This document outlines a peridynamic simulation of delamination propagation in fiber-reinforced composites. It introduces peridynamic theory and the bond-based peridynamic model for composites. It describes the micromodulus, critical stretch criteria, and energy-based approaches for modeling failure. It also discusses explicit and implicit solvers, as well as the use of GPU computing to simulate crack propagation examples like double cantilever beam and transverse crack tension tests.

1. College of Engineering
Department of Aerospace and Mechanical Engineering
Yile Hu
8/21/2014
Advisor: Prof. Erdogan Madenci
The University of Arizona
Peridynamic Simulation of
Delamination Propagation in Fiber-
Reinforced Composite

2. Page 2
Yile Hu
Peridynamic Simulation of Delamination
Propagation in Fiber-Reinforced Composite
Outline
• Introduction of Peridynamic Theory
• Equilibrium Equation
• Force Density Function
• Explicit & Implicit Solver
• Bond-Based Peridynamics for composites
• MicroModulus
• Critical Stretch
• Simulation Examples
• Crack Propagation (Mixed Implicit-Explicit)
• Double Cantilever Beam (Mode I)
• Transverse Crack Tension (Mode II)
• Future Work & Pending Publication
• Acknowledgements

3. Page 3
Yile Hu
1. Introduction of Peridynamic Theory
Peridynamic Theory [1] is a new nonlocal continuum
mechanics theory based on integral function.
“Peri” comes from a Greek root for “near”.
[1] Silling SA. Reformulation of elasticity theory for discontinuities and long-range forces[J]. Journal
of the Mechanics and Physics of Solids,2000,48:175-209.
• Allows for crack initiation in
unguided locations;
• Allows for damage propagation
along unguided paths;
• Allows for multi-crack emergence
and their interactions;
• No numerical difficulties after
discontinuities (cracks) occur;
Force Density
Function

4. Page 4
Yile Hu
1. Introduction of Peridynamic Theory
Relationship between local and nonlocal continuum models [2]
In Peridynamics, one material point interacts with other
material points located in its horizon with a radius δ,
through the prescribed force density function which
contains all of the constitutive information associated with
the material.
[2] Erdogan Madenci, Erkan Oterkus. Peridynamic Theory and Its Applications. Springer, 2014.

5. Page 5
Yile Hu
1. Introduction of Peridynamic Theory
3. Kalthoff-Winkler Experiment [2]
1. Diagonally Loaded Square Plate [3]
2. Compact Tension Test [3]
4. Open Hole Tension [3]
[3] Kyle Colavito. Peridynamics for failure and residual strength prediction of fiber-reinforced
composites. PhD dissertation, 2013, University of Arizona, Tucson.

6. Page 6
Yile Hu
1.1 Peridynamic Equilibrium Equation
Peridynamic Theory [4]:
In numerical implementation, the integral term is replaced by a finite summation:
Classical Continuum Mechanics [5]:
Equilibrium Equation:
     ', , ', , ', , (1)
x
xH
t t dV t  u x f u u x x b x&&
   '
1
, ', , ', , , , ,
i
x
N
x ij i j i j jH
j
t dV t dV

  f u u x x f u u x x
[4] Silling S.A., Askari E. A meshfree method based on the peridynamic model of solid
mechanics[J]. Computers and Structures, 2005, Vol.83, 1526-1535.
[5] Askari E. Xu JF. Peridynamic Analysis of Damage and Failure in Composites[J].In:44th AIAA
Aerospace Sciences Meeting and Exhibition,No.2006-88,Reno,Nevada,2006.
     , , , , (2)t t t   u x σ u x b x&&
Reference
Configuration
Deformed
Configurationi
j j
i

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Yile Hu
1.2 Force Density Function
Classification of Peridynamic methods [2]
Bond-Based
(Opposite direction and same magnitude)
Ordinary State-Based
(Opposite direction but different magnitude)
Nonordinary State-Based
(Different directions, different magnitude)

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Yile Hu
1.2 Force Density Function
Bond-Based force density function [4]:
 
'
, ', , ' (3)
'
c s

   

y y
f u u x x
y y
Where  is the status variable:
0
0
1, intact bond, s<s
(4)
0, damaged bond, s s


 

c is the micro-modulus (more later);
s is the stretch: (5)s


η ξ
ξ
'
'


y y
y y
defines the direction of force density.
Bond
Stretch
“Bond
Force”
S0
c
Critical Stretch
Micromodulus

9. Page 9
Yile Hu
1.3 Explicit Solver
 
1
, , , , (8)
iN
n n n
ij i j i j
n
j
n
i
j
it V

 u u x bf u x&&
 ,n n
i i tu u xWhere , use finite difference for the acceleration term:
1 1
2
( )
2
9
n n n
i i in
i
t
 
 


u u
u
u
&&
We can obtain an explicit form for displacement:
1
1
2
1
2 (10)
iN
n n n
i i i ij j
j
n
i
t
V

 

 
  
 
  u bu u f
Interior point: i
Boundary point: 0,n
i u&&i
1
(12)
iN
ij j
j
n
i V

 fb
0,n
i b 1
2
1
1
( 12 1 )
iN
n n n
i i i ij j
j
t
V

 


  u u u f
ib
Equilibrium Equation:

10. Page 10
Yile Hu
1.4 Implicit Solver
   
   
 2 2, ,
, , , (16)O
 
    

 
           
 
ψ u ψ u
ψ u u ψ u u u
u
Equilibrium Equation:
   
1
, , , , (13)
iN
n n n
ij i j i ji j i
j
n n
t V

  u f u u x x b b&& L u
Static Problem:
   , (14)   ψ u L u b 0
   , , (15)       ψ u u ψ u 0
Taylor Expansion:
   
T
, ,
(17)
 


 
       
 
ψ u ψ u
u K u b
u
On n+1 sub-step: 1 1
T (18)n n n 
  K u b Solve a linear system

11. Page 11
Yile Hu
1.4 Implicit Solver
   
1 1
1
1
1 1 1
1
T
,
| | | (19)
N
Ni
NM
j j
i i in n n n
j i j
M M M
j j M

   
   
 
 
 
     
            
 
 
   
   
 
L L L
u u u
L L Lψ u L u
K
u u uu u
L L L
u u u
L
O O O
L L
O O O
L
Peridynamic Tangent Stiffness (PTS) Matrix:
     
T T* * 1 * 1 * 1 * 1 * * 1 * 1 * 1
T T
1
min (20)
2
n n n n n n n n n
f      
          K u b u u K u u b
Convert to an optimization problem;
Solved by Conjugated Gradient Method:
• No inverse of PTS matrix;
• Maintain the sparseness of PTS matrix;
• Matrix-free algorithm.

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Yile Hu
2. Bond-Based Peridynamics for composites
Peridynamic model for composite laminate:
0°layer
45°layer
90°layer
Horizon
0°bond
45°bond
90°bond
Interlayer bond

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Yile Hu
2.1 Bond Model
0°bond
90°bond and
Interlayer bond
α°bond
Bond Stretch
“Bond
Force”
S0
c
Critical Stretch
Micromodulus
• Anisotropic properties;
• Brittle material;

14. Page 14
Yile Hu
2.2 Micromodulus
Micromodulus of composite lamina [7]:
11
(21)
Q
c e c
Q

 
 
 

ξ
Where  4 2 2 4
11 12 66 22cos 2 2 sin cos sin (22)Q Q Q QQ       
Fiber direction
0°bond
α°bond
90°bond
Horizon
2
3
x
x
x


 x
α
[7] Yile Hu, Yin Yu, Hai Wang. Peridynamic analytical method for progressive damage
in notched composite laminates. Composite Structures, 2014, Vol. 108, 801-810.

15. Page 15
Yile Hu
2.2 Micromodulus
2
11
cos (23)
ij
ij
ij j i
i R j R
Le cs V V E A
Q
Q

 
 
 
 
 
 
 
 
ξ
Micromodulus

16. Page 16
Yile Hu
2.3 Critical Stretch
Strength-Based Critical Stretch
T
ft
0
11
C
fc
0
11
T
mt
0
22
C
mc
0
22
, 0 fiber bonds
, 0 fiber bonds
(6)
, 0 matrix and interlayer bonds
, 0 matrix and interlayer bonds
X
s s
E
X
s s
E
Y
s s
E
Y
s s
E

 


 


  



 

nodes
( , )
=1 (7)
N
iR
i
i
t 




Damage Index:
Previous Approach

17. Page 17
Yile Hu
2.3 Critical Stretch
Energy-Based Critical Stretch
Benzeggah-Kenane Fracture Criteria [8]:
  (24)II
c Ic IIc Ic
T
G
G G G G
G

 
    
 
[8] Krueger, R., An approach to assess delamination propagation simulation
capabilities in commercial finite element codes. NASA/TM-2008-215123, 2008.
New Approach

18. Page 18
Yile Hu
2.3 Critical Stretch
Energy-Based Critical Stretch:
Mode I Mode IIMixed-Mode
cos cos cos
, ,
ij ij ij i j ij ij ij ij i j ij ij ij ij i j ijc s VV du c s VV dv c s VV dw
Gx Gy Gz
A A A
  
  
  
  
(25)II
T
G Gx Gy
G Gx Gy Gz


 
Substitute back to B-K criteria:
  (24)II
c Ic IIc Ic
T
G
G G G G
G

 
    
 
For each
material point

19. Page 19
Yile Hu
2.3 Critical Stretch
Energy contribution of each bond is
distributed according to the stiffness ratio:
1
(26)i
ijc
ij cN
ij
i
c
G G
c



For each bond
Critical Energy Release Rate cG
Material Point Bond
Stiffness Ratio
i
j

20. Page 20
Yile Hu
GPU computing
The technology
we are using.
Architecture of
CPU and GPU [6]
[6] NVIDIA, CUDA C Programming Guide. <http://developer.download.nvidia.com/
compute/DevZone/docs/html/C/doc/CUDA_C_Programming_Guide.pdf>

21. Page 21
Yile Hu
GPU computing
Code Structure:
Start
Read Input File
Discretization
Search Family Member
Compute Bond Properties
Apply Boundary Condition
If iteration <
total steps
No
End
Update Damage
Compute
Displacement
No
Output
Yes
Preprocess Interface
GPU Parallel Computing

22. Page 22
Yile Hu
3.1 Crack Propagation
Computation Model:
Number of family member
50 , 50 , 20
139400.0 , 10160.0 , 4600 , 0.33
2688 , 1488 , 91.8 , 291.0
:[90], t 0.125mm, 0.5
L T LT LT
T C T C
L mm W mm a mm
E MPa E MPa G MPa v
X MPa X MPa Y MPa Y MPa
Layup Mesh Spacing x mm
  
   
   
  
Node Number = 10,000
Bond Number = 135,698

23. Page 23
Yile Hu
3.1 Crack Propagation
Displacement X
Implicit:
Explicit:
Displacement Y

24. Page 24
Yile Hu
3.1 Crack Propagation
Explicit: Mixed Explicit-Implicit:
Explicit
Implicit

25. Page 25
Yile Hu
3.1 Crack Propagation
Force-Displacement Curve:
0.06125u mm 0.11u mm
0.1335u mm
0.1418u mm
Initial
Crack

26. Page 26
Yile Hu
3.2 Double Cantilever Beam (Mode I delamination)
11 22 12 12
2 2
139.4 , 10.16 , 0.30, 4.6
170.3 / , 493.6 / , 1.62Ic IIc
E GPa E GPa v G GPa
G J m G J m 
   
  
Computational Model:
T300/1076 Unidirectional Graphite/Epoxy Prepreg:
Dimension:
0
24
0 1
25.0 , 2 3.0 , 2 150.0 , 30.0
:[0]
6.0 , 20.0 , 0.5
B mm h mm L mm a mm
Layup
b mm b mm x mm
   
   
Node Number = 360,000
Bond Number = 14,520,420

27. Page 27
Yile Hu
3.2 Double Cantilever Beam (Mode I delamination)
Displacement X
Displacement Y
Deformation Contour
Displacement Z

28. Page 28
Yile Hu
3.2 Double Cantilever Beam (Mode I delamination)
Peridynamic resultsFE Benchmark [9]
[9] Krueger, R., Development of Benchmark Examples for Quasi-Static
Delamination propagation and Fatigue Growth Predictions. NIA,2012.
Energy/Area
in Z direction
2.936%II
T
G
G


29. Page 29
Yile Hu
3.2 Double Cantilever Beam (Mode I delamination)
Experiment [8]:
𝛿 2 = 1.2𝑚𝑚 𝛿 2 = 1.5𝑚𝑚 𝛿 2 = 1.8𝑚𝑚𝛿 2 = 0.9𝑚𝑚
Peridynamic Simulation

30. Page 30
Yile Hu
3.3 Transverse Crack Tension (Mode II delamination)
Node Number = 614,400
Bond Number = 24,786,160
Computational Model:
Dimension:
2 4 4 4 4 8
0 1
140.0 , 380.0 , 20.0 , 0.0
:[0 / 0 ] ,[0 / 0 ] ,[0 / 0 ]
2.0 , 20.0 , 0.5
s s s
L mm W mm b mm a mm
Layup
b mm b mm x mm
   
   
11 22 12 12
2 2
129.0 , 9.256 , 0.28, 5.0
170.0 / , 467.0 / , 1.47Ic IIc
E GPa E GPa v G GPa
G J m G J m 
   
  
T300/914C Unidirectional Carbon/Epoxy Prepreg:

31. Page 31
Yile Hu
3.3 Transverse Crack Tension (Mode II delamination)
Deformation Contour
Displacement X
Displacement Y
Displacement Z

32. Page 32
Yile Hu
3.3 Transverse Crack Tension (Mode II delamination)
[10] E.V. Gonzalez, P. Maimi, A. Turon, P.P. Camanho and J. Renart. Simulation of delamination by means of cohesive elements
using an explicit finite element code. CMC, 2009, vol.9, 51-92.
Energy/Area
in Y direction 97.88%II
T
G
G

Results in literature [10] Peridynamic prediction

33. Page 33
Yile Hu
3.3 Transverse Crack Tension (Mode II delamination)
Comparison with results from literature [11]
 
2
2
11
1
0.467 (N/ mm) (27)
4
c c
IIc
c
t P
G
b E t t t
 

Laminate Experimental
Pc (N)
Eq.27
Pc (N)
FE [11] Peridynamic
Pc (N) Error
(%)
Pc (N) Error
(%)
[02,04]s 8500 8502.45 8963 +5.42 0.519 8128.06 -4.40 0.427
[04,04]s 15219 13884.44 14380 +3.57 0.501 13639.27 -1.77 0.451
[04,08]s 14321 12024.28 13080 +8.78 0.553 12032.12 +0.07 0.468
IIcG IIcG

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Yile Hu
3.3 Transverse Crack Tension (Mode II delamination)
Experiment [11]:
[11] Ye, L.; Prinz, R.; Klose, R. (1990): Characterization of interlaminar shear fracturetoughness and delamination fatigue
growth of composite materials using TCT specimen. In: Report IB 131-90/15, DLR, Institute for Structural Mechanics.
Peridynamic Simulation
0.856mm 0.860mm 0.920mm0.852mm

35. Page 35
Yile Hu
Peridynamic Simulation of Delamination
Propagation in Fiber-Reinforced Composite
Acknowledgements
This work is supported by project “Holistic High-Fidelity
Modeling Strategy for Advanced Composite”, by National
Institute of Aerospace (NIA). Discussions with Nelson
Carvalho are gratefully acknowledged. The author also
wants to thank Professor Madenci and Professor Barut
from the University of Arizona for helpful discussion in
Peridynamic theory.