"A meshfree method based on the peridynamic model of solid mechanics"

aokomoriuta

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

- New physical theory
**Peridynamics**is for Solid Mechanics - This paper proposed
**numerical method**for Peridynamics - The method is
**meshfree**and**good at simulate cracks and failure** **Failure of glass**was computed and the result were shown- This paper is the
**first and original**numerical method for Peridynamics, so current research should be improve this method

New numerical method based on Peridynamics is expected simply to solve violent deformation and failure of solid body

- Title
- A meshfree method based on the peridynamic model of solid mechanics
- Author
- S.A. Silling, E. Askari
Department of Computational Physics, Sandia National Laboratories

- Journal
- Computers & Structures
- Year and pages
- 2005, Volume 83, Issues 17–18, pp.1526–1535
- DOI
- http://dx.doi.org/10.1016/j.compstruc.2004.11.026

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

Based on **Partial differential equations** (PDEs)

\[{\partial \mathbf{v}\over\partial t^2}= {\partial^2 \mathbf{K}\over\partial t} + {1\over \rho} ((\lambda+\mu)\nabla(\nabla\cdot \mathbf{v})+\mu\Delta \mathbf{v})\]

Focus on **local behavior at a point**

- PDEs need
**spatial derivatives** - Spatial derivatives need to assume
**continuous for any points** - However,
**cracks and failure are discontinuous**

**Spetial technique**

For example: Supply initial condition of cracks

- position of tip
- growth direction
- growth velocity
- ...and so on

**Peridynamics**

A new theory for Solid Mechanics

- proposed by Silling (2000)
- doesn't need spatial derivatives
- Details are described on next chapter

This paper shows **numerical method** for Peridynamics

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

\[ \begin{split} \rho \left(\mathbf{X} \right) \ddot{\mathbf{u}} \left( \mathbf{X}, t \right) =& \mathbf{b} \left(\mathbf{X}, t \right) \\ &+ \int_H \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \end{split} \]

Not partial differential but **integral** for space

\[ \rho \left(\color{red}{\mathbf{X}} \right) \ddot{\mathbf{u}} \left( \color{red}{\mathbf{X}}, t \right) = \mathbf{b} \left(\color{red}{\mathbf{X}}, t \right) + \int_H \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \]

\(\mathbf{X}\): in the reference configuration = initial position (**Material description**)

Do not confuse \(\mathbf{x}\): in the current configuration (**Spatial description**)

Why does this paper use lower case x for material description??

\[ \rho \left(\mathbf{X} \right) \ddot{\color{red}{\mathbf{u}}} \left( \mathbf{X}, t \right) = \mathbf{b} \left(\mathbf{X}, t \right) + \int_H \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \]

\(\mathbf{u}\): displacement whose initial position is \(\mathbf{X}\) (Material description)

Do not confuse flow velocity; used on Fluid Dynamics

\[ \rho \left(\mathbf{X} \right) \ddot{\mathbf{u}} \left( \mathbf{X}, t \right) = \mathbf{b} \left(\mathbf{X}, t \right) + \int_H \mathbf{f} \left(\color{red}{\mathbf{\xi}}, \color{red}{\mathbf{\eta}} \right) dV \]

\(\mathbf{\xi} = \mathbf{X}' - \mathbf{X}\): initial relative **position**

\(\mathbf{\eta} = \mathbf{u}' - \mathbf{u}\): relative **displacement**

\('\) means another point

Note: \(\mathbf{\xi} + \mathbf{\eta}\) = **current** relative position

\[ \rho \left(\mathbf{X} \right) \ddot{\mathbf{u}} \left( \mathbf{X}, t \right) = \mathbf{b} \left(\mathbf{X}, t \right) + \int_{\color{red}{H}} \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \]

\(H\) assumes **spherical** region

whose radius = \(\delta\) (**Horizon**)

for convinience

\[ \color{red}{\rho \left(\mathbf{X} \right) \ddot{\mathbf{u}} \left( \mathbf{X}, t \right)} = \color{red}{\mathbf{b}} \left(\mathbf{X}, t \right) + \int_H \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \]

\(\rho \ddot{\mathbf{u}}\):Inertia force

\(\mathbf{b}\):body force per volume

\[ \rho \left(\mathbf{X} \right) \ddot{\mathbf{u}} \left( \mathbf{X}, t \right) = \mathbf{b} \left(\mathbf{X}, t \right) + \int_H \color{red}{\mathbf{f}} \left(\mathbf{\eta}, \mathbf{\xi} \right) dV \]

Evaluate how the two points **interact** with each other

Remember: \(\mathbf{\xi} = \mathbf{x}' - \mathbf{x}, \mathbf{\eta} = \mathbf{u}' - \mathbf{u}\)

**Bond**: the interaction model

e.g. Bond "Spring" to simulate elastic body

Pairwise force function \(f\) specifies

the **numerical model** of the bond

e.g. for Bond "Spring" : \( \mathbf{f} = C \frac{\left| \mathbf{\xi} + \mathbf{\eta} \right| - \left| \mathbf{\xi} \right|}{\left| \mathbf{\xi} \right|} \frac{\mathbf{\xi} + \mathbf{\eta}}{\left| \mathbf{\xi} + \mathbf{\eta} \right|} \)

Conservation of momentum

- Linear
- \[ \mathbf{f} \left(-\mathbf{\eta}, -\mathbf{\xi} \right) = -\mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) \]
- Angular
- \[ \left(\mathbf{\xi} + \mathbf{\eta} \right) \times \mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) = \mathbf{0} \]

∴ **Force along current relative pos.**

Depends only on **distanece** \(y = \left|\mathbf{\xi} + \mathbf{\eta} \right|\)

The **microelastic** material: \[\mathbf{f} \left(\mathbf{\eta}, \mathbf{\xi} \right) = \frac{\mathbf{\xi} + \mathbf{\eta}}{\left| \mathbf{\xi} + \mathbf{\eta} \right|} f \]

\[f = \frac{\partial w}{\partial y} \left(y, \mathbf{\xi} \right) \]

\(w\) is **microptential**

\[f = cs\]

- \(c\): constant, similar to
**spring coef.** - \(s\):
**bond strech**, equivalent to strain \[s = \frac{y - \left| \mathbf{\xi} \right|}{\left| \mathbf{\xi} \right|} \]

**P**rotytype **M**icroelastic **B**rittle model: \[f = cs \cdot \mu \]

\(\mu\): the bonds exists or not

- \(0\), if \((s > s_0)\) =
**failure**has never happend - \(1\), otherwise

\(s_0\): "**critical streth**"

material constant for **failure criterion**

A-A' | 13 bonds | 13 bonds | 13 bonds |
---|---|---|---|

B-B' | 13 bonds | 11 bonds | 8 bonds |

state | Isotropic | Anisotropic | Failure |

\[\phi \left( \mathbf{X} \right) = \frac{\int_H \mu \left(\mathbf{X}\right) dV}{\int_H dV} \]

Num. of bonds(\(\mu\)) | All | less | nothing |
---|---|---|---|

\(\phi\) | 0 | 0<1 | 1 |

damage | Never | some | complete |

- spling-like constant: \(c\)
- \[c = \frac{18k}{\pi \delta^4}\]
- critical strech: \(s_0\)
- \[s_0 = \sqrt{\frac{5 G_0}{9 k \delta}} \]

- \(k\): bulk moduls
- \(G_0\): The work required to break all bonds per unit area

Models mentioned above is very basic, sightly **too simple**

Many **improved models** were proposed before and after this paper

Check them before you apply

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

\[ \rho \left(\mathbf{X} \right) \] | \[ \ddot{\mathbf{u}} \left( \mathbf{X}, t \right) \] | \[ = \] | \[ \mathbf{b} \left(\mathbf{X}, t \right) \] | \[ \int_H \] | \[ \mathbf{f} ( \] | \[ \mathbf{\eta}, \] | \[ \mathbf{\xi} \] | \[ ) \] | \[ dV \] |

↓ | |||||||||

point number \(i\), time step \(k\) | |||||||||

\[ \rho_i \] | \[ {\ddot{\mathbf{u}}_i}^k \] | \[ = \] | \[ {\mathbf{b}_i}^k \] | \[ \sum_j \] | \[ \mathbf{f} ( \] | \[ {\mathbf{\eta}_{ij}}^k, \] | \[ \mathbf{\xi}_{ij} \] | \[ ) \] | \[ V_j \] |

\[ {\mathbf{f}_{ij}}^k = \left( c {s_{ij}}^k \cdot {\mu_{ij}}^k \right) \mathbf{n} \]

- \( {s_{ij}}^k = \frac{\left| \xi_{ij} + {\eta_{ij}}^k \right| - \left| \xi_{ij} \right|}{\left| \xi_{ij} \right|} \)
- \( {\mu_{ij}}^k = \begin{cases} 1 & \forall \kappa < k \quad {s_{ij}}^\kappa < s_0 \\ 0 & \text{otherwise} \end{cases} \)
- \( {\mathbf{n}_{ij}}^k = \frac{\xi_{ij} + {\eta_{ij}}^k }{\left| \xi_{ij} + {\eta_{ij}}^k \right|} \)

From 1-dim analysys:

\[ \forall i \quad \Delta t < \sqrt{\frac{2 \rho_i}{\sum_j \frac{\partial \mathbf{f}_{ij}}{\partial \mathbf{\eta}_{ij}} V_j }} \]

Depends on \(\delta\) (\(\sum_j\)), not \(\Delta x\)

conditon | fixed value |
---|---|

force | \(\mathbf{b}\) |

displacement | \(\mathbf{u}\) |

- Arrange points along the body shape
- Calculate volume for a point, use
- Delaunay triangulation
- Voronoi diagram

Simply put, **no meshing**

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

- \(\rho = 8000 \)[kg/m
^{3}] - \(E = 192 \)[GPa]
- \(s_0 = 0.02 \)
- \(\delta = 1.6 \)[mm]

Material: glass

- \(\rho = 2200 \)[kg/m
^{3}] - \(k = 14.9 \)[GPa]
- \(s_0 = 0.0005 \)
- \(\Delta x = 0.5 \)[mm]
- \(\delta = 1.5 \)[mm]

Same material (glass)

- New physical theory
**Peridynamics**is for Solid Mechanics - This paper proposed
**numerical method**for Peridynamics - The method is
**meshfree**and**good at simulate cracks and failure** **Failure of glass**was computed and the result were shown- This paper is the
**first and original**numerical method for Peridynamics, so current research should be improve this method

New numerical method based on Peridynamics is expected simply to solve violent deformation and failure of solid body

No need of spatial derivatives

- No need of neighboring search
- No tensile instability

- Introduction
- Prehistory of Peridynamics
- Peridynamics theory
- Numerical method for Peridynamics
- Result and conclusion
- My opinion

- Is this "particle method"?
- How useful for fluid dynamics?

- Prof. Shibata (NIT, Gifu) said
- Not particle method but
**particle-based**method - My opinion
- Definitely "No"
- This is Lagrangian
**grid**method

- Grid Method
**Fixed connection**between computing points- Particle Method
**Changeable connection**between computing point at every time steps

- Eulerian method
**Fixed**computing points on space- Lagrangian Method
**Movable**computing points

- Computing points move when each time
- → Lagrangian
- Bonds (connection) are set up by initial condition
- → Grid

∴ Peridynamics is **grid** method (Lagrangian)

**D**iscrete **E**lement **M**ethod

= Particle method **in a broad sense**

connected-DEM | Peridynamics | |
---|---|---|

Shape of Element | Sphare | Not specified |

Interaction from | Collisioned | Bond |

Force model | Kelvin-Voigt | Pairwise force function |

Theory based on | discontinuous body | Continuum dynamics |

Instance | exists | not exists |

Peridynamics is **NOT** family of DEM

Bonds are set up when initial condition

→ cannot **unite** by violently deformation

∴ Peridynamics can be applied to **only Solid**

In Solid Dynamics;

**Failure**by violently deformation, not unite- Bonds and \(\mu\) can represent failue well

Peridynamics = Lagrangian

∴ seems good with **Particle method**

return 0;