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Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment
Composite Structures ( IF 6.3 ) Pub Date : 2021-01-09 , DOI: 10.1016/j.compstruct.2021.113552
Ismail Esen , Cevat Özarpa , Mohamed A. Eltaher

A mathematical model is developed, though this article, to investigate a vibrational behaviour of functionally graded (FG) cracked microbeam rested on elastic foundation and exposed to thermal and magnetic fields. The model includes a size scale effect and temperature dependent material properties, for the first time. The crack is modelled as a rotating spring, that is connecting the two parts of the microbeam at the crack’s position. The equation of motion of the FG microbeam is obtained by using the Euler-Bernoulli beam theory for kinematic assumption and nonlocal elasticity theory for size-dependency effects. The transverse Lorentz force induced from the magnetic field is derived using Maxwell's equations. By adding the effects of thermal loading and foundation parameters on the cracked micro beam, the motion equation of the entire system is obtained using the Hamilton’s principle and then solved with a Navier type solution method. Eight constraints equations are used to derived the frequency equation, which are boundary conditions at the end points and the displacement, slope, bending moment and transverse force continuity in the section where the crack is located. The resulting system of equations is solved sequentially, and natural frequencies and vibration modes of the cracked microbeam are obtained. The model is verified with previous published work. Numerical results are presented to illustrate influences of temperature, material composition, foundation parameters and magnetic field on the dynamics of the cracked FG microbeam.



中文翻译:

在热环境中嵌入弹性矩阵并暴露于磁场的破裂FG微束的自由振动

通过本文,开发了一个数学模型来研究功能梯度(FG)裂纹微梁的振动行为,该裂纹位于弹性基础上并暴露于热和磁场中。该模型首次包含了尺寸比例效应和与温度相关的材料特性。将裂纹建模为旋转弹簧,该弹簧在裂纹的位置连接微梁的两个部分。通过使用Euler-Bernoulli梁理论进行运动学假设,并使用非局部弹性理论进行尺寸依赖性效应,获得FG微束的运动方程。由磁场引起的横向洛伦兹力是使用麦克斯韦方程导出的。通过添加热载荷和基础参数对破裂的微梁的影响,使用汉密尔顿原理获得整个系统的运动方程,然后用Navier型解法求解。使用八个约束方程式导出频率方程式,这些方程式是端点的边界条件以及裂缝所在区域的位移,斜率,弯矩和横向力连续性。依次求解得到的方程组,并获得裂纹微束的固有频率和振动模式。该模型已通过先前发布的工作进行了验证。数值结果表明了温度,材料成分,基础参数和磁场对破裂的FG微束动力学的影响。使用八个约束方程式导出频率方程式,这些方程式是端点的边界条件以及裂缝所在区域的位移,斜率,弯矩和横向力连续性。依次求解得到的方程组,并获得裂纹微束的固有频率和振动模式。该模型已通过先前发布的工作进行了验证。数值结果表明了温度,材料成分,基础参数和磁场对破裂的FG微束动力学的影响。使用八个约束方程式导出频率方程式,这些方程式是端点的边界条件以及裂缝所在区域的位移,斜率,弯矩和横向力连续性。依次求解得到的方程组,并获得裂纹微束的固有频率和振动模式。该模型已通过先前发布的工作进行了验证。数值结果表明了温度,材料成分,基础参数和磁场对破裂的FG微束动力学的影响。依次求解得到的方程组,并获得裂纹微束的固有频率和振动模式。该模型已通过先前发布的工作进行了验证。数值结果表明了温度,材料成分,基础参数和磁场对破裂的FG微束动力学的影响。依次求解得到的方程组,并获得裂纹微束的固有频率和振动模式。该模型已通过先前发布的工作进行了验证。数值结果表明了温度,材料成分,基础参数和磁场对破裂的FG微束动力学的影响。

更新日期:2021-01-28
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