基于分数阶模型的牡蛎壳动力学特性研究

袁良柱; 陆建华; 苗春贺; 王鹏飞; 徐松林

doi:10.11883/bzycj-2022-0318

基于分数阶模型的牡蛎壳动力学特性研究

doi: 10.11883/bzycj-2022-0318

袁良柱^1,,
陆建华¹,
苗春贺¹,
王鹏飞¹,
徐松林^{1, 2, ,}

1.
中国科学技术大学中国科学院材料力学行为和设计重点实验室，安徽合肥 230027
2.
中国地震局地震预测研究所高压物理与地震科技联合实验室，北京 100036

基金项目: 国家自然科学基金（11672286, 11872361）；中央高校基本科研业务费专项资金（WK2480000008）；中石油与中科院重大战略合作项目（2015A-4812）；高压物理与地震科技联合实验室室开放基金（2019HPPES01）

详细信息

作者简介:
袁良柱（1998－　），男，博士研究生，ylzustcedu@mail.ustc.edu.cn

通讯作者:
徐松林（1971－　），男，博士，研究员，博士生导师，slxu99@ustc.edu.cn

中图分类号: O347
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- 被引次数: 0
出版历程
- 收稿日期: 2022-07-21
- 修回日期: 2022-10-11
- 网络出版日期: 2022-11-21
- 刊出日期: 2023-01-05

Dynamic properties of oyster shells based on a fractional-order model

1.
CAS Key Laboratory for Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, Anhui, China
2.
United Laboratory of High-Pressure Physics and Earthquake Science, Institute of Earthquake Forecasting, China Earthquake Administration, Beijing 100036, China

摘要

摘要: 贝壳、牡蛎等天然材料因其轻质高强的力学特性在材料设计等领域受到了广泛的关注，但由于材料本身结构的复杂性，对其力学行为的研究十分困难。近年来，分数阶模型在研究材料的力学特性上取得了成功，相比传统模型，分数阶模型可以更好地表征复杂介质的应力或应变与时间的关系。因此，本文从波传播理论出发，以分数阶模型作为材料本构，得到了复杂介质的波传播控制方程。通过Laplace变换得到了控制方程的解析解，并通过Laplace数值逆变换分析了波的衰减对分数阶模型中参量的敏感性，讨论了不同于材料弹性、黏性的材料“惯性”特性。接着，基于解析解和多种实验测试信号，给出了得到分数阶模型参数的拟合式子。以牡蛎材料作为研究对象，利用CO₂脉冲激光器进行小试样的冲击加载、应用两点激光干涉测速系统（laser interferometer velocimetry system, VISAR）对表面粒子的速度进行测量，得到了4种密度下不同厚度的牡蛎壳试样的粒子速度时程曲线，再结合上述理论方法分析得到了牡蛎壳试样的Abel模型和分数阶Maxwell模型的参数，模型参数反映了牡蛎壳试样的细微观结构特征。结果发现：牡蛎壳试样的密度越大，即在细微观上具有砖泥结构的珍珠层的占比越高，速度衰减越大，试样的黏性越大；这是由于CO₂激光脉冲器发射的激光波长与牡蛎壳试样珍珠层的砖泥结构间的缝隙尺寸相近，使得激光在冲击牡蛎壳试样中的珍珠层时发生较大的散射。
- 分数阶导数 /
- 本构模型 /
- 牡蛎壳 /
- 脉冲激光 /
- 动力学特性
Abstract: Natural materials such as shells and oysters have attracted extensive attention in the field of material design due to their lightweight and high-strength mechanical properties. However, due to the complex structure of shells, it is very difficult to study their mechanical behavior. In recent years, fractional-order models have been successful in studying the mechanical properties of materials. Compared with the traditional constitutive model, the fractional model can better characterize the relationship between the complex media’s stress or strain and time. Therefore, based on wave propagation theory and by using the time-dependent fractional-order model as the material constitutive model, the complex medium is simplified to the uniform medium, and its governing equation is obtained by then. The analytic solution of the governing equation which is a function of space coordinate x and Laplace variable s is obtained by the Laplace transform. It is hard to obtain the analytical solution of space coordinate x and time t directly through the inverse Laplace transform, so the numerical inverse Laplace transform is used to obtain the numerical solution of the governing equation in the time domain. The sensitivity of wave attenuation to parameters in the fractional model is analyzed. The inertial properties, which are different from the elastic and viscous properties of materials, are also discussed by analyzing the attenuation characteristics of stress waves when the order α is 0, 1.0, and 2.0 respectively. Then, based on the analytical solution of the governing equation and a variety of experimental test signals, a fitting formula is given to obtain the parameters of the fractional model. Oyster material with layered structure is taken as the research object. To obtain the local dynamic mechanical properties of oyster samples, the CO₂ pulse laser was used to carry out the impact loading of the small sample due to the high variability of the density distribution of oyster samples, and the two-point laser interferometer velocimetry system (VISAR) was used to measure the surface particle velocity. The particle velocity time history curve of the oyster sample with different densities and thicknesses was obtained. Combined with the above fitting formula, the parameters of the Abel model and Maxwell fractional differential model of oyster samples were obtained by fixing and unfixing the values of fractional order α, and the model parameters reflected the fine microstructure characteristics of oyster samples. The results show that the higher the density of the oyster sample is, the higher the proportion of nacre with brick and mortar structure in fine and micro, the greater the velocity attenuation, and the greater the viscosity of the oyster sample. The laser wavelength emitted by the CO₂ laser pulse is similar to the size of the gap between brick and mortar structures in the nacre of the oyster sample, so the laser has a large scattering when it impacts the nacre of the oyster sample, which causes the velocity attenuation. This study has a good reference significance for the study of the dynamic properties of meso-isomeric and macro-continuous complex media.
- fractional derivative /
- constitutive model /
- oyster shell /
- pulse laser /
- dynamic properties

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图 1 牡蛎壳材料、圆形实验试样、试样纵剖面电镜照片与局部放大

Figure 1. Oyster shell material, circular sample, longitudinal section electron microscopy (SEM) and local magnification

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图 2 CO₂脉冲加载与激光干涉测速系统(VISAR)

Figure 2. CO₂ pulse loading and laser interferometer velocimetry system (VISAR)

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图 3 几种分数阶本构模型

Figure 3. Several fractional differential constitutive models

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图 4 单个半脉宽正弦函数

Figure 4. Single half pulse width sine function

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图 5 正弦波在Abel黏壶模型介质的衰减

Figure 5. Attenuation of a single half pulse width sine wave in Abel model media

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图 6 当α分别为1.0和2.0时的波传播特性

Figure 6. The property of wave propagation when the order α is 1.0 and 2.0 , respectively

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图 7 参数α、η和E对幅值衰减的影响

Figure 7. Influence of parameters α, η and E on amplitude attenuation

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图 8 各个密度下不同厚度牡蛎试样的速度信号及拟合曲线

Figure 8. Velocity signals and fitting curves of oyster samples with different thicknesses and densities

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图 9 牡蛎试样分数阶模型的拟合曲线及参数与牡蛎试样密度的关系

Figure 9. The fitting curve of the fractional model of the oyster sample and relationship between parameters and oyster sample density

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表 1 牡蛎壳试样分数阶模型的拟合参数

Table 1. Fitting parameters of the fractional model of the oyster sample

材料本构	密度/(g·cm⁻³)	α	E/GPa	η/(Pa·s^α)
Abel模型	0.5	0.631	-	9.09×10⁴
	0.7	0.811	-	5.65×10³
	0.7	1.025	-	192.12
	1.2	1.181	-	17.84
	1.4	1.297	-	2.19
分数阶Maxwell模型	0.5	0.690	2.21	6.71×10⁴
	0.7	0.851	7.34	3.55×10³
	0.7	0.897	7.67	1.45×10³
	1.2	1.033	12.88	170.01
	1.4	1.224	15.40	6.76

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表 2 固定分数阶阶数情况下，牡蛎壳试样分数阶Maxwell模型的拟合参数

Table 2. Fitting parameters of fractional Maxwell model for oyster shell samples in the case of fixed fractional order

材料本构	密度/(g·cm⁻³)	α	E/GPa	η/(Pa·s^α)
分数阶Maxwell模型	0.5	0.94	2.44	1518.00
	0.7		7.67	929.35
	0.7		7.82	751.88
	1.2		13.15	687.85
	1.4		16.46	471.30

下载: 导出CSV

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