# Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework
## 2 Peridynamic Beam Theory
### Peridynamic Equations of Motion  
Unlike the CCM and the classical beam theory (CBT), where strains are defined in terms of derivatives with respect to the coordinates, non-local deformation measures are introduced in the PD framework. The PD deformations can be introduced as work conjugates to the bond force and moment densities considering the energy balance equation, e.g., [5; 16]. Assuming the linear-elastic material behavior, a class of PD theories for beams and plates can be derived with the correspondence of the strain energy density from the CCM as a function of the PD deformation measures [27; 28].  
Let \(\Gamma\) be the collection of points of the beam centerline. Introduce the PD horizon as an open ball centered at point \(x\) with the PD horizon size \(\delta\) defined as \(\forall x\in\Gamma\), i.e.,  
\[H_{x}:=\big{\{}x^{\prime}\in\Gamma:\big{|}x^{\prime}-x\big{|}<\delta\big{\}}. \tag{8}\]  
In PD, the strain energy density of any arbitrary material point \(x\) depends upon both the displacement of itself and the displacement fields within its horizon \(H_{x}\), i.e.,  
\[\bar{U}(x)=\bar{U}(w(H_{x})). \tag{9}\]  
To formulate the PD strain energy density, consider the following non-local curvature:  
\[\bar{\kappa}(x)=-\frac{1}{\delta}\int\limits_{x-\delta}^{x+\delta}\frac{w(x^{ \prime})-w(x)}{(x^{\prime}-x)^{2}}\mathrm{d}x^{\prime}. \tag{10}\]  
For smooth deflection function \(w(x)\), it can be shown that \(\bar{\kappa}(x)\) approaches the local curvature (7) \({}_{2}\) as \(\delta\to 0\). With Equations (7) \({}_{1}\) and (10), the PD bending strain energy density is formulated as follows:  
\[\bar{U}(x)=\frac{1}{2}\frac{EI}{\delta^{2}}\left[\int\limits_{x-\delta}^{x+ \delta}\frac{w(x^{\prime})-w(x)}{(x^{\prime}-x)^{2}}\mathrm{d}x^{\prime}\right] ^{2}. \tag{11}\]  
For elastic beams, the PD equation of motion can be derived using [PERSON]'s equation,  
\[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial\dot{w}}- \frac{\partial\mathcal{L}}{\partial w}=0, \tag{12}\]  
where \(\mathcal{L}=\mathcal{T}-\bar{U}+\mathcal{W}\) is the Lagrangian. The kinetic energy \(\mathcal{T}\) is computed as follows:  
\[\mathcal{T}=\frac{1}{2}\int\limits_{0}^{L}\rho A\dot{w}^{2}\mathrm{d}x \tag{13}\]  
in which \(\rho\) is the density and \(L\) is the length of the beam. The strain energy of the beam \(\bar{U}\) and the work of external forces \(\mathcal{W}\) are  
\[\bar{U}=\int\limits_{0}^{L}\bar{U}(x)\mathrm{d}x,\quad\mathcal{W}=\int\limits _{0}^{L}p(x)w(x)\mathrm{d}x, \tag{14}\]  
where \(p(x)\) is the distributed lateral load.  
With Equations (11)-(14), the following equation of motion is derived:  
\[\rho A\ddot{w}(x)=\frac{EI}{\delta^{2}}\int\limits_{x-\delta}^{x+\delta}\frac{ 1}{(x^{\prime}-x)^{2}}\Bigg{[}\int\limits_{x-\delta}^{x+\delta}\frac{w(\bar{U} )-w(x)}{(\bar{U}-x)^{2}}\mathrm{d}\bar{\zeta}-\int\limits_{x^{\prime}-\delta}^{ x^{\prime}+\delta}\frac{w(\bar{U})-w(x^{\prime})}{(\bar{U}-x)^{2}}\mathrm{d} \bar{\zeta}\Bigg{]}\mathrm{d}x^{\prime}+p(x). \tag{15}\]By changing the integration variables, Equation (15) can be put in the following form:  
\[\rho A\bar{w}(x)=\frac{EI}{\delta}\int\limits_{-\delta}^{\delta}\frac{1}{\xi^{2}} \left[\int\limits_{-\delta}^{\delta}\frac{w(x+\eta)-w(x)}{\eta^{2}}\mathrm{d} \eta-\int\limits_{-\delta}^{\delta}\frac{w(x+\xi+\eta)-w(x+\xi)}{\eta^{2}} \mathrm{d}\eta\right]\mathrm{d}\xi+p(x). \tag{16}\]  
For the static case, the PD governing equation for the Bernoulli-Euler beam (16) can be expressed as follows:  
\[\frac{EI}{\delta^{2}}\int\limits_{-\delta}^{\delta}\frac{1}{\xi^{2}}\left[ \int\limits_{-\delta}^{\delta}\frac{w(x+\eta)-w(x)}{\eta^{2}}\mathrm{d}\eta- \int\limits_{-\delta}^{\delta}\frac{w(x+\xi+\eta)-w(x+\xi)}{\eta^{2}}\mathrm{ d}\eta\right]\mathrm{d}\xi=-p(x). \tag{17}\]