1. Ph. D. Defense Computational Mechanics Laboratory
Numerical Analysis and Optimal Design of
Composite Thermoforming Process
by
Shih-Wei Hsiao
Department of Mechanical Engineering
and Applied Mechanics
The University of Michigan
March 25, 1997
2. Ph. D. Defense Computational Mechanics Laboratory
Outline
• Introduction
• FEM Analysis of Composite Thermoforming Process
– Rheological Characterization of Continuous fiber Composites
– Global FEM Analysis of Thermoforming Process
– Residual Stress Analysis
• Optimal Design of Composite Thermoforming Process
• Conclusions and Future Work
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Schematic of Thermoforming Process
Microstructure
Die
Blankholder
Before forming After forming
Laminate
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Motivations of this research
Deep drawing (stamping) of woven-fabric thermoplastic composites is a
mass production and precision shaping technology to produce composite
components.
Objectives of this research
• Develop a FEM model to analyze this thermoforming process.
• Develop an optimization algorithm based on this FEM model to
optimize this forming process.
5. Ph. D. Defense Computational Mechanics Laboratory
Why Thermoplastic Composites?
From the manufacturing viewpoints
• Thermosetting Resins
– Hand layed up into structural fiber preform and impregnation
after shaping
– Need chemical additives to cure after shaping and very long
cure cycle time
– Labor intense
• Thermoplastic Resins
– Shaping only depends on heat transfer and force without chemistry
– In a preimpregnated continuous tape form
– High processing rate
– Drawback: higher processing viscosity and forming
temperature (320~400 C), and higher equipment cost
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Advantages of the composite stamping process
u Deep drawing (stamping) of woven-fabric thermoplastic composites
is a mass production technology to produce composite components.
u This stamping process is also a precision shaping process.
u Woven-fabric composites possess a balanced drawability, and can
avoid the excessive thinning caused by the transverse intraply
shearing.
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Related work
• Under the kinematic consideration, the draping behavior of woven
fabrics over 3-D spherical, conical or arbitrary surfaces was
simulated by solving the intersection equations numerically.
• This approach provides a preliminary prediction of fiber buckling
and final fiber orientation.
• This approach can be used as a design tool fot selecting suitable
draped configurations for specific surfaces.
• Kinematic approach
Potter (1979)
Robertson et al. (1981)
Van West et al. (1990)
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• Diaphragm and blowing forming processes.
• Newtonian viscous flow formulation with the fiber inextensibility
constraint
• Inplane prediction of macroscopic stress and strain.
• Isothermal forming processes.
• Unidirectional composite sheets.
• FEM Analysis
O’Bradaigh and Pipes (1991)
O’Bradaigh et al. (1993)
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Goals of this FEM Analysis
A 3-D numerical modeling of thermoforming process on woven-fabric
thermoplastic composite laminates:
• Characterization of the processing rheology of woven-fabric
thermoplastic composite materials by the homogenization method.
• Coupled viscous flow and heat transfer FEM analyses for forming.
– Predictions of global and local stress, temperature and fiber
orientation distribution.
• Residual stress FEM analysis for cooling.
– Predictions of macroscopic and microscopic residual stress
and warpage after cooling.
10. Ph. D. Defense Computational Mechanics Laboratory
Governing equations for thermoforming process
Momentum and
continuity equations Thermal equation
Viscosity equation
Coupled
• Impossible to solve these equations at each microcell to obtain
obtain the global response.
• Homogenization method is used to overcome this difficulty.
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Homogenization Method for Composite Materials
X
Y
Γ
Ω
b
t
Γt
Γg
Unit cell
Unit cell
Review
• Under the assumption of periodic microstructures which can be
represented by unit cells.
• Using the asymptotic expansion of all variables and the average
technique to determine the homogenized material properties and
constitutive relations of composite materials.
• Capable of predicting microscopic fields of deformation mechanics
through the localization process.
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Digitized woven-fabric unit cell
Woven fabric laminate
Unit cell
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Implementation of Global FEA
• 3-D sheet forming FE analysis coupled with heat transferFE analysis
using ‘Viscous shell with thermal analysis’.
vz
vy
vx
ωx
ωy
ωz
x
y
z
Membrane element
+
Bending element
Shell element
X
Y
Z
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Viscous shell with thermal analysis
• Plane stress asumption-- the incompressibility constraint can be
achieved by adjusting the thickness of each shell element.
• Large deformation process divided into a series of small time step.
• Complicated geometry, friction and contact considerations.
v
(i)
→ Ý
ε
(i)
⇔ µ
(i)
⇔ T
(i)
At i-th time step
Coupled thermal analysis
Viscous shell
• Transient heat transfer FEM to solve temperature at each node.
are solved.
• At each step, solve nodal temperature and velocity iteratively
until convergence.
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Fiber Orientation Model
Purposes:
• Update the fiber intersection angle of each global finite element
by the global strain increment at every time step.
• Change material properties according to updated fiber orientation.
Assumptions:
• The fiber orientation of all the microstructures in one global finite
element is identical.
• The warp yarn and weft yarn of woven-fabric composites can be
represented by two unit fiber vectors.
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Schematic of lamina coordinate
X
Y Z
X
Y
X
Y
Layer A
Layer B
Fiber
Matrix
Lamina
Warp yarn
Weft yarn
1
2
1
2
a
b
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Updating Scheme
.
FEM mesh
Weft vector
Warp vector
X
Y
Z
∆t
∆ε
a
n
an+1
b
n+1
bn
φn
θ
n
θ
n+1
φn+1
x x
y y
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Global-local solution scheme
Global finite
element analysis
Fiber orientation
model
Obtain homogenized
material properties
Construct unit cells
Check if forming
is done
End
No
Yes
Local finite element
analysis (PREMAT)
Localization
(POSTAMT)
Begin
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Comparison with experiments
(cylindrical cup)
55
60
65
70
75
80
85
90
0 5 10 15 20 25
Simulation (Diagonal)
Simulation (Median)
Experiment (Diagonal)
Experiment (Median)
Fiber
intersection
angle
(deg)
Distance from the origin (mm)
Punch zone Free zone
Diagonal
Median
Median
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P
Laminate Unit Cell
Effective stress of laminate and unit cell at P.
Effective stress prediction
(square box)
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Temperature distribution
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Stamped body panel
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Residual Stress Analysis
• Three levels of residual stresses are generated during cooling
–Microscopic stress: Due to CTE mismatch between matrix and fiber
–Macroscopic stress: Due to stacking sequence of laminates
–Global stress: Due to thermal history along laminate thickness
• Warpage due to the release of residual stresses after demoulding.
• In this study, homogenization method based on incremental elastic
analysis with thermal history is adopted.
• Thermoelastic properties are dependent on temperature and crystallinity
from the thermal history.
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Curvature prediction compared with experimental data
[0/90] asymmetric laminate Compared with experimental data
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Stress prediction compared with experimental data
0
20
40
60
80
100
0 50 100 150 200 250 300
Residual
stress
(MPa)
Temperature ( C)
-40
-30
-20
-10
0.0
10
20
-1 -0.5 0 0.5 1
5 C/s
15 C/s
25 C/s
Global
residual
stress
(MPa)
Location (mm)
Macroscopic stress prediction for cross-ply laminate
compared with experimental data.
Global stress prediction for unidirectional
laminates with various cooling rates.
Macroscopic stress Global stress
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Warped shapes of square box and cylindrical cup
Deformed
Undeformed
Square box Cylindrical cup
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Warped body panel with its microscopic residual stress
Unit cell
Fiber part
Laminate
Deformed
Undeformed
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Conclusions
• The thermoforming process of woven-fabric thermoplastic
composite laminates was analyzed by the 3-D thermo-viscous flow
FEM.
• The constitutive and energy equations of the composites forming
were formulated by the Homogenization Method.
• The global-local analysis of the Homogenization Method enables
us to examine the macro and microscopic deformation mechanics.
• An optimization algorithm is developed to obtain uniform
distribution by adjusting preheating temperature field.
