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 引用本文: 张大羽, 罗建军, 王辉, 马小飞. ANCF/CRBF平面梁闭锁问题及闭锁缓解研究[J]. 力学学报, 2021, 53(3): 874-889.
Zhang Dayu, Luo Jianjun, Wang Hui, Ma Xiaofei. LOCKING PROBLEM AND LOCKING ALLEVIATION OF ANCF/CRBF PLANAR BEAM ELEMENTS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 874-889.
 Citation: Zhang Dayu, Luo Jianjun, Wang Hui, Ma Xiaofei. LOCKING PROBLEM AND LOCKING ALLEVIATION OF ANCF/CRBF PLANAR BEAM ELEMENTS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 874-889.

## LOCKING PROBLEM AND LOCKING ALLEVIATION OF ANCF/CRBF PLANAR BEAM ELEMENTS

• 摘要: 本文系统地研究了基于一致旋转场列式的绝对节点坐标 (ANCF consistentrotation-based formulation, ANCF/CRBF)平面梁单元的泊松闭锁问题及闭锁缓解技术.为了全面理解该类型单元的闭锁特性及明确单元的应用范围,文中首先开发了两种新的ANCF/CRBF刚性截面梁单元, 新单元在ANCF全参数梁的基础上,对梯度向量施加正交矩阵约束, 得到梯度与转角对时间导数之间的速度转换矩阵,从而引入转角参数. 新单元节点处完全消除了泊松闭锁和剪切效应,这是与传统ANCF/CRBF刚性截面梁单元的不同之处. 然后,对比分析了这三种ANCF/CRBF刚性截面梁单元泊松闭锁的特点.发现该类型单元对节点的横向梯度施加了运动学约束, 导致节点处截面不能变形,无法捕捉泊松效应, 但是单元内部能完全捕捉,这种不连续情况会加重单元整体的泊松闭锁问题. 并且发现对单元梯度约束的越多,闭锁问题越严重. 随后, 分别采用两种闭锁缓解技术, 弹性线方法和应变分解方法,进一步研究了单元的收敛性. 最终,通过多种静力学和动力学测试研究了泊松闭锁对ANCF/CRBF平面梁单元计算精度的影响及闭锁缓解技术在该类型单元上的缓解效果.

Abstract: In this paper, the Poisson locking on the consistent rotation-based formulation (CRBF) of the planar beam element based on the absolute nodal coordinate formulation (ANCF) kinematic description is discussed. First of all, in order to fully understand the locking problem of ANCF/CRBF element, two new ANCF/CRBF planar beams are developed by constraining all of the position-vector gradients of ANCF planar fully-parameterized beam with an orthogonal matrix. Using this orthogonal matrix, a nonlinear velocity transformation matrix is evaluated to write the time derivatives of the ANCF gradients at the nodes in terms of the time derivatives of the rotation parameter. Two new ANCF/CRBF beams have a rigid cross-section and no shear effect. One of the new ANCF/CRBF beams has two position vectors, one rotation angle and one axial extensibility parameter as the nodal coordinates, while the other does not have a longitudinal extensibility parameter. Then, the difference in the Poisson locking problem among the two new ANCF/CRBF beams and traditional ANCF/CRBF beam is discussed. It can be concluded that since the constraints imposed on the position-gradient vectors, ANCF/CRBF beam has an insufficient Poisson effect on the nodes and a sufficient Possion effect in the interior of the element. This inconsistent Poisson effect will lead to a more severe locking problem than ANCF fully-parameterized elements. Furthermore, the locking problem increases with the number of constraints on the nodal gradients, that is ANCF/CRBF new beams have a more serious locking effect compared to traditional ANCF/CRBF beam. Finally, in order to achieve a better convergence of the new elements, two locking alleviation methods, Elastic Center Line (ECL) and Strain Split Method (SSM), are used. The locking problem on the performance of ANCF/CRBF beams is tested using several examples that include static and dynamic examples, in order to identify the scope of applicability of such elements. The numerical results obtained from ANCF/CRBF beams are compared with the ANCF beam, and with the conventional beam implemented in a commercial finite element software LS-DYNA.

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