Skip to content

Commit

Permalink
docu
Browse files Browse the repository at this point in the history
  • Loading branch information
CWillberg committed Mar 26, 2024
1 parent 7a77a06 commit 918223b
Show file tree
Hide file tree
Showing 3 changed files with 63 additions and 3 deletions.
18 changes: 18 additions & 0 deletions docs/src/refs.bib
Original file line number Diff line number Diff line change
Expand Up @@ -157,6 +157,24 @@ @article{BrighentiR2021
doi = {10.1155/2021/7782326},
}

@article{TrageserJ2020,
author={Trageser, Jeremy
and Seleson, Pablo},
title={{Bond-Based Peridynamics: a Tale of Two Poisson's Ratios}},
journal={Journal of Peridynamics and Nonlocal Modeling},
year={2020},
month={Sep},
day={01},
volume={2},
number={3},
pages={278-288},
issn={2522-8978},
doi={10.1007/s42102-019-00021-x},
url={https://doi.org/10.1007/s42102-019-00021-x}
}



@inproceedings{LittlewoodDJ2013,
title={Estimation of the Critical Time Step for Peridynamic Models.},
author={David J. Littlewood and Jesse D. Thomas and Timothy Shelton},
Expand Down
44 changes: 43 additions & 1 deletion docs/src/theory/theory_bondbased.md
Original file line number Diff line number Diff line change
@@ -1,2 +1,44 @@
# Bond-based
TODO


# Bond-Based Peridynamics

Bond-based Peridynamics is a nonlocal extension of classical continuum mechanics, designed to address discontinuities like cracks within materials. Unlike traditional methods, which use partial differential equations (PDEs) and are based on local interactions, Peridynamics operates on integral equations and accounts for long-range forces.

## Fundamental Concepts

In bond-based Peridynamics, the material is considered as a continuum of particles that interact with each other through bonds. These interactions are not limited to immediate neighbors, allowing the framework to naturally handle the initiation and propagation of cracks.

### Equation of Motion

The equation of motion in Peridynamics is an integral equation, differing from the local PDEs in classical mechanics. For a particle at position $\mathbf{x}$, the equation is:

$$ \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x}, t) = \int_{\mathcal{H}} \mathbf{f}(\mathbf{x}', \mathbf{x}, t) \, dV + \mathbf{b}(\mathbf{x}, t) $$

where:
- $\rho(\mathbf{x})$ is the mass density at $\mathbf{x}$.
- $\ddot{\mathbf{u}}(\mathbf{x}, t)$ is the acceleration at point $\mathbf{x}$.
- $\mathcal{H}$ represents the horizon around $\mathbf{x}$, within which interactions occur.
- $\mathbf{b}(\mathbf{x}, t)$ is the body force term.


### Interaction

The fundamental interaction in bond-based Peridynamics is between pairs of points or particles within a certain horizon distance. The force vector between two points, $x$ and $x'$, is given by:

$$ \mathbf{f}(\mathbf{x}', \mathbf{x}) = \underline{\omega}\langle \boldsymbol{\xi} \rangle c \, (\mathbf{u}(\mathbf{x}') - \mathbf{u}(\mathbf{x})) $$

where:
- $\mathbf{f}(\mathbf{x}', \mathbf{x})$ is the force vector exerted by the particle at $\mathbf{x}'$ on the particle at $\mathbf{x}$.
- $c$ is a bond modulus representing the stiffness of the bond.
- $\underline{\omega}\langle \boldsymbol{\xi} \rangle$ is a bond-associated influence function.
- $\mathbf{u}(\mathbf{x})$ is the displacement of the particle at $\mathbf{x}$.

### Bond Moduli
From [TrageserJ2020](@cite) we get

- plane strain: $c = \frac{48 E}{\pi 5 \delta^3}$ with a fixed $\nu=0.25$
- plane stress: $c = \frac{9 E}{\pi \delta^3}$ with a fixed $\nu=1/3$
- 3D: $c = \frac{18 K}{\pi \delta^4}$ with a fixed $\nu=0.25$

4 changes: 2 additions & 2 deletions src/Physics/Material/BondBased/Bondbased_Elastic.jl
Original file line number Diff line number Diff line change
Expand Up @@ -87,9 +87,9 @@ function compute_forces(datamanager::Module, nodes::Union{SubArray,Vector{Int64}
E = material_parameter["Young's Modulus"]
for iID in nodes
if symmetry == "plane stress"
constant = 12.0 * E / (pi * (1 + 1.0 / 3) * horizon[iID]^3)
constant = 9.0 * E / (pi * horizon[iID]^3)
elseif symmetry == "plane strain"
constant = 12.0 * E / (pi * (1.25) * horizon[iID]^3)
constant = 48 * E / (5 * pi * horizon[iID]^3)
else
constant = 18.0 * K / (pi * horizon[iID]^4)
end
Expand Down

0 comments on commit 918223b

Please sign in to comment.