Numerical verification of the bond-based Peryidynamic
softening model against classical theory
P. Diehl, R. Lipton and M.A. Schweitzer
Abstract
Verification against classical theory
We introduce the bond-based Peridynamic Softening material model with
respect to small strain deformation. A specialty of this model is, that the
the material parameter are independent of the size of the neighborhood.
This model is derived from classical theory. Thus, the model parameters
can be obtained from material parameters. We verified the discrete version of the Softening model with the reproduction of the Poisson’s ratio
and the Young’s modulus and showed the conversion of the Peridynamic
energy density.
• Recovering the Poisson’s ratio ν = 0.25
Poisson:s ratio ν
1.0
ν
νx (1331)
0.8
νx (9261)
νx (68921)
0.6
0.4
0.2
0.0
Motivation
−0.2
0.0
0.2
0.4
0.6
Time steps normailzed
0.8
1.0
∆Ly (t)
νx (t) =
∆Lx (t)
• Recovering the Young’s modulus E = 1.6GPa
3.5
Young’s modulus E
×109
E
E(60621)
3.0
2.5
Fragmentation
[Pa]
2.0
Damage and fracture zones
1.5
1.0
0.5
Bond-based Peridynamics
0.0
0.0
PD equation of motion (Continuum)
Z
0
0
0
ρ(X)A(t, X) =
f (t, x(t, X ) − x(t, X), X − X) dX
0.2
0.4
0.6
Time steps normailized
0.8
1.0
F (t)Lx (0)
E(t) =
A∆Lx (t)
Bδ (X)
Verification against model
Discrete PD equation of motion (EMU)
ρ(Xi )A(t, Xi ) =
X
Xj ∈Bδ (Xi )
Xj − Xi
f (t, x(t, Xj ) − x(t, Xi ), Xj − Xi )
dXj
kXj − Xi k
Silling, S. & Askari, E: A meshfree method based on the peridynamic model of solid mechanics, Computer & Structures, 2005, 83, 1526–1535
Softening Model
Peridynamic energy WM : R3×3 → R obtained by the model

s 0
2
2
WM (F ) := 2µ|F | + λtr(F ) , with F := 0 0
0 0
3
X
Xj ∈Bδ (Xi )
3
Discrete Peridynamic energy W : [0, T ] × R × R → R
1
W (t, Xi , Xj ) :=
αβs(t, Xi , Xj )
δV3
Discrete bond-based softening model
ρ(Xi )A(t, Xi ) =
1
Vd

0
0
0
δ
Xj − Xi
f (t, x(t, Xj ) − x(t, Xi ), Xj − Xi )
dXj
kXj − Xi k
Comparison of the discrete and model Peridynamic energy density
Error :|Wδ (F)−Wδ (t,Xi ,Xj )|
0.006
0.005
0.004
0.003
0.002
0.001
0.000
0.001
10
Pair-wise force function f : [0, T ] × R3 × R3 → R
2
f (t, x(t, Xj ) − x(t, Xi ), Xj − Xi ) := αβs(t, x(t, Xj ) − x(t, Xi ), Xj − Xi )
δ
exp −αkXj − Xi ks2 (t, x(t, Xj ) − x(t, Xi ), Xj − Xi )
3
log(|B
δ (X)|)
Small strain stretch s : [0, T ] × R × R → R
R. Lipton, Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics, Journal of Elasticity, 2014, Volume 117, Issue 1, pp 21-50.
First Lamé parameter
8
35937
0 me
ti
]
0.0040
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
Conclusion & Outlook
Conclusion
• Material parameters are independent on δ
Material properties
Fracture Toughness
5 [10
s
ts ep
−
125
3
(Xj1 − Xi1 )2
s(t, Xi , Xj ) :=
kXj − Xi k2
0.0045
• Reproduce Young’s modulus E and Poisson’s ratio ν up to 10%
Model
Energy equivalence
6
G :=
β
16
1
λ :=
αβ
20
16KIc2
β(K, KIc ) :=
9K
K
α(K, β) := 12
β
All parameters are independent on the horizon δ!
• Recovering of the energy W δ (t, Xj , Xi ) up to 2%
Outlook
• Optimize simulations for Young’s modulus E and Poisson’s ratio ν
• Compare the energy around the fracture with the Griffith fracture
energy