Abstract:In order to describe the dynamic strain of rock better, lateral-axial strain fractional dashpot and viscoelastic lateral-axial strain combined models are put forward on the basis of viscoelastic stress-strain combined model theory. The fractional lateral-axial strain relationship and the fractional volume strain model are proposed based on the fractional order calculus, and the corresponding equations under dynamic loading and cyclic loading are derived. The analytic solution for the model of lateral-axial strain relationship is given theoretically. Moreover, a multi-functional material testing set-up is employed to measure the lateral-axial strain of a variety of rock, concrete and gypsum samples under the dynamic loading and cyclic loading. The parameters of the lateral-axial strain model are determined by fitting to the experimental results of lateral-axial strain of rock. In addition, a sensitivity study for the analytic solution of the lateral-axial strain model is carried out, showing the effects of confining pressure, strain level, fractional derivative order and model coefficient on axial strain of rock samples. Furthermore, it’s found that the new model can describe the volume strain phenomenon of negative and positive dilatancy and the variation law of deformation under dynamic loading and cyclic loading in rock.
何明明, 李宁, 陈蕴生, 朱才辉. 基于分数阶微积分岩石的动态变形行为研究[J]. 岩土工程学报, 2015, 37(zk1): 178-184.
HE Ming-ming, LI Ning, CHEN Yun-sheng, ZHU Cai-hui. Dynamic deformation behavior of rock based on fractional order calculus. Chinese J. Geot. Eng., 2015, 37(zk1): 178-184.
[1] WANG Qian, JI Shao-cheng, SUN Sheng-si, et al. Correlations between compressional and shear wave velocities and corresponding Poisson's ratios for some common rocks and sulfide ores[J]. Tectonophysics, 2009, 469(1): 61-72. [2] GERCEK H. Poisson’s ratio values for rocks[J]. International Journal of Rock Mechanics & Mining Sciences, 2007, 44(1): 1-13. [3] CAO Chen, REN Ting, COOK C. Calculation of the effect of Poisson's ratio in laboratory push and pull testing of resin-encapsulated bolts[J]. International Journal of Rock Mechanics & Mining Sciences, 2013, 64: 175-180. [4] TORVIK P J, BAGLEY R L. On the appearance of the fractional derivative in the behavior of real materials[J]. ASME J Appl Mech, 1984, 51(2): 294-298. [5] JESUS I S, MACHADO J A T. Implementation of fractional-order electromagnetic potential through a genetic algorithm[J]. Commun Nonlinear Sci Numer Simul, 2009, 14(5): 1838-1843. [6] SCHMIDT A, GAUL L. On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems[J]. Signal Proce, 2006, 86(10): 2592-2601. [7] ROSSIKHIN Y A, SHITIKOVA M V. A new method for solving dynamic problems of fractional derivative viscoelasticity[J]. Inte J Eng Sci, 2001, 39(2): 149-176. [8] PARK S W. Analytical modeling of viscoelastic dampers for structural and vibration control[J]. International Journal of Solids Structure, 2001, 38(44/45): 8065-8092. [9] YANG D, ZHU K. Start-up flow of a viscoelastic fluid in a pipe with a fractional Maxwell’s model[J]. Computers and Mathematics with Application, 2010, 60(8): 2231-2238. [10] WENCHANG T, WENXIAO P, MINGYU X. A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates[J]. International Journal of Non-Linear Mechanics, 2003, 38(5): 645-650. [11] 周宏伟, 王春萍, 段志强, 等. 基于分数阶导数的盐岩流变本构模型[J]. 中国科学 (物理学 力学 天文学), 2012, 42(9): 310-318. (ZHOU Hong-wei, WANG Chun-ping, DUAN Zhi-qiang, et al. Time-based fractional derivative approach to creep constitutive model of salt rock[J]. Scientia Sinica (Physica, Mechanica & Astronomica), 2012, 42(9): 310-318. (in Chinese)) [12] ZHOU H W, WANG C P, HAN B B, et al. A creep constitutive model for salt rock based on fractional derivatives[J]. International Journal of Rock Mechanics & Mining Sciences, 2011, 48(1): 116-121. [13] 殷德顺, 和成亮, 陈 文. 岩土应变硬化指数理论及其分数阶微积分理论基础[J]. 岩土工程学报, 2010, 32(5): 762-766. (YIN De-shun, HE Cheng-liang, CHEN Wen, et al. Theory of geotechnical strain hardening index and its rationale from fractional order calculus[J]. Chinese Journal of Geotechnical Engineering, 2010, 32(5): 762-766. (in Chinese)) [14] 殷德顺, 任俊娟, 和成亮, 等. 一种新的岩土流变模型元件[J]. 岩石力学与工程学报, 2007, 26(9): 1899-1903. (YIN De-shun, REN Jun-juan, HE Cheng-liang, et al. A new rheological model element for geomaterials[J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(9): 1899-1903. (in Chinese)) [15] 吴 斐, 谢和平, 刘建锋, 等. 分数阶黏弹塑性蠕变模型试验研究[J]. 岩石力学与工程学报, 2014, 33(5): 964-970. (WU Fei, XIE He-ping, LIU Jian-feng, et al. Experimental study of fractional viscoelastic-plastic creep model[J]. Chinese Journal of Rock Mechanics and Engineering, 2014, 33(5): 964-970. (in Chinese)) [16] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier, 2006. [17] SMIT W, DE VRIES H. Rheological models containing fractional derivat ives[J]. Rheologica Acta, 1970, 9(4): 525-534. [18] WONG M, PONTICIELLO M, KOVANEN V, et al. Volumetric changes of articular cartilage during stress relaxation in unconfined compression[J]. J Biomech, 2000, 33(9): 1049-1054. [19] 段晓梦, 殷德顺, 安丽媛, 等. 基于分数阶微积分的黏弹性材料变形研[J]. 中国科学 (物理学 力学 天文学), 2013, 43(8): 971-977. (DUAN Xiao-meng, YIN De-shun, AN Li-yuan, et al. The deformation study in viscoelastic materials based on fractional order calculus[J]. Scientia Sinica (Physica, Mechanica & Astronomica), 2013, 43(8): 971-977. (in Chinese))