均布荷载作用下简支磁电弹性梁的解析解

对磁电弹性材料构成的两端简支梁,在均布荷载作用下的弯曲问题按平面应力问题进行了研究.从横观各向同性磁电弹性体的三维基本方程出发,简化得到平面问题的基本方程,给出了用四个拟调和函数表达的四个特征根互异情况下的通解,进而以试凑法推导出了均布荷载作用下简支磁电弹性梁的位移、电势、磁势、应力、电位移和磁通密强度的解析解,所得解有易于理解、便于校对、形式统一简洁的特点.并且将计算结果与压电材料和弹性材料相应结果进行了分析、比较,发现应力σz与τxz材料常数无关,且与各向同性弹性梁结果一致;位移和电势受材料常数影响较大,而应力和电位移影响较小.所得结果为验证各种数值计算方法提供了参考依据.

Analytical solution of simply supported magnetoelectroelastic beam under uniform load

The bending of simply supported magnetoelectroelastic beam under uniform load treated as plane stress problem is investigated. The basic equations are generalized from those of the transversely isotropic magnetoelectroelastic media, and the general solution in forms of harmonic functions in the case of distinct eigenvalues are derived. Then, the analytical solution of simply supported magnetoelectroelastic beam under uniform load is obtained with the trial-and-error method, which is simply understood and concisely checked. Comparison of the numerical results of magnetoelectroelastic, piezoelectric and istropic elastic beam showed that the displacements and electric potential have greater differences than those of stress and electric displacements caused by material properties. The results can serve as benchmarks for numerical methods such as the finite element method, the boundary element method, etc.