Application of A Phenomenological Viscoplasticity Model to The Stress Relaxation Behavior of Unidirectional and

Angle-ply Laminates at High Temperature

Y. Masuko and M. Kawai

Institute of Engineering Mechanics and Systems,

University of Tsukuba, Tsukuba 305-8573 , Japan

Background

Objectives

Experimental results

Predicted results

Summary

Outline

Matrix-Dominated Behavior of PMCs

PMC Laminates:

Polymer Matrix:

Time dependent responses

Off-axis loadingShear loading

・ Creep

・ Stress relaxation

Unidirectional laminate

STRESS RELAXATION BEHAVIOR OF CFRP

Experimental Observation:

Unidirectional Laminates

Angle-Ply Laminates

Applicability of Viscoplasticity Model:Angle-ply laminate

Objectives

Ply Model

Laminate Model

T800H/Epoxy#3631 (Cure temperature: 180˚C, Tg = 215˚C)

Angle-ply specimens:

Fiber Orientation:

Off-axis specimens:

50 50100

1.70

201

Unit: mm

Material System

Specimens

[]12 = 0˚, 10˚, 30˚, 45˚, 60˚, 90˚ []3s = [0]3s, [45]3s, [0]3s

Experimental Procedure

Stress Relaxation Test (100˚C)

Time

a

c b

5 h

R

1 3

2

xf

xf

R 123

1 < 2 = xf < 3 )

a-b : Loading （ 1.0 mm/min; Stroke control ）

b-c : Relaxation Period （ 5 hours; Stroke control ）

・ Constant total strains for stress relaxation tests

Stroke control

Stress-Strain Curves for CFRP

0 0.5 1 1.5 2 2.5 30

100

200

300

400

500

600

= 0

= 30

= 10

= 45 = 90

Experimental (100°C)

T800H/3631

1 mm/min

Axi

al s

tres

s x,

MP

a

Axial strain x, %

Time

Dis

plac

emen

t

1.0 mm/min

Unidirectional laminate

xy

[±30]3s

0 2 4 6 8 10 120

100

200

300

400

500

600

Axi

al s

tres

s x,

MP

a

Axial strain x, %

Experimental (100°C)T800H/36311 mm/min

[±45]3s

[±60]3s

Angle-ply laminate

xy

Off-Axis Stress Relaxation of UD-CFRP

xy

xy

R = Const

R

0

50

100

150

200

250

300

0 1 2 3 4 5 6

Axi

al S

tres

s x,

MP

a

R = 0.63 %

R = 0.44 %

R = 0.22 %

ExperimentalT800H/3631

= 10 100C□,○ ,△

Time tR, h

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6

R = 1.08 %

R = 0.60 %

R = 0.34 %Axi

al S

tres

s x,

MP

a

ExperimentalT800H/3631

= 30 100C□,○ ,△

Time tR, h

Stress Relaxation of Angle-ply CFRP

xy

xy

R = Const

R

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6

[±30]3S

R = 1.73 %

R = 1.14 %

R = 0.65 %

ExperimentalT800H/3631

100C□,○ ,△

Axi

al S

tres

s x,

MP

a

Time tR, h

0

10

20

30

40

50

60

0 1 2 3 4 5 6

R = 1.17 %

R = 0.68 %

R = 0.41 %

[±60]3S

ExperimentalT800H/3631

100C□,○ ,△

Axi

al S

tres

s x,

MP

a

Time tR, h

Modeling of Time-Dependent Behavior (1/3)

ASSUMING Time-Dependent Elasticity:

Stress Relaxation Modulus

12

xy

0

2

4

6

8

10

12

10-2 10-1 100 101

Time tR, h

= 30

R = 1.08 %

R = 0.60 %R = 0.34 %

T800H/3631

Experimental (100C)

x/ R

×10

3 , M

Pa

○

△

□0

2

4

6

8

10

12

10-2 10-1 100 101

Experimental (100C) T800H/3631

[±30]3S

R = 1.73 %R = 1.14 %

R = 0.65 %

○

△

□

x/ R

×10

4 , M

Pa

Time tR, h

xy

C x

R

Modeling of Time-Dependent Behavior (1/3)

ASSUMING Time-Dependent Elasticity:

Nonlinear viscoelasticity (VE) modeling

Schapery model

Heredity integral form

Favored in polymer research

Modeling of Time-Dependent Behavior (2/3)

ASSUMING Time-Dependent Plasticity:

Loading-Unloading Behavior of UD-CFRP

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

100

200

300

400

500 = 10

Axi

al s

tres

s x,

MP

a

Axial Strain x, %

Experimental (100C)1 mm/min

T800H/3631

Monotonic○

0

30

60

90

120

150

0 0.5 1 1.5 2 2.5

= 30

Experimental (100C)1 mm/min

T800H/3631

Axi

al s

tres

s x,

MP

a

Axial Strain x, %

Monotonic○

1212

xy

xy

Modeling of Time-Dependent Behavior (2/3)

ASSUMING Time-Dependent Plasticity:

Nonlinear viscoplasticity (VP) modeling

Gates-Sun model

Nonlinear differential form

Technically, more profitable

Modeling of Time-Dependent Behavior (3/3)

Nonlinear VE + VP modeling

Ha-Springer modelTuttle et al. model

A difficulty in distinguishing between VE and VP components

ASSUMING Time-Dependent ElastoPlasticity:

Unidirectional Lamina

Angle-ply Laminate

Modified Gates-Sun model

Modified Gates-Sun model

+

Classical Lamination Theory

(CLT)

Viscoplasticity Modeling of Time-Dependent Behavior

Off-axis stress-plastic strain curves

0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

300

350

400

Axi

al s

tres

s x,

MP

a

Axial Plastic Strain %

xp,

Experimental (100C)1 mm/min

T800H/3631

= 90

■ = 10

▲ = 45 = 30◆

●

Sun-Chen Model (1989)

x

h 3

2sin4 2a66sin2 cos2

xp x

x

Ex

h xEffectiveStress

p xp h() Effective

Plastic-strain

a66 = 1.3

Effective stress - effective plastic strain curves

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3

a66 = 1.3Experimental (100C)

Eff

ecti

ve S

tres

s

, M

Pa

Effective Plastic Strain

p , %

= 90

■ = 10

▲ = 45 = 30◆

●

1 mm/minT800H/3631

Modified Gates-Sun Model

h x

Effective Stress:

0

20

40

60

80

100

120

140

0 0.005 0.01 0.015 0.02 0.025 0.03

Eff

ecti

ve S

tres

s

, M

Pa

Effective Plastic Strain

p, mm/mm

1 mm/min

Effective stress - effective internal strain curves

Effective Overstress:

H

r

Ý p K

1

mH

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3

a66 = 1.3Experimental (100C)

Eff

ecti

ve S

tres

s

, M

Pa

Effective Plastic Strain

p , %

= 90

■ = 10

▲ = 45 = 30◆

●

1 mm/minT800H/3631

Effective stress - effective plastic strain curves

r Qi 1 e b ip

i

Hardening Variable:

Effective Plastic Strain Rate:

r

H

Off-Axis Loading

where

h 1 1

m1K

1

m

h( ) 32

sin 4 2a66 sin2 cos2

Ý xp x r h()

1 m

Off-axis Specimen

Modified Gates-Sun Model

x

y

Off-Axis Creep Curves for UD-CFRP

1212

xy

xy

C = Const

C

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

C = 84 MPa

44 MPa

63 MPa

Time tc, h

Axi

al S

trai

n x

, % 100C

T800H/3631 □,○,△ = 30

Experimental

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6

51 MPa

35 MPa

C = 68 MPa

Time tc, h

Axi

al S

trai

n x

, % 100C

T800H/3631 □,○,△ = 45

Experimental

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

Time tc, h

Axi

al S

trai

n x

, %

Off-axis creep curve for = 10˚

Identification of Material Constants—1

p xp h()

r h()C

xp x 5h

C

Ex

x C

Effective stress - effective internal strain curves

0

20

40

60

80

100

120

140

0 0.005 0.01 0.015 0.02 0.025 0.03

Eff

ecti

ve S

tres

s

, M

Pa

Effective Plastic Strain

p, mm/mm

1 mm/min

r

Ý p H

K

1

m

r0

K

1

m

r

Q1 = 24 MPa

Q2 = 80 MPa

b1 = 750b2 = 45r0 = 17 MPa

C = 208 MPa = 10

■ Creep 5 h

= 90

■ = 10

▲ = 45 = 30◆

●

r

Ý p H

K

1

m

Effective stress - effective internal strain curves

0

20

40

60

80

100

120

140

0 0.005 0.01 0.015 0.02 0.025 0.03

Eff

ecti

ve S

tres

s

, M

Pa

Effective Plastic Strain

p, mm/mm

1 mm/min

Identification of Material Constants—2

H

r

r0

Effective overstress

H r r0

Effectiveplastic strain rate

Ý p 1

h()Ý x

Ý xEx

K = 79 MPa ･ minm

m = 0.205

Effective stress - effective plastic strain rate curves

10-3

10-2

10-1

100

101

102

103

104

105

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

Eff

ecti

ve O

ver

Str

ess

H,

MP

a

Effective Strain Rate , 1mm/mm/min

Ý p

T800H/3631Experimental (100°C)

○ Tension 1mm/min Base

= 90

□ = 10

△ = 45 = 30◇

○

Creep [5h] Base●

■◆

●▲

Predicted Off-Axis Stress-Stain Curves(Modified Gates-Sun Model)

50

150

250

350

450

0 0.5 1 1.5 2 2.5

Axi

al S

tres

s x,

MP

a

Axial Strain x, %

T800H/3631

Experimental (100C)○

= 90 = 45

= 30

= 10

1 mm/min

Predicted

Time

Dis

plac

emen

t

1.0 mm/min

Material Constants

a66 = 1.3Q1 = 24 MPaQ2 = 80 MPab1 = 750b2 = 45r0 = 17 MPaK = 79 MPa ･ minm

m = 0.205

Unidirectional laminate

x

y

0

10

20

30

40

50

60

0 1 2 3 4 5 6

R = 0.76 %

0.44 %

T800H/3631 = 45

Experimental○,△

100C

Axi

al S

tres

s x,

MP

a

Time tR, h

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6

R = 1.11%

0.70%

0.56%

= 60

Experimental□,○,△T800H/3631

100C

Axi

al S

tres

s x,

MP

a

Time tR, h

Predicted Off-Axis Stress Relaxation Curves

Predicted Predicted

1212

xy

xy

R = Const

R

0 0.5 1 1.5 20

50

100

150

R = 1.08 %

Axi

al S

tres

s x,

MP

a

Axial Strain x, %

DispExperimental (100C)T800H/3631

= 30

1 mm/minDisplacement

(100 mm)

Strain Gauge(2 mm)

GaugeR a tR b tR

2 c tR

3 d tR

4

Gauge

R = 0.87 %

StrokeControl

Disp R Constant x

y

Elastic Unloading due to Local Strain Recovery

Predicted Off-Axis Stress Relaxation Curveswith Strain Recovery

1212

xy

xy

R = R (tR)

R

0

10

20

30

40

50

60

0 1 2 3 4 5 6

R = 0.76 %

0.44 %

T800H/3631 = 45

Experimental○,△

100C

Axi

al S

tres

s x,

MP

a

Time tR, h

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6

R = 1.11%

0.70%

0.56%

= 60

Experimental□,○,△T800H/3631

100C

Axi

al S

tres

s x,

MP

a

Time tR, h

Predicted Predicted

0 2 4 6 8 10 12

0

50

100

150

200

250

Axi

al s

tres

s x,

MP

a

Axial strain x, %

[±45]3s Experimental (100C)

T800H/36311 mm/min

○

Predicted

0 0.5 1 1.5 2 2.50

20

40

60

80

100

Axi

al s

tres

s x,

MP

a

[±60]3s Experimental (100C)

T800H/36311 mm/min

Axial strain x, %

○

Predicted

Predicted Stress-Strain Curves for Angle-Ply Laminates

3S3S

xy

xy

Time

Dis

plac

emen

t

1.0 mm/min

0 2 4 6 8 10 120

50

100

150

200

250

Axi

al s

tres

s x,

MP

a

Axial strain x, %

[±45]3s Experimental (100C)

T800H/36311 mm/min

○

Predicted

ab

a’

b’

Fiber Rotation due to Deformation of Angle-Ply Laminate

x

y

tan b a

1 y

1 x

tan

(Sun, Herakovich, Wisnom)

0 2 4 6 8 10 12

0

50

100

150

200

250

Axi

al s

tres

s x,

MP

a

Axial strain x, %

[±45]3s Experimental (100C)

T800H/36311 mm/min

○

Predicted

0 0.5 1 1.5 2 2.50

20

40

60

80

100

Axi

al s

tres

s x,

MP

a

[±60]3s Experimental (100C)

T800H/36311 mm/min

Axial strain x, %

○

PredictedWith rotaion With rotaion

Predicted Stress-Strain Curves for Angle-Ply Laminateswith Fiber Rotation

3S3S

xy

xy

Time

Dis

plac

emen

t

1.0 mm/min

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6

Axi

al S

tres

s x,

MP

a

Time tR, h

0.65%

R =1.73%

1.14%

[±30]3S

Experimental□,○,△T800H/3631

100C0

10

20

30

40

50

60

0 1 2 3 4 5 6

0.68%

0.41%

R =1.17%

[±60]3S

T800H/3631

100C

Experimental□,○,△

Axi

al S

tres

s x,

MP

a

Time tR, h

Predicted Predicted

Predicted Stress Relaxation of Angle-Ply Laminates

3S3S

xy

xy

R = Const

R

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6

Axi

al S

tres

s x,

MP

a

Time tR, h

0.65%

R =1.73%

1.14%

[±30]3S

Experimental□,○,△T800H/3631

100C0

10

20

30

40

50

60

0 1 2 3 4 5 6

0.68%

0.41%

R =1.17%

[±60]3S

T800H/3631

100C

Experimental□,○,△

Axi

al S

tres

s x,

MP

a

Time tR, h

PredictedWith recovery

PredictedWith recovery

Predicted Stress Relaxation of Angle-Ply Laminates with Strain Recovery

3S3S

xy

xy

R = R (tR)

R

Stress relaxation effects at high temperature in unidirectional and angle-ply CFRP laminates were examined.

Simulation was also performed using a ply viscoplasticity model and CLT.

The stress relaxation effects are clearly observed in all specimens of unidirectional and angle-ply laminates.

The stress relaxation rate rapidly decreases to vanish in a short period, regardless of the ply orientations and the sustained strain levels.

Predictions using the ply viscoplasticity model and CLT together with a consideration of the local strain recovery agree well with the experimental results.

Conclusions

Good predictions of the stress relaxation behavior confirm that the stress relaxation behavior is consistent with the creep behavior.