Nonlinear Modeling of Fiber-Reinforced Elastomers and the Response of a Rubber Muscle Actuator Larry D. Peel, Ph.D.* Department of Mechanical & Industrial Engineering Texas A&M Univ. - Kingsville David Jensen, Ph.D. Center for Advanced Structural Composites Brigham Young University * speaker
Introduction to Research What are Fiber-Reinforced Elastomers (FRE)? Flexible elastomeric structures with long, oriented, embedded fibers (used to alter deformation, stiffness, damping, etc) Why conduct research? Increase awareness Resolve processing and experimental issues Improve predictive capability Create new applications Flexible underwater vehicles Re-shapeable aircraft surfaces Bio-mechanical devices, assistive devices Inflatable space structures
Objectives of Research Fabrication Fabricate specimens and simple application. (not discussed) Experiment Obtain material properties from test specimens. (not discussed) Theory Modify laminated plate model to include material and geometric nonlinearity. Predict response of test specimens. Predict response of FRE rubber muscle application. Use theory and models to develop new applications.
Previous Work Processing and Experimental Philpot et al., Krey, Chou, and Luo, Bakis & Gabrys. Crane has done some more recent work. Theoretical Lee et al. -- Conducted tire research (linear material, but highly complex viscoelastic models), Clark -- Used a bi-linear stress-strain model on tirecomposites. Chou, Luo -- Specimens had wavy fibers, model used quadratic material nonlinearity, considered strains up to 2%. Derstine & Crane - Micro-mechanics & Mooney-R model
Experimental - FRE Behavior Stress (MPa) 16 14 12 1 8 6 4 2 Urethane/Cotton u/c avg u/c 15 avg u/c 3 avg u/c 45 avg u/c 6 avg u/c 75 avg u/c 9 avg urethane rubber 2 15 1 5 Stress (psi) Stress (MPa) 12 1 8 6 4 2 Silicone/Glass s/g avg s/g 15 avg s/g 3 avg s/g 45 avg s/g 6 avg s/g 75 avg s/g 9 avg silicone rubber 16 14 12 1 8 6 4 2 Stress (psi).2.4.6.8 1 Strain (m/m)..5 1. 1.5 2. 2.5 Strain (m/m) V f = 62.4%, Urethane - linear and softening behavior V f = 12.1%, Silicone - stiffening, higher strain
Experimental - Material Properties 7 1 7 1 Shear Modulus (MPa) 6 5 4 3 2 1 Urethane/Fiberglass Urethane/Cotton Silicone/Fiberglass Silicone/Cotton 8 6 4 2 Shear Modulus (psi) Transverse Stiffness (kpa) 6 5 4 3 2 1 Urethane/fiberglass Urethane/cotton Silicone/fiberglass Silicone/cotton 8 6 4 2 Transverse Stiffness (psi).2.4.6.8 Strain (m/m)..5 1. 1.5 2. Strain (m/m) G 12 vs ε x E 2 vs ε x Nonlinearity a function of elastomer matrix. Magnitude a function of V f and fiber type.
Classical Laminated Plate Theory Assumes small strains and material properties are constant. 2 y 1 E 1 E 2, G 12, ν 12 stiffnesses Q ij. x (what if E 1 E 2, G 12, ν 12 are nonlinear?) Q ij rotated Q ij for each layer. Material and orientation can be different for each layer
Laminated Plate Theory - cont d Rotated stiffnesses assembled for each layer, become laminate stiffnesses A ij, B ij, and D ij. Laminate forces N i, and moments M i ; N i =[A ij ]{ε j }+[B ij ]{k j }, M i =[A ij ]{ε j }+[B ij ]{k j }, ε j - midplane strains, k j -curvatures. The modified theory includes nonlinear material properties as a function of strain and nonlinear straindisplacement theory in an iterative code.
Nonlinear Model - Material Ogden model σ x =Σ c j (a bj-1 -a -(1+.5bj) ) a(extension ratio) = ε x +1 Polynomial Model σ x = a 1 + a 2 ε x + a 3 ε x2 + a 4 ε x 3 Mooney-Rivlin Model (2-coefficient) ε x = axial strain σ x = 2(a-a -2 )(c 1 +c 2 a -1 ) a = ε x +1 Mooney-Rivlin Model (3-coefficient) σ x =2(c 1 a-c 2 /a 3 +c 3 (1/a 3 -a)) a = ε x +1
Nonlinear Model - Material Linear E 1 assumed, Nonlinear Ogden model chosen for E 2, G 12. Form: E 2, G 12 =dσ / da =Σ c j ((b j -1)a bj-2 +(1+.5b j )a -(2+.5bj) ) Results: 6 constants: c 1, c 2, c 3, b 1,b 2, b 3 for each material property. Shear Modulus (kpa) 15 125 1 75 5 25 s/c 45 G12 Ogden 6 3rd order polynomial Mooney-Rivlin 2 Mooney-Rivlin 3.1.3.5.7 Strain (m/m) 2 15 1 5 Shear Modulus (psi)
Nonlinear Model - Geometric Geometrically nonlinear straindisplacement relations. Includes high elongation terms, extra bending terms not needed. Addition of nonlinear components changes method of solution to iterative or incremental. Load is incrementally applied in form of strain. Fiber re-orientation is function of geometry.
Nonlinear Model - Predictions 16 25 Stress (psi) 14 12 1 8 6 s/g avg s/g predicted Stress (psi) 2 15 1 u/c avg u/c predicted 4 2 5.5 1 1.5 2 2.5 Strain (in/in).25.5.75 1 Strain (in/in) V f =12.1% V f =62.4% Predictions compare very well for most data points
Nonlinear Model - Poisson s Ratios Poisson's ratio, v xy 35 3 25 2 15 1 5 silicone/cotton silicone/glass urethane/cotton urethane/glass 15 3 45 6 75 9 Off-axis angle, θ Nonlinear model will predict Poisson s ratios at each angle, and as a function of strain. Poisson s ratios may be nonlinear.
Applications - Japan Flexible micro-actuators, rubber fingers, snakes were found at Toshiba, Okayama Univ., and Okayama Science Univ.
Rubber Muscle - Model Initially assumed to be cylindrical shape with no radial deflection at ends. 12 mm (.5 in) diameter, 25 cm (1 in) long. Incremental pressure applied. Fiber angle change calculated as a function of geometry only. At each step: material properties, geometry, fiber angle were recalculated until no change noted. After each step, geometry, fiber angles, and properties were updated.
Rubber Muscle - Predictions Force (N) 3 25 2 15 1 5 Pressure (psi) 2 4 6 Silicone/Cotton Silicone/Fiberglass Urethane/Cotton Urethane/Fiberglass Pressure * Area 6 5 4 3 2 1 Force (lbs.) Fiber Angle (degrees) 3 25 2 15 1 5 Pressure (psi) 2 4 6 Silicone/Cotton Silicone/Fiberglass Urethane/Cotton Urethane/Fiberglass 3 25 2 15 1 5 Fiber Angle (degrees) 1 2 3 4 Pressure (kpa) 1 2 3 4 Pressure (kpa)
Rubber Muscle - Predictions Cont d Force (N) 6 5 4 3 2 1 Displacement (in).1.2.3.4.5.6.7 s/c - 276 kpa (4 psi) s/c - 345 kpa (5 psi) s/c - 414 kpa (6 psi) u/c - 276 kpa (4 psi) u/c - 345 kpa (5 psi) u/c - 414 kpa (6 psi) 12 1 8 6 4 2 Force (lbs) 5 1 15 2 Displacement (mm) High force (12X), nonlinear actuator, Can be integral part of flexible structure.
Conclusions - Fabrication, Experimental Acquired good quality elastomer, fiber, and FRE stress-strain results and nonlinear properties. Rubber Muscle actuator fabricated, but not fully tested. High contractive forces exhibited. Need different end attachments / holding fixture
Conclusions - Nonlinear Model Incorporated material and geometric nonlinearity into a modified classical laminated plate model. Fiber reorientation is incorporated into a rubber muscle model. A six-coefficient Ogden rubber model used for nonlinear material properties - worked well. Extensional terms of Lagrangian strain-displacement tensor included - adequate for current test results. Nonlinear model provides good to excellent correlation with tensile stress-strain data. Rubber muscle model predicts high forces, fiber angle change, displacement, provides valuable insights into muscle behavior. Research provides new and valuable tools for FRE research.
Acknowledgements: This effort was sponsored in part by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number F4962-95-1-52, US-Japan Center of Utah.