应变梯度弹性理论下微构件尺寸效应的数值研究

NUMERICAL STUDY ON SIZE EFFECTS OF THE MICROSTRUCTURES BASED ON STRAIN GRADIENT ELASTICITY

根据位移二阶梯度分量的不同,应变梯度理论可以分为偶应力理论和全应变梯度理论。与偶应力理论相比较,全应变梯度理论增加了伸长梯度对应变能密度函数的贡献,因此,该理论预测的尺寸效应要强于偶应力理论预测的尺寸效应。基于建立的应变梯度弹性理论C1自然单元法,研究了微夹持器和拉伸微试件的尺寸效应现象。对于微夹持器,梳状静电驱动臂与固定端之间采用S形弹簧连接,降低了夹持臂的弯曲刚度,增加了夹持力的有效输出;当弹簧宽度接近材料的特征长度时,无量纲弯曲刚度值很大,微夹持器具有强烈的尺寸效应。对于拉伸微试件,当圆孔半径和椭圆孔长轴接近材料的特征长度时,无量纲应力集中系数很小,微试件尺寸效应明显;随着U槽端部半径的增加,微试件尺寸效应明显变弱;随着槽深的增加,微试件尺寸效应缓慢减弱。对于微构件的所有计算情况,全应变梯度理论下的尺寸效应强于偶应力理论下的尺寸效应,数值计算结果与理论预测相吻合。

Strain gradient theory can be classified into couple-stress theory and stretch-rotation gradient theory according to the difference in second-order gradients of the displacements. Compared with couple-stress theory, the stretch-rotation gradient theory involves the stretch gradients besides the rotation gradients; therefore, size effects predicted by stretch-rotation gradient theory are stronger than those predicted by couple-stress theory. Based on strain gradient elasticity C1 natural element method, size effects on microgripper and microspeciem are studied. For the microgripper, S-shaped spring structure adopted to link the electrostatic comb drive arm and the anchor reduces the bending stiffness and increases the gripping force; when the width of spring is close to the material characteristic length scales, the normalized bending stiffness is very large, which implies strong size effects in this case. For the microspeciem, size effects are obviously strong when the radius of circular perforation and the long axis of elliptical perforation are close to the material characteristic length scales. For the U-shaped notch, with the increase of notch radius, size effects become weak obviously; with the increase of the length ofnotch, size effects become weak slightly. Numerical results for all cases demonstrate that size effects in the stretch-rotation gradient theory are stronger than those in couple-stress theory, which agrees well with the theoretical predictions.