小梁骨具有不規則蜂窩微結構，在單軸拉伸和壓縮下表現出不對稱降伏。在有限元建模中，我們採用了沃羅諾伊蜂窩作為小梁骨的微結構，並使用了具有正交塑流模式的廣義 Hill 降伏函數；此外，有限元分析中的網格尺寸、單元數量和代表性區塊的大小是根據網格收斂性分析、邊界效應和尺寸效應進行確定的。為了研究小梁骨的初始和接續降伏面，我們設計了探測路徑以檢測降伏點，並設計了預加載路徑以研究加載路徑的影響。有限元模擬的降伏面檢測顯示出小梁骨的等向、走動、旋轉或扭曲硬化/軟化可以被識別。在分段線性預加載下觀察到降伏面的顯著扭曲。研究了加載路徑對降伏面縱橫比的影響，並顯示了包辛格效應的出現和消失。

Trabecular bone has non-periodic cellular microstructure and behaves the asymmetry yielding in uniaxial tension and compression. In our study, we explored the yield surface of trabecular bone in the two-dimensional stress space by using finite element simulations. In finite element modeling, we adopted the Voronoi honeycomb to be the microstructure of the trabecular bone and used the generalized Hill yield function with associated flow rule model. Furthermore, the mesh size, the cell number and the size of the representative block in finite element analysis were determined according to the analysis of the mesh convergence, the boundary effect, and the size effect. In order to investigate the two-dimensional initial and subsequent yield surface of trabecular bone, we designed probing paths to detect yield points and pre-loading paths to investigate the influence of loading paths. Finite element simulations of yield surface detection show the isotropic, kinematic, rotation or distortional hardening/softening of the trabecular
bone can be recognized. The significant distortion of the yield surfaces under piecewise-linear pre-loadings is observed. The influence of loading paths on the aspect ratio of yield surface was investigated and the appearance/disappearance of Bauschinger effect was demonstrated.

[1] M. Silva and L. Gibson, “Modeling the mechanical behavior of vertebral trabecular bone: Effects of age-related changes in microstructure,” Bone, vol. 21, no. 2, pp. 191–199, 1997.

[2] E. F. Morgan and T. M. Keaveny, “Dependence of yield strain of human trabecular
bone on anatomic site,” Journal of Biomechanics, vol. 34, no. 5, pp. 569–577, 2001.

[3] J. Wang, B. Zhou, X. S. Liu, A. J. Fields, A. Sanyal, X. Shi, M. Adams, T. M. Keaveny, and X. E. Guo, “Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone,” Bone, vol. 72, pp. 71–80, 2015.

[4] S. Raehtz, B. M. Hargis, V. A. Kuttappan, R. Pamukcu, L. R. Bielke, and L. R. McCabe;, “High molecular weight polymer promotes bone health and prevents
bone loss under salmonella challenge in broiler chickens,” Frontiers in Physiology,
vol. 9, no. 384, 2018.

[5] E. B. Long, M. M. Barak, and V. J. Frost, “The effect of staphylococcus aureus
exposure on white-tailed deer trabecular bone stiffness and yield,” Journal of the
Mechanical Behavior of Biomedical Materials, vol. 126, p. 105000, 2022.

[6] R. Belda, M. Palomar, J. L. Peris-Serra, A. Vercher-Martínez, and E. Giner, “Compression failure characterization of cancellous bone combining experimental testing, digital image correlation and finite element modeling,” International Journal of Mechanical Sciences, vol. 165, p. 105213, 2020.

[7] G. Schaffner, X.-D. E. Guo, M. J. Silva, and L. J. Gibson, “Modelling fatigue damage accumulation in two-dimensional voronoi honeycombs,” International Journal of Mechanical Sciences, vol. 42, no. 4, pp. 645–656, 2000.

[8] M. A. Hammond, J. M. Wallace, M. R. Allen, and T. Siegmund, “Mechanics
of linear microcracking in trabecular bone,” Journal of Biomechanics, vol. 83,
pp. 34–42, 2019.

[9] T. M. Keaveny, X. Guo, E. F. Wachtel, T. A. McMahon, and W. C. Hayes,
“Trabecular bone exhibits fully linear elastic behavior and yields at low strains,”
Journal of Biomechanics, vol. 27, no. 9, pp. 1127–1136, 1994.

[10] L. Gibson, M. Ashby, G. Schajer, and C. I. Robertson, “The mechanics of two-
dimensional cellular materials,” Proceedings of the Royal Society of London. A.
Mathematical and Physical Sciences, vol. 382, no. 1782, pp. 25–42, 1982.

[11] M. J. Silva, W. C. Hayes, and L. J. Gibson, “The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids,” International Journal of Mechanical Sciences, vol. 37, no. 11, pp. 1161–1177, 1995.

[12] I. Masters and K. Evans, “Models for the elastic deformation of honeycombs,” Composite Structures, vol. 35, no. 4, pp. 403–422, 1996.

[13] D.-A. Wang and J. Pan, “A non-quadratic yield function for polymeric foams,”
International Journal of Plasticity, vol. 22, no. 3, pp. 434–458, 2006.

[14] S. Malek and L. Gibson, “Effective elastic properties of periodic hexagonal honeycombs,” Mechanics of Materials, vol. 91, pp. 226–240, 2015.

[15] M. Alkhader and M. Vural, “A plasticity model for pressure-dependent anisotropic cellular solids,” International Journal of Plasticity, vol. 26, no. 11, pp. 1591–1605, 2010.

[16] X. Li, Z. Lu, Z. Yang, Q. Wang, and Y. Zhang, “Yield surfaces of periodic honeycombs with tunable poisson’s ratio,” International Journal of Mechanical Sciences, vol. 141, pp. 290–302, 2018.

[17] Y. Chiang, C.-C. Tung, X.-D. Lin, P.-Y. Chen, C.-S. Chen, and S.-W. Chang,
“Geometrically toughening mechanism of cellular composites inspired by fibonacci lattice in liquidambar formosana,” Composite Structures, vol. 262, p. 113349, 2021.

[18] M. Alkhader and M. Vural, “Mechanical response of cellular solids: Role of cellular topology and microstructural irregularity,” International Journal of Engineering Science, vol. 46, no. 10, pp. 1035–1051, 2008.

[19] C. Tekog ̃lu, L. Gibson, T. Pardoen, and P. Onck, “Size effects in foams: Exper-
iments and modeling,” Progress in Materials Science, vol. 56, no. 2, pp. 109–138,
2011.

[20] L.-W. Liu, C.-Y. Yang, and H.-G. Chen, “Finite element analysis on yield surface
evolution of cellular materials,” International Journal of Mechanical Sciences,
vol. 246, p. 108123, 2023.

[21] X.-D. E. Guo, T. A. McMahon, T. M. Keaveny, W. C. Hayes, and L. J. Gibson,
“Finite element modeling of damage accumulation in trabecular bone under cyclic loading,” Journal of Biomechanics, vol. 27, no. 2, pp. 145–155, 1994.

[22] A. S. Khan, R. Kazmi, A. Pandey, and T. Stoughton, “Evolution of subsequent
yield surfaces and elastic constants with finite plastic deformation. part-i: A very
low work hardening aluminum alloy (al6061-t6511),” International Journal of
Plasticity, vol. 25, no. 9, pp. 1611–1625, 2009. Exploring New Horizons of Metal
Forming Research.

[23] A. S. Khan, A. Pandey, and T. Stoughton, “Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. part ii: A very high work hardening aluminum alloy (annealed 1100 al),” International Journal of Plasticity, vol. 26, no. 10, pp. 1421–1431, 2010.

[24] A. S. Khan, A. Pandey, and T. Stoughton, “Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. part iii: Yield surface in tension–tension stress space (al 6061–t 6511 and annealed 1100 al),” International Journal of Plasticity, vol. 26, no. 10, pp. 1432–1441, 2010.

[25] J. Shi, B. Meng, C. Cheng, and M. Wan, “Size effect on the subsequent yield and hardening behavior of metal foil,” International Journal of Mechanical Sciences, vol. 180, p. 105686, 2020.

[26] C. M. A. Iftikhar, Y. L. Li, C. P. Kohar, K. Inal, and A. S. Khan, “Evolution
of subsequent yield surfaces with plastic deformation along proportional and non-
proportional loading paths on annealed aa6061 alloy: Experiments and crystal plasticity finite element modeling,” International Journal of Plasticity, vol. 143,
p. 102956, 2021.

[27] M. J. Michno and W. N. Findley, “An historical perspective of yield surface in-
vestigations for metals,” International Journal of Non-Linear Mechanics, vol. 11,
no. 1, pp. 59–82, 1976.
[28] J. Klintworth and W. Stronge, “Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs,” International Journal of Mechanical Sciences, vol. 30, no. 3, pp. 273–292, 1988.

[29] L. Gibson, M. Ashby, J. Zhang, and T. Triantafillou, “Failure surfaces for cellular
materials under multiaxial loads—i.modelling,” International Journal of Mechanical Sciences, vol. 31, no. 9, pp. 635–663, 1989.

[30] C. Chen, T. Lu, and N. Fleck, “Effect of imperfections on the yielding of two-
dimensional foams,” Journal of the Mechanics and Physics of Solids, vol. 47,
no. 11, pp. 2235–2272, 1999.

[31] N. Soltanihafshejani, T. Bitter, D. Janssen, and N. Verdonschot, “Development
of a crushable foam model for human trabecular bone,” Medical Engineering and
Physics, vol. 96, pp. 53–63, 2021.

[32] H. H. Bayraktar and T. M. Keaveny, “Mechanisms of uniformity of yield strains
for trabecular bone,” Journal of Biomechanics, vol. 37, no. 11, pp. 1671–1678,
2004.

[33] T. M. Keaveny, E. F. Wachtel, C. M. Ford, and W. C. Hayes, “Differences between
the tensile and compressive strengths of bovine tibial trabecular bone depend on
modulus,” Journal of Biomechanics, vol. 27, no. 9, pp. 1137–1146, 1994.

[34] E. F. Morgan, H. H. Bayraktar, O. C. Yeh, S. Majumdar, A. Burghardt, and T. M.
Keaveny, “Contribution of inter-site variations in architecture to trabecular bone
apparent yield strains,” Journal of Biomechanics, vol. 37, no. 9, pp. 1413–1420,
2004.

[35] A. Sanyal, A. Gupta, H. H. Bayraktar, R. Y. Kwon, and T. M. Keaveny, “Shear
strength behavior of human trabecular bone,” Journal of Biomechanics, vol. 45,
no. 15, pp. 2513–2519, 2012.