﻿ 一种双向曲率板成型应变分布的力学计算方法
 舰船科学技术  2017, Vol. 39 Issue (9): 6-11 PDF

Calculating strain field to forming a double-curved shape based on mechanic theory
HU Chang-cheng, ZHAO Yao, YUAN Hua
School of Naval Architecture and Ocean Engineering Huazhong University of Science and Technology, Wuhan 430074, China
Abstract: The determination of process paths and parameters according to the strain field that required to forming a double-curved desired shape is an important work during the process design of line-heating and rolling forming in the process design of line heating and rolling forming. Therefore, the precision of Calculated strain field will affect the accuracy of process paths and parameters. This paper presents a method to calculate the strain filed required to develop a desired shape to a planar shape depend on mechanic theory. Firstly, nodal displacements is applied to the flat plate via elastic-plastic finite element analysis. Owing to the spring back, the shape calculated by elastic-plastic finite element analysis is different between the required shape, then the deviation between calculated shape and required shape is calculated, and reverse deformation is added to nodal displacements, this process continues until the deviation can meet the requirement of precision to terminate the loop. Finally, the obtained strain field is applied to flat model as initial strain to verify the accuracy of this method, and results shows that it has very high calculation precision, this method provide the foundation of the determination of process paths and parameters.
Key words: double-curved plate     strain field     mechanics methods     in-plane strain     bending strain
0 研究背景

1 总体方案

1）首先根据所需的目标形状，对目标形状曲面进行三维曲面拟合，并根据拟合的结果计算有限元模型中每个节点的位移场；

2）将得到的位移场以节点位移载荷的形式施加到弹塑性模型上，由于平板模型在发生大挠度变形的同时会伴随着板长和板宽方向的位移，因此在施加节点位移时只施加板厚方向的位移，在板长和板宽方向则处于自由状态，同时对整块板进行刚体位移约束；

3）利用Abaqus有限元软件进行弹塑性计算，进行平衡迭代计算出平板的变形，在此步骤中会计算回弹对变形的影响，得到的应变场和变形场均为回弹后的结果；

4）将得到的变形结果与所需形状进行对比，由于两模型存在着坐标不一致等问题，本文利用基于最小二乘的原理对模型进行空间坐标变换，实现两模型坐标的统一，并计算两者之间的总体偏差；

5）若总体偏差满足精度要求，则直接从有限元计算结果中提取应变结果，若不满足精度要求，则根据步骤3中得到的变形与所需形状之间的偏差对位移场施加反向变形，重新进行步骤1～步骤4中的过程，直至偏差满足精度要求。

 图 1 计算应变分布的力学方法流程图 Fig. 1 The flow chart for calculating strain distributions based on mechanical methods
2 计算实例 2.1 模型建立

 图 2 有限元模型及网格划分 Fig. 2 Finite element model and grid generation
2.2 位移场施加

 图 3 所需的目标形状 Fig. 3 The desired shape

 图 4 节点位移场施加 Fig. 4 The nodal displacement applied to the plate
2.3 弹塑性计算

 图 5 回弹前后塑性最大主应变分布 Fig. 5 Maximum principal strain distribution before and after springback
2.4 结果对比

 $\sigma = \sqrt {\frac{{\sum\limits_{i = 1}^n {{D_i}^2} }}{n}},$ (1)

 图 6 变形结果与目标形状对比 Fig. 6 Comparison of deformation results with target shape
2.5 反变形迭代修正

 $\Delta {U_i}{\rm{ = }}R(x,y,z){\rm{ - }}{U_i}(x,y,z)\text{。}$ (2)

 ${C_{i + 1}}(x,y,z) = k \cdot \Delta {U_i},$ (3)

 \begin{aligned}{T_{i + 1}}(x,y,z)= & {T_i}(x,y,z) + {C_{i + 1}}(x,y,z)=\\& {T_i}(x,y,z) + k \cdot [R(x,y,z) - {U_i}(x,y,z)]\text{。}\end{aligned} (4)

 图 7 反变形迭代修正流程图 Fig. 7 The flow chart for iterative correction of inverse deformation

 图 8 成型形状与迭代次数之间关系 Fig. 8 The relationship between deformation and iteration number

 图 9 迭代计算完成后得到的塑性应变分布 Fig. 9 The plastic strain distribution after the iterative calculation
3 计算结果验证

 图 10 初应变法得到的形状与所需形状对比（单位：mm） Fig. 10 Comparison between the required shape and the deformation obtained by initial strain method
4 结　语

1）以节点位移的形式可以将目标形状作为位移载荷输入到有限元模型中，进行弹塑性计算可以得到应变和变形结果；

2）使用反变形迭代修正法可以减少回弹对变形的影响，提高应变分布的精度，随着迭代次数的增加，回弹后的形状逐步逼近于所需的形状，计算得到的应变场精度也随之提高；

3）通过初应变方法对计算得到的应变分布进行弹性计算可以得到应变分布对应的变形，结果表明本文介绍的方法可以精确地计算得到所需形状的应变场，为后续加工路径和工艺参数的确定提供了基础。

 [1] JIN. Process design of laser forming for three-dimensional thin plates[J]. Transactions of the ASME. Journal of Manufacturing Science and Engineering, 2004, 126 (2): 217–25. DOI: 10.1115/1.1751187 [2] XU Zhao-kang. Discuss on Forming Method for Complex Curved Plate and its Applicability[J]. Journal of Wuhan Institute of Shipbuilding Technology, 2006 (03): 9–12. [3] UEDA. Development of computer-aided process planning system for plate bending by line heating (report 1): Relation between the Final Form of Plate and the Inherent Strain(Machanics, Strength & Structural Design)[J]. Transactions of JWRI, 1991, 20 (2): 275–285. [4] UEDA. Development of computer-aided process planning system for plate bending by line heating (report 2). Practice for plate bending in shipyard viewed from aspect of inherent strain[J]. Journal of Ship Production, 1994, 10 (4): 239–239. [5] UEDA. Development of computer-aided process planning system for plate bending by line heating (report 3). Relation between heating condition and deformation[J]. Journal of Ship Production, 1994, 10 (4): 248–248. [6] LETCHER. LOFTING AND FABRICATION OF COMPOUND-CURVED PLATES[J]. Journal of Ship Research, 1993, 37 (2): 166–175. [7] SHIN. Nonlinear Kinematic Analysis of the Deformation of Plates for Ship Hull Fabrication[J]. Journal of Ship Research, 2000, 44 (3): 270–277. [8] YU. Optimal development of doubly curved surfaces[J]. Computer Aided Geometric Design, 2000, 17 (6): 545–577. DOI: 10.1016/S0167-8396(00)00017-0 [9] ZHANG Xue-biao. Research on ship-hull curved plate forming by pure heating [D]. Dalian: Dalian University of Trchnology, 2006. [10] LIU. FEM-Based Process Design for Laser Forming of Doubly Curved Shapes[J]. Journal of Manufacturing Processes, 2005, 7 (2): 109–121. DOI: 10.1016/S1526-6125(05)70088-8 [11] ZHANG Xue-chang, XI Jun-tong, YAN Jun-qi. Study of inspection technologies based on point cloud for complex surfaces and its system development [D]. Shanghai: Computer Integrated Manufacturing System, 2005(5): 770–774, 780.