Y. Masuko and M. Kawai Institute of Engineering Mechanics and Systems,

1. 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

2. Outline Background Objectives Experimental results Predicted results Summary

3. PMC Laminates: Shear loading Off-axis loading Polymer Matrix: Matrix-Dominated Behavior of PMCs ・Creep ・Stress relaxation Time dependent responses

4. Objectives STRESS RELAXATION BEHAVIOR OF CFRP Experimental Observation:  Unidirectional Laminates  Angle-Ply Laminates Unidirectional laminate Applicability of Viscoplasticity Model: Angle-ply laminate  Ply Model  Laminate Model

5. Unit: mm 20  50 100 50 1 1.70 Material System T800H/Epoxy#3631(Cure temperature: 180˚C, Tg = 215˚C) Specimens Fiber Orientation: Angle-ply specimens: Off-axis specimens: []12 = 0˚, 10˚, 30˚, 45˚, 60˚, 90˚ [±]3s = [±30]3s,[±45]3s, [±60]3s

6. e b c R a 5 h Time ・Constant total strains for stress relaxation tests  xf R  e1 e2 e3 xf 1< 2 = xf < 3) e e1 e2 e3 Experimental Procedure Stress Relaxation Test (100˚C) a-b : Loading （1.0 mm/min; Stroke control ） Stroke control b-c :Relaxation Period （5 hours; Stroke control）

7. Unidirectional laminate Angle-ply laminate Displacement 1.0 mm/min y y x x Time Stress-Strain Curves for CFRP

8. s eR = Const y y x x eR e Off-Axis Stress Relaxation of UD-CFRP  

9. s eR = Const y y x x eR e Stress Relaxation of Angle-ply CFRP  

10. Modeling of Time-Dependent Behavior (1/3) ASSUMING Time-Dependent Elasticity:

11.  y y x x Stress Relaxation Modulus 12

12. Schapery model Heredity integral form Modeling of Time-Dependent Behavior (1/3) ASSUMING Time-Dependent Elasticity: Nonlinear viscoelasticity (VE) modeling Favored in polymer research

13. Modeling of Time-Dependent Behavior (2/3) ASSUMING Time-Dependent Plasticity:

14. Loading-Unloading Behavior of UD-CFRP s y y x x e 12 12

15. Gates-Sun model Nonlinear differential form Modeling of Time-Dependent Behavior (2/3) ASSUMING Time-Dependent Plasticity: Nonlinear viscoplasticity (VP) modeling Technically, more profitable

16. Modeling of Time-Dependent Behavior (3/3) ASSUMING Time-Dependent ElastoPlasticity: Nonlinear VE + VP modeling Ha-Springer model Tuttle et al. model A difficulty in distinguishing between VE and VP components

17. Viscoplasticity Modeling of Time-Dependent Behavior Unidirectional Lamina Modified Gates-Sun model Angle-ply Laminate Modified Gates-Sun model + Classical Lamination Theory (CLT)

18. a66= 1.3 Effective Stress Effective Plastic-strain Effective stress - effective plastic strain curves Off-axis stress-plastic strain curves Sun-Chen Model (1989)

19. Modified Gates-Sun Model Effective Stress: Effective Overstress: Hardening Variable: r H H Effective stress - effective internal strain curves Effective Plastic Strain Rate: Effective stress - effective plastic strain curves

20. Modified Gates-Sun Model Off-Axis Loading x q y where Off-axis Specimen

21. y y x x Off-Axis Creep Curves for UD-CFRP 12 12 s C = Const C e

22. Identification of Material Constants—1 Q1=24 MPa Q2=80 MPa b1=750 b2=45 r0=17 MPa r r r Effective stress - effective internal strain curves Off-axis creep curve for q = 10˚

23. Identification of Material Constants—2 Effective plastic strain rate Effective overstress K=79 MPa･minm m=0.205 r Effective stress - effective plastic strain rate curves r0 Effective stress - effective internal strain curves

24. Predicted Off-Axis Stress-Stain Curves(Modified Gates-Sun Model) 1.0 mm/min y Displacement q x Time Material Constants a66=1.3 Q1=24 MPa Q2=80 MPa b1=750 b2=45 r0=17 MPa K=79 MPa･minm m=0.205 Unidirectional laminate

25. Predicted Off-Axis Stress Relaxation Curves s eR = Const y y x x eR e 12 12

26. Stroke Control x y Strain Gauge (2 mm) Displacement (100 mm) Elastic Unloading due to Local Strain Recovery

27. Predicted Off-Axis Stress Relaxation Curveswith Strain Recovery s eR = eR (tR) y y x x eR e 12 12

28. Predicted Stress-Strain Curves for Angle-Ply Laminates Displacement 1.0 mm/min y y x x Time 3S 3S

29. Fiber Rotation due to Deformation of Angle-Ply Laminate (Sun, Herakovich, Wisnom) x y b’ b a’ a q q

30. Predicted Stress-Strain Curves for Angle-Ply Laminateswith Fiber Rotation Displacement 1.0 mm/min y y x x Time 3S 3S

31. Predicted Stress Relaxation of Angle-Ply Laminates s eR = Const y y x x eR e 3S 3S

32. Predicted Stress Relaxation of Angle-Ply Laminates with Strain Recovery s eR = eR (tR) y y x x eR e 3S 3S

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