Analytical Studies on Mode III Fracture in Flexoelectric Solids

Abstract

Due to the stress concentration near crack tips, strong flexoelectric effect would be observed there, which might lead to new applications of flexoelectricity in material science and devices. However, different from the flexoelectric effect in cantilever beams or truncated pyramids, at the crack tip, multiple components of strain gradients with nonuniform distribution contribute to the flexoelectric effect, which makes the problem extremely complex. In this paper, with the consideration of both direct and converse flexoelectricity, the electromechanical coupling effect around the tip of a Mode III crack is studied analytically. Based on the Williams' expansion method, the displacement field, polarization field, strain gradient field along with the actual physical stresses field are solved. A path independent J-integral for Mode III cracks in flexoelectric solids is presented. Our results indicate that the existence of flexoelectricity leads to a decrease of both the J-integral and the out-of-plane displacement in Mode III cracks, which means that the flexoelectric effect around the tip of Mode III cracks enhances the local strength of materials.

Modeling mechanical waves propagation in flexoelectric solids

“…In equation ( 29), α 11 , α 22 , and α 12 are constant vectors to be determined from the requirement of the compatibility between the displacements and strains. Their general expressions are given as α 11 = α 11 1 , α Here, we use the collocation method at 9 Gaussian quadrature points (ξ G 1 , ξ G 2 ) in each element to satisfy the compatibility between the displacement and strain. Then, the following equation can be established:…”

“…9×9 are constant matrice associated with the coordinates of Gauss points. Thus, by using the collocation method at Gauss points for the strain, we establish the relation between α 11 , α 22 , α 12 , and the displacements at element nodes. Then, after substituting equation ( 36) into ( 29), we have…”

“…or pyramid [5][6][7], and introducing nano-cracks, dislocations, lattice mismatches, or nano-inclusions [8][9][10][11][12][13][14].…”

Extended Isogeometric Analysis of Cracked Piezoelectric Materials in the Presence of Flexoelectricity

“…[21] Since flexoelectricity is prominent in places with higher strain gradients, holes, cracks, thin films, and inclusions are naturally targeted for their presence. [22][23][24] Stress fields are singular at crack tips, and the high value of stress reduces at a minimal distance resulting in high strain gradients around the tip. The high strain gradient will allow flexoelectricity to act and generate electric potential proportional to it.…”

“…[40] Compared with piezoelectricity, fracture mechanics of domains with flexoelectricity is still a less explored area. [24,25] Gradients must be considered when modeling flexoelectricity, and XFEM's linear Lagrange-based finite elements cannot be employed. [41] Isogeometric analysis (IGA) using NURBS as smooth basis functions can interpolate gradients and higher-order fields.…”