A Multi-parameter Perturbation Solution For Functionally Graded Piezoelectric Cantilever Beams

In recent years,piezoelectric materials,as the representative of sensing and driving materials,have been widely used in smart structures.The mechanical study of piezoelectric components will contribute to the development of smart structures.At present,most of the piezoelectric components used in engineering are multi-layer structures.When some components or properties of the components change abruptly,the bonding layer will easily crack at low temperature and creep at high temperature,which will greatly reduce the reliability of materials.In order to improve the performance of materials,functionally graded piezoelectric materials(FGM)have emerged as a new type of composite materials.Functionally gradient piezoelectric composites combine the properties of piezoelectric materials and functionally gradient materials,which greatly improves the performance of piezoelectric materials in different environments and prolongs their service life.Therefore,the mechanical study of functionally gradient piezoelectric composites has become one of the hot topics that researchers at home and abroad pay close attention to.In this paper,the problem of functionally graded piezoelectric cantilever under composite loads is solved by using multi-parameter perturbation method.Firstly,the physical properties of piezoelectric materials and the gradient function of functionally graded materials are introduced.Then three piezoelectric coefficients are selected as perturbation parameters.In the process of calculation,two governing equations about stress function and potential function are deduced firstly,and then the perturbation expansion of stress function and potential function about the three selected perturbation parameters is carried out.Then,the perturbation expansion of stress function and potential function is substituted into the first two governing equations.Then,the same terms of the formulas are combined to obtain the perturbation equations of each order.By substituting the assumed basic forms of stress function and potential function,the differential equations of each order are obtained and the perturbation solutions of stress function and potential function are obtained by using the boundary conditions.Finally,the concrete expressions of the perturbation solutions of stress function and potential function are replaced by the perturbation expansions.The stress,displacement and electric displacement of the functionally graded piezoelectric cantilever beam are obtained by using the relevant basic equations.Finally,the validity of the solution is verified.Then the solutions of functionally graded piezoelectric beams under two-dimensional and one-dimensional pure bending are discussed,and the validity of one-dimensional solutions under pure bending is proved.At the end of the article,the effects of functionally gradient coefficient and piezoelectric coefficient on stress,displacement and electric displacement of cantilever beams are analyzed.The following conclusions are drawn from the calculation and discussion of this paper:(1)From the solutions of stress,displacement and electric displacement and the governing equations of stress function and potential function,it can be found that the stress function and potential function are coupled.(2)When the functional gradient coefficient ?(29)0,the maximum bending stress,shear stress,deflection and electric displacement increase with the increase of ?.When ?(27)0,the values listed above decrease with the increase of ?.However,the extrusion stress always decreases with the increase of ?.(3)The influence of piezoelectric coefficients in different directions on the solution of stress,displacement and electric displacement is different under static load,and the effect of piezoelectric effect on cantilever beam only produces perturbation,which also confirms the rationality of the original perturbation method.