Lifetime prediction of filled elastomers in consideration of multiaxial stress states London, 14 th of October 2010
Motivation Fatigue of elastomer products Failure of a clutch Firestone Case Fatigue tests for reliability and quality of elastomer products 2
Literature survey Concepts for lifetime estimation of rubber Crack nucleation maximum principal stress Wöhler (1867) Bathias et al. (1998) André et al. (1999) Flamm et al. (2004) Abraham et al. (2005) Saintier et al. (2006) maximum principal strain Cadwell et al. (1940) Fielding (1943) Robert & Benzies (1977) Roach (1982) Ro (1989) strain energy density Robert & Benzies (1977) Ro(1989) André et al. (1999) Abraham et al. (2005) Crack propagation energy release rate Inglis (1913) Griffith (1920) Thomas (1955) Paris et al. (1961) fracture mechanics Rivlin & Thomas (1953) Kadir & Thomas (1981) Mazich et al. (1989) Young (1990) Mars (2001) crack plane Mars & Fatemi (2002) Andriyana & Verron (2006) Verron & Andriyana. (2008) What kind of criterion is reliable? 3
Wöhler concept Wöhler concept Der Bruch des Materials läßt sich auch durch vielfach wiederholte Schwingungen, von denen keine die absolute Bruchgrenze erreicht, herbeiführen. Die Differenz der Spannungen, welche die Schwingung eingrenzen, sind daher für die Zerstörung des Zusammenhanges maßgebend. from A. Wöhler, Zeitschrift für Bauwesen, 1871 August Wöhler * 22. Juni 1819 in Soltau 21. März 1914 in Hannover Investigation of the cyclic resistance of materials (metals) Cyclic, sinusoidal loading till failure Logarithmic relation between load amplitude and resistance time 4 Wöhler curve
Wöhler concept Wöhler curve Cyclic loading at varied amplitudes until failure of the material Load [N] Miner s Rule Displacement [mm] Log S Palmgren, Langer and Miner (1924, 1937, 1945) R S LCF S HCF N LCF 5 10 4 N HCF Log N 5
Uniaxial fatigue behavior Cycled uniaxial tension test Load [N] Displacement [mm] 6
MORPH (MOdel for Rubber PHenomenology) Decompositon of stresses Auxiliary stress Basic stress Basic and cladding stresses Total stress Cladding stress History function 7
Uniaxial fatigue behavior Cyclic loading of a notched strip u = 2 x 20 mm 1 x 25 mm 2 x 20 mm Exp. Morph Yeoh 50 mm 8 Morph: p 1 = 0.062 p 5 = 0.00846 p 2 = 0.364 p 6 = 5.92 p 3 = 0.219 p 7 = 5.70 p 4 = 3.09 p 8 = 0.201 Yeoh: C 10 = 0.739 C 20 = 0.019 C 30 = 0.136 20 mm
Uniaxial fatigue behavior Cyclic loading of a strip with fillet Yeoh 5th loading cycle MORPH 9
Uniaxial fatigue behavior Variation of amplitude and preload Wöhler curve Log σ [MPa] Load [N] Displacement [mm] Abraham et al. (2005) Log N Load [N] Max. stress or strain criterion reliable? Dissipative energy seems to play an important role! 10 Displacement [mm]
Multiaxial fatigue behavior Deformation states varying loading conditions! 11
Multiaxial fatigue behavior Load types tension/compression Drawbacks: Repeating the same loading condition/direction Load-specific lifetime Dissipative energy is fully coupled to the loading process shear simple shear with rotating axes Advantages: Loading with changing directions Besides the dissipative energy, no further energy is inserted 12
Multiaxial fatigue behavior Simple shear with rotating axes 1. Double-Sandwich-layered specimen 3. Applying a continuous rotation 1. 2. F R 2. Initiation of simple shear by moving the middle part 3. F R 4. F R F U 4. Measuring the resulting force due to the restriction of the deflection 13
Multiaxial fatigue behavior Corresponding forces F R F U F R For rce [N] F U 14
Multiaxial fatigue behavior Response of the material until failure (experiments) 1400 1200 Extraction for Micro-CT F R Failure 1000 Lo oad [N] 800 600 400 200 F U 15 0-200 0 50000 100000 150000 200000 250000 300000 350000 Number of revolutions
Multiaxial fatigue behavior Micro-CT pictures of inner planes @ N 3 105 1st plane 2nd plane 3rd plane 4th plane already dead! 16
Summary Results Lifetime prediction of filled elastomers by using a hyperelastic material model without considering inelastic effects is not reliable Investigation of the influence of the loading direction on the lifetime by using simple shear with rotating axes A criterion based on the dissipative energy appears to be more sophisticated than stress or strain criteria The resulting crack geometry implies a dependence of the loading direction on the material failure Future work Combining the relation of dissipative energy and the loading direction into a new failure criterion 17