Online Parameter Identification of Linear Induction Motors Based on Improved Interconnected Full-Order Observer

FENG Fu; HU Hailin; ZHONG Deming; YANG Jie

doi:10.3969/j.issn.0258-2724.20230507

Volume 59 Issue 4

Jul. 2024

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Abstract

References

Journal of Southwest Jiaotong University > 2024 > 59(4): 776-785.

WEI Yapeng. Study on Uniaxial Tensile Test and Constitutive Model of Steel Structure Coating[J]. Journal of Southwest Jiaotong University. doi: 10.3969/j.issn.0258-2724.20230591

Citation:

FENG Fu, HU Hailin, ZHONG Deming, YANG Jie. Online Parameter Identification of Linear Induction Motors Based on Improved Interconnected Full-Order Observer[J]. Journal of Southwest Jiaotong University, 2024, 59(4): 776-785. doi: 10.3969/j.issn.0258-2724.20230507

WEI Yapeng. Study on Uniaxial Tensile Test and Constitutive Model of Steel Structure Coating[J]. Journal of Southwest Jiaotong University. doi: 10.3969/j.issn.0258-2724.20230591

Citation:

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Online Parameter Identification of Linear Induction Motors Based on Improved Interconnected Full-Order Observer

doi: 10.3969/j.issn.0258-2724.20230507

FENG Fu^{1, 2
,},
HU Hailin^{1, 3
,
,},
ZHONG Deming^{1, 2},
YANG Jie^{1, 2}

1.
School of Electrical Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou 341000, China
2.
Key Laboratory of Maglev Technology of Jiangxi Province, Jiangxi University of Science and Technology, Ganzhou 341000, China
3.
Key Laboratory of Railway Industry of Maglev Technology, Shanghai 710072, China

Received Date: 30 Sep 2023
Rev Recd Date: 05 Apr 2024

Available Online: 11 May 2024

Publish Date: 17 Apr 2024

Abstract

Abstract

Due to the special structure and dynamic end effect of linear induction motors, the change mechanism and law of their excitation inductance and secondary loss resistance are complicated. In order to improve the identification accuracy and performance of the observer for excitation inductance and secondary loss resistance, an online dual-parameter identification method of linear induction motors based on an improved interconnected full-order observer was proposed. Firstly, based on the T-type equivalent circuit of the linear induction motor considering dynamic end effects, the state space equations with dual-parameter changes were established, and the influence of parameter changes and coupling characteristics on motor poles was analyzed. Secondly, to reduce the impact of parameter coupling on identification accuracy, a low-coupling identification model with dual-parameter interconnection was established, and an interconnected full-order adaptive observer was designed. The adaptive laws for online identification of excitation inductance and secondary resistance were derived using Popov hyperstability theory, realizing online dual-parameter identification. Then, to improve the stability and convergence speed of the observer, a feedback gain matrix was derived and designed by using a novel pole configuration method. Finally, a simulation model and hardware-in-the-loop identification model were built for experimental verification. The results show that the new full-order adaptive observer achieves excitation inductance and loss resistance identification errors of around 0.01% during the startup acceleration phase and around 0.03% during dynamic loading.
- linear induction motor,
- full-order observer,
- parameter identification,
- pole configuration

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涂料被喷涂至钢桥构件的表面，并形成一定厚度的涂装层（涂层），直接将钢与腐蚀环境隔开，形成一道防腐蚀保护屏障，通过钝化缓蚀和阴极保护作用，推迟腐蚀介质与钢相接触时间. 目前，国内外钢桥的涂层体系多采用3层或4层体系，其中以富锌底漆、环氧中间漆和耐候性面漆组成的三层体系（重防腐涂装）最为常见^[1]，其中富锌底漆主要保障与钢基材的良好附着性，中间漆主要起增加厚度和提供柔韧性作用，面漆抵抗腐蚀介质和耐候性（包括美观）^[3].

钢结构桥梁涂装层厚度小且柔度高，具有明显的非线性特性,因此对涂层的本构模型、弹性模量以及泊松比等力学性能参数的研究对钢结构桥梁涂装层的设计具有重要意义. 目前国内外学者对涂层的力学性能展开了大量的试验研究和理论分析. 蔺鹏臻等^[4]利用有限元分析方法和断裂理论对重防腐涂层的界面应力与破坏准则进行研究，表明风沙冲蚀作用下涂层的界面应力破坏准则与摩尔-库伦准则相似. Hirokazu^[5]对PTFE（Polytetrafluoroethylene）膜材料进行了不同比例的双轴拉伸试验，利用分段线性函数拟合了PTFE膜的应力-应变关系曲线，通过最小二乘法求解了PTFE膜的弹性模量和泊松比等力学性能参数. Bogner^[6]通过应力-应变线性增量模型得到弹性模量计算公式. 许荔等^[7]通过对纳米涂层复合材料进行纳米划痕和纳米压痕试验得到，在试件表面进行纳米涂层喷涂可以改善试件的弹性模量和硬度. 陈辉等^[8-9]通过等粒子喷涂的方法制备了纳米涂层和普通涂层，并用试验的方法分别测量了其力学性能参数，表明纳米涂层的硬度、弹性模量、断裂韧性和结合强度均高于普通涂层. 易洪雷等^[10]通过对PES/PVC（Polyethersulfone/Polyvinyl Chloride ）膜同一平面内7个不同方向的单轴拉伸试验，证明了经过3次循环加载后的应力-应变曲线能较准确表征PES/PVC膜的弹性力学性质. 刘平等^[11]以SH-1050P膜材为试验对象进行了单、双轴拉伸试验，研究发现在单轴拉伸试验下SH-1050P膜满足双线性本构关系，并且可按应变的2%划分为弹性和塑性2个阶段，在应力比为1∶1的双轴拉伸试验条件下，SH-1050P膜的双轴弹性模量与单轴相同. 于沁灵等^[12]通过对Ferrari 1202S2膜进行多次双轴拉伸试验，利用不同的数据处理方式分别建立了双轴线性本构模型和双轴非线性本构模型. 王利钢等^[13]基于自主研制的拉伸试验机，研究了P-PVC-PVDF（Polyester fabric-polyvinyl chloride-polyvinylidene fluoride）膜材的剪切力学性能，并提出了一种改进的剪应力计算方法. 李阳^[14]利用图框法对PVC涂层膜材进行单轴拉伸试验，得出了聚酯织物膜材的剪切模量. 许珊珊等^[15-16]通过对PTFE膜材11个偏轴角度和6种拉伸速率工况下的试验研究，得出了3种破坏模式和不同工况下的膜材破坏机理.

综上所述，国内外学者对涂层的力学性能做了较多的工作，但对于钢结构桥梁底漆、中间漆和面漆复合涂层的本构模型及力学性能参数的研究还相对较少. 为此，本文以钢结构桥梁长效型涂装体系为试验对象分别进行单轴拉伸试验，研究了面漆、中间漆、底漆和复合涂层力学本构方程的统一表达形式，并针对每种漆膜的适用条件给出了具体的本构方程；得出了每种漆膜的弹性模量、泊松比、剪切模量等力学性能参数.

1. 试验材料与试验方法

1.1 试验材料与设备

本次试验采用钢结构桥梁长效型涂装体系，其底漆、中间漆和面漆的组分和配比见表1.

表 1 钢结构桥梁长效型涂装体系组分及配比

Table 1. Components and ratios of long-lasting coating system for steel bridge

对象	A组分	B组分	比例
底漆	H06-X环氧富锌底漆（含锌量80%）	H06-X固化剂	12.8∶1
中间漆	H06-C2环氧厚浆云母氧化铁中间漆	H06-C2固化剂	13.5∶1
面漆	E01-JY氟碳面漆	E01-JY固化剂	10∶1

下载: 导出CSV

| 显示表格

单轴拉伸试验机型号为WDW-J2，拉伸试验力量程为0～500 N，位移分辨率为0.01 mm，位移速度控制范围为0.01～500 mm/min. 采用高精度涂层测厚仪，量程为0～1250 μm. 采用聚酯膜制备长效型涂装体系的底漆膜、中间漆膜、面漆膜和复合涂层漆膜.

1.2 试验方法

按表1所示配比分别配制长效型涂装体系的底漆膜、中间漆膜、面漆膜及复合涂层漆膜，利用刮膜器在聚酯膜上制备以上4种漆膜，然后按照国际标准^[17]规定，每种漆膜制备3个试件，每个拉伸试件初始宽度l_b=20 mm，总长度l₀=150 mm，有效拉伸长度（标距）l=50 mm，夹持端长度取为50 mm，如图1所示. 试件在温度为25 ℃±2 ℃，相对湿度为65%±2%的条件下养护7 d. 利用高精度涂层测厚仪分别测试每个试件有效拉伸段的平均厚度t，然后利用单轴拉伸试验机以2 mm/min的拉伸速率对漆膜进行单向拉伸试验.

图 1 标准漆膜试件

Figure 1. Sample of standard coating film

下载: 全尺寸图片幻灯片

2. 涂层拉伸试验结果分析

2.1 拉伸试验的应力-应变曲线

将单轴拉伸试验荷载值T和漆膜纵向变形量$\Delta l$分别代入式（1）和式（2）中，求出每种漆膜所对应的拉伸应力σ和纵向拉伸应变ε（若无特别说明，下文中的应变均指纵向应变）.

$\sigma = \frac{T}{A},$

(1)

$\varepsilon = \frac{{\Delta l}}{l},$

(2)

式中： A为漆膜的横截面积，$A = l_{\mathrm{b}} t$.

图2给出了长效型涂装体系的底漆、中间漆、面漆以及复合涂层的应力-应变实测曲线，图中：ε_m为漆膜试件达到峰值应力σ_m时的纵向应变，ε_b为极限应变.

图 2 涂层拉伸的应力-应变实测曲线

Figure 2. Measured stress–strain curves for coatings in tension

下载: 全尺寸图片幻灯片

由图2可知：涂层在拉伸受力时，应力-应变曲线均包括明显的上升段和下降段，说明涂层材料均具有较好的韧性. 底漆和复合涂层的应力-应变曲线可分为弹塑性阶段、应变强化阶段和破坏阶段；中间漆可分为应变强化阶段和破坏阶段；面漆可分为近似线弹性阶段和破坏阶段.

2.2 涂层应力-应变曲线机理分析

涂层类高分子聚合物的应力-应变曲线依据应力转折点和应力峰值点可将其分为近似弹性阶段、弹塑性阶段、应变强化阶段和破坏阶段. 根据涂层的应力和应变特点以及涂层在达到转折点后的破环情况，可将涂层分为硬而脆、硬而韧、硬而强、软而韧、软而弱^[18].

底漆和复合涂层的弹性模量大而应变小，呈现出硬而脆的变形特点，在应力达到转折点之前，涂层先发生弹性变形后发生塑性变形，可认为涂层处于弹塑性阶段. 当涂层应力达到转折点之后，涂层原有的晶体结构被破坏，重组成强度更高，变形能力更强的微纤维束结构^[19]，随着变形的增加，应力也在逐渐增加，没有明显的屈服阶段，可认为涂层处于应变强化阶段. 在达到峰值应力后，漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力急剧下降，当应变达到极限应变时试样最终断裂破坏，可定义为破坏阶段.

面漆的弹性模量小而应变大，呈现出软而韧的变形特点，在涂层应力达到峰值应力之前，涂层发生近似弹性变形，近似满足胡克定律，可认为是近似弹性阶段. 在达到峰值应力后，漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力急剧下降，当应变达到极限应变时试样最终断裂破坏，可定义为破坏阶段.

中间漆具有较强的弹性模量和变形能力，呈现出硬而强的变形特点，涂层的弹塑性阶段非常短暂，迅速进入应变强化阶段，在达到峰值应力之后，进入破坏阶段.

2.3 涂层应力-应变曲线特征

1）底漆的应力-应变特征

由图2（a）可知，H06-X环氧富锌底漆（含锌量80%）的应力-应变曲线可分为3段. 在弹塑性阶段（a₁b₁段），漆膜应力增长较小，并且按照二次多项式规律先发生弹性变形，后发生塑性变形，b₁点为转折点，该点处应变ε₀=0.235ε_m，应力σ₀=0.077σ_m；然后漆膜进入应变强化阶段（b₁c₁段），漆膜的应力-应变曲线按照二次函数关系具有较大程度的增长，在c₁点达到峰值应力σ_m，与之相对应的应变ε_m为0.021；在达到峰值应力后，进入破坏阶段（c₁d₁段），漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力逐渐下降，当达到极限应变ε_b时漆膜最终断裂破坏. 因此，H06-X环氧富锌底漆（含锌量80%）上升段的本构方程按照最小二乘法拟合为

$\sigma=\left\{ \begin{aligned} &55246.647{\varepsilon }^{2}-99.227\varepsilon + 0.004,\\ &\quad R^2 = 0.998, \;\varepsilon \leqslant {\varepsilon }_{0},\;a_1b_1段,\\ & -18706.511{\varepsilon }^{2} + 1125.877\varepsilon -4.313,\\ &\quad R^2 = 0.985,\;{\varepsilon }_{0} < \varepsilon \leqslant {\varepsilon }_{\text{m}},\;b_1c_1段. \end{aligned}\right.$

(3)

2）中间漆的应力-应变特征

由图2（b）可知，H06-C2环氧厚浆云母氧化铁中间漆的应力-应变曲线可分为2段. 在拉伸过程中漆膜的弹塑性阶段非常短暂，迅速进入应变强化阶段（a₂b₂段），漆膜的应力-应变曲线按二次多项式规律具有较大程度的增长，在b₂点达到峰值应力σ_m，与之对应的应变ε_m=0.075；在达到峰值应力后，进入破坏阶段（b₂c₂段），漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力逐渐下降，当应变达到极限应变ε_b时漆膜最终断裂破坏. 因此，H06-C2环氧厚浆云母氧化铁中间漆上升段的本构方程按照最小二乘法拟合为

$\begin{split} &\sigma =-\text{597}\text{.587}{\varepsilon }^{2}\text{+116}\text{.028}\varepsilon -\text0.0\text{30}, \\ &\quad R^2 = 0.999,\;a_2b_2段\text{ ， }\varepsilon \leqslant {\varepsilon }_{\text{m}}. \end{split}$

(4)

3）面漆的应力-应变特征

由图2（c）可知，E01-JY氟碳面漆的应力-应变曲线可分为两段. 在近似线弹性阶段（a₃b₃段），应力-应变曲线按线性规律变化，在b₃点达到峰值应力σ_m，与之相对应的应变ε_m为0.248；在达到峰值应力后，进入破坏阶段（b₃c₃段），漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力急剧下降，当应变达到极限应变ε_b时试样最终断裂破坏. 因此，E01-JY氟碳面漆上升段的本构方程按照最小二乘法拟合为

$\sigma =\text{9}\text{.76}\varepsilon -0.053,\quad R^2 =0.998,\; a_3b_3段\text{ , }\varepsilon \leqslant {\varepsilon }_{\text{m}}.$

(5)

4）复合涂层的应力-应变特征

由图2（d）可知，由H06-X环氧富锌底漆（含锌量80%）、H06-C2环氧厚浆云母氧化铁中间漆和E01-JY氟碳面漆组成的复合涂层的应力-应变曲线可分为三段. 在弹塑性阶段（a₄b₄段），漆膜应力增长较小，并且按照二次多项式规律先发生弹性变形，后发生塑性变形，b₄点为转折点，在该阶段应变ε₀为0.163ε_m，应力σ₀=0.225σ_m；然后漆膜进入应变强化阶段（b₄c₄段），应力-应变曲线按照二次函数关系具有较大程度的增长，在c₄点达到峰值应力σ_m，与之相对应的应变ε_m=0.049；在达到峰值应力后，进入破坏阶段（c₄d₄段），漆膜的有效拉伸段开始逐渐出现裂纹，随着应变的增加应力逐渐下降，当达到极限应变ε_b时漆膜最终断裂破坏. 因此，长效型复合涂层上升段的本构方程按照最小二乘法拟合为

$\sigma =\left\{\begin{aligned} &\sigma =\text{16 970}\text{.810}{\varepsilon }^{2}-\text{3}\text{.729}\varepsilon + \text{4}\text{.602},\\ &\quad R^2 = 0.998,\; a_4b_4段,\;\varepsilon \leqslant {\varepsilon }_{0},\\ & \sigma =-\text{2 023}\text{.240}{\varepsilon }^{2} + 1\text{99}\text{.523}\varepsilon -\text{0}\text{.296 },\\ &\quad R^2 = 0.999,\; b_4c_4段,\;{\varepsilon }_{0} < \varepsilon \leqslant {\varepsilon }_{\text{m}}. \end{aligned}\right.$

(6)

3. 涂层拉伸试验结果分析

3.1 拉伸试验的应力-应变曲线

为获得涂层的弹性模量、泊松比（υ）、剪切模量、单轴拉伸强度及拉伸断裂应变等力学性能参数，依据2.1节的涂层应力-应变实测曲线，采用峰值应力σ_m处的割线模量E_t作为涂层弹性模量E的代表值，见式（7）. H06-X环氧富锌底漆（含锌量80%）、H06-C2环氧厚浆云母氧化铁中间漆、E01-JY氟碳面漆和复合涂层的弹性模型值见表2.

表 2 涂层力学性能参数

Table 2. Mechanical property parameters for coatings

涂层种类	E/MPa	υ	G/MPa	σ_m /MPa	ε_m	ε_b
H06-X底漆	535.7	0.20	223.2	11.0	0.021	0.026
H06-C2中间漆	71.5	0.06	33.8	5.4	0.075	0.086
E01-JY面漆	9.9	0.03	4.8	2.5	0.248	0.289
复合涂层	95.1	0.13	41.9	4.7	0.049	0.062

下载: 导出CSV

| 显示表格

$E = {E_{\text{t}}} = \frac{{{\sigma _{\text{m}}}}}{{{\varepsilon _{\text{m}}}}}.$

(7)

通过试验测得当拉伸应力达到峰值应力时漆膜试件的宽度为l_bm，然后利用式（8）计算出H06-X环氧富锌底漆（80%锌含）、H06-C2环氧厚浆云母氧化铁中间漆、E01-JY氟碳面漆和复合涂层的泊松比值，见表2.

$\upsilon =\frac{\left|{\varepsilon }_{{\mathrm{h}}}\right|}{{\varepsilon }_{\text{m}}}=\frac{\left|{l}_{\text{bm}}-l_{\mathrm{b}}\right|}{{\varepsilon }_{\text{m}}},$

(8)

式中：ε_h为漆膜的横向应变；l_b为漆膜试件的初始宽度.

H06-X环氧富锌底漆（含锌量80%）、H06-C2环氧厚浆云母氧化铁中间漆、E01-JY氟碳面漆和复合涂层的剪切模量G可利用式（9）得出，结果见表2.

$G=\frac{E}{2(1 + \upsilon )}.$

(9)

依据图2所示的涂层应力-应变实测曲线，定义峰值应力σ_m为涂层的单轴拉伸强度，极限应变ε_b为涂层的拉伸断裂应变，则底漆、中间漆、面漆和复合涂层的单轴拉伸强度σ_m、单轴拉伸强度σ_m所对应的应变ε_m以及涂层的拉伸断裂应变ε_b见表2.

3.2 涂层力学本构模型

对H06-X环氧富锌底漆（含锌量80%）、H06-C2环氧厚浆云母氧化铁中间漆、E01-JY氟碳面漆和复合涂层的应力-应变曲线的上升段进行对比分析，以期获得统一形式的上升段本构方程. 首先，将式（3）～（6）进行无量纲化处理，然后，将4种漆膜上升段的本构方程统一归纳成式（10）.

$\frac{\sigma }{{{\sigma _{\text{m}}}}} = \alpha {\left( {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} \right)^2} + \beta {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} + \gamma.$

(10)

1）对于H06-X环氧富锌底漆（含锌量80%）和复合涂层，即当ε₀>0时，有

$\frac{\sigma }{{{\sigma _{\text{m}}}}} =\left\{ \begin{aligned} \alpha {\left( {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} \right)^2} + \beta {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} + \gamma,\quad {\text{ 0 < }}\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}} \leqslant \frac{{{\varepsilon _0}}}{{{\varepsilon _{\text{m}}}}}, \\ \eta {\left( {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} \right)^2} + \kappa {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} + \lambda,\quad {\text{ }}\frac{{{\varepsilon _0}}}{{{\varepsilon _{\text{m}}}}}{\text{ < }}\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}} \leqslant 1 . \end{aligned}\right.$

(11)

2）对于H06-C2环氧厚浆云母氧化铁中间漆，即当ε₀=0时，有

$\frac{\sigma }{{{\sigma _{\text{m}}}}} = \alpha {\left( {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} \right)^2} + \beta {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} + \gamma，\quad {\text{ 0 = }}\frac{{{\varepsilon _0}}}{{{\varepsilon _{\text{m}}}}}{\text{ < }}\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}} \leqslant 1.$

(12)

3）对于E01-JY氟碳面漆，即当ε₀=0且a=0时，有

$\frac{\sigma }{{{\sigma _{\text{m}}}}} = \beta {\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}}} + \gamma，\quad {\text{ 0 = }}\frac{{{\varepsilon _0}}}{{{\varepsilon _{\text{m}}}}}{\text{ < }}\frac{\varepsilon }{{{\varepsilon _{\text{m}}}}} \leqslant 1,$

(13)

式（10）～（13）中：α和η为二次项系数，β和κ为一次项系数，γ和λ为常数项.

由图2可知：底漆、中间漆、面漆和复合涂层的应力-应变曲线的下降段相差较大，数据也比较离散，暂无明显规律可循. 因此为获得较为可靠的涂层应力-应变曲线下降段的变化特征，还需在确定标准试验方法的基础上进行大量试验研究.

3.3 涂层抗拉强度及变形能力对比分析

为了探究不同漆膜的单轴拉伸强度和变形能力，对钢结构桥梁长效型涂装体系的单轴拉伸强度值σ_m及其对应的应变ε_m进行了对比分析，如图3.

图 3 长效型涂装体系σ_m和ε_m对比分析

Figure 3. Comparative analysis of σ_m and ε_m of long-lasting coating system

下载: 全尺寸图片幻灯片

由图3（a）可知：H06-X环氧富锌底漆（含锌量80%）、H06-C2环氧厚浆云母氧化铁中间漆、长效型复合涂层的单轴拉伸强度值分别为E01-JY氟碳面漆的450.4%、220.0%、102.2%和189.7%. 因此，钢结构桥梁涂装体系的单轴拉伸强度底漆最强、中间漆次之，面漆最差；长效型复合涂层的单轴拉伸强度接近中间漆.

由图3（b）可知：H06-C2环氧厚浆云母氧化铁中间漆、E01-JY氟碳面漆和长效型复合涂层的单轴拉伸强度所对应的应变ε_m分别为H06-X环氧富锌底漆（含锌量80%）的365.9%、1205.1%和237.3%. 因此，钢结构桥梁涂层的变形性能面漆最强、中间漆次之，底漆最差；长效型复合涂层的变形性能接近中间漆.

4. 结　论

1） H06-X环氧富锌底漆（含锌量80%）和长效型复合涂层的应力-应变关系分为弹塑型阶段、应变强化阶段和破坏阶段，并且在弹塑性阶段和应变强化阶段的表达式均满足二次多项式.

2） H06-C2环氧厚浆云母氧化铁中间漆的应力-应变曲线分为应变强化阶段和破坏阶段，并且在应变强化阶段的表达式可用一元二次函数表征.

3） E01-JY氟碳面漆的应力-应变曲线分为近似线弹性阶段和破坏阶段，并且近似线弹性阶段的表达式满足线性关系.

4）将长效型涂装体系上升段的本构方程归纳成了统一形式，并且针对每种漆膜的适用条件给出了具体的表达式.

5）得出了底漆、中间漆、面漆和复合涂层的弹性模量、泊松比、剪切模量、单轴拉伸强度和拉伸断裂应变等力学性能参数.

6钢结构桥梁长效型涂装体系单轴拉伸强度底漆最强、中间漆次之，面漆最差，复合涂层的单轴拉伸强度接近中间漆；变形性能面漆最强、中间漆次之，底漆最差，复合涂层的变形性能接近中间漆.

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