An electroelastic solution for functionally graded piezoelectric material beams with different moduli in tension and compression

Abstract

In this study, we derived an electroelastic solution of functionally graded piezoelectric beams with bimodular effect under the combined action of uniformly distributed loads, concentrated loads, and bending moments. We first presented the simplified mechanical model established in previous study, and this model is based on a complete partition of tension and compression. Based on the theory of piezoelectric elasticity and functional graded property, we assumed that material parameters vary exponentially along the thickness, and the exponential functions adopt two different forms in the tensile zone and the compressive zone. Using stress function method, we obtained an analytical solution of the problem in which unknown constants are determined by boundary conditions and continuity conditions on the neutral layer. The results indicate that the introduction of exponential functions makes us be able to effectively distinguish the relative magnitude of the tensile modulus, the compressive modulus and the modulus on the neutral layer. Thus, the variation in the stress and the displacement including the electric displacement with the thickness are discussed in detail. The results obtained are helpful to refined analysis and design of functionally graded piezoelectric materials and structures with obvious bimodular effect.

A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads

“…Su et al [ 28 ] dealt with the electro-mechanical vibration characteristics of FGPM rectangular plates with different boundary conditions based on first-order shear deformation theory. More recently, He et al [ 29 ] presented an electroelastic solution for FGPM beams with different moduli in tension and compression. Given that there are many relative works in this field, we do not review them in detail.…”

An Electroelastic Solution for Functionally Graded Piezoelectric Circular Plates under the Action of Combined Mechanical Loads

“…Suppose that [ 32 , 33 ] in which and are also looked at as the displacement functions, is also looked at as the potential functions, and they depend only on z . The detailed reason for the assumption of Equation (8) is shown in the Appendix A , which includes some results from functionally graded piezoelectric beams [ 34 , 35 ]. Substituting Equation (8) into Equation (5), it gives Substituting Equations (6), (8), and (9) into Equation (4), we can obtain Then, substituting Equation (10) into Equations (2) and (3), respectively, we can also obtain …”

“…Substituting Equations (5), (6), and (A3) into (A6), we get From Equation (A7), we can obtain From Equation (A8) we can see that all the even items of in the expression of are zero and all the odd items of in the expression of are zero. Then, according to the boundary conditions of simply supported circular plates, we can finally get the forms of the displacement and the electric potential as follows (i.e., Equation (8); the more detailed derivation can be found in [ 32 ]): In addition, similar expressions for displacement and potential function may be found in the analysis of functionally graded piezoelectric beams [ 34 , 35 ], in which the stress function and the potential functions were expressed in the form in which x represents the longitudinal direction of the beam (similar to the radial direction r in the plate problem) and z stands for the thickness direction (similar to the transverse direction z in the plate problem). From the similarities of two sets of expression of beams and plates, we may find some consistencies in the analyses of beams and plates.…”

“…In addition, similar expressions for displacement and potential function may be found in the analysis of functionally graded piezoelectric beams [ 34 , 35 ], in which the stress function and the potential functions were expressed in the form in which x represents the longitudinal direction of the beam (similar to the radial direction r in the plate problem) and z stands for the thickness direction (similar to the transverse direction z in the plate problem). From the similarities of two sets of expression of beams and plates, we may find some consistencies in the analyses of beams and plates.…”

Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

“…The result also indicates that the deformation magnitude of piezoelectric plates is smaller than that of plates without piezoelectricity, due to the well-known piezoelectric stiffening effect.With the increasing application of functionally-graded piezoelectric materials, precise characterization of their mechanical properties is urgently needed. A great deal of research has been done on the mechanical properties of functionally-graded piezoelectric materials and structures: for example, FGPM cantilever beams [8][9][10][11][12], FGPM plates [13][14][15][16][17], and FGPM shells [18][19][20][21]. At the same time, the generation of new problems also puts forward greater requirements for the corresponding solving methods.…”