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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 , Japan

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Background Objectives Experimental results Predicted results Summary Outline

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

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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

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T800H/Epoxy#3631 ( Cure temperature: 180˚C, T g = 215˚C) Angle-ply specimens : Fiber Orientation: Off-axis specimens : Unit: mm Material System Specimens [ ] 12 = 0˚, 10˚, 30˚, 45˚, 60˚, 90˚ [ ] 3s = [ 0] 3s, [ 45] 3s, [ 0] 3s

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Experimental Procedure Stress Relaxation Test ( 100˚C ) Time a c b 5 h R x f x f R < 2 = x f < 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

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

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Off-Axis Stress Relaxation of UD-CFRP x y x y R = Const R

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Stress Relaxation of Angle-ply CFRP x y x y R = Const R

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

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Stress Relaxation Modulus 12 x y x y

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

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Modeling of Time-Dependent Behavior (2/3) ASSUMING Time-DependentPlasticity:

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Loading-Unloading Behavior of UD-CFRP 12 x y x y

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

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

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

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Off-axis stress-plastic strain curves Sun-Chen Model (1989) Effective Stress Effective Plastic-strain a 66 = 1.3 Effective stress - effective plastic strain curves

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

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

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Off-Axis Creep Curves for UD-CFRP 12 x y x y C = Const C

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Off-axis creep curve for = 10˚ Identification of Material Constants1 Effective stress - effective internal strain curves r r Q 1 = 24 MPa Q 2 = 80 MPa b 1 = 750 b 2 = 45 r 0 = 17 MPa r

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Effective stress - effective internal strain curves Identification of Material Constants2 r r0r0 Effective overstress Effective plastic strain rate K = 79 MPa min m m = Effective stress - effective plastic strain rate curves

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Predicted Off-Axis Stress-Stain Curves (Modified Gates-Sun Model) Time Displacement 1.0 mm/min Material Constants a 66 = 1.3 Q 1 = 24 MPa Q 2 = 80 MPa b 1 = 750 b 2 = 45 r 0 = 17 MPa K = 79 MPa min m m = Unidirectional laminate x y

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Predicted Off-Axis Stress Relaxation Curves 12 x y x y R = Const R

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

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Predicted Off-Axis Stress Relaxation Curves with Strain Recovery 12 x y x y R = R (t R ) R

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Predicted Stress-Strain Curves for Angle-Ply Laminates 3S x y x y Time Displacement 1.0 mm/min

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

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Predicted Stress-Strain Curves for Angle-Ply Laminates with Fiber Rotation 3S x y x y Time Displacement 1.0 mm/min

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Predicted Stress Relaxation of Angle-Ply Laminates 3S x y x y R = Const R

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Predicted Stress Relaxation of Angle-Ply Laminates with Strain Recovery 3S x y x y R = R (t R ) R

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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.

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