工程科学与技术   2021, Vol. 53 Issue (3): 166-172

Calculation Model of Residual Stress During Quenching and Straightening of High-strength Plate
WANG Xiaogang, ZHU Xiaoyu, HAN Peisheng
Eng. Research Center of Heavy Machinery for Ministry of Education, Taiyuan Univ. of Sci. and Technol., Taiyuan 030024, China
Abstract: In order to reduce the large residual stresses, which is generated in the production of high-strength plates during the quenching process and results in plate defects and product quality degradation. Numerical calculations of residual stresses, experiments and thermodynamic coupling simulations were carried out to study the changes of residual stresses during the whole process of straightening of Q960E high-strength steel. After quenching and heat treatment, a compressive stress of about 276.3 MPa was formed on the surface of the high-strength plate, and a tensile stress was formed in the opposite direction. The results showed that the roll straightening process could effectively reduce the residual stress of the sheet, and had a greater effect on the surface residual stress and a smaller effect on the core residual stress. This result also verifies the correctness of the calculation model for calculating the residual stress, which provides an effective calculation method for the straightening process to reduce the residual stress of the sheet.
Key words: roller straightening    residual stress    numerical calculation

1 矫直过程残余应力计算模型 1.1 淬火后残余应力计算

 $\left\{\!\!\!\! \begin{array}{l} k\dfrac{{{\partial ^2}T}}{{\partial {y^2}}} = \rho c\dfrac{{\partial T}}{{\partial t}} ; \\ T = {T_2},\;y = \pm \dfrac{h}{2}; \\ \dfrac{{\partial T}}{{\partial t}} = 0,\;y = 0; \\ T = {T_1},\;t = 0 \end{array} \right.$ (1)

 \begin{aligned}[b] T =& {T_2} + \frac{{4 \Delta T}}{\text{π} }\bigg( {{{\rm{e}}^{ - {p_1}t}}\cos \frac{{\text{π} y}}{h}} \bigg. - \\ &\bigg. {\frac{{{{\rm{e}}^{ - {p_3}t}}}}{3}\cos \frac{{3\text{π} y}}{h} + \frac{{{{\rm{e}}^{ - {p_5}t}}}}{5}\cos \frac{{5\text{π} y}}{h} + \cdots } \bigg) \end{aligned} (2)

 ${\varepsilon _x} = {\varepsilon _{\textit{z}}} = {\varepsilon _0} + \frac{y}{\rho } - \alpha T$ (3)
 ${\sigma _x} = {\sigma _{\textit{z}}} = \frac{E}{{1 - {\nu ^2}}}\left( {{\varepsilon _{xx}} + \nu {\varepsilon _{\textit{zz}}}} \right)$ (4)

 $\left\{\!\!\!\! {\begin{array}{*{20}{l}} {\displaystyle\int_{ - \frac{h}{2}}^{\frac{h}{2}} {{\sigma _x}} {\rm{d}}y = 0}, \\ {\displaystyle\int_{ - \frac{h}{2}}^{\frac{h}{2}} {{\sigma _x}} y{\rm{d}}y = 0} \end{array}} \right.$ (5)

 {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}[b] {\sigma _x} =& {\sigma _{\textit{z}}} = - \frac{{\alpha E}}{{1 - \nu }} + \frac{{\alpha E}}{{2\left( {1 - \nu } \right)h}}\int_{ - \frac{h}{2}}^{\frac{h}{2}} T {\rm{d}}y + \\ & \frac{{2y\alpha E}}{{2{h^3}\left( {1 - \nu } \right)}}.\int_{ - \frac{h}{2}}^{\frac{h}{2}} T y{\rm{d}}y \end{aligned}} (6)

 ${\sigma _x} = {\sigma _{\textit{z}}} = - \frac{{4\alpha E\Delta T{{\rm{e}}^{ - {p_1}{t_2}}}}}{{\text{π} \left( {1 - \nu } \right)}}\left( {\frac{2}{\text{π} } - \cos \frac{{\text{π} y}}{h}} \right)$ (7)

1.2 矫直过程计算模型 1.2.1 研究单元体划分

 图1 矫直区间几何划分 Fig. 1 Geometric division of straightening section

1）弯曲前后板料的横截面依旧保持平面，并且仍然垂直变形后的中轴线。2）矫直过程中，拉伸与压缩区域的应力应变对应关系一致。3）材料为均质且连续的材质，符合Hooke定律，应力中心层与应变中心层相互重合。4）矫直过程塑性变形前后体积保持不变。5）变性材料符合Von−Mises屈服条件。

1.2.2 矫直过程中表面受力分析

 图2 板材弯曲状态示意图 Fig. 2 Schematic diagram of plate bending state

 ${\varepsilon _{wx}} = \left\{\!\!\!\! {\begin{array}{*{20}{l}} {1 + \lambda \left( {\dfrac{{2yC}}{H} - 1} \right){\mkern 1mu} {\kern 1pt} ,\;\dfrac{{{H_{\rm{t}}}}}{2} \le y \le \dfrac{H}{2}};\\ {\dfrac{{2Cy}}{H} ,\;\; - \dfrac{{{H_{\rm{t}}}}}{2} \le y \le \dfrac{{{H_{\rm{t}}}}}{2}};\\ { - 1 - \lambda \left( {\dfrac{{2yC}}{H} + 1} \right),\; - \dfrac{H}{2} \le y \le - \dfrac{{{H_{\rm{t}}}}}{2}} \end{array}} \right.$ (8)

 $f\left( {\sigma _{ij}^\prime } \right) = J_2^\prime = C$ (9)

 ${\varepsilon _x^{n + 1}} + {\varepsilon _y^{n + 1}} + {\varepsilon _{\textit{z}}^{n + 1}} = 0$ (10)

 $\left\{\!\!\!\! {\begin{array}{*{20}{l}} {\displaystyle\int_{ - h/2}^{h/2} {\int_{ - b}^b {{\sigma _x}{\rm{d}}{\textit{z}}{\rm{d}}y = 0} } }, \\ {\displaystyle\int_{ - h/2}^{h/2} {\int_0^l {{\sigma _{\textit{z}}}{\rm{d}}x{\rm{d}}y = 0} } } \end{array}} \right.$ (11)

 $\left\{\!\!\!\! {\begin{array}{*{20}{l}} {\displaystyle\int_{ - h/2}^{h/2} {\int_{ - b}^b {{\sigma _x}y{\rm{d}}{\textit{z}}{\rm{d}}y = 0} } },\\ {\displaystyle\int_{ - h/2}^{h/2} {\int_0^l {{\sigma _{\textit{z}}}y{\rm{d}}x{\rm{d}}y = 0} } } \end{array}} \right.$ (12)

 图3 矫直过程残余应力计算 Fig. 3 Calculation of residual stress in the straightening process

2 计算结果分析

 图4 淬火后厚度方向残余应力分布 Fig. 4 Distribution of residual stress in thickness direction after quenching

 图5 矫直过程中残余应力计算结果 Fig. 5 calculation results of residual stress in straightening process

3 实验及模拟验证 3.1 验证方法及流程

 图6 验证方法及流程 Fig. 6 Verification method and process

3.2 实验设备

3.3 模拟结果

 图7 矫直过程整个板材残余应力变化过程 Fig. 7 Change process of residual stress in the whole plate straightening process

 图8 矫直后上下表面残余应力状态 Fig. 8 Residual stress state of upper and lower surfaces after straightening

3.4 计算、模拟与实验对比分析

 图9 计算与模拟结果对比 Fig. 9 Comparison of calculation and simulation results

4 结　论

1）本文计算模型与实验误差在15%以下，在可忽略的误差范围以内，实验和模拟也验证了计算模型的实用性。

2）计算、实验、模拟结果表明，辊式矫直过程是降低残余应力的有效方法。矫直前上下表面xz方向都处于压应力–276 MPa左右。矫直后上表面残余应力x方向减少到42～79 MPa，z方向减少到–89～–122 MPa；下表面x方向减少到–56～–115 MPa，z方向减少到23～58 MPa。

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