering, vol. 58, no. 11, pp. 1713–1748, 2003. View at Publisher · View at Google Scholar · View at Scopus
  27. S. Rajendran, “A technique to develop mesh-distortion immune finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 17–20, pp. 1044–1063, 2010. View at Publisher · View at Google Scholar · View at Scopus
  28. E. T. Ooi, S. Rajendran, and J. H. Yeo, “A 20-node hexahedron element with enhanced distortion tolerance,” International Journal for Numerical Methods in Engineering, vol. 60, no. 15, pp. 2501–2530, 2004. View at Publisher · View at Google Scholar · View at Scopus
  29. E. T. Ooi, S. Rajendran, and J. H. Yeo, “Remedies to rotational frame dependence and interpolation failure of US-QUAD8 element,” Communications in Numerical Methods in Engineering, vol. 24, no. 11, pp. 1203–1217, 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. Y. Long, L. Juxuan, Z. Long, and C. Song, “Area co-ordinates used in quadrilateral elements,” Communications in Numerical Methods in Engineering, vol. 15, no. 8, pp. 533–543, 1999. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. Q. Long, S. Cen, and Z. F. Long, Advanced Finite Element Method in Structural Engineering, Springer/Tsinghua University Press, Berlin, Germany, 2009.
  32. Z. F. Long, J. X. Li, S. Cen, and Y. Q. Long, “Some basic formulae for area coordinates used in quadrilateral elements,” Communications in Numerical Methods in Engineering, vol. 15, no. 12, pp. 841–852, 1999. View at Publisher · View at Google Scholar
  33. Z.-F. Long, S. Cen, L. Wang, X.-R. Fu, and Y.-Q. Long, “The third form of the quadrilateral area coordinate method (QACM-III): theory, application, and scheme of composite coordinate interpolation,” Finite Elements in Analysis and Design, vol. 46, no. 10, pp. 805–818, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. G. R. Liu, K. Y. Dai, and T. T. Nguyen, “A smoothed finite element method for mechanics problems,” Computational Mechanics, vol. 39, no. 6, pp. 859–877, 2007. View at Publisher · View at Google Scholar · View at Scopus
  35. Y. Peng and Y. Liu, “Base force element method of complementary energy principle for large rotation problems,” Acta Mechanica Sinica, vol. 25, no. 4, pp. 507–515, 2009. View at Publisher · View at Google Scholar · View at Scopus
  36. Y. Liu and Y. Peng, “Base force element method (BFEM) on complementary energy principle for linear elasticity problem,” Science China: Physics, Mechanics and Astronomy, vol. 54, no. 11, pp. 2025–2032, 2011. View at Publisher · View at Google Scholar · View at Scopus
  37. Y. Peng, Z. Dong, B. Peng, and Y. Liu, “Base force element method (BFEM) on potential energy principle for elasticity problems,” International Journal of Mechanics and Materials in Design, vol. 7, no. 3, pp. 245–251, 2011. View at Publisher · View at Google Scholar · View at Scopus
  38. Y. Peng, Z. Dong, B. Peng, and N. Zong, “The application of 2D base force element method (BFEM) to geometrically non-linear analysis,” International Journal of Non-Linear Mechanics, vol. 47, no. 3, pp. 153–161, 2012. View at Publisher · View at Google Scholar · View at Scopus
  39. Y.-J. Peng, J.-W. Pu, B. Peng, and L.-J. Zhang, “Two-dimensional model of base force element method (BFEM) on complementary energy principle for geometrically nonlinear problems,” Finite Elements in Analysis and Design, vol. 75, pp. 78–84, 2013. View at Publisher · View at Google Scholar · View at Scopus
  40. Y. J. Peng, N. N. Zong, L. J. Zhang, and J. W. Pu, “Application of 2D base force element method with complementary energy principle for arbitrary meshes,” Engineering Computations, vol. 31, no. 4, pp. 1–15, 2014. View at Google Scholar
  41. Y. Peng, L. Zhang, J. Pu, and Q. Guo, “A two-dimensional base force element method using concave polygonal mesh,” Engineering Analysis with Boundary Elements, vol. 42, pp. 45–50, 2014. View at Publisher · View at Google Scholar · View at Scopus
  42. Y. Peng, Y. Liu, J. Pu, and L. Zhang, “Application of base force element method to mesomechanics analysis for recycled aggregate concrete,” Mathematical Problems in Engineering, vol. 2013, Article ID 292801, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  43. Y. Liu, Y. Peng, L. Zhang, and Q. Guo, “A 4-mid-node plane model of base force element method on complementary energy principle,” Mathematical Problems in Engineering, vol. 2013, Article ID 706759, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  44. C. Y. Dong and G. L. Zhang, “Boundary element analysis of three dimensional nanoscale inhomogeneities,” International Journal of Solids and Structures, vol. 50, no. 1, pp. 201–208, 2013. View at Publisher · View at Google Scholar · View at Scopus
  45. C. Y. Dong and E. Pan, “Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects,” Engineering Analysis with Boundary Elements, vol. 35, no. 8, pp. 996–1002, 2011. View at Publisher · View at Google Scholar · View at Scopus
  46. S. S. Chen, Q. H. Li, Y. H. Liu, and Z. Q. Xue, “A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures,” Engineering Analysis with Boundary Elements, vol. 37, no. 2, pp. 273–279, 2013. View at Publisher · View at Google Scholar · View at Scopus
  47. S. Chen, Y. Liu, J. Li, and Z. Cen, “Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures,” European Journal of Mechanics, A/Solids, vol. 30, no. 2, pp. 183–194, 2011. View at Publisher · View at Google Scholar · View at Scopus
  48. S. Cen, X.-R. Fu, and M.-J. Zhou, “8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 29–32, pp. 2321–2336, 2011. View at Publisher · View at Google Scholar · View at Scopus
  49. S. Cen, G.-H. Zhou, and X.-R. Fu, “A shape-free 8-node plane element unsymmetric analytical trial function method,” International Journal for Numerical Methods in Engineering, vol. 91, no. 2, pp. 158–185, 2012. View at Publisher · View at Google Scholar · View at Scopus
  50. H. A. F. A. Santos, “Complementary-energy methods for geometrically non-linear structural models: an overview and recent developments in the analysis of frames,” Archives of Computational Methods in Engineering, vol. 18, no. 4, pp. 405–440, 2011. View at Publisher · View at Google Scholar · View at Scopus
  51. H. A. F. A. Santos and C. I. Almeida Paulo, “On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables,” International Journal of Non-Linear Mechanics, vol. 46, no. 2, pp. 395–406, 2011. View at Publisher · View at Google Scholar · View at Scopus
  52. Y. C. Gao, “A new description of the stress state at a point with applications,” Archive of Applied Mechanics, vol. 73, no. 3-4, pp. 171–183, 2003. View at Publisher · View at Google Scholar · View at Scopus
  53. Y. C. Gao, “Asymptotic analysis of the nonlinear Boussinesq problem for a kind of incompressible rubber material (compression case),” Journal of Elasticity, vol. 64, no. 2-3, pp. 111–130, 2001. View at Publisher · View at Google Scholar · View at Scopus
  54. Y. C. Gao and T. J. Gao, “Large deformation contact of a rubber notch with a rigid wedge,” International Journal of Solids and Structures, vol. 37, no. 32, pp. 4319–4334, 2000. View at Publisher · View at Google Scholar · View at Scopus
  55. Y. C. Gao and S. H. Chen, “Analysis of a rubber cone tensioned by a concentrated force,” Mechanics Research Communications, vol. 28, no. 1, pp. 49–54, 2001. View at Publisher · View at Google Scholar · View at Scopus
  56. Y.-C. Gao, M. Jin, and G.-S. Dui, “Stresses, singularities, and a complementary energy principle for large strain elasticity,” Applied Mechanics Reviews, vol. 61, no. 3, Article ID 030801, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus