A Distributed-order Maxwell Constitutive Model for Vibration Isolation SR and its Shock Response
-
摘要: 为精确描述隔振硅橡胶冲击力学行为,基于高应变率下硅橡胶呈阶段性力学响应,且分数阶Maxwell模型中阶数α可体现材料粘弹性分布,将阶数α分布取值,构建了Maxwell分布阶本构模型。采用SHPB技术开展了高应变率下的隔振硅橡胶冲击力学性能实验,对模型进行验证。结果表明:高应变率下隔振硅橡胶的应力-应变实验曲线可分为4个阶段,即弹性阶段、软化阶段、硬化阶段和失效阶段;随应变率的增大,屈服应变、屈服应力及割线模量均增大;软化阶段内曲线的上凸曲率增大;硬化阶段内分子链运动能力越差,硬化效应越显著;分析了Maxwell分布阶本构模型对实验数据的表征能力,结果表明不同应变率下该模型的RMSE均值仅为0.087 2。Maxwell分布阶本构模型对粘弹性冲击力学行为的表征具有较好的综合优势,即精度高、物理含义明确,可在较宽应变率范围准确描述隔振硅橡胶的冲击力学行为特性。
-
关键词:
- 硅橡胶 /
- 分布阶Maxwell模型 /
- 冲击性能 /
- SHPB实验
Abstract: Under shock loading with variable amplitude and frequency, the viscoelastic damping structure is always in the multifactorial dynamic state, and its shock response is obviously different from that at the low strain rate. Based on the periodic mechanical response for SR at high strain rates and the fractional order α of the fractional Maxwell model reflecting the elasticity-viscosity distribution, a distributed-order Maxwell constitutive model is constructed to accurately simulate the shock response of vibration isolation SR. To verify the proposed model, the shock experiments for different high strain rates are performed by SHPB system. The results indicate that experimental stress-strain curves could be divided into four stages: the linear stage, strain-softening stage, strain-hardening stage and failure stage. With increase of the strain rate, the yield strain, the yield stress, the secant modulus and the curvature in strain-softening stage all increase, and the hardening effect in the strain-hardening stage tends to stronger, which should embody the rate-dependent properties of SR. Besides, the representational ability of the proposed model on experimental data is further analyzed. The RMSE values at different strain rates are all small. For all these reasons, the distributed-order Maxwell constitutive model could accurately describe the shock response of vibration isolation SR in a wider range of strain rates with the advantages of higher fitting precision, and clear physical meaning.-
Key words:
- silicon rubber (SR) /
- distributed-order Maxwell model /
- shock response /
- SHPB experiment
-
图 4 应力波的传播过程[16]
表 1 不同应变率下应力-应变曲线的特征参数值
εⅠ εⅡ εⅢ εⅣ εs σs/MPa E/MPa 2500 0.013 0.259 - 0.315 0.259 4.383 18.837 2800 0.013 0.329 - 0.370 0.323 5.437 19.147 3200 0.013 0.258 0.385 0.428 0.385 6.830 20.155 3450 0.021 0.247 0.415 0.468 0.415 8.322 21.938 3600 0.013 0.244 0.416 0.489 0.416 9.068 22.481 表 2 高应变率下Maxwell分布模型参数值
软化阶段 硬化阶段 阶段Ⅱa 阶段Ⅱb c, k, β α c, k, β α α, β ke 2 500 0.431 0.414 - 2 800 c=4×105 0.428 c=4×105 0.411 α=0
β=0- 3 200 k=6×106 0.420 k=6×106 0.409 1.370×107 3 450 β=0.100 0.435 β=0.100 0.422 1.603×107 3 600 0.445 0.428 2.004×107 -
[1] MORI K, KONO D, YAMAJI I, et al. Modelling of viscoelastic damper support for reduction in low frequency residual vibration in machine tools[J]. Precision Engineering, 2017, 50:313-319 doi: 10.1016/j.precisioneng.2017.06.004 [2] GHAEMMAGHAMI A R, KWON O S. Nonlinear modeling of MDOF structures equipped with viscoelastic dampers with strain, temperature and frequency-dependent properties[J]. Engineering Structures, 2018, 168:903-914 doi: 10.1016/j.engstruct.2018.04.037 [3] NAKRA B C. Vibration control in machines and structures using viscoelastic damping[J]. Journal of Sound & Vibration, 1998, 211(3):449-465 [4] LEWANDOWSKI R. Influence of temperature on the dynamic characteristics of structures with viscoelastic dampers[J]. Journal of Structural Engineering, 2019, 145(2):04018245 doi: 10.1061/(ASCE)ST.1943-541X.0002238 [5] 胡洋, 赵祺, 芦艾, 等.苯基硅橡胶泡沫的制备及阻尼性能[J].材料导报, 2019, 33(10):1752-1755 doi: 10.11896/cldb.18090240HU Y, ZHAO Q, LU A, et al. Preparation and damping properties of phenyl silicone rubber foam[J]. Materials Review, 2019, 33(10):1752-1755 (in Chinese) doi: 10.11896/cldb.18090240 [6] SASSO M, CHIAPPINI G, ROSSI M, et al. Visco-hyper-pseudo-elastic characterization of a fluoro-silicone rubber[J]. Experimental Mechanics, 2014, 54(3):315-328 doi: 10.1007/s11340-013-9807-5 [7] 涂春潮, 王珊, 任玉柱, 等.阻尼硅橡胶/丁苯橡胶共混胶动态性能[J].航空材料学报, 2014, 34(2):58-62 https://www.cnki.com.cn/Article/CJFDTOTAL-HKCB201402012.htmTU C C, WANG S, REN Y Z, et al. Dynamic mechanical properties of damping silicon rubber/styrene-butadiene rubber blend[J]. Journal of Aeronautical Materials, 2014, 34(2):58-62 (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-HKCB201402012.htm [8] ILG J, RUPITSCH S J, SUTOR A, et al. Determination of dynamic material properties of silicone rubber using one-point measurements and finite element simulations[J]. IEEE Transactions on Instrumentation and Measurement, 2012, 61(11):3031-3038 doi: 10.1109/TIM.2012.2203449 [9] 米志安, 黄艳华, 苏正涛, 等.周期载荷下的阻尼硅橡胶动态粘弹性研究[J].特种橡胶制品, 2011, 32(1):33-35 doi: 10.3969/j.issn.1005-4030.2011.01.007MI Z A, HUANG Y H, SUN Z T, et al. Study on dynamic-viscoelastic properties of dampening silicone rubber under periodical loading[J]. Special Purpose Rubber Products, 2011, 32(1):33-35 (in Chinese) doi: 10.3969/j.issn.1005-4030.2011.01.007 [10] 刘占芳, 励凌峰, 胡文军.多孔硅橡胶有限变形的粘弹性行为[J].固体力学学报, 2002, 32(3):347-353 doi: 10.3969/j.issn.0254-7805.2002.03.015LIU Z F, LI L F, HU W J. Viscoelastic behaviors of porous silicone rubbers under finite deformation[J]. Acta Mechanica Solida Sinica, 2002, 32(3):347-353 (in Chinese) doi: 10.3969/j.issn.0254-7805.2002.03.015 [11] 郭玲梅, 汪洋, 徐伟芳.硅橡胶拉伸力学的应变率相关性研究[J].中国测试, 2018, 44(10):85-88 doi: 10.11857/j.issn.1674-5124.2018.10.014GUO L M, WANG Y, XU W F. The study on strain-rate dependence of tension behavior of filled silicone rubber[J]. China Measurement & Test, 2018, 44(10):85-88 (in Chinese) doi: 10.11857/j.issn.1674-5124.2018.10.014 [12] 胡时胜, 王正道, 赵立中.泡沫硅橡胶动态力学性能的实验研究[J].高分子材料科学与工程, 1999, 15(2):113-115 doi: 10.3321/j.issn:1000-7555.1999.02.032HU S S, WANG Z D, ZHAO L Z. Experimental study of dynamic mechanical behaviors of silicone rubber foam[J]. Polymer Materials Science and Engineering, 1999, 15(2):113-115 (in Chinese) doi: 10.3321/j.issn:1000-7555.1999.02.032 [13] 林玉亮, 卢芳云, 卢力.高应变率下硅橡胶的本构行为研究[J].高压物理学报, 2007, 21(3):289-294 doi: 10.3969/j.issn.1000-5773.2007.03.012LIN Y L, LU F Y, LU L. Constitutive behaviors of a silicone rubber at high strain rates[J]. Chinese Journal of High Pressure Physics, 2007, 21(3):289-294 (in Chinese) doi: 10.3969/j.issn.1000-5773.2007.03.012 [14] LI Z L, SUN D G, QIN Y, et al. Stiffness-damping matching modelling for vibration isolation system of roadheader ECB[J]. International Journal of Acoustics and Vibration, 2020, 25(1):54-61 doi: 10.20855/ijav.2020.25.11514 [15] LI Z L, QIN Y, SUN B, et al. A fractional approach to the time-temperature dependence of dynamic viscoelastic behavior[J]. Journal of Mechanical Science and Technology, 2019, 33(1):139-147 doi: 10.1007/s12206-018-1214-5 [16] BAI Y, LIU C M, HUANG G Y, et al. A hyper-viscoelastic constitutive model for polyurea under uniaxial compressive loading[J]. Polymers, 2016, 8(4):133 doi: 10.3390/polym8040133 [17] 卢芳云, CHEN W, FREW D J.软材料的SHPB实验设计[J].爆炸与冲击, 2002, 22(1):15-19 https://www.cnki.com.cn/Article/CJFDTOTAL-BZCJ200201002.htmLU F Y, CHEN W, FREW D J. A design of SHPB experiments for soft materials[J]. Explosion and Shock Waves, 2002, 22(1):15-19 (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-BZCJ200201002.htm [18] SONG B, CHEN W. One-dimensional dynamic compressive behavior of EPDM rubber[J]. Journal of Engineering Materials and Technology, 2003, 125(3):294-301 doi: 10.1115/1.1584492 [19] DAVIES E D H, HUNTER S C. The dynamic compression testing of solids by the method of the split Hopkinson pressure bar[J]. Journal of the Mechanics and Physics of Solids, 1963, 11(3):155-179 doi: 10.1016/0022-5096(63)90050-4 [20] LEE O S, CHO K S, KIM S H, et al. Dynamic deformation behavior of soft material using SHPB technique and pulse shaper[J]. International Journal of Modern Physics B, 2006, 20(25-27):3751-3756