A NEW MESHLESS FRAGILE POINTS METHOD (FPM) WITH MINIMUM UNKNOWNS AT EACH POINT, FOR FLEXOELECTRIC ANALYSIS UNDER TWO THEORIES WITH CRACK PROPAGATION I: THEORY AND IMPLEMENTATION

“Flexoelectricity” refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM) is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. The first theory is fully nonlinear and is recommended at the nanoscale, while the second theory is linear and is sufficient at the microscale. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a bonding-energy-rate (BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal as well as mixed FPM weak forms is trivially simple; and FPM can be easily employed in crack development simulations without remeshing or trial function enhancement. In this first part of the two-part series, we focus on the theoretical formulation and implementation of the proposed primal as well as mixed FPM. Numerical results and validation are then presented in Part II of the present paper.