Numerical Analysis and Optimal Design of Composite ...

Ph. D. Defense Computational Mechanics Laboratory

Numerical Analysis and Optimal Design ofComposite Thermoforming Process

by

Shih-Wei Hsiao

Department of Mechanical Engineering and Applied Mechanics The University of Michigan March 25, 1997


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


Schematic of Thermoforming Process

Microstructure

Die

Blankholder

Before forming After forming

Laminate


Motivations of this research

Deep drawing (stamping) of woven-fabric thermoplastic composites is amass production and precision shaping technology to produce compositecomponents.

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.


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


Advantages of the composite stamping process

u Deep drawing (stamping) of woven-fabric thermoplastic compositesis 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 canavoid the excessive thinning caused by the transverse intraplyshearing.


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)


• 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)


Goals of this FEM Analysis

A 3-D numerical modeling of thermoforming process on woven-fabricthermoplastic 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.


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.


Homogenization Method for Composite Materials

X

Y

ΓΩb

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.


Digitized woven-fabric unit cell

Woven fabric laminateUnit cell


Thermal conductivity prediction for unidirectional composites vs volume fraction

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80

The

rmal

con

duct

ivit

y (W

/m-K

)

Fiber volume fraction (%)

Axial conductivity

Transverse conductivity

Experimental (axial)

Experimental (transverse)


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

yz

Membrane element

+

Bending element

Shell elementX

Y

Z


Viscous shell with thermal analysis

• Plane stress asumption-- the incompressibility constraint can beachieved 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.


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.


Schematic of lamina coordinate

X

Y ZX

Y

X

Y

Layer A

Layer B

Fiber

Matrix

Lamina

Warp yarn

Weft yarn1

2

1

2

a

b


Updating Scheme

.

FEM mesh

Weft vector

Warp vector

X

Y

Z

∆ t

∆ε

a n

a n+1

b n+1

b n

φnθn

θn+1

φn+1

x x

y y


Global-local solution scheme

Global finiteelement analysis

Fiber orientationmodel

Obtain homogenizedmaterial properties

Construct unit cells

Check if forming is doneEnd

No

Yes

Local finite elementanalysis (PREMAT)

Localization(POSTAMT)

Begin


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)

Fib

er in

ters

ecti

on a

ngle

(de

g)

Distance from the origin (mm)

Punch zone Free zone

Diagonal

MedianMedian


P

Laminate Unit Cell

Effective stress of laminate and unit cell at P.

Effective stress prediction(square box)


Temperature distribution


Stamped body panel


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 elasticanalysis with thermal history is adopted.

• Thermoelastic properties are dependent on temperature and crystallinityfrom the thermal history.


Thermal history along thickness

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6

Tem

pera

ture

( C

)

Location (mm)

10 sec

20 sec

40 sec

60 sec

90 sec

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6

Location (mm)

10 sec

20 sec

30 sec

40 sec50 sec

60 sec

65.1 sec

Vol

ume

frac

tion

cry

stal

linit

y

Z

oSym.

Laminate


Curvature prediction compared with experimental data

[0/90] asymmetric laminate Compared with experimental data


Stress prediction compared with experimental data

0

20

40

60

80

100

0 50 100 150 200 250 300

Res

idua

l str

ess

(MP

a)

Temperature ( C)

-40

-30

-20

-10

0.0

10

20

-1 -0.5 0 0.5 1

5 C/s15 C/s25 C/s

Glo

bal r

esid

ual s

tres

s (M

Pa)

Location (mm)

Macroscopic stress prediction for cross-ply laminatecompared with experimental data.

Global stress prediction for unidirectionallaminates with various cooling rates.

Macroscopic stress Global stress


Warped shapes of square box and cylindrical cup

DeformedUndeformed

Square box Cylindrical cup


Warped body panel with its microscopic residual stress

Unit cell

Fiber part

Laminate

DeformedUndeformed


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.

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