Development of Anisotropic Hyperelastic Model Considering Stress Softening

Transcription

1 Development of Anisotropic Hyperelstic Model Considering Stress Softening Akihiro Mtsud University of Tsukub ECCMR205, Prgue

2 Contents Introduction Objective Anisotropic Hyperelstic model Mteril Tests FEM Anlysis Summerly

3 Stress Introduction Fiber-Reinforced Rubber FRR hve, Anisotropic mechnicl chrcteristics Stiffness softening ccording to mximum stretchmullis effect Lrge deformtion gives dmge to the internl structures Mullins effect First tension Second tension FRR Stretch

4 Objective Develop nisotropic hyperelstic model considering stress softeningmullins effect Evlute pplicbility of proposed model to FE nlysis

5 Hyperelstic Modeling Anisotropic hyperelstic model The mtrix bout the stress S is obtined from strin energy function S W W 2 C W iso C W C, M ni Isotropic component Anisotropic component Mooney-Rivlin Model ws pplied to isotropic prt W iso c I 3 c2 I2 3

6 Anisotropic Prt Introduce informtion of directions of reinforced fibers Here, = mens wrp fiber, =2 mens weft fiber, respectively n is unit vector of which direction is sme s reinforced fibers mens squre of fiber stretch Angle between n nd n 2 is 90 Orthotropic Another invrints re introduced nd K re pplied to stored energy 2 5 : : M C M C F n F n n n : cof I I K M C

7 Anisotropic Prt Use following model for nisotropic component exp ln n n n K K d K K d ni d d c c c c K d c d c W K : Tensile deformtion, M C, K M C : Sher deformtion Mteril Constnts Yrn s xil stress Yrn s sher stress The stress in the smll stretch ni weft wrp W W W 2 2

8 Stress Modeling of Stress-Softening Anisotropic prt ws modified by Stress-Softening function W ni S f W wrp S 2 f W weft S f exp mx,2 C Wni, mx W C ni First tension, mx Second tension c c : constnt Stretch c

9 Nominl Stress MP Bixil loding test for rubber mtrix W iso c I 3 c2 I2 3 c c Nitrile rubber Experiment Theory Tension side Fixed side Tensile direction Nominl Stretch The stress of tension side nd tht of fixed side Bixil testing mchine nd nitrile rubber

10 Unixil cyclic tensile loding test Specimen Nitrile rubber reinforced by cotton fiberuse sme nitrile rubber to bixil loding test 20mm 0mm 2mm Fiber orienttion ngle θ = 0, 5, 90 Test condition Velocity ws 0.% / sec 5 cycles of tensile deformtion were given in ech mximum stretch Rnge of tension for ech ngle Fiber Mximum stretch orienttion - θ= θ= θ= Tensile direction Reltionship between time nd stretch

11 Unixil cyclic tensile loding test 50 Nominl Stress MP θ = 0 Mullins effect θ = 90 θ = Nominl Stretch.3.

12 Identifiction of mteril constnts c d c K d K c n d n α γ Nominl Stress MP Experiment Theory Nominl Stress MP Experiment Theory Nominl Stretch.0.05 Reltionships between stress nd strin θ = Nominl Stretch Reltionships between stress nd strin θ = 90

13 Finite Element Method Size nd mesh 20mm 0mm 2mm 3-dimentionl 8 node element Boundry condition Top nd bottom surfce were constrined Tensionl displcement were pplied

14 Finite Element Method θ = 0.0 θ = 5.30 θ = Experienced the stretch of Experienced the stretch of.0.20 Experienced the stretch of.25.5 Experienced the stretch of.0 Experienced the stretch of.30 Experienced the stretch of.20 MP MP MP

15 Results of FE simultion Results of FEM simultions re compred with tensile loding test results The proposed model is ble to represent nisotropic nd stress softening of fiber reinforced rubber Nominl Stress MP Experiment Simultion Nominl Stress MP Experiment Simultion Nominl Stress MP Experiment Simultion Nominl Stretch Nominl Stretch Nominl Stretch..5 θ = 0 θ = 5 θ = 90 Comprison of the experimentl dt nd FEM results dt

16 Stbility of FE simultion FEM code ws developed 3-dimensionl solid element Displcement/pressure mixed method Newton-Rphson method for itertion y z x Boundry condition FEM model is 20mmx20mmx20mm Upper nd bottom surfce were constrined Cyclic 0% of sher deformtion ws pplied Wrp θ 20 mm 20mm 20mm

17 Sher Deformtion Wrp 90 Wrp α=90 20% st 0% st 20% 2nd 0% 2nd 5 α=5 20% st 0% st 20% 2nd 0% 2nd

18 Conclusion A stress-softening model for nisotropic hyperelsticity ws shown in this study Anisotropic mechnicl chrcteristics nd stress softening Mullins effect of FRR were ble to be represented by proposed model Proposed model is possible to pply to FEM nlysis nd FEM simultion shows high robustness

19 Thnk you for your ttention

20 Stress of Hyperelsticity Stress of isotropic hyperelsticity is given by S: 2 nd Piol Kichhoff stress C: Right-Green deformtion tensor :Stored energy function I S 2, I3 2, I2, I3 /3 I3 I 2 3 I2 I I, I C : st, 2 nd nd 3 rd prime invrint of C tensor : Modified st invrint 2/3 I I : Modified 2 nd invrint

21 Fiber Reinforced Rubber FRR Fiber reinforced rubberfrr is composite of rubber mtrix nd reinforcing fibers FRR shows high strength nd high flexibility Anisotropic mechnicl chrcteristics in lrge deformtion

22 Fiber Reinforced Rubber FRR Fiber reinforced rubberfrr is composite of rubber mtrix nd reinforcing fibers FRR shows high strength nd high flexibility Anisotropic mechnicl chrcteristics in lrge deformtion

23 Tensile Tests Bi-directionl tensile loding test To evlute the dmge independence of wrp yrn nd weft yrn Unixil cyclic tensile loding test To evlute the dmge effect t ech stretch To do the identifiction of mteril constnts

24 Bi-directionl tensile loding test Nitrile rubber reinforced by cotton fiber θ = 0, 5, 90θ mens fiber orienttion ngle. Loding up to certin stress vlue in orthogonl direction 2. Providing the relxtion time for n hour 3. Cutting the specimen in rectngulr shpe nd loding θ Wrp yrn Weft yrn Specimen Min direction Cut specimen

25 Orthogonl direction Min direction Bi-directionl tensile loding test For exmple, the specimen of θ = 0 Loding up to certin stress Loding up to mteril filure Cut Weft yrn is stretched Wrp yrn Weft yrn Wrp yrn is stretched

26 Bi-directionl tensile loding test Nominl Stress MP Loding up to 0 MP in orthogonl direction Loding up to 0 MP in orthogonl direction Loding up to 20 MP in orthogonl direction Loding up to 30 MP in orthogonl direction Nominl Stress MP Loding up to 0 MP in orthogonl direction Loding up to 0 MP in orthogonl direction Loding up to 20 MP in orthogonl direction Loding up to 30 MP in orthogonl direction Nominl Stress MP Nominl Stretch Loding up to 0 MP in orthogonl direction Loding up to 30 MP in orthogonl direction Nominl. Stress : Nominl Stretch :.2.3. Nominl Stretch θ = 0 θ = Lod Initil cross-section re Lengt Initil h length 0

27 Bi-directionl tensile loding test Considertion Wrp yrn nd weft yrn shows dmge independence. The reltionship between the wrp direction nd the weft direction. The structure of weve yrn in the specimen Tensio n Structure of weve yrn Wrp yrn Weft yrn

28 Fiber Reinforced Rubber FRR Fiber reinforced rubberfrr is composite of rubber mtrix nd reinforcing fibers FRR shows high strength nd high flexibility Anisotropic mechnicl chrcteristics in lrge deformtion

29 Stress of Hyperelsticity Stress of isotropic hyperelsticity is given by S: 2 nd Piol Kichhoff stress C: Right-Green deformtion tensor :Stored energy function I S 2, I3 2, I2, I3 /3 I3 I 2 3 I2 I I, I C : st, 2 nd nd 3 rd prime invrint of C tensor : Modified st invrint 2/3 I I : Modified 2 nd invrint

30 Invrints for Anisotropic Hyperelsticity Introduce informtion of directions of reinforced fibers Here, =,2, respectively n is unit vector of which direction is sme s reinforced fibers mens squre of fiber stretch Angle between n nd n 2 is 90 Orthotropic Another invrints re introduced nd K re pplied to stored energy 2 5 : : M C M C F n F n n n : cof I I K M C

31 Stored Energy of Anisotropic Hyperelsticity Stored energy function of nisotropic hyperelsticity iso I, I2 ni, K vol I3 Isotropic prtrubber mtrix iso I 3 C I 3 I, I2 C 2 2 The Mooney-Rivline model ws pplied to rubber mtrix

32 Mteril Test of Rubber Mtrix Bixil loding test for Nitric rubber mtrix Rubber sheet specimen0mmx0mmx3mm One side ws loded to 200% stretch. Other side ws kept initil length Two stress σ nd σ 2 were pproximted by Mooney Rivline model C =.28MP, C 2 =0.5MP

33 Tensile Loding Test of FRR Tensile loding tests of fiber reinforced rubber were conducted Tensile loding test specimen20mmx20mmx3mm Aluminum pltes20mmx20mmx2mm were bonded t fixing prt α is ngle between wrp nd tensile direction Tensile direction Tensile tests ws conducted under the condition of α=0, 5, 30, 5, 60, 75, 90

34 Tensile Test Results Nominl StressMP θ=0 θ=5 θ=30 θ=5 Nominl StressMP θ=90 θ=75 θ=60 θ= Nominl Stretch Nominl Stretch α=0, 5, 30, 5 α=5, 60, 75, 90 Stiffness in the direction of wrp nd weft show different Stress-strin curve of α=30, 5, 60 shows softening between.05~.2 of stretch

35 Stored Energy of Anisotropic Prt Anisotropic prt ws modified to consider different strength of wrp nd weft ni, K dd We pply dditionl function to the Asi nd the Itskov models for better pproximtion in smll deformtion dd 2 Asi Its exp d C ' d Itskov,M. et l., Interntionl ournl of Solids nd Structures, Vol.200 Asi, M. et l. ournl of Applied Mechnics, SCE, Vol. 2008, pp.67

36 Tensile Test nd Proposed Model θ=0 θ=5 Experiment Experiment Simultion Simultion θ=30 θ=5 Experiment Experiment Simultion Simultion θ=90 θ=75 Experiment Experiment Simultion Simultion θ=60 θ=5 Experiment Experiment Simultion Simultion Nominl StressMP Nominl StressMP Nominl Stretch Nominl Stretch Behvior in smll stretch region is improved by the proposed model Propose model is pplicble to the reinforced rubber for sels in genertor

37 Stbility of FEM Code FEM code ws developed 3-dimensionl solid element Displcement/pressure mixed method Newton-Rphson method for itertion Wrp y z x Boundry condition FEM model is 20mmx20mmx20mm Upper nd bottom surfce were constrined θ 20 mm 20mm 20mm

38 θ=0 θ=5 θ=30 θ=5 Wrp Tensile Deformtion Weft Men stress distributions of tensile deformtion of.5 stretch Men stress distribution of cross section

39 Sher Deformtion Wrp θ Initil α=0 α=5 α=30 α=5 α=60 α=75 α=90

40 THANK YOU FOR YOUR ATTENTION 2 2 ln ' ' C C C K d C d C K K d K K d si K Asi model Itskov model 2 2 ln I Its K C