A Deformation-Based Peridynamic Model: Theory and Application
Abstract
1. Introduction
2. Materials and Methods: Deformation-Based PD Formulation
2.1. A Brief Introduction to PD
2.2. Virtual Internal Bond Method
2.3. Two Parameter PD
Failure Criteria
2.4. Stress-Based Correction Factor
3. Numerical Experiment and Results
3.1. Implementation of Cython
3.2. 2D Plate Under Uniaxial Loading
3.3. Rectangular Plate with a Hole
3.4. Uniaxial Compression Test
3.5. Comparison with Existing Method
4. Discussion
5. Conclusions
- Individual PD bonds are treated as virtual internal bonds, with equivalent normal and tangential stiffness derived through strain energy equivalence with a classical continuum model. Relationships between these stiffness parameters and traditional elastic constants are established. The resulting PD equations support materials with Poisson’s ratios up to 1/3 for plane-stress conditions and up to 1/4 for plane-strain conditions.
- The proposed stress-based correction method effectively mitigates skin effects. The absolute errors introduced are of the same magnitude as those calculated using the energy-based correction method. The accuracy of node displacement compared to analytical displacements improves as Poisson’s ratio decreases, indicating the correction’s dependence on the sample’s Poisson’s ratio.
- Before correction, PD equations using bond strain conditions yield lower absolute displacement errors along the loading axis compared to those using bond deformation. However, after correction, both methods produce similar errors along this axis. Along the other axis, although the absolute errors are higher before correction for PD equations formulated using bond deformation, they tend to be lower near the boundary after correction compared to those formulated using bond strain conditions. This implies that the PD model developed based on bond deformation could potentially be more effective compared to the model derived from bond strain.
- Deformation-based PD has proven effective in accurately capturing the stress concentration factor, demonstrating its potential for application in modeling complex geometries.
- A mixed-mode failure criterion is derived for deformation-based PD. X-shaped conjugate shear failure is observed in granite specimens, resembling the failure of intact rocks under uniaxial compression testing. Additionally, tangential bond failure occurs before normal bond failure in the uniaxial test.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PD | peridynamics |
BBPD | bond-based peridynamics |
FEM | finite element method |
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Poisson’s Ratio | (%) | (%) | Computed Poisson’s Ratio | Remarks |
---|---|---|---|---|
0.0 | - | 22.074 | 0.0 | Uncorrected |
0.0 | - | 8.482 | 0.0 | Corrected |
0.1 | 23.973 | 22.674 | 0.124 | Uncorrected |
0.1 | 5.144 | 7.658 | 0.104 | Corrected |
0.2 | 24.983 | 23.468 | 0.249 | Uncorrected |
0.2 | 3.680 | 11.545 | 0.205 | Corrected |
0.3 | 26.611 | 24.587 | 0.378 | Uncorrected |
0.3 | 7.224 | 16.532 | 0.311 | Corrected |
E (GPa) | (kg/m3) | UCS (MPa) | (Jm2) | (Jm2) | (°) | |
---|---|---|---|---|---|---|
21 | 2800 | 0.23 | 112 | 93 | 170 |
Parameters | Values |
---|---|
material spacing | 0.0002 m |
horizon | 0.0006 m |
critical tensile deformation | µm |
critical compressive deformation | µm |
critical tangential deformation | µm |
loading rate | m/s |
time step | 1 s |
Before Correction | After Correction | |||||||
---|---|---|---|---|---|---|---|---|
Poisson’s Ratio | Deformation | Stretch | Deformation | Stretch | ||||
0 | - | 22.074 | - | 17.700 | - | 8.482 | - | 8.251 |
0.1 | 23.973 | 22.674 | 17.204 | 17.924 | 2.840 | 9.682 | 3.667 | 9.419 |
0.2 | 24.983 | 23.468 | 17.504 | 18.193 | 3.680 | 11.545 | 4.984 | 11.237 |
0.3 | 26.611 | 24.587 | 18.254 | 18.568 | 7.224 | 16.532 | 10.111 | 16.072 |
After Correction | ||||
---|---|---|---|---|
Poisson’s Ratio | Deformation | Stretch | ||
0.0 | - | 5.550 | - | 5.417 |
0.1 | 2.840 | 6.305 | 2.010 | 6.142 |
0.2 | 3.680 | 7.421 | 2.505 | 7.219 |
0.3 | 5.672 | 10.247 | 4.7971 | 9.908 |
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Adhikari, B.; Li, D.; Han, Z. A Deformation-Based Peridynamic Model: Theory and Application. Buildings 2025, 15, 1931. https://doi.org/10.3390/buildings15111931
Adhikari B, Li D, Han Z. A Deformation-Based Peridynamic Model: Theory and Application. Buildings. 2025; 15(11):1931. https://doi.org/10.3390/buildings15111931
Chicago/Turabian StyleAdhikari, Bipin, Diyuan Li, and Zhenyu Han. 2025. "A Deformation-Based Peridynamic Model: Theory and Application" Buildings 15, no. 11: 1931. https://doi.org/10.3390/buildings15111931
APA StyleAdhikari, B., Li, D., & Han, Z. (2025). A Deformation-Based Peridynamic Model: Theory and Application. Buildings, 15(11), 1931. https://doi.org/10.3390/buildings15111931