﻿ 非规则颗粒与滚筒球磨机相互作用的离散元–有限元耦合分析
 上海理工大学学报  2023, Vol. 45 Issue (6): 543-551, 601 PDF

A DEM-FEM coupled analysis for the interaction between the irregular granular materials and tumbling mill
LI Tianze, WANG Siqiang, YANG Dongbao, JI Shunying
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
Key words: discrete element method     finite element method     coupled algorithm     superquadric element     irregular granular materials

1 超二次曲面DEM-FEM耦合算法

1.1 基于超二次曲面方程的非规则颗粒构造

 ${f\left(x,y,z\right)=\left({\left|\frac{x}{a}\right|}^{{n}_{2}}+{\left|\frac{y}{b}\right|}^{{n}_{2}}\right)}^{{n}_{1}/{n}_{2}}+{\left|\frac{z}{c}\right|}^{{n}_{1}}-1=0$ (1)

 $\left\{\begin{array}{l}{\alpha }_{1}={\left(\left|b{n}_{y}\right|/\left|a{n}_{x}\right|\right)}^{1/\left({n}_{2}-1\right)}\\ {\gamma }_{1}={\left(1+{\alpha }_{1}^{{n}_{2}}\right)}^{{n}_{1}/{n}_{2}-1}\\ {\beta }_{1}={\left({\gamma }_{1}\left|{n}_{z}c\right|/\left|{n}_{x}a\right|\right)}^{1/\left({n}_{1}-1\right)}\\ x=a\Big/{\left[{\left(1+{\alpha }_{1}^{{n}_{2}}\right)}^{{n}_{1}/{n}_{2}}+{\beta }_{1}^{{n}_{1}}\right]}^{1/{n}_{1}}{\text{·}} {\rm{sign}}\left({n}_{x}\right)\\ y={\alpha }_{1}b\left|x\right|/a{\text{·}} {\rm{sign}}\left({n}_{y}\right)\\ z={\beta }_{1}c\left|x\right|/a{\text{·}} {\rm{sign}}\left({n}_{z}\right)\end{array}\right.$ (5)

 ${{{\boldsymbol{\delta}} }}_{\mathrm{n}}=\left[\left({{{\boldsymbol{x}}}}_{A}-{{\boldsymbol{x}}}\right){\text{·}} {{{\boldsymbol{n}}}}_{\mathrm{w}}\right]{\text{·}} {{{\boldsymbol{n}}}}_{\mathrm{w}}$ (6)

1.3 超二次曲面颗粒的非线性接触力模型

 ${{\boldsymbol{F}}}_{\mathrm{n}}^{\mathrm{e}}=\frac{4}{3}{E}^{*}\sqrt{{r}^{*}}{\left({{\boldsymbol{\delta }}}_{\mathrm{n}}\right)}^{\frac{3}{2}}$ (8)
 ${{\boldsymbol{F}}}_{\mathrm{n}}^{\mathrm{d}}={C}_{\mathrm{n}}{\left({8m}^{*}{E}^{*}\sqrt{{r}^{*}\left|{{\boldsymbol{\delta }}}_{\mathrm{n}}\right|}\right)}^{\frac{1}{2}}\cdot {{{\boldsymbol{v}}}}_{\mathrm{n}}$ (9)

 $\begin{split} r=&1\Bigg/\left[\nabla {\boldsymbol{F}}^{\mathrm{T}}{\text{·}} {\nabla }^{2}F{\text{·}} \nabla F-{\left|\nabla F\right|}^{2}\right.\\ &\left.\left(\frac{{\partial }^{2}F}{\partial {\boldsymbol{x}}^{2}}+\frac{{\partial }^{2}F}{\partial {y}^{2}}+\frac{{\partial }^{2}F}{\partial {z}^{2}}\right)\right]\Bigg/2{\left|\nabla F\right|}^{3} \end{split}$ (10)

 $\begin{split} {{\boldsymbol{F}}}_{\mathrm{t}}^{\mathrm{e}}=&{\mu }_{{\rm{s}}}\left|{{\boldsymbol{F}}}_{{\rm{n}}}^{{\rm{e}}}\right|\left\{1-{\left[1-\mathrm{m}\mathrm{i}\mathrm{n}\left({{\boldsymbol{\delta }}}_{\mathrm{t}},{{\boldsymbol{\delta }}}_{\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}\right)\Big/{{\boldsymbol{\delta }}}_{\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}\right]}^{\tfrac{3}{2}}\right\}\cdot\\& {{\boldsymbol{\delta }}}_{\mathrm{t}}\Big/\left|{{\boldsymbol{\delta }}}_{\mathrm{t}}\right| \end{split}$ (11)
 $\begin{split} {{\boldsymbol{F}}}_{\mathrm{t}}^{\mathrm{d}}=&{C}_{\mathrm{t}}\left(6{\mu }_{\mathrm{s}}{{\rm{m}}}^{*}\left|{{\boldsymbol{F}}}_{{\rm{n}}}^{{\rm{e}}}\right|\cdot \right. \\& \left. \sqrt{1-\mathrm{m}\mathrm{i}\mathrm{n}\left({{\boldsymbol{\delta }}}_{\mathrm{t}},{{\boldsymbol{\delta }}}_{\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}\right)/{{\boldsymbol{\delta }}}_{{\rm{t}},\mathrm{m}\mathrm{a}\mathrm{x}}}\right)^{\tfrac{1}{2}}\cdot {{{\boldsymbol{v}}}}_{\mathrm{t}} \end{split}$ (12)

1.4 超二次曲面颗粒的信息更新

 $m\frac{\mathrm{d}{{\boldsymbol{v}}}_{t}}{\mathrm{d}t}=\sum _{i=1}^{N}\left({{\boldsymbol{F}}}_{\mathrm{n},i}+{{\boldsymbol{F}}}_{\mathrm{t},i}\right)+m{{\boldsymbol{g}}}$ (13)
 ${{\boldsymbol{J}}}\frac{{\mathrm{d}{{\boldsymbol{\omega}} }}_{t}}{\mathrm{d}t}=\sum _{i=1}^{N}\left({{{\boldsymbol{M}}}}_{\mathrm{n},i}+{{{\boldsymbol{M}}}}_{\mathrm{t},i}+{{{\boldsymbol{M}}}}_{\mathrm{r},i}\right)$ (14)

 ${{{\boldsymbol{e}}}}_{\mathrm{l}}^{\mathrm{T}}={{\boldsymbol{R}}}{{{\boldsymbol{e}}}}_{\mathrm{g}}^{\mathrm{T}}$ (15)

 ${{\boldsymbol{M}}}{\ddot{{{\boldsymbol{u}}}}}_{t}+{{\boldsymbol{C}}}{\dot{{{\boldsymbol{u}}}}}_{t}+{{\boldsymbol{K}}}{{{\boldsymbol{u}}}}_{t}={{\boldsymbol{F}}}_{t}$ (18)

2 DEM-FEM耦合算法的理论验证

 图 10 球体颗粒填充下提升条位移随旋转角度的变化关系 Fig. 10 Relationship between the displacement of lifting bar and rotation angle under spherical granular filling

 图 12 立方体颗粒填充下提升条位移随旋转角度的变化关系 Fig. 12 Relationship between the displacement of lifting bar and rotation angle under cubic granular filling
4 结束语

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