For a clamped-clamped bar subjected to some arbitrary loading, the boundary conditions are given as (3)u(−ξ)=−u(ξ)  and  u(L+ξ)=−u(L−ξ)∀ξ∈[0,δ]where L is the length of the bar.

Selecting the trial function as follows: (4)u(x)=∑n=1∞ansin⁡nπxLwhere a_n is an unknown coefficient and substituting Eq. (4) into Eq. (1) gives (5)c∑n=1∞an∫−δδ1|ξ|(1−cos⁡nπξL)dξsin⁡nπxL=b(x)

Unknown coefficients in Eq. (5) can be determined using the orthogonality condition as follows: (6)an=2cL∫0Lb(x)sin⁡nπxLdx∫−δδ1|ξ|(1−cos⁡nπξL)dξ

Substituting Eq. (6) back into Eq. (4) results in the peristatic analytical solution for a clamped-clamped bar as follows: (7)u(x)=2cL∑n=1∞∫0Lb(x)sin⁡nπxLdx∫−δδ1|ξ|(1−cos⁡nπξL)dξsin⁡nπxL

For a bar that is only clamped at the left end, the boundary conditions can be described as follows: (8)u(−ξ)=−u(ξ)and u(L+ξ)=u(L−ξ)∀ξ∈[0,δ]

In this case, the trial function can be chosen as follows: (9)u(x)=∑n=1∞bnsin⁡(2n−1)πx2Lwhere b_n is an unknown coefficient. Substituting Eq. (9) into Eq. (1) and after simplification yields (10)c∑n=1∞bn∫−δδ1|ξ|[1−cos⁡(2n−1)πξ2L]dξsin⁡(2n−1)πx2L=b(x)

Coefficients in Eq. (10) can be obtained using the orthogonality condition as follows: (11)bn=2cL∫0Lb(x)sin⁡(2n−1)πx2Ldx∫−δδ1|ξ|[1−cos⁡(2n−1)πξ2L]dξ

Therefore, the peristatic analytical solution for this case can be written as: (12)u(x)=2cL∑n=1∞∫0Lb(x)sin⁡(2n−1)πx2Ldx∫−δδ1|ξ|[1−cos⁡(2n−1)πξ2L]dξsin⁡(2n−1)πx2L

For the clamped-free case, comparing Eq. (17) with Eq. (1) yields: (18)c∫−δδU(x+ξ,s)−U(x,s)|ξ|dξ=−{ρv0(x)+du0(x)+B(x,s)−s2ρU(x,s)−s[dU(x,s)−ρu0(x)]}c∫−δδu(x+ξ)−u(x)|ξ|dξ=−b(x)

Considering U(x,s) as an analogue to u(x) leads to: (19)U(x,s)=∑n=1∞An(s)sin⁡n^x   n^=(2n−1)π2L

Referring to Eq. (10), substituting Eq. (19) into Eq. (18a) yields: (20)∑n=1∞An(s)[c∫−δδ1|ξ|(1−cos⁡n^ξ)dξ+s2ρ+sd]sin⁡n^x=ρv0(x)+du0(x)+B(x,s)+sρu0(x)

Coefficients in Eq. (20) can be determined using the orthogonality condition as follows: (21)An(s)=2L1ρρv^0(n)+du^0(n)+∫0LB(x,s)sin⁡n^xdx+sρu^0(n)(s−α1(n))(s−α2(n))where u^0(n)=∫0Lu0(x)sin⁡n^xdx  and  v^0(n)(x)=∫0Lv0sin⁡n^xdxwith (22)α1(n)=−d+d2−4ρc∫−δδ1|ξ|(1−cos⁡n^ξ)dξ2ρ=−ηnωn+ωn(ηn)2−1α2(n)=−d−d2−4ρc∫−δδ1|ξ|(1−cos⁡n^ξ)dξ2ρ=−ηnωn−ωn(ηn)2−1 ωn=cρ∫−δδ1|ξ|(1−cos⁡n^ξ)dξ  and  ηn=d2ρωn

It can be rewritten using partial fractions in Eq. (21) as (23)An(s)=2L1ρ{ρv^0(n)+du^0(n)α1(n)−α2(n)(1s−α1(n)−1s−α2(n))+ρu^0(n)α1−α2(α1(n)s−α1(n)−α2(n)s−α2(n))+∫0LB(x,s)sin⁡n^xdx(s−α1(n))(s−α2(n))}

Recalling the following and applying the inverse Laplace transform to Eq. (23) yields Eq. (25). (24)L−1(1s−α)=eαt  and  L−1[F(s)G(s)]=(f∗g)(t)=∫0tf(u)g(t−u)du (25)an(t)=2L1ρ1α1(n)−α2(n){(ρv^0(n)+du^0(n))(eα1(n)t−eα2(n)t)+ρu^0(n)(α1eα1(n)t−α2eα2(n)t)+∫0t(eα1(n)u−eα2(n)u)[∫0Lb(x,t−u)sin⁡n^xdx]du}where the following is the solution to this case: (26)u(x,t)=L−1[U(x,s)]=∑n=1∞an(t)sin⁡n^x

Substituting Eq. (25) into Eq. (26) yields: (27)u(x,t)=2L∑n=1∞1α1(n)−α2(n){[(v^0(n)+dρu^0(n))+u^0α1]eα1(n)t−[(v^0(n)+dρu^0(n))+u^0α2]eα2(n)t+1ρ∫0t(eα1(n)u−eα2(n)u)[∫0Lb(x,t−u)sin⁡(n^x)dx]du}sin⁡(n^x)where (28)u^0(n)=∫0Lu0(x)sin⁡(n^x)dxv^0(n)(x)=∫0Lv0sin⁡(n^x)dxα1(n)=−ηnωn+ωn(ηn)2−1α2(n)=−ηnωn−ωn(ηn)2−1ωn(δ)=cρ∫−δδ1|ξ|[1−cos⁡(n^ξ)]dξηn=d2ρωnn^=(2n−1)π2Lc=2EAδ2

In particular, for a damped free system, i.e., d=0 and Eqs. (27) and (28) reduce to (29)u(x,t)=2L∑n=1∞1ωn{v^0(n)sin⁡ωnt+u^0(n)ωncos⁡ωnt+1ρ∫0tsin⁡ωnu[∫0Lb(x,t−u)sin⁡(n^x)dx]du}sin⁡(n^x)in which (30)u^0(n)=∫0Lu0(x)sin⁡(n^x)dxv^0(n)(x)=∫0Lv0sin⁡(n^x)dxωn(δ)=cρ∫−δδ1|ξ|[1−cos⁡(n^ξ)]dξn^=(2n−1)π2Lc=2EAδ2

For a bar with clamped-free boundary conditions, the analytical solution has the same form except n^=nπL other than n^=(2n−1)π2L for Eqs. (27) to (29). Particular to the resonance condition, consider a damped-free bar subjected to some harmonic load at an arbitrary point x0, as shown in Fig. 1.

If the loading frequency coincides with some natural frequencies of the bar, such that (31)b(x,t)=b0δ(x−xo)sin⁡ωmtm=1,2,3,…

Substituting Eq. (31) into Eq. (27) results in the following: (32)u(x,t)=2L∑n=1∞1ωn{v^0(n)sin⁡ωnt+u^0(n)ωncos⁡ωnt+1ρ∫0tsin⁡ωnu[∫0Lb0δ(x−xo)sin⁡ωm(t−u)sin⁡(n^x)dx]du}sin⁡(n^x)=2L∑n=1∞1ωn{v^0(n)sin⁡ωnt+u^0(n)ωncos⁡ωnt+1ρb0sin⁡(n^x0)∫0tsin⁡(ωnu)sin⁡[ωm(t−u)]du}sin⁡(n^x)=2L{∑n=1n≠m∞1ωn[v^0(n)sin⁡ωnt+u^0(n)ωncos⁡ωnt+b0ρsin⁡(n^x0)ωmsin⁡ωnt−ωnsin⁡ωmtωm2−ωn2]sin⁡n^x+1ωm[v^0(m)sin⁡ωmt+u^0(m)ωmcos⁡ωmt+b0ρsin⁡(m^x0)sin⁡ωmt−ωmtcos⁡ωmt2ωm]sin⁡m^x}

As t→∞, Eq. (32) diverges.

First, the free vibration behavior of a damped system is investigated. Suppose the initial conditions are given as follows: (33)u0(x)=0.01xv0(x)=10x(x−1)

In this case, a bar that is initially at rest is subjected to a harmonic loading condition exerted at the free end (Fig. 4): (34)b(x,t)=b0δ(x−L)sin⁡ω¯tω¯=ω12  b0=1000

For this particular loading condition, an oscillatory behavior is observed, as shown in Fig. 5, since there is damping in the system, and there is a very good agreement between peridynamic and classical results.

This case is similar to the previous case by including damping in the system, as shown in Fig. 6. (35)b(x,t)=b0δ(x−L)sin⁡ω¯tω¯=ω12b0=1000  and  d=1000 Ns/m2

Variation of the displacement at the center of the bar, i.e., x = 0.5 m, as time progresses, as shown in Fig. 7. Different from the previous case, the amplitude of oscillations decreases due to damping. There is a very good agreement between peridynamic and classical results.

The above results indicate that PD agrees with classical theory in oscillation behaviors. However, those solutions are obtained for a small horizon size (δ≪L) so that the nonlocal character does not play a significant role. It can compare the natural frequencies between peridynamics and classical theory by considering different horizon sizes to investigate the subtlety of the nonlocal effect. The natural frequencies from peridynamics are obtained from Eq. (30).

In this case, the horizon size, δ, is varied from 0.0001 to 0.1m. The natural frequencies of the first four modes (n = 1, 2, 3, 4) are shown in Fig. 8. Each mode corresponds to an independent motion of the structure.

Wang et al. [33] reported that as the horizon size converges to 0 the peridynamic solution should converge to the classical continuum mechanics solution for the condition without the existence of damage in the structure, and nonlocal effects are insignificant. In addition, as indicated in [34], peridynamics can also represent wave dispersion, especially at short wavelengths, which is a phenomenon observed in real materials. For such conditions, the horizon can be determined by comparing the peridynamic wave dispersion curves against those obtained from lattice dynamics. As expected, peridynamic results converge to classical results as the horizon size approaches zero and diverge as the horizon size increases, which can represent non-classical nonlocal behavior that can occur, especially at small scales (see Fig. 8).

In the last numerical case, a bar subjected to a harmonic load at the free end is considered without the damping effect (Fig. 9). If the loading frequency coincides with the first natural frequency, such as b(x,t)=b0δ(x−L)sin⁡ω1tb0=100

And the bar is initially at rest, i.e., u(x,0)=0v˙(x,0)=0

The natural frequencies in PD are functions of the horizon size δ. In this study, two different scenarios for external loading are chosen as follows: ω1(δ=0.001m)=cρ∫−0.0010.0011|ξ|[1−cos⁡(π2Lξ)]dξω1(δ=0.5m)=cρ∫−0.50.51|ξ|[1−cos⁡(π2Lξ)]dξ

The displacement variation at the right edge of the bar, i.e., x = 1 m, for two different horizon sizes, 0.001 and 0.5 m, as the time progresses obtained by PD and classical theory are shown in Fig. 10. It can observe that both PD and classical theory results show resonant oscillation condition as the amplitude of oscillations increases as the time increases. In particular, for a loading frequency corresponding to a smaller horizon size, the resonance behavior obtained from the PD model and that of Classical Continuum Mechanics (CCM) is similar. In contrast, for a larger horizon size, the resonance of classical theory behaves more weakly than PD. This is expected since natural frequencies of PD converge to classical results for small horizon sizes and deviate from classical results as the horizon size increases, as indicated in Section 4.4.

This section presents the convergence analysis of the PD series solution for the free vibration of a fixed-free rod by considering the following initial conditions: u0(x)=0.01xv0(x)=0

The corresponding CCM solution truncated at the first 100 terms is chosen as the base reference: uCCM(x,t)=∑n=1100[2L∫0L0.01αsin⁡(2n−12Lπα)dα]cos⁡(2n−12LEρπt)sin⁡(2n−12Lπx)

A small horizon of δ=10−4m is selected for the peridynamic solution. The variation of the displacement of the material point at the right edge of the bar with respect to the number of terms in the PD series solution at different times, 0.01, 0.015, 0.02, and 0.025 s, is shown in Fig. 11.

Fig. 11 indicates that the PD solution converges to the CCM solution by using more than 10 terms in the PD series solution.