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Two-dimensional double horizon peridynamics for membranes

  • In this study, a two-dimensional "double-horizon peridynamics" formulation was presented for membranes. According to double-horizon peridynamics, each material point has two horizons: inner and outer horizons. This new formulation can reduce the computational time by using larger horizons and smaller inner horizons. To demonstrate the capability of the proposed formulation, various different analytical and numerical solutions were presented for a rectangular plate under different boundary conditions for static and dynamic problems. A comparison of peridynamic and classical solutions was given for different inner and outer horizon size values.

    Citation: Zhenghao Yang, Erkan Oterkus, Selda Oterkus. Two-dimensional double horizon peridynamics for membranes[J]. Networks and Heterogeneous Media, 2024, 19(2): 611-633. doi: 10.3934/nhm.2024027

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  • In this study, a two-dimensional "double-horizon peridynamics" formulation was presented for membranes. According to double-horizon peridynamics, each material point has two horizons: inner and outer horizons. This new formulation can reduce the computational time by using larger horizons and smaller inner horizons. To demonstrate the capability of the proposed formulation, various different analytical and numerical solutions were presented for a rectangular plate under different boundary conditions for static and dynamic problems. A comparison of peridynamic and classical solutions was given for different inner and outer horizon size values.



    As a new continuum-mechanics formulation, peridynamics [1] has been introduced by considering the challenges that classical continuum mechanics (CCM) is facing. One major challenge of CCM is discontinuities such as cracks in the solution domain. In this case, spatial derivatives in governing equations of CCM cannot be defined along discontinuities, which makes governing equations. In addition, CCM does not have a length scale parameter, which makes it difficult to make predictions for some emerging areas such as nanoengineering. Peridynamics overcomes the first issue by utilising integrations rather than spatial derivatives in its governing equation. Moreover, by incorporating a length scale parameter, the horizon, it can capture physical behaviours seen at small scales.

    There has been a significant development on peridynamics, especially during recent years. Amongst these, De Meo et al. [2] demonstrated the pit-to-crack process using peridynamics, starting from crack initiation to crack propagation phases by considering the microstructural features. Yin et al. [3] used peridynamics for large deformations and hyper-elastic materials. Liu et al. [4] utilized peridynamics to investigate fracture characteristics observed in zigzag graphene sheets. Peridynamics has also been used for the analysis of functionally graded materials [5] and composite materials [6]. Chen et al. [7] developed a fully coupled thermo-mechanical peridynamic formulation to analyse concrete cracking. Qin et al. [8] developed a peridynamic model for hydraulic fracturing of layered rock mass systems. Lakshmanan et al. [9] performed three-dimensional crystal plasticity simulations. Yan et al. [10] developed a coupled water flow and chemical transport peridynamic model suitable for unsaturated porous media. Wang et al. [11] presented a mixed-mode peridynamic fatigue model by employing ordinary-state-based peridynamics. Peridynamic formulations for beam [12,13] and plate [14,15] structures are also available in the literature. Peridynamics has also been utilized to free vibration [16] and buckling [17] analysis of cracked plates.

    Although they are limited, analytical solutions to peridynamic equations for some problems are available [18,19]. Numerical implementation of peridynamics is usually based on a meshless approach rather than mesh-based or semi-analytical approaches [20]. Ni et al. [21] coupled the finite element method and ordinary state-based peridynamics. Pagani and Carrera [22] coupled three-dimensional peridynamic formulation and higher-order one-dimensional finite elements. Xia et al. [23] coupled isogeometric analysis and peridynamics for the analysis of cracks. Liu et al. [24] coupled peridynamics and updated Lagrangian particle hydrodynamics to simulate ice–water interactions. Wang et al. [25] developed three-dimensional conjugated bond pair–based peridynamic formulation. Diana et al. [26] introduced anisotropic peridynamics suitable for homogenised micro-structured materials. Mikata [27] presented peridynamic formulations for fluid mechanics and acoustics. Another new peridynamic concept is "peridynamic differential operator", which is mainly used to convert differentiations to their corresponding integral form [28]. Ren et al. [29] developed a higher-order nonlocal operator method for the solution of boundary value problems. In another study, Ren et al. [30] proposed a nonlocal operator method applicable to solving partial differential equations of mechanical problems. Zhuang et al. [31] presented a nonlocal operator method for dynamic fracture exploiting an explicit phase-field model.

    Yang et al. [29] introduced the concept of "double-horizon peridynamics" with an intention to reduce computational time of peridynamic simulations. Double-horizon peridynamics is different than the dual-horizon peridynamics developed by Ren et al [30]. Derivation of dual-horizon peridynamics based on Euler-Lagrange formulation is presented in Wang et al [31]. In double-horizon peridynamics, each material point has two horizons, whereas, in dual-horizon peridynamics, each interacting material point has a different horizon and size. More information about peridynamics literature can be found in Javili et al [32].

    In this study, the double-horizon peridynamics formulation presented by Yang et al. [29] for one-dimensional structures is extended to two-dimensional structures, especially for a membrane. First, the details of the formulation are given. Next, the treatment of boundary conditions in the double-horizon peridynamics framework is provided. Then, analytical solutions for different boundary conditions are given. Finally, several numerical cases are presented by considering different boundary conditions for static or dynamic problems.

    The equation of motion (EOM) for a two-dimensional membrane in classical continuum mechanics (CCM) can be written as:

    2wt2(x,y,t)=c2(2wx2(x,y,t)+2wy2(x,y,t))+f(x,y,t). (2.1)

    Taylor expansion can be used to convert the Laplace term into nonlocal form as:

    w(x+ξ)w(x)=wxi|xξni+122wxixj|xξ2ninj+13!3wxixjxk|xξ3ninjnk+O(ξ4). (2.2)

    As shown in Figure 1, ξ=||ξ|| represents the distance between two material points, and ni represents the component of unit orientation vector such that

    ni={n1n2}={cosφsinφ} (2.3)

    and O(ξ4) denotes the truncation error.

    Figure 1.  Peridynamic horizon in classical PD formulation.

    Considering x as fixed, multiplying each term of Eq (2.2) by an attenuated kernel function 1ξ and integrating over the PD horizon gives

    2π0δ0w(x+ξ)w(x)ξξdξdφ=wxi|x2π0δ0niξdξdφ+122wxixj|x2π0δ0ξninjξdξdφ+13!3wxixjxk|x2π0δ0ξ2ninjnkξdξdφ+2π0δ0O(ξ3)ξdξdφ (2.4)

    which implies

    2wxixi=6πδ32π0δ0w(x+ξ)w(x)ξξdξdφO(δ4) (2.5)

    Plugging Eq (2.5) back into (2.1) yields:

    2wt2(x,y,t)=c26πδ32π0δ0w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ+f(x,y,t)O(δ4) (2.6)

    Eq (2.6) reduces to the classical PD equation of motion if we neglect the residual term:

    2wt2(x,y,t)=6c2πδ32π0δ0w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ+f(x,y,t) (2.7)

    One can observe from Eq (2.5) that there exists a truncation error between the classical Laplace expression and the PD Laplace expression. As a consequence, the PD equation of motion, Eq (2.7), differs from the corresponding CCM, Eq (2.1), on the order of O(δ4). One can reduce this error by choosing a small horizon size, δ. However, this will weaken the nonlocal characteristic of PD. On the other hand, with a large horizon size, δ, enhances PD nonlocal characteristics but can have negative effects on solution accuracy. Such contradiction is a common issue in most PD studies and can be overcome by the double-horizon peridynamics formulation. Moreover, double-horizon peridynamics can provide a computational advantage by utilising two horizons for each material point. In the double-horizon peridynamics formulation, a smaller inner horizon is introduced inside the original horizon, as shown in Figure 2. In this section, details of double-horizon peridynamics are presented for two-dimensional membranes.

    Figure 2.  Inner and outer horizons in the double-horizon peridynamics formulation.

    First, we know from Eq (2.5) that the nonlocal Laplace term within the inner horizon, ΩI can be expressed as

    2wxixi=6πε32π0ε0w(x+ξ)w(x)ξξdξdφO(ε4) (2.8)

    Note that the inner horizon size can be chosen as arbitrarily small so that 0O(ε4)O(δ4) so that the residual is negligible.

    Next, let us consider the nonlocal Laplace term over the outer horizon, ΩO. Multiplying each term in Eq (2.2) by 1ξ gives

    w(x+ξ)w(x)ξ=wxi|xni+122wxixj|xξninj+13!3wxixjxk|xξ2ninjnk+O(ξ3) (2.9)

    Note that when ξ varies over the outer horizon, the truncation error ranges as

    |O(ε3)||O(ξ3)||O(δ3)| (2.10)

    In order to minimize the truncation error so that it levels with that of the inner horizon, we can multiply Eq (2.9) by ε3ξ3:

    ε3ξ3w(x+ξ)w(x)ξ=wxi|xε3ξ3ni+122wxixj|xε3ξ2ninj+13!3wxixjxk|xε3ξninjnk+O(ε3) (2.11)

    Integrating Eq (2.11) over the outer horizon gives

    ΩOε3ξ3w(x+ξ)w(x)ξdΩO=wxi|xΩOε3ξ3nidΩO+122wxixj|xΩOε3ξ2ninjdΩO+13!3wxixjxk|xΩOε3ξninjnkdΩO+ΩOO(ε3)dΩO (2.12)

    Ignoring the residual results in the nonlocal Laplace term with respect to outer horizon as

    2wxixi=2πε3ln(δ/ε)2π0δεε3ξ3w(x+ξ)w(x)ξξdξdφ (2.13)

    Introducing two weight functions ωI and ωO for inner and outer horizon, respectively, such that

    ωI+ωO=1 (2.14)

    Coupling Eq (2.8) and (2.14) with the introduction of weight functions gives

    2wxixi=ωI6πε32π0ε0w(x+ξ)w(x)ξξdξdφ+ωO2πε3ln(δ/ε)2π0δεε3ξ3w(x+ξ)w(x)ξξdξdφ (2.15)

    in which the weight functions can be chosen by considering each area in proportion to the total horizon area as

    ωI=ε2δ2andωO=δ2ε2δ2 (2.6)

    Coupling Eq (2.15) with (2.16) and substituting back into (2.1) yields the refined PD equation of motion for membrane structure as

    2wt2(x,y,t)={kI2π0ε0w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ+kO2π0δεε3ξ3w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ}+f(x,y,t) (2.17a)

    In particular, it reduces to static case when eliminating the inertia term:

    kI2π0ε0w(x+ξ1,y+ξ2)w(x,y)ξξdξdφ+kO2π0δεε3ξ3w(x+ξ1,y+ξ2)w(x,y)ξξdξdφ+f(x,y)=0 (2.17b)

    where

    kI=c2ε2δ26πε3andkO=c22(δ2ε2)δ2ln(δ/ε)1πε3 (2.18)

    which represent PD parameters with respect to inner and outer horizons, respectively.

    Note that when the inner horizon radius is equal to the outer horizon radius, ε=δ, Eq (2.17a) and (2.17b) reduce to the traditional PD form.

    Eq (2.17) holds if and only if each integration domain is intact. In other words, the total PD horizon of each material point is completely embedded in the body. However, for some material points adjacent to the boundary whose PD horizon is incomplete, we can introduce a fictitious region with a width of δ outside the body to ensure that Eq (2.17) holds for the entire body, as shown in Figure 3. The displacement field of the fictitious region is related to the real body to achieve a different kind of boundary conditions, and two common cases are explained below.

    Figure 3.  Real and fictitious regions for peridynamic solution domain.

    Consider a body subjected to a fixed constrain at the edge x=x. Geometrically, this implies a zero curvature during deformation such that

    w(x,y,t)=0 (3.1a)
    2wx2|(x,y,t)=0 (3.1b)

    By performing central difference

    w(xξ,y,t)2w(x,y,t)+w(x+ξ,y,t)ξ2 (3.2)

    and substituting Eq (3.1a) back into Eq (3.2) yields:

    w(xξ,y,t)=w(x+ξ,y,t)ξ[0,δ] (3.3a)

    Similarly, a fixed boundary along the edge y=y satisfies the following:

    w(x,yξ,t)=w(x,y+ξ,t)ξ[0,δ] (3.3b)

    One can observe that a fixed boundary manipulates an anti-symmetric displacement relationship between real and fictitious regions with respect to the boundary.

    Consider a body subjected to free boundary at the edge x=x and geometrically, which implies a zero slope about y axis such that

    wx|(x,y,t)=0 (3.4)

    Performing central difference yields

    w(xξ,y,t)w(x+ξ,y,t)2ξ=0w(xξ,y,t)=w(x+ξ,y,t)ξ[0,δ] (3.5a)

    Similarly, for free edge along y=y, the PD boundary condition will be

    w(x,yξ,t)=w(x,y+ξ,t)ξ[0,δ] (3.5b)

    One can observe that a free boundary manipulates a symmetric displacement relation between real and fictitious regions with respect to the boundary.

    Consider a rectangular membrane with four fixed (clamped) edges (CCCC) subjected to some arbitrarily distributed load, as shown in Figure 4.

    Figure 4.  Rectangular membrane with four fixed edges (CCCC).

    As explained above, the PD boundary conditions can be given as

    {w(ξ,y)=w(ξ,y)w(a+ξ,y)=w(aξ,y)w(x,ξ)=w(x,ξ)w(x,b+ξ)=w(x,bξ)ξ[0,δ] (4.1)

    If we periodically extend such relationship over the entire xy-plane, one can obtain a periodic displacement field with respect to x-direction and it is skew-symmetric about (x,y)=(2na,y) for any integers n, with period of 2a. With respect to y-direction, it is skew-symmetric about (x,y)=(x,2nb) for any integers n, with period of 2b.

    Therefore, an admissible function can be chosen as

    w(x,y)=m=1n=1Amnsin(ˉmx)sin(ˉny) (4.2)

    where

    ˉm=mπaandˉn=nπb (4.3)

    Substituting Eq (4.25) into Eq (2.17b) yields

    m=1n=1Amn[kI2π0ε01cosˉmξ1cosˉnξ2ξξdξdφ+kO2π0δεε3ξ31cosˉmξ1cosˉnξ2ξξdξdφ]sinˉmxsinˉny=f(x,y) (4.4)

    where the coefficients can be obtained based on orthogonality conditions as

    Amn=4abb0a0f(x,y)sinˉmxsinˉnydxdykI2π0ε01cosˉmξ1cosˉnξ2ξξdξdφ+kO2π0δεε3ξ31cosˉmξ1cosˉnξ2ξξdξdφ (4.5)

    Substituting Eq (4.5) into (4.2) results in the analytical solution to PD double-horizon model as

    w(x,y)=m=1n=14ab[b0a0f(x,y)sin(ˉmx)sin(ˉny)dxdy]sin(ˉmx)sin(ˉny)kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ31cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ (4.6)

    In particular, when ε=δ, Eq (4.6) reduces to the solution for the classical PD model:

    w(x,y)=4ab1c2πδ36m=1n=1[b0a0f(x,y)sin(ˉmx)sin(ˉny)dxdy]sin(ˉmx)sin(ˉny)2π0δ01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ (4.7)

    Regarding boundary conditions of CCFF and CCCF, the corresponding solutions can be obtained from Eq (4.6) by letting:

    CCFF:ˉm=(2m1)π2aandˉn=(2n1)π2b (4.8)
    CCCF:ˉm=mπaandˉn=(2n1)π2b (4.9)

    Consider a rectangular membrane with two opposite edges clamped and others free subjected to some arbitrarily distributed load, as shown in Figure 5.

    Figure 5.  Rectangular membrane with mixed boundary conditions (CFCF).

    As explained above, the PD boundary conditions can be written as

    {w(ξ,y)=w(ξ,y)w(a+ξ,y)=w(aξ,y)w(x,ξ)=w(x,ξ)w(x,b+ξ)=w(x,bξ)ξ[0,δ] (4.10)

    Obeying such relationship and periodically extending the displacement field over the entire xy-plane, one admissible solution can be selected as

    w(x,y)=m=1n=0Amnsin(ˉmx)cos(ˉny) (4.11)

    where

    ˉm=mπaandˉn=nπb (4.12)

    Substituting Eq (4.11) back into Eq (2.17b), one can obtain

    m=1n=0Amn[kI2π0ε01cosˉmξ1cosˉnξ2ξξdξdφ+kO2π0δεε3ξ31cosˉmξ1cosˉnξ2ξξdξdφ]sinˉmxcosˉny=f(x,y) (4.13)

    where the coefficients can be obtained as

    {Amn=4abb0a0f(x,y)sin(ˉmx)cos(ˉny)dxdykI2π0ε01ξ(1cosˉmξ1cosˉnξ2)ξdξdφ+kO2π0δεε3ξ31ξ(1cosˉmξ1cosˉnξ2)ξdξdφm,n1Am0=2abb0a0f(x,y)sin(ˉmx)dxdykI2π0ε01ξ(1cosˉmξ1)ξdξdφ+kO2π0δεε3ξ31ξ(1cosˉmξ1)ξdξdφm1,n=0 (4.14)

    Substituting Eq (4.14) in Eq (4.11) yields

    w(x,y)=m=1{2abb0a0f(x,y)sin(ˉmx)dxdykI2π0ε01ξ(1cosˉmξ1)ξdξdφ+kO2π0δεε3ξ31ξ(1cosˉmξ1)ξdξdφ+n=1[4abb0a0f(x,y)sin(ˉmx)cos(ˉny)dxdykI2π0ε01cosˉmξ1cosˉnξ2ξξdξdφ+kO2π0δεε3ξ31cosˉmξ1cosˉnξ2ξξdξdφ]cosˉny}sinˉmx (4.15)

    Again, letting ε=δ reduces to the solution for the classical PD model as

    w(x,y)=4ab1c2πδ36m=1{12b0a0f(x,y)sinˉmxdxdy2π0δ01cosˉmξ1ξξdξdφ+n=1b0a0f(x,y)sinˉmxcosˉnydxdy2π0ε01cosˉmξ1cosˉnξ2ξξdξdφcosˉny}sinˉmx (4.16)

    Regarding boundary conditions of CFFF, the corresponding solution can be obtained from Eq (4.15) by letting:

    ˉm=(2m1)π2aandˉn=nπb (4.17)

    Consider an a×b rectangular membrane with four edges being clamped. The equation of motion and initial conditions are given as

    2wt2(x,y,t)=kI2π0ε0w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ+kO2π0δεε3ξ3w(x+ξ1,y+ξ2,t)w(x,y,t)ξξdξdφ (5.1)
    w(x,y,0)=w0(x,y)andwt(x,y,0)=v0(x,y) (5.2)

    Let us separate the variables as

    w(x,y,t)=W(x,y)T(t) (5.3)

    and substituting back into Eq (5.1):

    W(x,y)d2T(t)dt2=T(t){kI2π0ε0W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ+kO2π0δεε3ξ3W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ} (5.4)

    which yields

    1T(t)d2T(t)dt2=1W(x,y){kI2π0ε0W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ+kO2π0δεε3ξ3W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ}=λ (5.5)

    where λ ∈ ℝ is a constant independent of x,y and t. Two characteristic functions can be written as

    d2T(t)dt2=λT(t) (5.6a)

    and

    kI2π0ε0W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ+kO2π0δεε3ξ3W(x+ξ1,y+ξ2)W(x,y)ξξdξdφ=λW(x,y) (5.6b)

    Comparing Eqs (5.6b) with (2.17b), if we consider W(x,y) as an analogue to w(x,y) and λw(x,y) as an analogue to f(x,y), the following can be obtained by utilising Eqs (4.2) and (4.5) as

    m=1n=1Amn{kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ31cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ}sin(ˉmx)sin(ˉny)=m=1n=1λAmnsin(ˉmx)sin(ˉny) (5.7)

    By comparing corresponding coefficients on both sides, the eigenvalues can be obtained as

    λmn=kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ41cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ (5.8)

    The general solution to Eq (5.6a) can be written as

    Tmn(t)=Amncos(λmnt)+Bmnsin(λmnt) (5.9)

    According to the superposition principle, the general solution to Eq (5.3) can be written as a linear combination of each mode as

    w(x,y,t)=m=1n=1[Amncos(λmnt)+Bmnsin(λmnt)]sin(ˉmx)sin(ˉny) (5.10)

    and initial conditions can be written as

    w0(x,y)=m=1n=1Amnsin(ˉmx)sin(ˉny) (5.11a)
    v0(x,y)=m=1n=1Bmnλmnsin(ˉmx)sin(ˉny) (5.11b)

    where the coefficients can be obtained as

    Amn=4abb0a0w0(x,y)sin(ˉmx)sin(ˉny)dxdy (5.12a)
    Bmn=4ab1λmnb0a0v0(x,y)sin(ˉmx)sin(ˉny)dxdy (5.12b)

    In summary, the complete solution for CCCC is:

    w(x,y,t)=m=1n=1[Amncos(λmnt)+Bmnsin(λmnt)]sin(ˉmx)sin(ˉny)Amn=4abb0a0w0(x,y)sin(ˉmx)sin(ˉny)dxdyBmn=4ab1λmnb0a0v0(x,y)sin(ˉmx)sin(ˉny)dxdyλmn=kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ41cos(ˉmξ1)cos(ˉnξ2)ξξdξdφξ1=ξcosφξ2=ξsinφˉm=mπaˉn=nπbkI=c2ε2δ26πε3kO=c22(δ2ε2)δ2ln(δ/ε)1πε3 (5.13)

    In particular, by setting ε=δ, Eq (5.13) reduces to the solution of the classical PD model. Regarding boundary conditions of CCFF and CCCF, the corresponding solutions can be obtained from Eq (5.13) by letting:

    CCFF:ˉm=(2m1)π2aandˉn=(2n1)π2b (5.14a)
    CCCF:ˉm=mπaandˉn=(2n1)π2b (5.14b)

    Comparing Eq (5.6b) with (2.17b), if we consider W(x,y) as an analogue to w(x,y) and λw(x,y) as an analogue to f(x,y), the following can be obtained by utilising Eqs (4.11) and (4.13) as

    m=1n=0Amn{kI2π0ε01cosˉmξ1cosˉnξ2ξξdξdφ+kO2π0δεε3ξ31cosˉmξ1cosˉnξ2ξξdξdφ}sinˉmxcosˉny=m=1n=0λAmnsinˉmxcosˉny (5.15)

    By comparing corresponding coefficients on both sides, the eigenvalues can be obtained as

    λmn=kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ31cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ (5.16)

    The general solution to Eq (5.6a) is:

    Tmn(t)=Amncos(λmnt)+Bmnsin(λmnt) (5.17)

    According to the superposition principle, the general solution to Eq (5.3) can be written as a linear combination of each mode:

    w(x,y,t)=m=1n=0[Amncos(λmnt)+Bmnsin(λmnt)]sinˉmxcosˉny (5.18)

    and initial conditions can be written as

    w0(x,y)=m=1n=0Amnsinˉmxcosˉny (5.19a)
    v0(x,y)=m=1n=0Bmnλmnsinˉmxcosˉny (5.19b)

    According to orthogonality, one can obtain:

    Amn=4abb0a0w0(x,y)sinˉmxcosˉnydxdym,n1Am0=2abb0a0w0(x,y)sinˉmxdxdym1,n=0Bmn=4ab1λmnb0a0v0(x,y)sinˉmxcosˉnydxdym,n1Bm0=2ab1λmnb0a0v0(x,y)sinˉmxdxdym1,n=0 (5.20)

    In summary, the complete solution for CFCF is:

    w(x,y,t)=m=1n=0[Amncos(λmnt)+Bmnsin(λmnt)]sinˉmxcosˉnyAmn=4abb0a0w0(x,y)sinˉmxcosˉnydxdym,n1Am0=2abb0a0w0(x,y)sinˉmxdxdym1,n=0Bmn=4ab1λmnb0a0v0(x,y)sinˉmxcosˉnydxdym,n1Bm0=2ab1λmnb0a0v0(x,y)sinˉmxdxdym1,n=0λmn=kI2π0ε01cos(ˉmξ1)cos(ˉnξ2)ξξdξdφ+kO2π0δεε3ξ31cos(ˉmξ1)cos(ˉnξ2)ξξdξdφξ1=ξcosφξ2=ξsinφˉm=mπaˉn=nπbkI=c2ε2δ26πε3kO=c22(δ2ε2)δ2ln(δ/ε)1πε3 (5.21)

    In order to validate the capability of the current formulation, several numerical cases are considered and compared against the corresponding classical solutions. The numerical solution of equations of peridynamics is usually done by using the meshless approach [36]. Lopez and Pellegrino [37] implemented a spectral method for the space discretization based on the Fourier expansion of the solution while considering the Newmark-β method for the time marching. Jafarzadeh et al. [38] introduced an efficient boundary-adapted spectral method for peridynamic transient diffusion problems with arbitrary boundary conditions. Without loss of generality, membrane dimensions a × b = 1 m × 1 m and parameter c = 1 Nm/kg are chosen throughout this section. The outer horizon size of δ=0.1m and varying inner horizon sizes of ε=δ50,δ10,δ5 and ε=δ (i.e., classical PD model) are considered.

    In this first numerical case, a rectangular plate with four fixed edges (CCCC) subjected to a loading of f(x,y)=0.05sinπxasinπyb is considered under static conditions. The deflection of the plate along the central x-axis for different inner horizon values is given in Figure 6a. As shown in Figure 6b, as the inner horizon size decreases, the peridynamic solution approaches the classical solution.

    Figure 6.  (a) Comparison of PD deflection results with CCM results along x-axis; (b) zoomed view.

    In the second numerical case, for the same loading condition f(x,y)=0.05sinπxasinπyb, the rectangular plate is subjected to mixed boundary conditions (CFCF). Deflection values along the central x- and y-axes are given in Figure 7a, c. As in the previous case, the peridynamic solution converges to a classical solution as the inner horizon size decreases (see Figures 7b, d).

    Figure 7.  (a) Comparison of PD deflection results with CCM results along x-axis; (b) zoomed view; (c) comparison of PD deflection results with CCM results along y-axis; (d) zoomed view.

    For the next numerical case, the dynamic behavior of the rectangular plate is investigated. The plate has fixed boundaries and is subjected to initial displacement and velocity conditions:

    ICs:u0(x,y)=0.05x(xa)y(yb)v0(x,y)=sinπxasinπyb

    Figure 8a demonstrates the variation of the deflection at the center of the rectangular plate as time progresses. Similar to the static cases, the peridynamic solution converges to the classical solution as the inner horizon size decreases as shown in Figure 8b.

    Figure 8.  (a) Comparison of PD deflection results with CCM results at the center of the rectangular plate as time progresses; (b) zoomed view.

    For the second dynamic case, the rectangular plate is subjected to the same initial displacement and velocity conditions:

    ICs:u0(x,y)=0.05x(xa)v0(x,y)=sinπxasinπyb

    but the edges are subjected to mixed boundary conditions (CFCF). For this boundary condition, the variation of deflection at the center of the plate as time progresses is given in Figure 9a. As shown in Figure 9b, peridynamic and classical solutions become closer as the inner horizon size decreases.

    Figure 9.  (a) Comparison of PD deflection results with CCM results at the center of the rectangular plate as the time progresses; (b) zoomed view.

    For the final numerical case, for the mixed boundary condition (CFCF) for the rectangular membrane, vibrational frequencies of the first 6 modes are compared in Figures 10af. According to these figures, it can be seen that as the outer horizon decreases, peridynamic solutions converge to classical solutions and, after certain outer horizon values, these two solutions start diverging due to nonlocal effects. On the other hand, a decreasing inner horizon size can allow representation of the classical behaviour although the outer horizon size is relatively large.

    Figure 10.  Comparison of PD deflection results with CCM results as the outer horizon size increases. (a) Mode m = 1, n = 0; (b) mode m = 1, n = 1; (c) mode m = 2, n = 1; (d) mode m = 2, n = 2; (e) mode m = 3, n = 2; (f) mode m = 3, n = 3.

    In the study, a newly proposed double-horizon peridynamics formulation was presented for two-dimensional membranes subjected to fixed or mixed boundary conditions. Both analytical and numerical solutions are presented. According to the numerical results, it was shown that as the inner horizon size decreases, peridynamic solutions converge to a classical solution for different boundary conditions and both static and dynamic problems. Moreover, it was also demonstrated that for relatively large outer horizon sizes, the inner horizon can allow capturing classical behavior as the inner horizon size decreases. Therefore, it can be concluded that double-horizon peridynamics can provide an alternative platform to reduce computational time by allowing larger horizon sizes but still capturing classical behavior by reducing the inner horizon sizes.

    Zhenghao Yang: Conceptualization, Methodology, Formal Analysis, Software, Validation, Visualization, Writing-Original Draft Preparation, Erkan Oterkus: Conceptualization, Methodology, Writing-Review & Editing, Selda Oterkus: Conceptualization, Methodology, Writing-Review & Editing.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare that there is no conflict of interest.



    [1] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J Mech Phys Solids, 48 (2000), 175–209. https://doi.org/10.1016/S0022-5096(99)00029-0 doi: 10.1016/S0022-5096(99)00029-0
    [2] D. De Meo, L. Russo, E. Oterkus, Modeling of the onset, propagation, and interaction of multiple cracks generated from corrosion pits by using peridynamics, J Eng Mater Techn, 139 (2017), 041001. https://doi.org/10.1115/1.4036443 doi: 10.1115/1.4036443
    [3] B. B. Yin, W. K. Sun, Y. Zhang, K. M. Liew, Modeling via peridynamics for large deformation and progressive fracture of hyperelastic materials, Comput. Methods Appl. Mech. Eng., 403 (2023), 115739. https://doi.org/10.1016/j.cma.2022.115739 doi: 10.1016/j.cma.2022.115739
    [4] X. Liu, X. He, J. Wang, L. Sun, E. Oterkus, An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets, Proc. Math. Phys. Eng. Sci., 474 (2018), 20180019. https://doi.org/10.1098/rspa.2018.0019 doi: 10.1098/rspa.2018.0019
    [5] M. Dorduncu, Stress analysis of sandwich plates with functionally graded cores using peridynamic differential operator and refined zigzag theory, Thin Wall Struct, 146 (2020), 106468. https://doi.org/10.1016/j.tws.2019.106468 doi: 10.1016/j.tws.2019.106468
    [6] A. Kutlu, M. Dorduncu, T. Rabczuk, A novel mixed finite element formulation based on the refined zigzag theory for the stress analysis of laminated composite plates, Compos Struct, 267 (2021), 113886. https://doi.org/10.1016/j.compstruct.2021.113886 doi: 10.1016/j.compstruct.2021.113886
    [7] W. Chen, X. Gu, Q. Zhang, X. Xia, A refined thermo-mechanical fully coupled peridynamics with application to concrete cracking, Eng. Fract. Mech., 242 (2021), 107463. https://doi.org/10.1016/j.engfracmech.2020.107463 doi: 10.1016/j.engfracmech.2020.107463
    [8] M. Qin, D. Yang, W. Chen, S. Yang, Hydraulic fracturing model of a layered rock mass based on peridynamics, Eng. Fract. Mech., 258 (2021), 108088. https://doi.org/10.1016/j.engfracmech.2021.108088 doi: 10.1016/j.engfracmech.2021.108088
    [9] A. Lakshmanan, J. Luo, I. Javaheri, V. Sundararaghavan, Three-dimensional crystal plasticity simulations using peridynamics theory and experimental comparison, Int J Plasticity, 142 (2021), 102991. https://doi.org/10.1016/j.ijplas.2021.102991 doi: 10.1016/j.ijplas.2021.102991
    [10] H. Yan, M. Sedighi, A. P. Jivkov, Peridynamics modelling of coupled water flow and chemical transport in unsaturated porous media, J Hydrol, 591 (2020), 125648. https://doi.org/10.1016/j.jhydrol.2020.125648 doi: 10.1016/j.jhydrol.2020.125648
    [11] H. Wang, S. Tanaka, S. Oterkus, E. Oterkus, Study on two-dimensional mixed-mode fatigue crack growth employing ordinary state-based peridynamics, Theor. Appl. Fract. Mech., 124 (2023), 103761. https://doi.org/10.1016/j.tafmec.2023.103761 doi: 10.1016/j.tafmec.2023.103761
    [12] Z. Yang, S. Oterkus, E. Oterkus, Peridynamic formulation for Timoshenko beam, Procedia Struct. Integr, 28 (2020), 464–471. https://doi.org/10.1016/j.prostr.2020.10.055 doi: 10.1016/j.prostr.2020.10.055
    [13] Z. Yang, E. Oterkus, S. Oterkus, Peridynamic higher-order beam formulation, J Peridyn Nonlocal Model, 3 (2021), 67–83. https://doi.org/10.1007/s42102-020-00043-w doi: 10.1007/s42102-020-00043-w
    [14] Z. Yang, B. Vazic, C. Diyaroglu, E. Oterkus, S. Oterkus, A Kirchhoff plate formulation in a state-based peridynamic framework, Math Mech Solids, 25 (2020), 727–738. https://doi.org/10.1177/1081286519887523 doi: 10.1177/1081286519887523
    [15] Z. Yang, E. Oterkus, S. Oterkus, Peridynamic formulation for higher-order plate theory, J Peridyn Nonlocal Model, 3 (2021), 185–210. https://doi.org/10.1007/s42102–020–00047–6 doi: 10.1007/s42102–020–00047–6
    [16] J. Heo, Z. Yang, W. Xia, S. Oterkus, E. Oterkus, Free vibration analysis of cracked plates using peridynamics, Ships Offshore Struct, 15 (2020), S220–S229. https://doi.org/10.1080/17445302.2020.1834266 doi: 10.1080/17445302.2020.1834266
    [17] J. Heo, Z. Yang, W. Xia, S. Oterkus, E. Oterkus, Buckling analysis of cracked plates using peridynamics, Ocean Eng., 214 (2020), 107817. https://doi.org/10.1016/j.oceaneng.2020.107817 doi: 10.1016/j.oceaneng.2020.107817
    [18] Z. Yang, C. C. Ma, E. Oterkus, S. Oterkus, K. Naumenko, Analytical solution of 1–dimensional peridynamic equation of motion, J Peridyn Nonlocal Model, 5 (2023), 356–374. https://doi.org/10.1007/s42102–022–00086–1 doi: 10.1007/s42102–022–00086–1
    [19] Z. Yang, C. C. Ma, E. Oterkus, S. Oterkus, K. Naumenko, B. Vazic, Analytical solution of the peridynamic equation of motion for a 2–dimensional rectangular membrane, J Peridyn Nonlocal Model, 5 (2023), 375–391. https://doi.org/10.1007/s42102–022–00090–5 doi: 10.1007/s42102–022–00090–5
    [20] E. Oterkus, E. Madenci, M. Nemeth, Stress analysis of composite cylindrical shells with an elliptical cutout, J mech mater struct, 2 (2007), 695–727.
    [21] T. Ni, M. Zaccariotto, Q. Z. Zhu, U. Galvanetto, Coupling of FEM and ordinary state–based peridynamics for brittle failure analysis in 3D, Mech Adv Mater Struc, 28 (2021), 875–890. https://doi.org/10.1080/15376494.2019.1602237 doi: 10.1080/15376494.2019.1602237
    [22] A. Pagani, E. Carrera, Coupling three‐dimensional peridynamics and high‐order one‐dimensional finite elements based on local elasticity for the linear static analysis of solid beams and thin‐walled reinforced structures, Int J Numer Meth Eng, 121 (2020), 5066–5081. https://doi.org/10.1002/nme.6510 doi: 10.1002/nme.6510
    [23] Y. Xia, X. Meng, G. Shen, G. Zheng, P. Hu, Isogeometric analysis of cracks with peridynamics, Comput Method Appl M, 377 (2021), 113700. https://doi.org/10.1016/j.cma.2021.113700 doi: 10.1016/j.cma.2021.113700
    [24] R. Liu, J. Yan, S. Li, Modeling and simulation of ice–water interactions by coupling peridynamics with updated Lagrangian particle hydrodynamics, Comput. Part. Mech., 7 (2020), 241–255. https://doi.org/10.1007/s40571–019–00268–7 doi: 10.1007/s40571–019–00268–7
    [25] Y. Wang, X. Zhou, Y. Wang, Y. Shou, A 3–D conjugated bond–pair–based peridynamic formulation for initiation and propagation of cracks in brittle solids, International Journal of Solids and Structures, 134 (2018), 89–115. https://doi.org/10.1016/j.ijsolstr.2017.10.022 doi: 10.1016/j.ijsolstr.2017.10.022
    [26] V. Diana, A. Bacigalupo, M. Lepidi, L. Gambarotta, Anisotropic peridynamics for homogenized microstructured materials, Comput Method Appl M, 392 (2022), 114704. https://doi.org/10.1016/j.cma.2022.114704 doi: 10.1016/j.cma.2022.114704
    [27] Y. Mikata, Peridynamics for fluid mechanics and acoustics, Acta Mech, 232 (2021), 3011–3032. https://doi.org/10.1007/s00707–021–02947–0 doi: 10.1007/s00707–021–02947–0
    [28] E. Madenci, A. Barut, M. Dorduncu, Peridynamic differential operator for numerical analysis, Berlin: Springer International Publishing, 2019,978–980. https://doi.org/10.1007/978–3–030–02647–9
    [29] H. Ren, X. Zhuang, T. Rabczuk, A higher order nonlocal operator method for solving partial differential equations, Comput Method Appl M, 367 (2020), 113132. https://doi.org/10.1016/j.cma.2020.113132 doi: 10.1016/j.cma.2020.113132
    [30] H. Ren, X. Zhuang, T. Rabczuk, A nonlocal operator method for solving partial differential equations, Comput Method Appl M, 358 (2020), 112621. https://doi.org/10.1016/j.cma.2019.112621 doi: 10.1016/j.cma.2019.112621
    [31] X. Zhuang, H. Ren, T. Rabczuk, Nonlocal operator method for dynamic brittle fracture based on an explicit phase field model, Eur J Mech A-solid, 90 (2021), 104380. https://doi.org/10.1016/j.euromechsol.2021.104380 doi: 10.1016/j.euromechsol.2021.104380
    [32] Z. Yang, E. Oterkus, S. Oterkus, C. C. Ma, Double horizon peridynamics, Math Mech Solids, 28 (2023), 2531–2549. https://doi.org/10.1177/10812865231173686 doi: 10.1177/10812865231173686
    [33] H. Ren, X. Zhuang, Y. Cai, T. Rabczuk, Dual‐horizon peridynamics, International Journal for Numerical Methods in Engineering, 108 (2016), 1451–1476. https://doi.org/10.1016/j.cma.2016.12.031 doi: 10.1016/j.cma.2016.12.031
    [34] B. Wang, S. Oterkus, E. Oterkus, Derivation of dual-horizon state-based peridynamics formulation based on Euler-Lagrange equation, Continuum Mech Therm, 35 (2023), 841–861. https://doi.org/10.1007/s00161–020–00915–y doi: 10.1007/s00161–020–00915–y
    [35] A. Javili, R. Morasata, E. Oterkus, S. Oterkus, Peridynamics review, Math Mech Solids, 24 (2019), 3714–3739. https://doi.org/10.1177/1081286518803411 doi: 10.1177/1081286518803411
    [36] E. Madenci, E. Oterkus, Peridynamic Theory and Its Applications, New York: Springer New York, 2013. https://doi.org/10.1007/978–1–4614–8465–3
    [37] L. Lopez, S. F. Pellegrino, A space-time discretization of a nonlinear peridynamic model on a 2D lamina, Comput Math Appl, 116 (2022), 161–175. https://doi.org/10.1016/j.camwa.2021.07.004 doi: 10.1016/j.camwa.2021.07.004
    [38] S. Jafarzadeh, A. Larios, F. Bobaru, Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods, J Peridyn Nonlocal Model, 2 (2020), 85–110. https://doi.org/10.1007/s42102–019–00026–6 doi: 10.1007/s42102–019–00026–6
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