基于复虚场方法的亚克力薄板黏弹性本构参数识别

doi:10.12382/bgxb.2024.0051

摘要/Abstract

摘要：

为研究在材料参数反演领域的新方法,提出一种求解正交各向异性复合材料弹性常数的虚场方法。为奠定正交各向异性本构参数反演的基础,以各向同性材料聚甲基丙烯酸甲酯(Polymethyl Methacrylate,PMMA)材料为实验试样,基于薄板理论和光偏折法,通过对薄板试样进行振动加载试验,获取试件表面的全场变形;基于C³连续性的Hermite有限单元对测得的变形场进行处理,得到薄板试样的离面位移场和曲率场;利用复虚场法计算材料的黏弹性常数,将反演结果与三点弯曲实验测得的弹性参数和悬臂梁锤击实验测得的阻尼系数进行了对比。以上实验结果表明:弹性模量与阻尼系数的测量误差均在0.3%以内。研究结果验证了该方法在各向同性本构参数反演中的可行性,为后续在各向异性材料本构参数反演奠定了基础。

关键词: 虚场方法, 光偏折法, 数字图像相关方法, 薄板振动, 黏弹性本构

Abstract:

To investigate new methods in the field of material parameter inversion,a complex virtual fields method is proposed for solving the elastic constants of orthogonal anisotropic composite materials.To lay the groundwork for the inversion of orthogonal anisotropic constitutive parameters,an isotropic material,polymethyl methacrylate (PMMA),is initially used as an experimental specimen.Based on thin plate theory and deflectometry,a vibration loading test is conducted on a thin plate specimen to acquire the full-field deformation of the specimen surface.The measured deformation field is processed using C³ continuity Hermite finite elements,resulting in the out-of-plane displacement and curvature fields of the thin plate specimen.The complex virtual fields method is then used to calculate the viscoelastic constants of material.The inversion results are compared with the elastic parameters measured by the three-point bending test and the damping coefficients measured by the cantilever beam hammer impact test.The experimental results show that the measured errors of the elastic modulus and damping coefficients are all within 0.3%.This confirms the feasibility of the proposed method in the inversion of isotropic constitutive parameters and lays the foundation for the subsequent inversion of anisotropic material constitutive parameters.

Key words: virtual fields method, deflectometry, digital image correlation, thin plate vibration, viscoelastic constitutive parameter

中图分类号:

O348.9
O329

王岗亭, 郭保桥, 刘函, 栾可迪, 谷源森, 陈鹏万, 周江帆, 刘战伟. 基于复虚场方法的亚克力薄板黏弹性本构参数识别[J]. 兵工学报, 2025, 46(2): 240051-.

WANG Gangting, GUO Baoqiao, LIU Han, LUAN Kedi, GU Yuansen, CHEN Pengwan, ZHOU Jiangfan, LIU Zhanwei. Identification of Viscoelastic Constitutive Parameters of Acrylic Thin Plates Using Complex Virtual Fields Method[J]. Acta Armamentarii, 2025, 46(2): 240051-.

图/表 24

图1 Kelvin-Voigt黏弹性模型

Fig.1 Kelvin-Voigt viscoelastic model

图2 三点弯曲实验装置图

Fig.2 Experimental set-up of three-point bending

图3 PMMA三点弯曲载荷-位移曲线

Fig.3 Load-displacement curve for PMMA in three-point bending

表1 PMMA弹性模量参数

Table 1 Elastic modulus parameters of PMMA

样本	弹性模量E_f/MPa
1	1904
2	1908
3	1906
4	1903
均值	1905
均方差	2.2

图4 阻尼装置示意图

Fig.4 Schematic diagram of damping device

表2 PMMA的阻尼系数

Table 2 Damping parameters of PMMA

样本	阻尼参数ζ
1	0.0356
2	0.0344
3	0.0348
4	0.0342
5	0.0357
均值	0.0349
标准差	0.0006

图5 光偏折法原理图

Fig.5 Schematic diagram of deflectometry

图6 试样尺寸示意图

Fig.6 Schematic diagram of specimen dimensions

图7 薄板的相关计算区域及镜面反射效果

Fig.7 Relevant calculation area and mirror reflection effect of thin plate

图8 振动加载实验平台

Fig.8 Vibration loading experimental platform

图9 振动同步原理图

Fig.9 Schematic diagram of vibration synchronization

图10 50Hz下转角场

Fig.10 Slope fields at 50Hz

图11 150Hz下转角场

Fig.11 Slope fields at 150Hz

图12 200Hz下转角场

Fig.12 Slope fields at 200Hz

图13 500Hz下转角场

Fig.13 Slope fields at 500Hz

图14 800Hz下转角场

Fig.14 Slope fields at 800Hz

图15 1000Hz下转角场

Fig.15 Slope fields at 1000Hz

图16 矩形的Hermite有限单元示意图

Fig.16 Schematic diagram of rectangular Hermite finite element

图17 3×3均匀Hermite有限单元网格

Fig.17 3×3 Hermite finite element mesh

图18 重构后的PMMA薄板1000Hz频率下变形场结果

Fig.18 Deformation fields of reconstructed PMMA thin plate at 1000Hz frequency

图19 重构的4组虚拟变形场图像

Fig.19 Reconstructed images of 4 sets of virtual deformation fields

表3 虚场反演4组黏弹性本构参数

Table 3 Four sets of inverted viscoelastic constitutive parameters

样本	频率/Hz	E₁₁/Gpa	E₂₂/Gpa	ν₁₂	G₁₂/Gpa
1	50	1.900	1.906	0.358	0.701
2	150	1.905	1.900	0.357	0.701
3	200	1.910	1.912	0.361	0.702
4	500	1.898	1.898	0.355	0.700
5	800	1.902	1.907	0.364	0.698
6	1000	1.903	1.904	0.350	0.705
均值	-	1.905	1.906	0.358	0.700
标准差	-	0.002	0.002	0.005	0.003

表4 虚场反演阻尼参数

Table 4 Inverted damping parameters for the virtual fields method

样本	η₁₁	η₂₂
1	0.0342	0.0345
2	0.0346	0.0348
3	0.0352	0.0343
4	0.0343	0.0350
5	0.0348	0.0352
6	0.0351	0.0347
均值	0.0349	0.0348
标准差	0.0003	0.0005

表5 复虚场方法结果与参考结果对比

Table 5 Comparison of the results obtained by complex virtual fields method and the reference results

样本	弹性模量/GPa	阻尼
反演参数	1.9055	0.0348
实验	1.9052	0.0349
相对误差/%	0.016	0.29

参考文献 22

[1]	WANG Q K, CHEN Y, MIAO L L, et al. Wide spectral and wavelength-tunable dissipative soliton fiber laser with topological insulator nano-sheets self-assembly films sandwiched by PMMA polymer[J]. Optics Express, 2015, 23(6):7681-7693. doi: 10.1364/OE.23.007681 pmid: 25837106
[2]	ALI U, KARIM K J B A, BUANG N A. A review of the properties and applications of poly (methyl methacrylate) (PMMA)[J]. Polymer Reviews, 2015, 55(4):678-705.
[3]	MOUSA W F, KOBAYASHI M, SHINZATO S, et al. Biological and mechanical properties of PMMA-based bioactive bone cements[J]. Biomaterials, 2000, 21(21):2137-2146. doi: 10.1016/s0142-9612(00)00097-1 pmid: 10985486
[4]	ZHAO J Y, LIU Z W, GUO B Q. Three-dimensional digital image correlation method based on a light field camera[J]. Optics and Lasers in Engineering, 2019, 116:19-25.
[5]	SUTTON M, WOLTERS W, PETERS W. Determination of displacements using an improved digital correlation method[J]. Image and Vision Computing, 1983, 1(3):133-139.
[6]	PAN B, ASUNDI A, XIE H M, et al. Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements[J]. Optics and Lasers in Engineering, 2009, 47(7-8):865-874.
[7]	陈新, 许巍, 何玉怀. 基于DIC的超高频振动疲劳试样动态全场应变测量方法[J]. 实验力学, 2021, 36(5):677-685.
	CHEN X, XU W, HE Y H. Ultra-high frequency vibration fatigue dynamic strain field measurement based on digital image correlation method[J]. Journal of Experimental Mechanics, 2021, 36(5):677-685. (in Chinese)
[8]	黄心畏, 单晓锋, 高红俐, 等. 基于边缘检测和数字图像相关法的疲劳裂纹长度测量方法[J]. 兵工学报, 2022, 43(4):940-951. doi: 10.12382/bgxb.2021.0268
	HUANG X W, SHAN X F, GAO H L, et al. Fatigue crack length measurement method based on edge detection and DIC[J]. Acta Armamentarii, 2022, 43(4):940-951. (in Chinese) doi: 10.12382/bgxb.2021.0268
[9]	HILD F, ROUX S. Digital image correlation:from displacement measurement to identification of elastic properties-a review[J]. Campus Scientifique, 2006, 42(2):69-80.
[10]	PIERRON F, GREDIAC M. The virtual fields method.Extracting constitutive mechanical parameters from full-field deformation measurements[M].New York,NY, US:Springer, 2012.
[11]	AVRIL S, BONNET M, BRETELLE A S, et al. Overview of identification methods of mechanical parameters based on full-field measurements[J]. Experimental Mechanics, 2008, 48(4):381-402.
[12]	AVRIL S, GREDIAC M, PIERRON F. Sensitivity of the virtual fields method to noisy data[J]. Computational Mechanics, 2004, 34(6):439-452.
[13]	NERI P, PAOLI A, RAZIONALE A V, et al. Low-speed cameras system for 3D-DIC vibration measurements in the kHz range[J]. Mechanical Systems & Signal Processing, 2022,162.
[14]	WARBURTON J R, LU G, BUSS T M, et al. Digital image correlation vibrometry with low speed equipment[J]. Experimental Mechanics, 2016, 56(7):1219-1230.
[15]	单晓锋, 高红俐, 黄心畏, 等. 基于显微DIC的谐振疲劳短裂纹尖端变形场演化规律[J]. 兵工学报, 2023, 44(5):1482-1492.
	SHAN X F, GAO H L, HUANG X W, et al. Evolution law of deformation field of short crack tip under high-frequency resonant loading using a microscope DIC system[J]. Acta Armamentarii, 2023, 44(5):1482-1492. (in Chinese)
[16]	MIAO C Y, TIPPUR H V. A simplified reflection-mode digital gradient sensing technique for measuring surface slopes,curvatures and topography[J]. Optics and Lasers in Engineering, 2020, 124:105843.
[17]	GUAN J T, FEI J J, LI W, et al. Defect classification for specular surfaces based on deflectometry and multi-modal fusion network[J]. Optics and Lasers in Engineering, 2023, 163:107488.
[18]	ROBIN O, O’DONOUGHUE P, BERRY A, et al. Full field vibration measurements on a cantilever beam under impact using visible and infrared deflectometry[J]. Applied Acoustics, 2021, 183:108294.
[19]	SADATI M, LUAP C, KROGER M, et al. Smooth full field reconstruction of velocity and its gradients from noisy scattered velocimetry data in a cross-slot flow[J]. Journal of Rheology, 2011, 55(2):353-377.
[20]	ZHAO J Q, ZENG P, PAN B, et al. Improved hermite finite element smoothing method for full-field strain measurement over arbitrary region of interest in digital image correlation(Article)[J]. Optics and Lasers in Engineering, 2012, 50(11):1662-1671.
[21]	GREEDIAC M, TOUSSAINT E, PIERRON F. Special virtual fields for the direct determination of material parameters with the virtual fields method.2-Application to in-plane properties[J]. International Journal of Solids and Structures, 2002, 39(10):2707-2730.
[22]	王岗亭, 郭保桥, 栾可迪, 等. 基于光偏折法的薄板振动变形场测量技术[J]. 实验力学, 2023, 38(3):347-359.
	WANG G T, GUO B Q, LUAN K D, et al. Measurement technology of thin plate vibration deformation field based on deflectometry[J]. Journal of Experimental Mechanics, 2023, 38(3):347-359. (in Chinese)