This example shows that the simulations recovers the discplacement predicted by classical continnums mechanics for a given load in force. The full example is shown in the following publication:

  • Diehl, P., Jha, P.K., Kaiser, H. et al. An asynchronous and task-based implementation of peridynamics utilizing HPX—the C++ standard library for parallelism and concurrency. SN Appl. Sci. 2, 2144 (2020). https://doi.org/10.1007/s42452-020-03784-x.

Generating the mesh

The file input_mesh.yaml in the example folder will generate the mesh as shown in the following figure.

setup

Model

The quasi-static models as described in [1] is used to assemble the tangent stiffness matrix and obtain the solution by solving Newton steps.

Model: 
  Dimension: 2 
  Discretization_Type:
    Spatial: finite_difference
    Time: quasi_static
  Final_Time: 1 
  Time_Steps: 1
  Horizon: 0.5
  Horizon_h_Ratio: 5

Material model

The state-based elastic material models as described in [2] is implemented for the ElasticState material.

Material: 
  Type: ElasticState 
  Density: 1200
  Compute_From_Classical: true 
  K: 4000.0 
  G: 1500.0
  Is_Plane_Strain: False
  Influence_Function: 
    Type: 1

Applying boundary conditions

Displacement boundary conditions

The following code applies a fixed displacement to the first layer of nodes on the right-hand side in the first figure in both directions.

Displacement_BC: 
  Sets: 2 
  Set_1:  
    Location:   
      Line: [1.55, 1.65]
    Direction: [1] 
    Time_Function: 
      Type: constant 
      Parameters: 
        - 0.0 
    Spatial_Function: 
      Type: constant 
  Set_2:  
    Location:   
      Line: [1.55, 1.65]
    Direction: [2] 
    Time_Function: 
      Type: constant 
      Parameters: 
        - 0.0 
    Spatial_Function: 
      Type: constant

Force boundary conditions

The following code applies a external body force density to the last node on the left-hand side in the first figure.

Note that in this example we want to apply a force $F=-40N$, however, the body force density has the units $\frac{N}{mm^2}$. Thus, the force is devided by the area $1.6\times 0.1$, where $1.6$ is the length of the plate and $0.1$ is the mesh width. This results in a body force density $-250$.

Force_BC:
  Sets: 1
  Set_1:
    Location:
      Line: [-0.1, 0.05]
    Direction: [1]
    Time_Function:
      Type: linear
      Parameters:
        - 1
    Spatial_Function:
      Type: constant
      Parameters:
        - -250.0

Solver

Solver:
  Type: BiCGSTAB
  Max_Iteration: 200
  Tolerance: 1e-6
  Perturbation: 1e-7

Validation

The prediction of the displacement field in $x$-direction is shown in second figure. For the derivation of the displacement field, we refer to [3].

setup

The third figure shows the obtained displacement field by the code using the above configuration.

setup

References

  1. Littlewood, David J. “Roadmap for peridynamic software implementation.” SAND Report, Sandia National Laboratories, Albuquerque, NM and Livermore, CA (2015).
  2. Silling, Stewart A., et al. “Peridynamic states and constitutive modeling.” Journal of Elasticity 88.2 (2007): 151-184. 3.