How to simulate fracture mode and crack propagation path in a plate with multiple cracks is an attractive but difficult issue in fracture mechanics. Peridynamics is a recently developed nonlocal continuum formulation that can spontaneously predict the crack nucleation, branch and propagation in materials and structures through a meshfree discrete technique. In this paper, the peridynamic motion equation with boundary traction is improved by simplifying the boundary transfer functions. We calculate the critical cracking load and the fracture angles of the plate with multiple cracks under uniaxial tension. The results are consistent with those predicted by classical fracture mechanics. The fracture mode and crack propagation path are also determined. The calculation shows that the brittle fracture process of the plate with multiple cracks can be conveniently and correctly simulated by the peridynamic motion equation with boundary conditions.

Peridynamics (PD) is a reformulation of the classical continuum mechanics [

The interest in PD has been growing due to its unparalleled ability to analyze various complex crack problems [

Shou et al. proposed an extended non-ordinary state-based peridynamics to investigate the initiation, propagation and coalescence of 3D pre-existing flaws in PMMA specimens subjected to uniaxial compressive loads [

Peridynamics has been used to simulate the failure of concrete structures. Zhang et al. proposed a novel coupled axial-shear interaction bond-slip model to simulate the damage behavior of reinforced concrete structures [

In present PD simulations, the external loads leading to crack initiation and growth are given by velocities or displacements of particles, rather than traction. This is due to the fact that it is very difficult to directly handle the traction and constraint imposed on the boundary surface of a body in peridynamics [

Although such treatment can obtain good results, some unphysical artifacts often appear in computation owing to the approximate feature of the fictitious boundary layer [

Parks et al. [

Recently, Huang [

The outline of the paper is as follows. In

The peridynamic motion equation advanced by Silling [

where

Meanwhile, it has been certified that the traction

When the displacement boundary condition _{p} = 0. Here,

If the traction boundary condition

Different from the existing models, here _{p}. Through the transfer functions, the effects of

By inserting

where

Let _{p} is the boundary surface prescribed by traction.

Combining

As a result, we acquire the peridynamic motion equation with the mixed boundary condition. For simplicity, we further assume that

_{B})/_{B} and

When ∂_{p} = ∂

The bond-based constitutive models have been established by Silling (BPD) [

Here,

In

The transfer functions of boundary constraints contain the transfer function of the boundary displacement constraint

In order to introduce as few undetermined functions as possible to represent

Peridynamics describes the deformation and motion due to the interaction between material points. Material points are connected to each other by bonds. The bond failure between material points is described by the bond stretch s, which is defined by [

When the deformation stretch exceeds the critical bond stretch

According to the peridynamic theory, the critical bond stretch s_{0} is related to the critical energy release rate G_{0} of materials [_{0} and G_{0} is represented as

As thus, the damage of the bond can be characterized by the scalar-valued function

By

Consequently, local damage

Note that the local damage ranges from 0 to 1. When

The peridynamic motion equation is an integral equation, to find the analytical solution of which is very difficult [

The adaptive dynamic relaxation method [

In the explicit central difference form, the acceleration

Take a square plate with the side length of 50 mm, as show in ^{3}, Poisson’s ratio

The square plate is discreated into a set of particles equally spaced from each other. In order to investigate the character of

In the elasticity, the displacements of the plate in

^{*}_{x} and ^{*}_{y} are two displacement components. We compare the displacements given by

The fracture of a square plate with small circular hole are also used to demonstrate the convergence [_{0} of bond failure is taken as 0.003.

The square plate is discretized into 40000 (200 × 200) collocation points with the fixed grid spacing size Δx = 0.25 mm. The time step is set to Δt = 1 s.

As shown in

Take a square plate with the side length of 50 mm. Let the Young’s modulus of the plate E = 30 GPa, the mass density ρ = 2460 kg/m^{3}, the Possion’s ratio v = 1/3 and the critical energy release rate G_{0} = 110 J/m^{2} [_{0} corresponding to G_{0} = 110 J/m^{2} is 0.0026.

During the simulation, the square plate is discretized into 40000 (200 × 200) collocation points with the spacing of Δx = 0.25 mm. The horizontal size is specified as δ = 3.015Δx. The time step is set to Δt = 1 s. It should be emphasized that the surface correction and artificial boundary layer in the numerical algorithm are no longer needed due to the introduction of the transfer functions of boundary constraints. Next, we will calculate the fracture of the plate when different cracks are preset in the plate.

As illustrated in

The plate preset an edge crack is shown in

As illustrated in

As illustrated in

Two cracks are preset in the plate in two different configurations. One configuration is that two equal length parallel cracks are arranged in the plate at an angle of 45°, as shown in

The crack propagation of the second configuration is shown in

Two configurations of three cracks in the plate are illustrated in

The propagation process of cracks in the first configuration is shown in

As shown in

Under the tension of 16.2 MPa, the propagation of the cracks in

The propagation of the cracks in

By simplification, the transfer functions of the boundary traction are determined. On this basis, the peridynamic motion equation with the boundary traction is improved and used to solve multiple crack propagation in an elastic-brittle square plate subjected to uniaxial tension on the boundary surface. The critical cracking load, fracture angle and propagation path of the crack are investigated. The conclusions are summarized as follows:

The surface correction and artificial boundary layer in the numerical algorithm can be cancelled due to the introduction of the transfer functions of boundary traction.

The brittle fracture can be well characterized by the prototype microelastic model.

The fracture angle can be directly given by peridynamic calculation without relevant criteria.

The fracture mode and propagation path of solids with multiple cracks can be conveniently determined in terms of peridynamics.

The work was supported by the National Nature Science Foundation of China through the Grant Nos. 12072145 and 11672129.

The authors declare that they have no conflicts of interest to report regarding the present study.