# Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework
## 1 Introduction  
In recent years, non-local theories of solid mechanics have become of increasing interest, due to many applications such as analysis of micro- and nanometer-scale components, laminates with thin layers, composite materials with extreme differences in properties of constituents, among others. One way to develop a non-local theory is to extend the classical continuum mechanics (CCM) by introducing higher-order strain gradients as arguments of the strain energy density [1]. By applying a variational principle, the governing equations of motion and boundary conditions can be derived. For linear-elastic material behavior, and in the case of states, a strain gradient theory of the \(n\)th order provides a \(2\)\(n\)th-order system of linear partial differential equations, e.g., [2]. In addition, higher-order stress tensors are introduced, and many boundary conditions are required to pose the boundary-value problem. Strain gradient theories were recently applied to the analysis of micro- and nanometer structures in, e.g., [3, 4] (see also the works cited therein). Alternatively, non-local continuum theories can be formulated within the peridynamics (PD) framework [5, 6]. Here, the balance equations for long-range internal forces are introduced. The deformation gradient, higher strain gradients, or gradients of internal state variables are not required. peridynamic theories for three-dimensional solids are discussed in [5]. Thin-walled structural components can be modeled by using classical three-dimensional continuum mechanics or alternatively by applying classical theories of beams, plates, and shells, as discussed in many textbooks, e.g., [7]. Within the framework of peridynamics, a three-dimensional theory can be used in the analysis of thin-walled structures, e.g., [8]. However, a very high computational effort is required for the numerical solution, as the thickness direction must be discretized by many node layers to to provide satisfactory accuracy. Furthermore, for components with a very low thickness-to-span-length ratio, for example, soft polymeric layers in laminated glass [9], the PD horizon size may be comparable to or even larger than the layer thickness. This would lead to non-physical long-range force interactions between the layers in a laminate. Therefore, robust peridynamic theories for beams, plates, and shells are required to analyze slender components. Similarly to classical theories, PD theories for thin-walled structures can be derived from balance laws for a deformable line in the case of a beam, a deformable plane for a plate, and a deformable surface for a shell. These types of theories can be referred to as direct theories by analogy to the classical continuum mechanics. A direct PD beam theory proposed in [10] assumes the strain energy density to be a function of a non-local curvature. Discretized PD equations of motion with respect to the displacements of a straight beam are derived in [11]. For initially curved beams, PD equations of motion are presented in [12]. Discretized forms of PD equations of motion for plates and shells are derived in [13] based on the Lagrange variational principle. Peridynamic theories for membranes and fibers are presented in [14]. A direct PD shell theory in terms of displacements and linearized rotations is proposed in [15]. In [16], a non-linear direct PD plate theory is developed. Points of the plate midplane are assumed to have the rigid body-type degrees of freedom: displacements and finite rotations. Long-range force and moment interactions between the points of a plate are introduced. Governing balance laws are formulated directly for the plate. Alternatively, applying the through-the-thickness approximations for the kinematic variables, a three-dimensional PD theory can be reduced to one- or two-dimensional cases. In [17], a PD plate theory is developed from the three-dimensional bond-based PD theory by applying asymptotic analysis.