Structures and Multiaxial Fatigue Analysis. Timothy Langlais

Transcription

1 Computational Methods for Multiaxial Fatigue 1 Structures and Multiaxial Fatigue Analysis Timothy Langlais University of Minnesota langlais@me.umn.edu Advisers: J. H. Vogel and T. R. Chase langlais/

2 Computational Methods for Multiaxial Fatigue 2 Outline 1. undergraduate work on SAE Mini-Baja 2. graduate work on multiaxial fatigue (a) project outline (b) strain-based approach to multiaxial fatigue (c) overview of contributions to process (d) building an empirical plasticity model

3 Computational Methods for Multiaxial Fatigue 3 SAE Mini-Baja Competition 9-month project to design and build an off-road vehicle with the following constraints: unmodified Briggs & Stratton Engine rollcage safety requirements $1500 cost limit Project culminates with competition that includes acceleration braking durability flotation

4 Computational Methods for Multiaxial Fatigue 4 Mini-Baja Design Observation: nearly all Mini-Baja designs use a tubular skeleton design fitted with caulked/riveted sheet aluminum skin for flotation skin adds approximately 30 lbs. of extra weight Idea: build main hull using a structural skin aluminum honeycomb panel

5 Computational Methods for Multiaxial Fatigue 5 Mini-Baja Testing? How does one connect the panels? P create a baseline 90 o each connection design specimen for test in static 3-pt bending measure failure load

6 Computational Methods for Multiaxial Fatigue 6 Mini-Baja Testing Test several designs: cut inside panel sheet, fold outside panel sheet rivet aluminum doublers to the panels bond aluminum doublers to the panels Final Design: bond and rivet aluminum doublers to the panels

7 Computational Methods for Multiaxial Fatigue 7 Graduate Work on Multiaxial Fatigue Goal: design and create a validated multiaxial fatigue analysis tool program funded by Deere & Co. application to isotropic steels used in axles, rods, etc. focus on variable-amplitude service histories with many thousands of samples

8 Computational Methods for Multiaxial Fatigue 8 Why Is Fatigue Important? Fact: more than 80% of all failures in the ground vehicle industry are fatigue-related Thus: industry must understand and be able to predict fatigue in order to design for product life cycle

9 Computational Methods for Multiaxial Fatigue 9 Why Computational Multiaxial Fatigue Methods? 1. Computation is much cheaper/faster than experimentation. But answers are only as good as the underlying models. 2. Most components are subjected to multiple loads, leading to a multiaxial σ-ɛ state (e.g., tractor axle in bending and torsion). Can only be partially accounted for using equivalent uniaxial methods. 3. Many components are subjected to multiple loads with varying phase. Cannot be accounted for using uniaxial or equivalent methods.

10 Computational Methods for Multiaxial Fatigue 10 Loads, P (t) Strains, ɛ(t) The Computational Approach to Fatigue stress concentrations notch correction σ + ɛ plasticity σ(t), ɛ(t) ɛ cycle counting t Ci(σ, ɛ), i =0...n N i, i =0...n material properties ɛ γ damage model n i=0 f(n i) material properties summation N f

11 Computational Methods for Multiaxial Fatigue 11 Project Contributions 1. infinite-surface plasticity model 2. combined notch correction and plasticity models 3. multiaxial cycle counting algorithm 4. robust numerical implementation of damage models 5. experiments on multiaxial behavior under constrained plasticity 6. empirical plasticity modeling

12 Computational Methods for Multiaxial Fatigue 12 Infinite-Surface Plasticity Model s α active s u Model based on the work of Mróz and Chu each surface represents a unique value of the plastic modulus, H model captures material memory behavior geometric implementation reduces system to single tensor differential equation model inaccurate for repeated nonproportional cycling

13 Computational Methods for Multiaxial Fatigue 13 Combining Notch Correction and Plasticity σ Notch Problem: Given nominal strains, e, find notch σ and ɛ. e ɛ p e ɛ e e ɛ Köttgen s Hypothesis: The governing equations of plasticity can be used as a structural model to relate elastically-calculated strains ( e ɛ = K t e) to nonlinear notch stresses (σ). It is possible to simultaneously solve Köttgen s structural model and the material model

14 Computational Methods for Multiaxial Fatigue 14 Multiaxial Cycle Counting strain stress Sample Number uniaxial methods assume that all channels are in-phase only need to identify reversals on one channel uniaxial rainflow methods can only count cycles on peaks and valleys intermediate samples must be removed uniaxial methods fail to identify important peaks and valleys on other channels

15 Computational Methods for Multiaxial Fatigue 15 Numerical Implementation of Damage Models Usual numerical implementation establishes an explicit relation between the damage parameter and the life: P = σ f E (2N f ) b + ɛ f (2N f ) c requires re-fit of material properties σ f, b, ɛ f,andc for each damage parameter assumes a relationship between the ɛ N f and σ ɛ material properties

16 Computational Methods for Multiaxial Fatigue 16 Numerical Implementation of Damage Models Instead, establish implicit relation P = f ( N f ; σ f,b,ɛ f,c ) requires only one fit of the material properties the uniaxial ɛ N f properties will do robust: assumes nothing about how ɛ N f and σ ɛ material properties were fit

17 Computational Methods for Multiaxial Fatigue 17 Multiaxial Experiments collect ɛ-gage data in area of constrained plasticity near hole measure load input, P attempt to predict ɛ response using P input

18 Computational Methods for Multiaxial Fatigue 18 Outline Empirical Plasticity Model Goal describe the process for building an empirical plasticity model 1. Plasticity Models 2. Building an Empirical Plasticity Model 3. Preliminary Results

19 Computational Methods for Multiaxial Fatigue 19 What Is a Plasticity Model? A plasticity model is used to compute nonlinear stresses from measured strains via a set of governing differential equations 3τ σ = f ( ɛ,a,σ,h) ȧ = µ ˆβ σ 2 y = (σ a) :(σ a) σ von Mises yield criterion: a center a σ y σ σ y radius kinematic hardening: the yield surface may move but cannot grow during loading

20 Computational Methods for Multiaxial Fatigue 20 What Defines a Plasticity Model ˆn σ a σ ȧ direction of yield surface motion ˆβ = ȧ ȧ magnitude of yield surface motion or H = f ( ȧ ) µ = ȧ Note ˆβ and H or µ are both free parameters

21 Computational Methods for Multiaxial Fatigue 21 Experiments Behind Multiaxial Plasticity Modeling T τ,γ σ,ɛ cannot measure yield surface motion ( ˆβ and H or µ) directly thin-walled tube experiments can measure ɛ = ( ɛ, γ/ 3 ) P using strain gages can measure σ = ( σ, 3τ ) from loads

22 Computational Methods for Multiaxial Fatigue 22 Conventional Method for Building a Plasticity Model 1. propose functions for ˆβ and H or µ based on theory or experimental observations 2. program a plasticity model based on those functions 3. compare plasticity model s predicted stresses against measured stresses for strain-controlled histories

23 Computational Methods for Multiaxial Fatigue 23 Determination of Yield Surface Motion from Experimental Data 1. use curve fits to find derivatives 2. Hooke s Law: ɛ p = ɛ ɛ e 3. Normality: 4. Kinematic Hardening: ˆn = ɛp ɛ p a = σ σ y ˆn H = σ :ˆn ɛ p :ˆn ˆβ = ȧ ȧ

24 Computational Methods for Multiaxial Fatigue 24 Building an Empirical Model Find: functions or state variables that correlate the experimentally-derived values of H or µ and ˆβ tensor that correlates the direction, ˆβ ˆβ = f(?) scalar variable/function that correlates the magnitude, H or µ H = f(?) µ = f(?)

25 Computational Methods for Multiaxial Fatigue 25 Correlating Direction of Yield Surface Motion Axial Stress Rate, σ Torsional Stress Rate, 3 τ Axial Backstress Rate, ˆβ a Torsional Backstress Rate, ˆβ t Conclusion: Yield surface center moves in the direction of the stress rate, ˆβ σ

26 Computational Methods for Multiaxial Fatigue 26 Correlating Magnitude of Motion Mróz Active Surface σ σ act 3 τ σ Plastic Modulus, H 1e uniaxial proportional nonproportional H is a function of the size of the largest loading surface in contact with stress point, σ act Mroz Active Surface

27 Computational Methods for Multiaxial Fatigue 27 Correlating Magnitude of Motion Dafalias-Popov 2-Surface Distance 1e+06 3 τ β δ σ β in σ Plastic Modulus, H uniaxial proportional nonproportional H is a nonlinear function of β δ and β in Dafalias-Popov 2 Surface

28 Computational Methods for Multiaxial Fatigue 28 Correlating Magnitude of Motion Bannantine 2-Surface Distance 3 τ 1e+06 D σ H is a function of the distance to the limit surface, D σ Plastic Modulus, H uniaxial proportional nonproportional Bannantine 2 Surface

29 Computational Methods for Multiaxial Fatigue 29 Correlating Magnitude of Motion McDowell/Dafalias-Popov Accumulated Plastic Strain 3 τ 1e+06 uniaxial proportional nonproportional σ σ Plastic Modulus, H H is a function of the accumulated plastic strain, ɛ p dt McDowell/Dafalias-Popov Accumulated Plastic Strain

30 Computational Methods for Multiaxial Fatigue 30 Conclusions computational analysis is an inexpensive way to evaluate fatigue it is possible to determine plasticity model parameters using thin-walled tube data the direction of the yield surface motion roughly follows the motion of the stress point, ˆβ = ȧ ȧ σ the magnitude of the yield surface motion, H, is best correlated by the accumulated plastic strain