Timothy Langlais - University of Minnesotalanglais/research/canned.pdf · Computational Methods for Multiaxial Fatigue 2 Outline 1. undergraduate work on SAE Mini-Baja 2. graduate

Computational Methods for Multiaxial Fatigue 1

Structures and Multiaxial Fatigue Analysis

Timothy LanglaisUniversity of [email protected]

Advisers: J. H. Vogel and T. R. Chasehttp://www.menet.umn.edu/˜langlais/


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


SAE Mini-Baja Competition

9-month project to design and build an off-road vehicle with thefollowing constraints:

❖ unmodified Briggs & Stratton Engine

❖ rollcage safety requirements

❖ $1500 cost limit

Project culminates with competition that includes

❖ acceleration

❖ braking

❖ durability

❖ flotation


Mini-Baja Design

Observation: nearly all Mini-Baja designs use a tubular skeletondesign fitted with caulked/riveted sheet aluminum skin forflotation

❖ skin adds approximately 30 lbs. of extra weight

Idea: build main hull using a structural skin—aluminumhoneycomb panel


Mini-Baja Testing

?How does one connect the panels?

P

❖ create a baseline 90o specimen foreach connection design

❖ test in static 3-pt bending

❖ measure failure load


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 alu-minum doublers to the panels


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 manythousands of samples


Why Is Fatigue Important?

Fact: more than 80% of all failures in the ground vehicle industryare fatigue-related

Thus: industry must understand and be able to predict fatigue inorder to design for product life cycle


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 amultiaxial σ-ε state (e.g., tractor axle in bending and torsion).

❖ Can only be partially accounted for using equivalentuniaxial methods.

3. Many components are subjected to multiple loads with varyingphase.

❖ Cannot be accounted for using uniaxial or equivalentmethods.


The Computational Approach to Fatigue

notch correction

cycle counting damage model

material properties

stressconcentrations

material properties

summation

+

plasticity

Loads, P (t)

Strains, ε(t)σ

ε

ε

εγ

σ(t

),ε(

t)

t

∑ni=0 f(Ni)

Ci(

σ,ε

),i=

0...n

Ni, i = 0...n

Nf


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


Infinite-Surface Plasticity Model

s

suαactive

Model based on the work of Mrozand Chu

❖ each surface represents a uniquevalue of the plastic modulus, H

❖ model captures material mem-ory behavior

❖ geometric implementation re-duces system to single tensordifferential equation

❖ model inaccurate for repeatednonproportional cycling


Combining Notch Correction and Plasticity

eε

σ

eεp eεe

Notch Problem: Given nominalstrains, e, find notch σ and ε.

Kottgen’s Hypothesis: The gov-erning equations of plasticity canbe used as a structural modelto relate elastically-calculatedstrains (eε = Kte) to nonlinearnotch stresses (σ).

❖ It is possible to simultaneously solve Kottgen’s structuralmodel and the material model


Multiaxial Cycle Counting

Sample Number

strainstress

❖ uniaxial methods assume that allchannels are in-phase

➠ only need to identify reversals onone channel

❖ uniaxial rainflow methods can onlycount cycles on peaks and valleys

➠ intermediate samples must be re-moved

❖ uniaxial methods fail to identify im-portant peaks and valleys on otherchannels


Numerical Implementation of Damage Models

Usual numerical implementation establishes an explicit relationbetween the damage parameter and the life:

P =σ′f

E(2Nf )b + ε′f (2Nf )c

❖ requires re-fit of material properties σ′f , b, ε′f , and c for each

damage parameter

❖ assumes a relationship between the ε−Nf and σ − ε materialproperties


Numerical Implementation of Damage Models

Instead, establish implicit relation

P = f(Nf ; σ′

f , b, ε′f , c)

❖ requires only one fit of the material properties—the uniaxialε−Nf properties will do

❖ robust: assumes nothing about how ε−Nf and σ − ε materialproperties were fit


Multiaxial Experiments

❖ collect ε-gage data in area of con-strained plasticity near hole

❖ measure load input, P

❖ attempt to predict ε response us-ing P input


Outline—Empirical Plasticity Model

Goal describe the process for building an empirical plasticitymodel

1. Plasticity Models

2. Building an Empirical Plasticity Model

3. Preliminary Results


What Is a Plasticity Model?

A plasticity model is used to compute nonlinear stresses frommeasured strains via a set of governing differential equations

σ = f (ε, a, σ, H)

a = µβ

σ2y = (σ − a) : (σ − a)

σ

√3τ

a

σ

σy

❖ von Mises yield criterion:

a− center

σy− radius

❖ kinematic hardening: the yieldsurface may move but cannotgrow during loading


What Defines a Plasticity Model

σ

a

a

σn

❖ direction of yield surface mo-tion

β =a

‖ a ‖❖ magnitude of yield surface

motion

H = f (‖ a ‖)or

µ =‖ a ‖

Note β and H or µ are both free parameters


Experiments Behind Multiaxial Plasticity Modeling

P

T σ,ετ ,γ

❖ cannot measure yield surfacemotion (β and H or µ) di-rectly

❖ thin-walled tube experiments

❖ can measure ε =(ε, γ/

√3)

using strain gages

❖ can measure σ =(σ,√

3τ)

from loads


Conventional Method for Building

a Plasticity Model

1. propose functions for β and H or µ based on theory orexperimental observations

2. program a plasticity model based on those functions

3. compare plasticity model’s predicted stresses against measuredstresses for strain-controlled histories


Determination of Yield Surface Motion

from Experimental Data

1. use curve fits to find derivatives

2. Hooke’s Law:εp = ε− εe

3. Normality:

n =εp

‖ εp ‖4. Kinematic Hardening:

a = σ − σyn

H =σ : n

εp : n

β =a

‖ a ‖


Building an Empirical Model

Find: functions or state variables that correlate theexperimentally-derived values of H or µ and β

❖ tensor that correlates the direction, β

β = f(?)

❖ scalar variable/function that correlates the magnitude, H or µ

H = f(?)

µ = f(?)


Correlating Direction of Yield Surface Motion

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Axial Backstress Rate, βa

Axi

alSt

ress

Rat

e,σ

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Torsional Backstress Rate, βtTor

sion

alSt

ress

Rat

e,√ 3τ

Conclusion: Yield surface center moves in the direction of thestress rate, β ∝ σ


Correlating Magnitude of Motion

Mroz Active Surface

σ

σact

σ

√3 τ

❖ H is a function of thesize of the largest load-ing surface in contactwith stress point, σact

10000

100000

1e+06

100 150 200 250 300 350 400 450 500 550

Plas

tic M

odul

us, H

Mroz Active Surface

uniaxialproportional

nonproportional



Dafalias-Popov 2-Surface Distance

βin

βδ

σ

σ

√3 τ

❖ H is a nonlinear func-tion of β

δand β

in

10000

100000

1e+06

0 2 4 6 8 10 12 14

Plas

tic M

odul

us, H

Dafalias-Popov 2 Surface


nonproportional



Bannantine 2-Surface Distance

D

σ

σ

√3 τ

❖ H is a function of thedistance to the limitsurface, D

10000

100000

1e+06

400 600 800 1000 1200

Plas

tic M

odul

us, H

Bannantine 2 Surface


nonproportional



McDowell/Dafalias-Popov Accumulated Plastic Strain

σ

σ

√3 τ

❖ H is a function ofthe accumulated plas-tic strain,

∫ ‖ εp ‖ dt

10000

100000

1e+06

0 0.002 0.004 0.006 0.008 0.01

Plas

tic M

odul

us, H

McDowell/Dafalias-Popov Accumulated Plastic Strain


nonproportional


Conclusions

❖ computational analysis is an inexpensive way to evaluatefatigue

❖ it is possible to determine plasticity model parameters usingthin-walled tube data

❖ the direction of the yield surface motion roughly follows themotion of the stress point, β = a

‖a‖ ≈ σ

❖ the magnitude of the yield surface motion, H, is best correlatedby the accumulated plastic strain

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Timothy Langlais - University of Minnesotalanglais/research/canned.pdf · Computational Methods for Multiaxial Fatigue 2 Outline 1. undergraduate work on SAE Mini-Baja 2. graduate