Peridynamics (PD) is a non-local mechanics theory that overcomes the limitations of classical continuum mechanics (CCM) in predicting the initiation and propagation of cracks. However, the calculation efficiency of PD models is generally lower than that of the traditional finite element method (FEM). Structural idealization can greatly improve the calculation efficiency of PD models for complex structures. This study presents a PD shell model based on the micro-beam bond via the homogenization assumption. First, the deformations of each endpoint of the micro-beam bond are calculated through the interpolation method. Second, the micro-potential energy of the axial, torsional, and bending deformations of the bond can be established from the deformations of endpoints. Finally, the micro moduli of the shell model can be obtained via the equivalence principle of strain energy density (SED). In addition, a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation. Numerical examples of crack propagation are provided, and the results demonstrate the effectiveness of the proposed PD shell model.

Peridynamic (PD) theory was proposed by Silling [

However, the calculation efficiency of PD models is usually lower than that of other traditional numerical methods, such as the finite element method (FEM), especially for complex structures. The coupling method is a strategy to improve calculation efficiency. Han et al. [

The structural idealization strategies for PD plate and shell models proposed above can be divided into two types: BBPD model and SBPD model. Compared with the SBPD model, the BBPD model has the advantages of easy programming, simple calculation, and high calculation efficiency and has a wide range of application prospects. In the study, the in-plane deformation part of the PD shell in our previous work [

The rest of this paper is organized as follows.

As shown in

In the local coordinate system of the micro-beam bond, the relationship between the force/moment densities and displacements of endpoints of the micro-beam bond can be expressed as

In

As shown in

The in-plane deformation only includes the translational displacement of the

The in-plane displacement vector at any position in the square element can be expressed by the Allman interpolation function [

According to

In the local coordinate system of the micro-beam bond, the in-plane displacement vector at endpoints

Through coordinate transformation, the in-plane displacement vector at endpoints

As shown in

Substituting

According to the homogenization assumption, the strain in the square element is assumed to be equal everywhere and is

Substituting

The micro moduli matrix of the micro-beam bond corresponding to axial and transversal deformations is defined as

Therefore, the micro-potential energy of the micro-beam bond can be expressed by

Then, the in-plane SED of point

The in-plane SED of point

By setting

The bending deformation only includes the transversal displacement along the

The bending displacement vector at any position of the square element can be expressed by a cubic interpolation function [

According to

In the local coordinate system of the micro-beam bond, the bending displacement vector at endpoints

Through coordinate transformation, the bending displacement vector at endpoints

As shown in

Substituting

According to the homogenization assumption, the strain in the square element is assumed to be equal everywhere and is

Substituting

The micro moduli matrix of the micro-beam bond corresponding to bending and torsional deformations is defined as

Therefore, the micro-potential energy of the micro-beam bond can be expressed by

Then, the bending SED of point

The bending SED of point

By setting

So far, the four micro moduli of various deformations for the PD shell model in

By comparing the relationship between the elongation rate

Assuming that the micro-beam bond is uniformly deformed in the neighborhood horizon of point

Then, substituting

Finally, the energy failure criterion of a micro-beam bond can be expressed by

A soda-lime brittle glass plate is selected to demonstrate that the proposed shell model can simulate its crack propagation. The geometry and dimensions of the glass specimen are shown in

Numerical results are shown in

As shown in

Four discrete mesh sizes of 1.25, 1, 0.8, and 0.5 mm are selected to investigate the effect of mesh refinement on cracks, respectively. The crack propagation paths of the plate using four uniform grids at 47

A cylindrical shell [

In this example, an S-shaped curved shell with pre-notches is selected to verify that the proposed PD shell model is applicable to any shell structure and to deal with thin shells with arbitrary cracks. The geometry and dimensions of the long side of the S-shaped curved shell are shown in

The damage diagrams of the shell structure at different loading times for two cases are shown in

In this study, a PD shell model based on the micro-beam bond via the homogenization method is presented. First, the deformations of each endpoint of the micro-beam bond are calculated using the interpolation method. Second, the micro-potential energy of the axial, torsional, and bending deformations of the bond can be established from the deformations of endpoints. Finally, the micro moduli of the shell model can be obtained via the equivalence principle of SED. The most critical aspect is the establishment of the conversion relationship between strains of the micro-beam bond and strains of the square element, which can significantly simplify the process of deriving the micro moduli. Different deformations of the micro-beam bond are considered; thus, it is not restricted by material parameters. On the basis of elasticity and small deformations, the 2D plane stress model and the bending model are combined to form a general PD shell model, which is a combination of axial, torsional, and bending deformations. The effectiveness of the proposed PD shell model is demonstrated by several numerical examples. The strain homogenization method used to construct the PD shell model in this study will be applied to the construction of other models in the future, such as anisotropic and large deformation models.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

The parameters of the bottom right corner mentioned in