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Peridynamics has a great advantage over modeling the damage process of rock-like materials, which is assumed to be in a continuum interaction with each other across a finite distance. In the paper, an approach to incorporate classical elastic damage model in the nonordinary, state-based peridynamics is introduced. This method can model the dynamic damage process and stress change of rock-like materials. Then two instances about three-point bend experiment are simulated in the rock-like materials. Finally the conclusions are drawn that numerical results are close to the experimental results. So the method has a great predictable value in the geotechnical engineering.

The rock-like material is a quasibrittle material; it is widely applied to the geotechnical engineering, for example, tunnels and lots of other underground buildings. Especially for the mountain area, the research on the damage process of rock-like materials is essential, so the prediction for the damage process of rock-like materials becomes a focus problem gradually. But it is difficult to model because the mechanical character of rock-like materials is very complex and the precise damage prediction of rock-like materials is elusive. In the geotechnical engineering, the rock-like mass includes lots of tiny cracks, even before the load is applied. The damage process of the rock-like materials is often caused by tension, comparison, the change of the temperature, and so on, and the stress and displacement fields are influenced by the propagation and coalescence of the crack. Even through the damage process of rock-like materials has been investigated for many years, the damage mechanism and prediction of rock-like material are still not understood. In fact, the damage process of rock-like materials includes the initiation, propagation, and coalescence of the crack, which will lead to a sudden collapse due to brittle damage. How to model the damage character of the rock-like materials has brought great challenge.

Over the past decades, many methods are put forward to model the damage process of rock-like materials. In the finite element-based method, singular crack-tip elements are frequently encountered [

The peridynamic theory is a nonlocal meshless method; it is put forward by Silling [

After this theory is put forward, it has been widely applied to model the damage process of different materials. Firstly, the theory is applied to model the fracture process of composite material; for example, the fracture processes in laminated composites subjected to low-velocity impact and in woven composites subject to static indentation are predicted by Askari et al. [

The paper is organized as follows. In Section

For completeness, nonordinary state-based peridynamics is reviewed briefly; it includes a summary of basic peridynamic equation, the idea of constitutive relation, and its property.

Peridynamic theory is put forward by Silling [

The deformation of PD material point.

In the neighborhood of point

The nonlocal deformation gradient

Alternatively, this integral can be defined without recourse to peridynamic states as follows:

The Green-Lagrange strain sensor can be expressed as

The elastic strain energy

From (

And Cauchy stress is found:

The first Piola-Kirchhoff stress can be written from the Cauchy stress as

Equation (

Peridynamic force vector state

Substituting (

From equations (

To describe the damage correspondence within the peridynamic framework, the influence function

To model the problem of damage, the concept of local damage value is introduced; it is defined as

The region is discretized into nodes, each with a known volume in the reference configuration; taken together, the nodes form a grid. So the deformation gradient

Likewise, the discretization form of basic equation in the nonordinary, state-based peridynamics is shown as follows:

The acceleration

To evaluate the functionality of the proposed state-based peridynamic formulation, the discretized equations were implemented in a Fortran computer code for two-dimensional simulation. The following two examples are investigated, respectively.

Three-point bend test [^{3}, and

The geometric configuration.

The damage contour of the beam.

The maximum princess stress

The experimental result of three-point bend in the beam [

Load versus crack mouth displacement (CMD) curves obtained from the extended nonordinary, state-based peridynamic model and experiment.

It can be seen from Figure

This test consists of a circular rock sample with a preexisting vertical crack, its geometric configuration is shown in Figure ^{3}, and

The geometric configuration of rock sample.

The numerical contour about the failure of rock sample.

The maximum horizontal princess stress

The experimental result of the rock sample [

It can be seen from Figure

An extension of the state-based peridynamic constitutive correspondence framework to incorporate into elastic damage has been proposed. Then three-point bending simulation is performed with damage process. In two examples, the numerical results are compared with the experimental observation. In conclusion, the suggested method provides a new thought for the damage process prediction of the rock-like materials.

The density of material point

The applied body force density of material point

The displacement of material point

The relative origin position between material points

The force density vector of material point

The force density vector of material point

The relative position of points

The deformation state of the bond between material points

The deformation gradient of material point

The influence function of the bond

The volume of material point

The elastic strain energy

The second Piola-Kirchhoff stress

The horizontal radius of material point

A scalar function of the bond between material points

The stretch of the bond

The volume modulus

The critical energy dissipating ratio

The local damage value

The neighborhood of material points

Poisson’s ratio

The Green-Lagrange strain sensor

The elastic tensor

The first Piola-Kirchhoff stress

The critical stretch of the bond

The shear modulus

The fracture toughness

The acceleration of the node

The authors declare that they have no competing interests.

This work is supported by the first batch of Talent Introduction Projection in 2016 in Sichuan University of Science & Engineering (no. 2016RCL19), the projection in the Sichuan Provincial Department of Science (no. 2016GFW0137), and financial contributions from the Sichuan Province Science and Technology Plan Project (2013JY0119).