STUDY OF WAVE DISPERSION AND PROPAGATION IN PERIDYNAMICS
-
摘要: 近场动力学是一类基于非局部思想的新固体力学方法, 其采用积分形式的控制方程, 自然地适用于极端载荷下材料破碎和裂纹发展的模拟, 被广泛用于国防安全等领域的研究. 但是, 非局部性会引入色散效应, 对波的传播产生不利影响, 制约其对断裂等固体行为的捕捉能力. 为此, 采用谱分析方法, 对近场动力学系统的色散行为进行了全面的研究. 发现相比于低频成分, 高频成分的色散关系呈现出振荡趋势和零能模式, 色散问题更为严重. 高频域的色散行为还随波的传播方向发生改变, 呈现出沿45°的对称性. 而近场动力学系统本身缺乏数值耗散, 无法抑制色散问题带来的不利影响. 因此, 从引入数值耗散的角度出发, 在合理保留传统近场动力学理论框架的基础上, 建立了黏性引入的控制方程. 并考虑固体中常见的体积变形和对高频成分的选择性抑制, 构造了相应的黏性力态. 最后, 在数值研究中模拟了极端载荷下激波的产生, 以探究波的间断性对色散行为的影响. 发现间断性强的波表现出更为显著的色散行为, 呈现出Gibbs不稳定性. 这些均能有效地被黏性力态所抑制, 验证了所提方法的正确性. 这为在近场动力学系统中实现对波传播过程的正确捕捉, 获得正确的固体行为提供了重要参考, 从而为国防安全领域研究提供了技术支撑和借鉴.Abstract: Periydnamics (PD) is a new nonlocal method reformulated from solid mechanics. It adopts the integral form of governing equation and is naturally suitable to model fragments and cracks under extreme events, thus widely applied in the field of national defense security. However, the nonlocality in PD introduces the dispersion effect and imposes adverse effect on wave propagation, which will greatly restrict its capability in capturing solid behaviors, especially the fractures. For this purpose, we employ the spectral analysis method to study the dispersion behavior of PD system comprehensively. It is found that compared to the low frequencies, the dispersion relation of high frequency components shows an oscillation trend and zero-energy modes, leading to more serious dispersion problems. The dispersion behavior of high frequencies changes with the wave propagation direction and shows 45° symmetry in the spatial wave propagation. As the PD system itself is non-dissipative, the adverse effect of the dispersion problem can not be suppressed. As a result, the simulation accuracy may be greatly influenced. To introduce the numerical dissipation for dispersion effect suppression, the governing equation of viscosity introduction is proposed as a minimum variation of conventional PD. Both the typical deformation in solids and the selective suppression on high frequencies are considered then the corresponding viscous force state is constructed. Finally, a numerical study is conducted to model the shock waves under extreme events and investigate the influence of wave discontinuity. It is indicated that the wave discontinuity aggravates the dispersion problem and shows Gibbs instability in the wave propagation. These can be effectively suppressed by the viscous force state, which verifies the proposed method. This provides an important reference to reproduce the correct wave propagation process and obtain the reasonable solid behavior in PD, thus helps to support and guide the research of national defense security field.
-
Key words:
- peridynamics /
- wave dispersion /
- high frequency /
- wave discontinuity /
- viscosity
-
图 9 带理论解的一维正弦波传播结果对比
Figure 9. Comparative study of 1D sine wave propagation problem with theoretical result
图 10 带理论解的一维方波传播结果对比
Figure 10. Comparative study of 1D square wave propagation problem with theoretical result
图 11 一维波传播算例示意图 (单位: mm)
Figure 11. The schematic of numerical example for 1D wave propagation (unit: mm)
图 20 二维波传播算例示意图 (单位: mm)
Figure 20. The schematic of numerical example for 2D wave propagation (unit: mm)
-
[1] 高光发. 波动力学基础. 北京: 科学出版社, 2019Gao Guangfa. Fundamentals of Wave Dynamics. Beijing: National Scientific Press, 2019 (in Chinese) [2] 李永池. 波动力学. 合肥: 中国科学技术大学出版社, 2015Li Yongchi. Wave Dynamics. Hefei: University of Science and Technology of China Press, 2015 (in Chinese) [3] Cox BN, Gao H, Gross D, et al. Modern topics and challenges in dynamic fracture. Journal of the Mechanics and Physics of Solids, 2005, 53(3): 565-596 doi: 10.1016/j.jmps.2004.09.002 [4] Freund LB. Dynamic Fracture Mechanics. Cambridge: Cambridge University Press, 1998 [5] Ravi-Chandar K. Dynamic Fracture. New York: Elsevier, 2004 [6] Liang Y, Zhang X, Liu Y. Extended material point method for the three-dimensional crack problems. International Journal for Numerical Methods in Engineering, 2021, 122(12): 3044-3069 doi: 10.1002/nme.6653 [7] Wu J, Wang D, Lin Z, et al. An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture. Computational Particle Mechanics, 2020, 7(2): 193-207 doi: 10.1007/s40571-019-00240-5 [8] Imtiaz H, Liu B. An efficient and accurate framework to determine the failure surface/envelop in composite lamina. Composites Science and Technology, 2021, 201: 108475 doi: 10.1016/j.compscitech.2020.108475 [9] Bobaru F, Foster JT, Geubelle PH, et al. Handbook of Peridynamic Modeling. CRC Press, 2016 [10] Hu Y, Feng G, Li S, et al. Numerical modelling of ductile fracture in steel plates with non-ordinary state-based peridynamics. Engineering Fracture Mechanics, 2020, 225: 106446 doi: 10.1016/j.engfracmech.2019.04.020 [11] Yang D, He X, Liu X, et al. A peridynamics-based cohesive zone model (PD-CZM) for predicting cohesive crack propagation. International Journal of Mechanical Sciences, 2020, 184: 105830 doi: 10.1016/j.ijmecsci.2020.105830 [12] Xia Y, Meng X, Shen G, et al. Isogeometric analysis of cracks with peridynamics. Computer Methods in Applied Mechanics and Engineering, 2021, 377: 113700 doi: 10.1016/j.cma.2021.113700 [13] Silling SA. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175-209 doi: 10.1016/S0022-5096(99)00029-0 [14] Yu H, Chen X, Sun Y. A generalized bond-based peridynamic model for quasi-brittle materials enriched with bond tension–rotation–shear coupling effects. Computer Methods in Applied Mechanics and Engineering, 2020, 372: 113405 doi: 10.1016/j.cma.2020.113405 [15] Silling SA, Epton M, Weckner O, et al. Peridynamic states and constitutive modeling. Journal of Elasticity, 2007, 88(2): 151-184 doi: 10.1007/s10659-007-9125-1 [16] Behzadinasab M, Foster JT. A semi-lagrangian constitutive correspondence framework for peridynamics. Journal of the Mechanics and Physics of Solids, 2020, 137: 103862 doi: 10.1016/j.jmps.2019.103862 [17] Zhang H, Qiao P. A two-dimensional ordinary state-based peridynamic model for elastic and fracture analysis. Engineering Fracture Mechanics, 2020, 232: 107040 doi: 10.1016/j.engfracmech.2020.107040 [18] Zhou XP, Wang YT, Shou YD, et al. A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads. Engineering Fracture Mechanics, 2018, 188: 151-183 doi: 10.1016/j.engfracmech.2017.07.031 [19] Zhu QZ, Ni T. Peridynamic formulations enriched with bond rotation effects. International Journal of Engineering Science, 2017, 121: 118-129 doi: 10.1016/j.ijengsci.2017.09.004 [20] Gu X, Zhang Q, Madenci E. Non-ordinary state-based peridynamic simulation of elastoplastic deformation and dynamic cracking of polycrystal. Engineering Fracture Mechanics, 2019, 218: 106568 doi: 10.1016/j.engfracmech.2019.106568 [21] Liu Z, Bie Y, Cui Z, et al. Ordinary state-based peridynamics for nonlinear hardening plastic materials' deformation and its fracture process. Engineering Fracture Mechanics, 2019, 223: 106782 [22] 王涵, 黄丹, 徐业鹏等. 非常规态型近场动力学热黏塑性模型及其应用. 力学学报, 2018, 50(4): 810-819 (Wang Han, Huang Dan, Xu Yepeng, et al. Non-ordinary state-based peridynamic thermal-viscoplastic model and its application. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(4): 810-819 (in Chinese) doi: 10.6052/0459-1879-18-113 [23] Wang H, Xu YP, Hang D. A non-ordinary state-based peridynamic formulation for thermo-visco-plastic deformation and impact fracture. International Journal of Mechanical Sciences, 2019, 159: 336-344 doi: 10.1016/j.ijmecsci.2019.06.008 [24] 胡祎乐, 余音, 汪海. 基于近场动力学理论的层压板损伤分析方法. 力学学报, 2013, 45(4): 624-628 (Hu Yile, Yu Yin, Wang Hai. Damage analysis method for laminates based on peridynamic theory. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(4): 624-628 (in Chinese) doi: 10.6052/0459-1879-12-368 [25] 章青, 顾鑫, 郁杨天. 冲击载荷作用下颗粒材料动态力学响应的近场动力学模拟. 力学学报, 2016, 48(1): 56-63 (Zhang Qing, Gu Xin, Yu Yangtian. Peridynamics simulation for dynamic response of granular materials under impact loading. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1): 56-63 (in Chinese) doi: 10.6052/0459-1879-15-291 [26] Wu P, Yang F, Chen Z, et al. Stochastically homogenized peridynamic model for dynamic fracture analysis of concrete. Engineering Fracture Mechanics, 2021, 253: 107863 doi: 10.1016/j.engfracmech.2021.107863 [27] Huang XP, Kong XZ, Chen ZY, et al. Peridynamics modelling of dynamic tensile failure in concrete. International Journal of Impact Engineering, 2021, 155: 103918 doi: 10.1016/j.ijimpeng.2021.103918 [28] Bažant ZP, Luo W, Chau VT, et al. Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models. Journal of Applied Mechanics, 2016, 83(11): 111004 doi: 10.1115/1.4034319 [29] Butt SN, Timothy JJ, Meschke G. Wave dispersion and propagation in state-based peridynamics. Computational Mechanics, 2017, 60(5): 725-738 doi: 10.1007/s00466-017-1439-7 [30] Wildman RA. Discrete micromodulus functions for reducing wave dispersion in linearized peridynamics. Journal of Peridynamics and Nonlocal Modeling, 2019, 1(1): 56-73 doi: 10.1007/s42102-018-0001-0 [31] Gu X, Zhang Q, Huang D, et al. Wave dispersion analysis and simulation method for concrete SHPB test in peridynamics. Engineering Fracture Mechanics, 2016, 160: 124-137 doi: 10.1016/j.engfracmech.2016.04.005 [32] Li S, Jin Y, Lu H, et al. Wave dispersion and quantitative accuracy analysis of bond-based peridynamic models with different attenuation functions. Computational Materials Science, 2021, 197: 110667 doi: 10.1016/j.commatsci.2021.110667 [33] Wildman RA, Gazonas GA. A finite difference-augmented peridynamics method for reducing wave dispersion. International Journal of Fracture, 2014, 190(1): 39-52 [34] Alebrahim R, Packo P, Zaccariotto M, et al. Improved wave dispersion properties in 1D and 2D bond-based peridynamic media. Computational Particle Mechanics, 2021, 9: 597-614 [35] Zhou G, Hillman M. A non-ordinary state-based Godunov-peridynamics formulation for strong shocks in solids. Computational Particle Mechanics, 2020, 7(2): 365-375 doi: 10.1007/s40571-019-00254-z [36] Ren XD, Zhu JG. Temporally stabilized peridynamics methods for shocks in solids. Computational Mechanics, 2021, 69: 489-504 [37] Zhu F, Zhao J. Peridynamic modelling of blasting induced rock fractures. Journal of the Mechanics and Physics of Solids, 2021, 153: 104469 doi: 10.1016/j.jmps.2021.104469 [38] 黄丹, 章青, 乔丕忠等. 近场动力学方法及其应用. 力学进展, 2010, 40(4): 448-459 (Huang Dan, Zhang Qing, Qiao Pizhong, et al. A review of peridynamics (PD) method and its application. Advances in mechanics, 2010, 40(4): 448-459 (in Chinese) doi: 10.6052/1000-0992-2010-4-J2010-002 [39] Jia F, Gao Z, Don WS. A spectral study on the dissipation and dispersion of the WENO schemes. Journal of Scientific Computing, 2015, 63(1): 49-77 doi: 10.1007/s10915-014-9886-1 [40] Ren B, Fan H, Bergel GL, et al. A peridynamics–SPH coupling approach to simulate soil fragmentation induced by shock waves. Computational Mechanics, 2015, 55(2): 287-302 doi: 10.1007/s00466-014-1101-6 [41] Monaghan JJ. On the problem of penetration in particle methods. Journal of Computational physics, 1989, 82(1): 1-15 doi: 10.1016/0021-9991(89)90032-6 [42] Silling SA, Parks ML, Kamm JR, et al. Modeling shockwaves and impact phenomena with Eulerian peridynamics. International Journal of Impact Engineering, 2017, 107: 47-57 doi: 10.1016/j.ijimpeng.2017.04.022 [43] Lai X, Liu L, Li S, et al. A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. International Journal of Impact Engineering, 2018, 111: 130-146 doi: 10.1016/j.ijimpeng.2017.08.008 [44] Caramana EJ, Shashkov MJ, Whalen PP. Formulations of artificial viscosity for multi-dimensional shock wave computations. Journal of Computational Physics, 1998, 144(1): 70-97 doi: 10.1006/jcph.1998.5989 [45] Landshoff R. A numerical method for treating fluid flow in the presence of shocks. Los Alamos National Lab NM, 1955 [46] Silling SA, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, 2005, 83(17-18): 1526-1535 [47] Achenbach J. Wave Propagation in Elastic Solids. New York: Elsevier, 2012 -
下载:
点击查看大图
图(21)
计量
- 文章访问数: 385
- HTML全文浏览量: 119
- PDF下载量: 105
- 被引次数: 0
