Finite Element and Dynamic Stiffness Methods of two-phase peridynamic elastic Euler-Bernoulli beams with exponential kernels

This paper deals with the conventional Finite Element and exact Finite Element analyses of two-phase peridynamic Euler-Bernoulli beams (TPPEBB), both in statics and in dynamics, applied to single- and multi-span beams. The integro-differential equation of the two-phase peridynamic Euler-Bernoulli beam is reformulated as a higher-order differential problem discretized through Finite Element and Finite Difference Methods. Two new beam Finite Elements with polynomial interpolation functions are introduced with 6 and 10 degrees of freedom (DOF), respectively (associated with quintic and nonic interpolation fields). The first Element considers the deflection, the slope, and the curvature of both end nodes as DOF (C2 Finite Element). A more accurate Finite Element with 10 degrees of freedom is also introduced (C4 Finite Element). In addition to these polynomial Finite Elements which estimate approximately the displacement field, the exact displacement field is also utilized to find the exact stiffness matrix for static analysis and the exact dynamic stiffness matrix for vibration analysis of TPPEBB, correspondingly. In this article, these two afore-mentioned matrices are extracted based on Eisenberger’s methodology, and the closed-form expressions for matrices’ components are presented thoroughly. These exact matrices can provide exact results using only one exact Finite Element. The static and dynamic analysis of multi-span TPPEBB is performed by three methods, i.e., analytical approach, exact FE and conventional FE. It is shown that the results obtained by these three methods are consistent, the first and second methods which give exact solutions, and the third method with a convergence order that depends on the considered Finite Element (with convergence of order 6 and 10). Moreover, the Finite Difference (FD) procedure introduced by Challamel and Zingales, 2025b from the discretized integro-differential formulation, is here completed with the application of the FD scheme applied to the equivalent higher-order differential equation. A comprehensive study on the experimental order of convergence for both numerical methods, i.e., FEM and FDM, is conducted. It is concluded from this study that the order of convergence for the presented FEM is greater than the presented FDM schemes.