Canada (NSERC). ⢠University of British Columbia. Graduate Fellowship (Patrick. David Campbell Endowment). ⢠The Boeing Company. Acknowledgements ...
A Hierarchical Finite Element Approach to Modeling Process-Induced Deformations in Composite Structures presented at
Composites Testing and Model Identification Conference (CompTest 2006) April 10-12, 2006, Porto, Portugal
A. Arafath, R. Vaziri and A. Poursartip Composites Group Civil Engineering / Materials Engineering The University of British Columbia Vancouver, Canada
Outline • Introduction and Background • Using Standard Solid Elements to Model Shell-Like Structures • Hierarchical Element Approach • Summary and Conclusions
Acknowledgements • Natural Sciences and Engineering Research Council of Canada (NSERC) • University of British Columbia Graduate Fellowship (Patrick David Campbell Endowment) • The Boeing Company
Process Modelling is for Large/Complex Structures • From raw material to finished structure in one step • Realistic structures are large and complex • Mesh creation time, run-time, and ability to easily make changes are all important
• UBC process model family, COMPRO, used heavily in the Boeing 787 program • Has provided significant insight into what is needed for the next round of improvements
Using Solid Elements to Model Shell-Like Structures Current approach Large aspect ratio
Locking Problems
Possible Remedies
• Shear locking
• Reduced integration method
• Volumetric locking
• Assumed natural strain method
• Trapezoidal locking
• Higher order shape functions
• ……………….
24-node Solid Element Thickness direction 15
12
11
23 20
16 24
22 6 13 10
9
21
17
18 5
1
2
y x
3
14
7
4
8
z
19
Why our own element? • Every code uses its own patches for locking ¾ No cross-platform uniformity • Flexibility to add features for easy modelling of composite structures
Main Features of the Element • Selective Reduced Integration 7
4
3 hn
3
η
Layer n
4
8
6
ξ
ζ h2 Layer 2
1
2 h1 Layer 1
1
5
2
Gauss Quadrature (in-plane)
Simpson’s Rule (through-thickness)
Variable layer thickness t13
t14
t11
Layer 1
Layer thicknesses are defined at 4 corners of the element and interpolated linearly to the integration points Additional layer rotation w.r.t. axis - 3'
t12
Layer orientation w.r.t. element material orientation
3' 2'
[0] [90]
1'
3' 2' 1' [0] [90]
Honeycomb
z
[90] [0]
[90] [0] 3
3
2
2 Global coordinate x
y
1
1 Element material orientation w.r.t. global coordinates
Verification Example Cantilever Beam under end shear Rectangular shaped elements P
t = 10 mm L = 1000 mm
Trapezoidal shaped elements
Parallelogram shaped elements
E = 200 GPa ν = 0.0
PL3 wmax = − 3EI
Sensitivity of aspect ratio Normalized Maximum Warpage Element Type
Skew Sensitivity F – Full integration R – Reduced Integration 20-node and 27-node elements are ABAQUS built-in elements
Normalized Maximum Warpage Element Type
Parallelogram
Trapezoid
Skew angle (Deg)
Skew angle (Deg)
0º
45º
75º
0º
45º
75º
20-node-F
0.998
0.988
0.510
0.998
0.917
0.502
27-node-F
0.998
0.992
0.857
0.998
0.994
0.829
20-node-R
1.000
0.997
0.726
1.000
0.984
0.356
24-node-R
1.000
1.000
0.978
1.000
1.000
0.878
27-node-R
1.000
1.000
0.978
1.000
1.000
0.878
Warpage Phenomenon Autoclave Pressure Part Tool
Low CTE part is stretched by higher CTE tooling due to interfacial shear stress
Autoclave Pressure Part Tool
Shear deformation within the part results in non-uniform stress distribution through part thickness. Upon curing this is locked in a residual stress. Residual stress distribution causes a net bending moment ⇒ warped part
Warpage of a Flat Part Interface
y
t2
Part
x
Tool
t1
2L
y
y
x
x L
Boundary conditions before tool removal
Boundary conditions after tool removal
Temperature (ºC)
180
Cure temperature cycle
20 80
220
300
320
Time (min)
n elements
Finite element model
m elements
Dimensions
Material Part = T300/3900-2 Tool = Aluminum
Interface condition
Fully bonded Length = 1200 mm Part thickness = 1.6 mm (8 layers) Lay-up = Unidirectional Tool thickness = 5.0 mm
Mesh Size
Warpage (mm)
Time (min)
10*1
73.77
1.46
20*1
73.71
2.93
30*1
73.70
4.27
30*2
47.16
5.87
30*4
37.56
9.73
30*8
35.52
18.7
30*10
35.38
23.43
30*12
35.32
25.43
30*16
35.29
35.38
60*16
35.29
64.13
300*8
35.52
170.37
Not very sensitive to number of elements in in-plane direction
Very sensitive to number of elements in thickness direction
Need large number of elements in thickness direction even though the part thickness is very small (1.6 mm)
• How to decide on the number of elements – Mesh refining – Costly and time consuming, particularly for realistic structures
• Need to understand the physics of the problem so that the number of elements can be selected prior to the analysis
Closed-Form Solution τt
y u =0
t σ =0
x
τb
l Exponential function
σ x = F ( x) G ( β , y ) β=
πt 2l
Ex Gx y Material dependent Geometry dependent
Material Dependency 6
6 150
Resin modulus (GPa)
4.5
Temperature 100
3
Resin modulus 50
1.5
5 4 1
3
2
0
0 0
50
100
150
200
250
300
350
Time (min) 1.6
4 Thickness coord. (mm)
Temperature (ºC)
200
cure stage - 6
3
cure stage - 5 cure stage - 4
5,6
cure stage - 3 cure stage - 2
0.8
2
cure stage - 1
1
0 -50
0
50
100
150
200
250
300
Axial stress (MPa)
350
400
450
500
• Stress gradient in the thickness direction reduces with the curing of the material ¾ Large number of elements may be necessary only at the beginning of the cure cycle
• Adaptive finite element techniques may be the best option to change the number of elements dynamically based on the material properties ¾ h – method ¾ p – method (or hierarchical – method)
h- vs. p-Refinement
e r h
nt e em n i f
p-r e
fin em e
nt
Hierarchical Finite Element Formulation Thickness direction Top surface
z
Mid surface y
x
Bottom surface
• Similar to the 24-node element introduced earlier • The element has three surfaces • The in-plane shape functions for each surface are similar to the standard elements
7
4
3
η 8
1
ξ
5
6
2
• Only the through-thickness interpolation function is changed hierarchically ζ ζ=+1 R2 R3 ζ=0 ξ, η
How to select the number of terms? • From the closed-form solution:
u ≈ eβ ζ ⎛E π t ⎜ β= 2l ⎜⎝ G
⎞ ⎟ ⎟ 13 ⎠
11
• By Taylor series expansion:
βζ β ζ u ≈ 1+ + ⎛⎜ ⎝
⎞⎟ ⎠
2 +
⎛⎜ ⎝
βζ
3
⎞⎟ ⎠
+"
1! 2! 3! ≈ 1+ C ζ + C ζ 2 + C ζ 3 +" 1 2 3
• The basic element has a quadratic shape function in the thickness direction • The additional order of the shape function is decided based on the ratio of (Cn+1/Cn) • If this ratio is less than a certain threshold value, then the order of the shape function is n • Currently the threshold value is set to 1.0
(Cn+1/Cn) ratio for the flat composite part at the initial stages of cure Symmetric line
1.6 mm Part Tool
5 mm
300 mm
4
3
2
11 terms
1
0
C3 C2
C4 C3
C5 C4
C6 C5
C7 C6
C8 C7
C9 C8
C 10 C 11 C 9 C 10
C 12 C 13 C 11 C 12
C 14 C 13
C 15 C 16 C 14 C 15
Axial stress variation in the thickness direction – standard vs. hierarchical method 1.6
8 Elements Thickness coord (mm)
2 Terms 4 Terms 10 Terms 0.8
0 -100
0
100
200
300
Axial stress (MPa)
400
500
Comparison of warpage prediction – standard vs. hierarchical
Normalized warpage
2.5
2
1.5
1
0.5
0 0
2
4
6
Number of terms
8
10
12
Implementation in a FE code • The FE code needs to allow the degrees of freedom at a node to change dynamically • This can also be done by constraining the unnecessary degrees of freedom dynamically
200
6
150
4.5
100
3
Temperature Resin modulus
Resin modulus (GPa)
Temperature (ºC)
• Currently this cannot be done in ABAQUS, which is our current FE platform
1.5
50
12 Terms
2 Terms 0
0 0
50
100
150
200
Time (min)
250
300
350
Warpage of a Flat Composite Part Dimensions
Material
Interface condition
Length = 1200 mm Part thickness = 1.6 mm (8 layers) Lay-up = unidirectional Tool thickness = 5.0 mm
Part = T300/3900-2 Tool = Aluminum
Fully bonded
A work-around to achieve reasonable run-time efficiency is to run the problem in two steps, with more terms early in the cure cycle Warpage (mm)
Run Time (min)
Standard
Mesh (60*16)
35.29
46.0
Hierarchical (12 Terms)
Fixed
35.29
119.0
Changing
35.29
53.0
Thus one can get an accurate answer, with equivalent efficiency even with an un-optimized FE platform, with the benefit of:
- much more efficient meshing - no requirement for mesh density evaluation
Summary and Conclusions • From an industrial perspective there is a need for a process model that is both accurate and efficient • Understanding the physics of the problem helps make the numerical model more efficient • In this study, a hierarchical finite element method is developed to meet this need
