1 Introduction
Classical continuum mechanics has achieved great success in describing the macro-scale properties of solid material based on the continuous medium hypothesis that the material is a continuous mass rather than as discrete particles. The assumption indicates that the substance of the object completely fills the space it occupies, without considering the inherent micro-structure of the material. Such a continuous medium hypothesis is not always valid in solid medium. Over the years, researchers found that many phenomena, such as size effect [
1], length scale effect [
2], skin/edge effect [
3], can not be well predicted by traditional continuum mechanics. These phenomena may be attributed to the nonlocal effect in the solid. In contrast with local theory whose mathematical language is partial differential derivatives defined at an infinitesimal point, nonlocal theory is formulated as integral form in a domain.
Classical continuum mechanics is regarded as a local theory. For solid mediums of multiple materials with a material interface or discontinuity such as fracture, the partial differential operator is no longer well defined. Around the fracture front tip, the stress singularity happens for local theory. To model fracture and its evolution, various local theories have been proposed, for example, finite element method (FEM) [
4], extended finite element method [
5], phase-field fracture method [
6‐
8], cracking particle method [
9,
10], extended finite element method [
11], numerical manifold method [
12], extended isogeometric analysis (XIGA) for three-dimensional crack [
13], meshfree methods [
14‐
16]. Another approach for fracture modeling is the nonlocal method. Compared with classical continuum mechanics without length scale, nonlocal theory takes into account the length scale explicitly and it is less sensitive to the inhomogeneity/discontinuity encountered in the materials due to its integral form.
Two general theories to account for the length scale of solid material, are the gradient elasticity [
1,
17‐
20] and the nonlocal elasticity [
21‐
24]. The gradient elasticity theory can be traced back to Cosserat theory in 1909 [
25]. It incorporates the length scale and higher order derivative of the displacement field. A variety of gradient elasticity theories have been proposed such as Mindlin solid theory [
2,
17], couple stress theory [
1,
26], modified couple stress [
18,
27] and second-grade materials [
19]. In nonlocal elasticity, the stress tensor is based on the integral of the “local” stress field in a domain, in contrast with the local elasticity defining the stress based on the strain field at a point. Under certain circumstances, the nonlocal elasticity can be transformed into gradient elasticity [
23,
28].
Among various nonlocal elasticity theories, peridynamics (PD) [
29,
30] has attracted the attention of the researchers in the fracture mechanics field. PD is based on the integral form well defined in domain with/without discontinuity. This salient feature enables PD a versatile method for fracture modeling [
31‐
34]. The origin of PD is the bond-based PD (BB-PD) with the Poisson ratio restriction. BB-PD can model 2D elasticity with Poisson ratio of 1/3 and 3D elasticity with Poisson ratio of 1/4. Many efforts have been dedicated to overcome this restriction, for example, PD with shear deformation [
35], bond-rotation effect by [
36], PD with micropolar deformation [
37]. The further development of PD is the state-based PD [
30,
38]. Several treatments are developed to overcome the instability issue in non-ordinary state-based PD (NOSBPD), including, bond-associated higher-order stabilized model [
39], higher-order approximation [
40], stabilized non-ordinary state-based PD [
41,
42], sub-horizon scheme [
43] and stress point method [
44].
In the spirit of nonlocality, PD has been extended in many directions, for example, dual-horizon PD [
45,
46], peridynamic plate/shell theory [
47‐
50], mixed peridynamic Petrov-Galerkin method for compressible and incompressible hyperelastic material [
51,
52], phase-field-based peridynamic damage model for composite structures [
53], wave dispersion analysis of PD [
54], damage mechanism in PD [
55], coupling scheme for state-based PD and FEM [
56,
57], higher-order peridynamic material models for elasticity [
58] and Peridynamic differential operator (PDDO) [
59‐
61] for solving partial differential equations, to name a few. PDDO has greatly extended the power of peridynamics and was applied to numerous challenging problems including fluid flow coupled with heat transfer [
62] and fracture evolution in batteries [
63] among others.
Dual-horizon PD overcomes the restriction of constant horizon in PD, without introducing side effects for variable horizons. Dual-horizon peridynamic formulation can be derived from the Euler–Lagrange equations [
64]. Based on the concept in nonlocal theory, we developed the Nonlocal Operator Method (NOM) as the generalization of dual-horizon PD. NOM uses the nonlocal operators of integral form to replace the local partial differential operators of different orders. There are three versions of NOM, first-order particle-based NOM [
65,
66], higher-order particle-based NOM [
67] and higher-order NOM based on numerical integration [
68]. The particle-based version can be viewed as a special case of NOM with numerical integration when nodal integration is employed. The nonlocal operators can be viewed as an alternative to the partial derivatives of shape functions in FEM. Combined with a variational principle or weighted residual method, NOM obtains the residual vector and tangent stiffness matrix in the same way as in FEM. NOM has been applied to the solutions of the Poisson equation in high dimensional space, von-Karman thin plate equations, fracture problems based on phase field [
67], waveguide problem in electromagnetic field [
66], gradient solid problem [
68] and Cahn–Hilliard equation [
69].
Although much progress in nonlocal methods has been achieved in the above mentioned literatures, the derivations for many physical problems remain cumbersome and complicated, see for example [
48,
58,
70,
71]. In local theory, the local differential operator is a fundamental element for describing physical problems. In analogy, the nonlocal operators would be very beneficial for developing nonlocal theoretical models. The power of NOM in deriving nonlocal models remains largely unexplored. In addition, NOM based on implicit algorithms is relatively complicated in implementation and in this paper, we explore the explicit algorithm in solving the nonlocal models. Furthermore, we propose an instability criterion of the nonlocal gradient operator for the purpose of fracture modeling.
The remaining of the paper is outlined as follows. In Sect.
2, the second-order NOM in 2D/3D is formulated in detail. In Sect.
3, we apply the NOM scheme combined with variational principle/weighted residual method to derive the nonlocal governing equations for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture model. The correspondence between local form and nonlocal form for higher-order problems is discussed. In Sect.
4, an instability criterion of nonlocal gradient is presented in the fracture modeling of linear elastic solid. The implementation of nonlocal solid and nonlocal thin plate is discussed in Sect.
5. Several numerical examples for solid and thin plate are used to demonstrate the accuracy and efficiency of the current method in Sect.
6. Last but not the least, some concluding remarks are presented.
4 Instability criterion for fracture modelling
Typical methods for fracture modelling are either based on diffusive crack domain in phase-field methods or on direct topological modification on meshes in XFEM or bonds in PD. Direct topological modification on meshes often leads to instability issues. For example, in NOSBPD, the breakage of a bond based on the quantities derived from stress state or strain state often introduces too much perturbation to the scheme, which may abort the calculation because of the singularity in shape tensors. These criteria include critical stretch [
29], energy based [
31] or stress-based criterion [
33,
34]. Another issue in NOSBPD is that the strain energy carried by a bond is closely related to other bonds. It also depends on the direction, the length of the bond, the choice of influence functions. Removing one neighbour often gives rise to catastrophic results on the calculation. A criterion on how to remove the neighbours safely from the neighbour list remains unclear.
Damage is a process deviated from the robust mathematical expression, where the transition happens in a very narrow zone, such as the crack tip front. It is observed that around the crack tip, the gradient or strain undergoes a sharp transition within a very small zone. Most conventional numerical methods for fracture modelling focus on accurate description of the singularity occurring around the crack tip, such a description is very hard to tackle and its evolution is inconvenient to update. This dilemma can be handled when something different from continuous function is introduced.
In NOM, the gradient operator is defined in a “redundant” way. Around the crack tip, the deformation is irregular and the part due to hourglass energy is comparable to the strain energy carried by a particle. More specifically, the operator energy in nonlocal operator method describes the irregularity of a function around the crack tip. The irregularity is the part that cannot be described by the continuous function. For continuous domain, the strain energy density is much larger than the operator energy density. However, for particles around the crack tip, the operator energy density is far from zero and the irregularity due to the singularity around the crack tip increases comparably to the strain energy density. In this sense, the operator energy density can be viewed as an indicator for the crack tip.
Unlike the strain energy density, the hourglass energy density describes the irregular deformation around the crack tip. It depends on the penalty for the strain energy. Larger penalty improves the continuity of deformation, but the extent of hourglass energy compared with the strain energy density is hard to estimate. In this paper, we propose a special manner to estimate the critical hourglass strain. Let the critical bond strain be denoted by
\(s_{max}\), which may depend on the characteristic length scale of the support, critical energy release rate and the elastic modulus. When the maximal strain reached
\(s_{max}\), the damage process is activated and the critical hourglass strain
\(s^{hg}_{max}\) is set as the maximal hourglass strain
\(s^{hg}_{ij}\) for all bonds in the computational model. In the sequential calculation, when the hourglass strain of a bond is larger than
\(s^{hg}_{max}\), the damage on that bond occurs, which is mathematically described as
$$\begin{aligned} d_{ij}={\left\{ \begin{array}{ll} 0 \text{ if } s^{hg}_{ij}(t)>s^{hg}_{max}, t\in [0,T]\\ 1 \text{ otherwise } \end{array}\right. } \end{aligned}$$
(87)
where
\(d_{ij}\) denotes the damage status between particle
i and particle
j.
The damage of a particle is calculated as
$$\begin{aligned} d_{i}=\frac{\int _{{\mathcal {S}}_i} d_{ij} \, \mathrm{d}V_j}{\int _{{\mathcal {S}}_i} \, \mathrm{d}V_j}. \end{aligned}$$
(88)
Every time one particle is removed from the neighbour list, the nonlocal gradient for the central particle should be recalculated based on the remaining “healthy” neighbour. We will apply this rule to model fractures in 2D and 3D linear elastic material.
5 Numerical implementation
We have applied NOM to derive the nonlocal strong forms for the traditional continuum model in Sect.
3. Two representative nonlocal theories, the dual-horizon peridynamics by Eq.
44 for fracture modeling and the nonlocal thin plate by Eq.
60, are selected for numerical test. For the DH-PD, the focus is on the test of instability criterion for quasi-static fracture modeling by explicit time integration algorithm. The nonlocal thin plate is compared with the finite element method. The nonlocal derivatives can be viewed as a generalization of the local derivatives, and the nonlocal derivatives recover the local derivatives when the size of the support degenerates to zero. The range of nonlocality depends on the choice of the weighting functions and the size of the supports. One obstacle of the nonlocal models is the verification since the exact solutions of the nonlocal model is rare. For simplicity of verification, we aim at solving the local problems with nonlocal forms where the nonlocal effect is reduced by selecting certain weighting functions.
The primary step in the implementation is the calculation of internal force based on the governing equations. In the first step, the computational domain is discretized into particles.
$$\begin{aligned} \Omega =\sum _{i=1}^N \Delta V_i \end{aligned}$$
(89)
where
N is the number of particles in the domain. Then the support of each particle is represented by a list of particle indices,
$$\begin{aligned} {\mathcal {S}}_i=\{j_1,j_2,\ldots ,j_{n_i}\} \end{aligned}$$
(90)
where
j is the global index of the particle and
\(n_i\) is the number of particles in
\({\mathcal {S}}_i\).
The gradient
\({\varvec{g}}_{ij}\) and Hessian
\({\varvec{h}}_{ij}\) for two particles
i,
j can be assembled by collecting terms in
\({\varvec{K}}_i \cdot {\varvec{p}}_{ij}\) according to Eqs.
21 or
23, where
$$\begin{aligned} {\varvec{K}}_i=\Bigg (\sum _{{\mathcal {S}}_i} \omega (\varvec{r}_{ij}){\varvec{p}}_{ij}\otimes {\varvec{p}}_{ij}^T \Delta V_j\Bigg )^{-1} \end{aligned}$$
(91)
with weight function
\(\omega (\varvec{r}_{ij})=1/|\varvec{r}_{ij}|^2\).
The nonlocal differential derivatives at point
i can be calculated as
$$\begin{aligned} {\tilde{\partial }}u_i= \sum _{j\in {\mathcal {S}}_i}\omega (\varvec{r}_{ij}){\varvec{K}}_i\cdot {\varvec{p}}_{ij}u_{ij} \Delta V_j \end{aligned}$$
(92)
The nonlocal operators in
\({\tilde{\partial }}u_i\) can be used to define the strain tensor, stress tensor, bending moment and others.
In discrete form, Eqs.
44 and
58 become
$$\begin{aligned}&\sum _{{\mathcal {H}}_i} \omega (\varvec{r}_{ij}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij}\Delta V_j\Delta V_i\nonumber \\&\quad -\sum _{{\mathcal {H}}_i'} \omega (\varvec{r}_{ji}) {\varvec{P}}_j \cdot {\varvec{g}}_{ji}\Delta V_j\Delta V_i +{\varvec{b}}_i\Delta V_i=\rho \Delta V_i \ddot{{\varvec{u}}}_i \end{aligned}$$
(93)
$$\begin{aligned}&\sum _{{\mathcal {S}}_i} \omega (\varvec{r}_{ij}) {\varvec{M}}_i: {\varvec{h}}_{ij} \Delta V_j\Delta V_i\nonumber \\&\quad - \sum _{{\mathcal {S}}_i'} \omega (\varvec{r}_{ij}) {\varvec{M}}_j: {\varvec{h}}_{ji}\Delta V_j\Delta V_i+q_i \Delta V_i=t \rho \Delta V_i \ddot{ w}_i \end{aligned}$$
(94)
In Eqs.
93 and
94, the volume of particle
i is multiplied on both sides of the equations. It is not required to calculate the internal forces from the dual-support. Let
\({\varvec{f}}_i=\varvec{0}, 1\le i\le N\) denote the initial internal force on particle
i. For each particle, one only needs to focus on the support, calculating the forces and adding the force to the particle internal force
$$\begin{aligned} \sum _{j\in {\mathcal {S}}_i} \omega (\varvec{r}_{ij}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij}\Delta V_j\Delta V_i&\rightarrow {\varvec{f}}_i \nonumber \\ -\omega (\varvec{r}_{ij_1}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij_1}\Delta V_{j_1}\Delta V_i&\rightarrow {\varvec{f}}_{j_1}\nonumber \\ -\omega (\varvec{r}_{ij_2}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij_2}\Delta V_{j_2}\Delta V_i&\rightarrow {\varvec{f}}_{j_2}\nonumber \\ \cdots&\nonumber \\ -\omega (\varvec{r}_{ij_{n_i}}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij_{n_i}}\Delta V_{j_{n_i}}\Delta V_i&\rightarrow {\varvec{f}}_{j_{n_i}} \end{aligned}$$
(95)
where
\(a\rightarrow b\) denotes the addition of
a to
b. The process of adding force
\( -\omega (\varvec{r}_{ij_1}) {\varvec{P}}_i \cdot {\varvec{g}}_{ij}\Delta V_{j}\Delta V_i\) to
\({\varvec{f}}_{j}\) is equivalent to accumulating the internal forces from particle
j’s dual-support.
For the calculating of internal force of thin plate, the same applies
$$\begin{aligned} \sum _{j\in {\mathcal {S}}_i} \omega (\varvec{r}_{ij}) {\varvec{M}}_i : {\varvec{h}}_{ij}\Delta V_j\Delta V_i&\rightarrow {\varvec{f}}_i \nonumber \\ -\omega (\varvec{r}_{ij_1}) {\varvec{M}}_i : {\varvec{h}}_{ij_1}\Delta V_{j_1}\Delta V_i&\rightarrow {\varvec{f}}_{j_1}\nonumber \\ -\omega (\varvec{r}_{ij_2}) {\varvec{M}}_i : {\varvec{h}}_{ij_2}\Delta V_{j_2}\Delta V_i&\rightarrow {\varvec{f}}_{j_2}\nonumber \\ \cdots&\nonumber \\ -\omega (\varvec{r}_{ij_{n_i}}) {\varvec{M}}_i : {\varvec{h}}_{ij_{n_i}}\Delta V_{j_{n_i}}\Delta V_i&\rightarrow {\varvec{f}}_{j_{n_i}} \end{aligned}$$
(96)
To maintain the stability of the nonlocal operator, the discrete form of Eq.
39 is
$$\begin{aligned}&\sum _{{\mathcal {S}}_i'} \omega (\varvec{r})\frac{p^{hg}}{m_j}\big (\varvec{u}_{ji}-\varvec{p}_i^T{\tilde{\partial }}\varvec{u}_j \big )\Delta V_j\Delta V_i\nonumber \\&\quad -\sum _{{\mathcal {S}}_i} \omega (\varvec{r})\frac{p^{hg}}{m_i}\big (\varvec{u}_{ij}-\varvec{p}_j^T{\tilde{\partial }}\varvec{u}_i\big ) \Delta V_j \Delta V_i. \end{aligned}$$
(97)
For particle
i with support
\({\mathcal {S}}_i\), the hourglass force is calculated as follows
$$\begin{aligned} \sum _{j\in {\mathcal {S}}_i} \omega (\varvec{r}_{ij})\frac{p^{hg}}{m_i}\big (\varvec{u}_{ij}-\varvec{p}_j^T{\tilde{\partial }}{\varvec{u}}_i\big )\Delta V_{j}\Delta V_i&\rightarrow {\varvec{f}}_i \nonumber \\ -\omega (\varvec{r}_{ij_1})\frac{p^{hg}}{m_i}\big (\varvec{u}_{ij_1}-\varvec{p}_{j_1}^T{\tilde{\partial }}{\varvec{u}}_i\big )\Delta V_{j_{1}}\Delta V_i&\rightarrow {\varvec{f}}_{j_1}\nonumber \\ -\omega (\varvec{r}_{ij_2})\frac{p^{hg}}{m_i}\big (\varvec{u}_{ij_2}-\varvec{p}_{j_2}^T{\tilde{\partial }}{\varvec{u}}_i\big )\Delta V_{j_{2}}\Delta V_i&\rightarrow {\varvec{f}}_{j_2}\nonumber \\ \cdots&\nonumber \\ -\omega (\varvec{r}_{ij_{n_i}})\frac{p^{hg}}{m_i} \big (\varvec{u}_{ij_{n_i}}-\varvec{p}_{j_{n_i}}^T{\tilde{\partial }}{\varvec{u}}_i\big )\Delta V_{j_{n_i}}\Delta V_i&\rightarrow {\varvec{f}}_{j_{n_i}} \end{aligned}$$
(98)
When the internal force is attained and the contribution of the external force boundary condition or body force is accumulated, the basic Verlet algorithm [
78] outlined as follows is used to update the displacement
$$\begin{aligned} {\varvec{u}}_i(t+\Delta t)&={\varvec{u}}_i(t)+{\varvec{v}}_i(t) \Delta t+\frac{1}{2} {\varvec{a}}_i(t)\Delta t^2 \end{aligned}$$
(99)
$$\begin{aligned} {\varvec{v}}_i(t+\Delta t)&={\varvec{v}}_i(t)+\frac{1}{2} \Big ({\varvec{a}}_i(t)+{\varvec{a}}_i(t+\Delta t)\Big )\Delta t \end{aligned}$$
(100)
where
\({\varvec{u}}_i\) denotes the displacement or deflection,
\({\varvec{v}}_i\) the velocity and
\({\varvec{a}}_i=\frac{{\varvec{f}}_i}{m_i}\) the acceleration for particle
i with mass
\(m_i\) subject to net force
\({\varvec{f}}_i\). For the detailed implementation and the numerical examples, the reader can find the open source code on Github
https://github.com/hl-ren/Nonlocal_elasticity, and
https://github.com/hl-ren/Nonlocal_thin_plate.
7 Conclusion
In this paper, we employ the recently proposed NOM to derive the nonlocal strong forms for various physical models, including elasticity, thin plate, gradient elasticity, electro-magneto-elastic coupled model and phase-field fracture model. These models require a second-order partial derivative at most and we make use of the second-order NOM scheme, which contains the nonlocal gradient and nonlocal Hessian operator. Considering the fact that most physical models are compatible with the variational principle/weighted residual method, we start from the energy form/weak form of the problem, by inserting the nonlocal expression of the gradient/Hessian operator into the weak form, based on the dual property of the dual-support in NOM, the nonlocal strong form is obtained with ease. Such a process can be extended to many other physical problems in other fields. The derived strong forms are variationally consistent and allow elegant description for inhomogeneous nonlocality in both theoretical derivation and numerical implementation.
We also propose an instability criterion in nonlocal elasticity or dual-horizon state-based peridynamics for the fracture modeling. The criterion is formulated as the functional of nonlocal gradient in support, which minimizes the zero-energy deformation that cannot be described by the nonlocal gradient. Such an operator functional approaches zero for continuous fields but has comparable value to the strain energy density for the deformation around the crack tip. During the fracture modeling by removing particles from the neighbor list, it is safer to delete the particle with larger zero-energy deformation. The numerical examples for 2D/3D fracture modeling confirm the feasibility and robustness of this criterion. The instability criterion is applicable for anisotropic elastic material and hyperelastic materials.
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