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Element failure correction method for UH model
LUO Ting, SHU Wenjun, YAO Yangping
School of Transportation Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-04-20; Accepted: 2016-07-22; Published online: 2016-08-23 10:21
Foundation item: National Basic Research Program of China (2014CB047001); National Natural Science Foundation of China (51579005, 11272031)
Corresponding author. LUO Ting, E-mail:tluo@buaa.edu.cn
Abstract: The problem of some elements' damage often appears when finite element calculation of using UH model UMAT is conducted, including tension failure and shear failure. The failure stress state not only makes the results unreasonable, but also reduces the stability of calculation. To solve the problem generated by unreasonable failure stress state when the UH model's UMAT is used to conduct finite element analysis, based on certain assumptions and combined with stress transformation relationship under different coordinates, three-dimensional element failure correction formulas for UH model can be elicited. Then FORTRAN language is used to write the subroutine of element failure correction, and it is embedded in the UH model's UMAT to eliminate the unreasonable failure stress state and improve the stability of finite element calculation. Finally, an example of foundation pit excavation is used to verify the validity and rationality of this method.
Key words: UH model     UMAT     finite element     tension failure     shear failure     failure correction

1 UH模型及其有限元实现 1.1 UH模型

 图 1 UH模型的当前屈服面与参考屈服面 Fig. 1 Current yield surface and referential yield surface of UH model

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1.2 变换应力方法

UH模型UMAT中采用姚仰平等[11-12]提出的变换应力方法进行三维化 (见图 2)，变换应力与真实应力之间的映射关系为

 图 2 变换应力空间 Fig. 2 Transformed stress space
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1.3 UH模型的UMAT

UH模型的UMAT[13-14]是在UH模型理论基础上，引入了变换应力三维化方法，并加入黏聚力c和弹塑性刚度矩阵对称化内容，采用FORTRAN语言编写而成的，UH模型UMAT采用半隐式回映应力更新算法，具体的计算流程如图 3所示。

 图 3 UH模型UMAT计算流程图 Fig. 3 Computation flowchart of UH model's UMAT
2 单元拉裂修正

2.1 平面问题的单元拉裂修正

 图 4 平面问题的单元拉裂修正 Fig. 4 Element tension failure correction of plane problem
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2.2 适用于UH模型的三维问题单元拉裂修正

2.2.1 主应力与应力分量之间的变换关系

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 坐标轴 x y z x′ l1 m1 n1 y′ l2 m2 n2 z′ l3 m3 n3

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2.2.2 拉裂修正原理

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3 单元剪坏修正

3.1 平面问题的单元剪坏修正

 图 5 平面问题的单元剪坏修正 Fig. 5 Element shear failure correction of plane problem
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3.2 适用于UH模型的三维问题单元剪坏修正

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4 破坏修正的有限元验证

UH模型UMAT的单元验证可以参见参考文献[6]，本文不再赘述。本文将采用一个有限元计算基坑开挖的例子来说明拉裂修正和剪坏修正的有效性。UH模型UMAT是基于三维单元编写的，这里通过对边界条件的特殊设置来实现用三维单元描述平面问题。基坑的各个部分几何尺寸如图 6所示。土材料的本构模型选用UH模型，模型参数取值如表 2所示。表中：ν为泊松比；Ne-lnp空间中正常压缩线的截距 (即当p=1时的孔隙比)。支护桩采用线弹性模型，弹性模量E=30 GPa，泊松比ν=0.1。土体与支护桩之间采用简单的绑定约束。图 7为计算模型的网格划分，土单元与墙单元均采用8节点六面体实体单元，单元总数为1 380，开挖区土单元数为150，墙单元数为20。

 图 6 计算模型的几何尺寸 Fig. 6 Geometric dimension of computation model
 图 7 计算模型网格划分 Fig. 7 Computation model meshing

 参数 M ν κ λ e0 N c/kPa 数值 0.984 0.3 0.016 0.09 0.6 1.15 10

 图 8 有无破坏修正的土体等值云图 Fig. 8 contour of soils with and without failure correction

 变量 1 244号单元第1积分点 18号单元第4积分点 修正前 修正后 修正前 修正后 32.53 32.53 42.11 42.11 7.39 23.97 5.07 23.93 -14.16 21.44 -7.47 21.44 σx/kPa -11.16 3.51 -15.61 2.86 σy/kPa -35.600 0.007 -28.910 0 σz/kPa 8.19 10.10 19.92 20.30 τxz/kPa -8.03 -2.74 -5.24 -2.57 4.714 0.387 3.373 0.670 1.908 1.908 1.801 1.801 -0.223 29.290 -0.444 10.830

 图 9 有无破坏修正的土体的等值云图 Fig. 9 contour of soils with and without failure correction
 图 10 有无破坏修正的土体的等值云图 Fig. 10 contour of soils with and without failure correction

 图 11 有无破坏修正的土体的等值云图 Fig. 11 contour of soils with and without failure correction

 图 12 有无破坏修正的土体的竖向位移等值云图 Fig. 12 Vertical displacement contour of soils with and without failure correction

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5 结论

1) 针对在有限元计算过程中出现的局部单元破坏问题，本文介绍了一种适用于UH模型的三维问题单元破坏修正方法，该方法不仅可以消除有限元计算中出现的不合理的破坏应力状态，以及由于这种应力状态而导致的计算稳定性问题，而且该方法并没有改变原有UH模型UMAT的算法流程，不会由于这种修正而导致原有UH模型的UMAT产生新的理论上的问题。

2) 本文所述的单元破坏修正方法对于有限元计算收敛性是存在影响的，由于修正过程强制修改了应力分量，会产生了新的不平衡力，理论上会增加有限元迭代的步数。但一方面这种修正一般只是对局部少数单元的修正，因此产生的不平衡力一般不会很大；另一方面，修正之后的应力如果变得更为合理，也有可能降低了迭代的不平衡力，降低有限元计算整体的迭代步数，提高计算的收敛性。

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#### 文章信息

LUO Ting, SHU Wenjun, YAO Yangping

Element failure correction method for UH model

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(4): 667-675
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0321