Some analytical solutions to peridynamic beam equations

Peridynamics (PD) has been introduced to account for long range internal force/moment interactions and to extend the classical continuum mechanics (CCM). PD equations of motion are derived in the form of integro‐differential equations and only few analytical solutions to these equations are presented in the literature. The aim of this paper is to present analytical solutions to PD beam equations for both static and dynamic loading conditions. Applying trigonometric series, general solutions for the deflection function are derived. For several examples in the static case including simply supported beam and cantilever beam, the coefficients in the series are presented in a closed analytical form. For the dynamic case, the solution is derived for a simply supported beam applying the variable separation with respect to the time and the axial coordinate. Several numerical cases are presented to illustrate the derived solutions. Furthermore, PD results are compared against results obtained from the classical beam theory (CBT). A very good agreement between these two different approaches is observed for the case of the small horizon sizes (HSs), which shows the capability of the current approach.


INTRODUCTION
Peristatic and PD solutions for one-dimensional rods are presented in Refs. [23][24][25][26]. For beams under static loading several truncated PD solutions are derived in Chen [9]. The deflection function is expanded in the Taylor series about a point of a beam. The PD integral equation for the bending moment is then evaluated approximately by keeping two terms of the series expansion. This approach leads to a non-local beam theory with higher derivatives of the unknown deflection function.
The aim of this study is to derive analytical solutions to PD beam equations for both static and dynamic loading conditions. By use of trigonometric series representation, general solutions for the deflection function are derived. For several boundary conditions in the static case and for a simply supported beam in the dynamic case, the coefficients in the series are evaluated in a closed analytical form. The results are validated by comparing against solutions to the classical beam theory (CBT).

Equation of motion
The PD equation of motion for the Bernoulli-Euler beam can be derived by using Lagrange's equatioṅ where is the deflection and = − represents the Lagrangian. The total kinetic energy, , of the system can be written as follows: in which is the density, is the cross sectional area and is the length of the beam. The potential energy is defined as follows: where PD is the PD strain energy density and ( ) is the distributed transverse load. PD as a non-local theory, whose strain energy density function has non-local form such that for any certain material point, it not only depends on its own deflection but also that of its finite neighbourhood. In order to obtain the PD strain energy density function, it can be achieved by converting the local term in corresponding strain energy density function in CBT into non-local form. For the Bernoulli-Euler beam, it takes the form where is the Young's modulus and is the second moment of area of the beam cross section. Performing Taylor expansion on the displacement function about and ignore the higher order terms results in Considering as a fixed point, multiplying each term in Equation (5) by 1∕ 2 and integrating over the PD domain gives

F I G U R E 1 Real and fictitious domains for the beam
where is the finite size neighbourhood of the material point , which is called the horizon. Note that the kernel function 1∕ 2 in Equation (6) ensures the dimension of the integrand to be consistent with that of curvature. Plugging Equation (6) back into Equation (4) arises the PD strain energy density as With Equations (2), (3) and (7), Equation (1)

Peridynamic boundary conditions
Equation of motion (8) is valid only if the material points are completely embedded in its PD influence domain. However, for material points adjacent to the boundary whose PD influence domain is defective, it is necessary to introduce fictitious material region for the sake of ensuring the validation of the PD formulation. In this study, the width of the fictitious region is suggested as the double of the PD HS, 2 , as shown in Figure 1. Let us explain how to implement boundary supports in PD framework.

Simple support
Suppose the beam is subjected to simply supported boundary condition at the left end. From geometrical point of view, the deflection and curvature of the end point must be zero, that is (10) Replacing Δ by and associating with Equation (9) 1 gives the PD simply supported condition as It indicates that simply supported end enforces in the fictitious region an anti-symmetric relation with respect to the real material domain, as shown in Figure 2.

Clamped support
Suppose the beam is clamped at the left end. In this case, both the deflection and cross section rotation must be zero, that is Applying central difference with respect to the slope relation gives Replacing Δ by PD notation and associating with Equation (12) 1 gives the PD clamped boundary condition as Equation (14) implies that clamped boundary in the fictitious region provides a symmetric relation with respect to the real material domain, as shown in Figure 3a.

Free boundary condition
The free boundary condition imposes both the zero curvature and shear force, that is Applying the central difference scheme with respect to zero curvature expression gives The central difference scheme applied to zero shear force gives Coupling Equation (17) with Equation (18) yields It can be claimed that the relation given in Equation (17) automatically satisfies zero shear force boundary conditions. Thus, the PD free boundary condition can be summarized as In geometrical point of view, this enforces a skew symmetric relation about point ( , ) = ( , ( )) between fictitious region and real material vicinal to free end, as shown in Figure 4.

Simply supported beam
The PD governing equation for the Bernoulli-Euler beam (8) in the static case can be expressed as with the following PD boundary conditions The BCs (22) periodically extend the displacement field oddly with period of 2L, as shown in Figure 5.
With consideration of oddity and periodicity, the displacement field function can be expressed in terms of Fourier series as follows: Consequently one may compute Likewise it follows:

F I G U R E 6 Solution decomposition for a cantilever beam
Plugging Equations (24) and (25) into Equation (21) and simplifying results in Invoking trigonometric orthogonality implies Plugging Equation (27) into Equation (23) gives the PD analytical solution for the simply supported beam as follows:

Cantilever beam
Suppose a cantilever beam subjected to arbitrary load, the solution to cantilever beam can be identically decomposed by following process (Figure 6): 1. Transit the origin to level with the free end then subtract the corresponding rigid body motion 2. Release the vertical displacement constrain and replace it by the corresponding shear force The PD governing equation for the new configuration becomes where Δ( − 0) represents the Dirac Delta function, and the shear force can be casted as The corresponding PD boundary conditions are Based on the PD boundary stipulation in Equation (32), ( ) can be evenly extended with period of 4 , as shown in Figure 7. Thus, ( ) can be expressed by using Fourier series as follows: Likewise it follows Plugging Equations (36) and (37) into Equation (30) gives in which the Fourier series coefficients can be obtained as Plugging Equation (39) back into Equation (35) implies Substituting Equation (40) into Equation (29) results in the PD solution for cantilever beam as in which

ANALYTICAL SOLUTION FOR FREE VIBRATION OF SIMPLY SUPPORTED BEAM
By using variable separation method, the displacement function can be expressed as After inserting Equation (43) into Equation (8), we obtain By isolating variables, it follows Thus, we obtain and̈( By comparing Equation (46) with Equation (21), we can consider ( ) as an analogue to ( ) and ( ) to ( )∕ . Thus, associating with Equations (23) and (26), one can obtain and Therefore, the PD eigenvalue can be casted as The general solution to Equation (47) can be expressed as With Equations (48) and (51), the general PD solution can be formulated as follows: in which the undetermined coefficients and depend upon the initial conditions. Suppose the initial conditions are By associating Equation (52), we have and Utilizing orthogonal properties gives By collecting derivation above, the PD analytic solution for free vibration of Euler beam can be summarized as

NUMERICAL RESULTS
In order to demonstrate the validation of the PD analytical solution, various numerical cases are presented comparing against analytical solution in CBT for both static and dynamic condition. Different boundary conditions are considered including simply supported beam and cantilever beam.

Static solution
Consider a beam with length of = 1 m, thickness of ℎ = 0.01 m, width of = 0.05 m and Young's modulus of = 200 GPa subjected to simply supported and clamped free boundary condition, respectively. The PD HS is chosen as = 0.001 m. A distributed loading of ( ) = 1 N/m is applied for both cases, as shown in Figure 8a,b. The variation of deflection along the beam from PD is calculated and compared against CCM solution. As shown in Figure 9a,b, a very good agreement is observed between PD and CBT solutions.
The variation of deflection from PD analytical solution is obtained at two different locations = 0.25 m and = 0.5 m. As shown in Figure 10, a very good agreement is obtained between PD and CBT solutions.

Analysis of PD horizon size effect
The horizon as an important parameter in PD, which is considered as a length-scale parameter and gives non-local property to the system. The size of the PD horizon determines the non-locality of the PD governing equation. If the size is chosen tending to zero, the PD solution should converge to that of CBT. It can be seen from Equation (57) that the natural frequencies obtained by PD solution are depended upon the HS . In order to investigate the effect of HS, vibrational behaviour of simply supported beam with various of hHS are compared against the corresponding solution in CBT. A very good agreement is observed between PD and CBT when the HSs are small, as illustrated in Figures 11 and 12. Let us note that as the HS approaches zero, the non-locality disappears and equations of the CBT follow from the PD beam equations. However, as the HS increases, solutions of PD diverge from CBT solution.

CONCLUSIONS
In this study, new closed-form analytical solutions to PD beam equations are presented. By the use of trigonometric series, representation of the deflection function of the axial coordinate general solutions for the static case is derived. For several examples including simply supported beam and cantilever beam, the coefficients in the series are presented in a closed analytical form. For the dynamic case, the solution is derived for a simply supported beam. By the use of the variable separation, the deflection is represented as a function of the time and the axial coordinate. Numerical examples are presented to illustrate and to compare results of PD against solutions according to the CBT. A very good agreement between these two different approaches is observed for the case of the small HSs, which shows the capability of the current approach.
The derived analytical solutions can be applied in the future to verify the numerical methods and codes to solve PD equations.