· a useful concept in multiaxial fatigue and especially in plasticity is that of deviatoric...

Why do Multiaxial Fatigue Calculations?

* Fatigue analysis is an increasingly important part of the design and development process

* Many components have multiaxial loads, and some of those have multiaxial loading in critical locations

* Uniaxial methods may give poor answers needing bigger safety factors

MEASURED STRAINS

STRESS & STRAIN

COMPONENTS LIFE

Plasticity Modelling

Damage Model

Constitutive Model and Notch Rule

ELASTIC STRAINS FROM

FEA

Life Prediction Process: E-N Approach

x

y

τxy

τyx

τxy

τyx

σxx

σyy

σyy

σxx

2-D Stress State

z

x

y

σxx σxx

σyy

σzz

σzz

σyy

τxy

τxz

τyz τyx

τzx

τzy

3-D Stress State

σ τ ττ σ ττ τ σ

xx xy xz

yx yy yz

zx zy zz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Tensor Representation of Stress State

* Stresses can be represented as a tensor.

* Diagonal terms are direct stresses

* Other terms are shear stresses.

* For equilibrium purposes it must be symmetric.

* On free surface (z is surface normal) all terms with “z” become zero.

* Can be written sij

ε ε εε ε εε ε ε

xx xy xz

yx yy yz

zx zy zz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Strain Tensor

* Strains can also be represented by tensors.

* Diagonal terms are the direct strains and the other terms are shear strains.

* For equilibrium the matrix is symmetric.

* Shear strains, e.g. exy are half the engineering shear strain gxy

* Can be written eij

X

Y

Z

Y’

Z’

X‘

Transformation of Stress / Strain

=~

Stress Tensor Rotation

* Stress or strain tensors can be rotated to a different coordinate system by a transformation matrix.

* The matrix contains the direction cosines of the new co-ordinate axes in the old system.

* The tensor is pre-multiplied by the matrix and post-multiplied by its transpose. l11, l12, l13 are the direction

cosines of the X’ axis in the original system and so on.

T

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

l l l l l ll l l

11 12 13

21 22 23

31 32 33

=

Principal Stresses (& Strains)

* The principal stress axes are the set in which the diagonal terms disappear. In these directions the direct stresses reach their extreme values

* The maximum shear strains occur at 45 degrees to the principal axes.

* The principal stresses can be calculated from:

2xyz

2xzy

2yzxyzxzxyzyx3

2yz

2xz

2xyzyzyyx2

zyx1

322

13

2

where0

τσ−τσ−τσ−τττ+σσσ=

τ−τ−τ−σσ+σσ+σσ=

σ+σ+σ=

=−σ+σ−σ

I

I

I

CIII

σ

τ

σx

σy

τxy

τxy

σ1 σ2

τmax

2θ

MOHR’S Circle for 2-D Stress

σ

τ

σ1 σ2

τmax

σ3

MOHR’S Circle for 3-D Stress

Generalized HOOKE’S LAW for 3-D (Strains)

( )

( )

( )

( )v2EGwhere

GGG

Ev

E

Ev

E

Ev

E

zxzx

yzyz

xyxy

yxz

z

xzy

y

zyx

x

+=

τ=γ

τ=γ

τ=γ

σ+σ−σ

=ε

σ+σ−σ

=ε

σ+σ−σ

=ε

1

, ,

( )( ){ }

( )( ){ }

( )( ){ } zzzzyyxxzz

yyzzyyxxyy

xxzzyyxxxx

vE

vvvE

vE

vvvE

vE

vvvE

ε+

+ε+ε+ε−+

=σ

ε+

+ε+ε+ε−+

=σ

ε+

+ε+ε+ε−+

=σ

1211

1211

1211

Generalized HOOKE’S LAW for 3-D (Stress)

x y

z

Stress state on free surface is biaxial - principal stresses σ1 and σ2

(where | σ1 |>| σ2 |) lie in the x-y plane

Free Surface Stresses

Ratio of Principals or Biaxiality Ratio:

* Stress state can be characterised by ratio of principal stresses and their orientation (angle)

*  If orientation and ratio are fixed, loading is proportional.

* Otherwise loading is non-proportional

* Biaxiality analysis: •  ae = -1 Pure Shear

•  ae = +1 Equi-Biaxial

•  ae = 0 Uniaxial

ae =σσ

2

1

Multiaxial Assessment

0 2 4 6 8 10 12-392.3

1301Strain(UE) S131A.DAC

Seconds

Sample = 409.6Npts = 9446Max Y = 1301Min Y = -392.3

0 2 4 6 8 10 12-284.3

121.1Strain(UE) S131B.DAC

Seconds

Sample = 409.6Npts = 9446Max Y = 121.1Min Y = -284.3

0 2 4 6 8 10 12-298.7

2663Strain(UE) S131C.DAC

Seconds

Sample = 409.6Npts = 9446Max Y = 2663Min Y = -298.7

Screen 1

Example of Near Proportional Loading

-1 -0.5 0 0.5 1-1000

0

1000

2000

3000

4000

5000

S131.ABSStrainUE

Biaxiality Ratio (No units)

Time range : 0 secs to 23.06 secs

Screen 1-50 0 50

-1000

0

1000

2000

3000

4000

5000

S131.ABSStrainUE

Angle (Degrees)


Screen 1

Biaxiality Ratio vs. σ1 Orientation of σ1 vs. σ1

Example of Near Proportional Loading, cont.

* The left plot indicates that the ratio of the principal stresses is nearly fixed at around 0.4, especially if the smaller stresses are ignored.

* The right hand plot shows that the orientation of the principal stresses is more or less fixed.

* This is effectively a proportional loading (these calculations assume elasticity)

Example of Near Proportional Loading, cont.

0 50 100 150-81.32

161.4GAGE 1X( uS) GAGE103.DAC

Sample = 200Npts = 3.672E4Max Y = 161.4Min Y = -81.32

0 50 100 150-274.6

559.5GAGE 1Z( uS) GAGE102.DAC

Sample = 200Npts = 3.672E4Max Y = 559.5Min Y = -274.6

0 50 100 150-651

716.2GAGE 1Y( uS) GAGE101.DAC

Sample = 200Npts = 3.672E4Max Y = 716.2Min Y = -651

Screen 1

Example of Non-Proportional Loading

-1 -0.5 0 0.5 1-200

-100

0

100

200

GAGE1.ABSStressMPa

Biaxiality Ratio (No units)


Screen 1-50 0 50

-200

-100

0

100

200

GAGE1.ABSStressMPa

Angle (Degrees)


Screen 1

Both the ratio and orientation of σ1 and σ2 vary considerably: non-proportional loading.

Example of Non-Proportional Loading, cont.

Uniaxial

Proportional Multiaxial

Non-Proportional Multiaxial

Increasing Difficulty

(and Rarity)

Decreasing Confidence

OK

Need ae

Tricky

φp ae

φp constant

φp constant

φp may vary

ae = 0

-1 < ae < +1

ae may vary

Effect of Multiaxiality on Plasticity, Notch Modelling and Damage

Deviatoric Stresses

S PS PS P

x x hy y hz z h

= −= −= −

σσσ

The shear stresses are unchanged.

The deviatoric stresses Sx,y,z are given by:

A useful concept in multiaxial fatigue and especially in plasticity is that of deviatoric stresses. The deviatoric stresses are the components of stress that deviate from the hydrostatic stress.

( )zyxh σ+σ+σ=31P

The hydrostatic stress Ph is an invariant:

τσ σ σ σ σ σ σ

max max , ,=− − −⎡

⎣⎢⎢

⎤

⎦⎥⎥=1 2 2 3 3 1

2 2 2 2y

( )S S Sy

12

22

32

232+ + =σ

or ( ) ( ) ( )12 1 2

22 3

23 1

2σ σ σ σ σ σ σ− + − + − = y

Yield Criteria

When the stress state is not uniaxial, a yield point is not sufficient. A multiaxial yield criterion is required. The most popular criterion is the von Mises yield criterion. All common yield theories assume that the hydrostatic stress has no effect, ie., the yield criterion is a function of the deviatoric stresses. The von Mises criterion - based on distortion energy - can be expressed in terms of principal stresses:

An alternative, the Tresca Criterion can be expressed as:

23σ y

von Mises

Tresca

S1

S2 S3

VON MISES & TRESCA in Deviatoric Space

σ1

σ2

von Mises Tresca

VON MISES & TRESCA in Principals

*  EQUIVALENT STRESS AND STRAIN METHODS Extension of the use of yield criteria to fatigue under combined stresses

Some Equivalent Stress / Strain Criteria

* Maximum Principal Stress

* Maximum Principal Strain

* Maximum Shear Stress (Tresca Criterion)

* Shear Strain (Tresca)

* von Mises stress

* von Mises strain

Note that ν can be found from:

σ σ1 = eq

ε ε1 = eq

σ στ

σ1 3

2 2−

= =eqeq

( )ε ε γ ν ε1 3

2 21

2−

= =+max eq

( ) ( ) ( )12 1 2

22 3

23 1

2σ σ σ σ σ σ σ− + − + − = eq

( ) ( ) ( )11 2 1 2

22 3

23 1

2

( )+− + − + − =

νε ε ε ε ε ε εeq

νν ε ν ε

ε ε=

+

+e e p p

e p

( )bff N22

σσ

′=Δ

( )bff N22

1 σσ

′=Δ

( )bffVM N2

2σ

σ′=

Δ

( )bff N2

22max στ ′

=Δ

S-N Methods with Equivalent Stress

* Basquin equation for uniaxial

* Using (Abs) Max Principal

* Using Max Shear

* Using von Mises

* Coffin-Manson-Basquin equation for uniaxial

* Using (Abs) Max Principal

* Adapted for Torsion

* But if we assume the principal stress/strain criterion:

( ) ( )Δε σε2 2 2= +

ff

bf f

c

E N N'

'

( ) ( )σ τ εγ γ σ

ε1 1 2 2 2 2 2= = = +and so G N Nf

fb

f fc

,'

'Δ

( ) ( )Δε σε1

2 2 2= +f

fb

f fc

E N N'

'

( ) ( )Δγ τγ2 2 2= +

ff

bf f

c

G N N'

'

S-N Methods with Equivalent Strain

* Tresca criterion

* von Mises Criterion

* which is the same as...

( ) ( )cffpb

ff NN

G2)1(2

22'

'

ενσγ

++=Δ

( ) ( )cf'fp

bf

'fe N2)1(N2

E)1(

2εν++

σν+=

γΔ

( ) ( )Δγ ν σε2

2 1

32 3 2=

++

( ) ''e f

fb

f fc

EN N

S-N Methods with Equivalent Strain, cont.

0 1 2 3 -250

250 Stress(MPa) maximum principal

Seconds

0 1 2 3 -250

250 Stress(MPa) minimum principal

Seconds

0 1 2 3 -250

250 Stress(MPa) absolute maximum principal

Seconds

0 1 2 3 -250

250 Stress(MPa) von Mises stress

Seconds

0 1 2 3 -250

250 Stress(MPa) maximum shear stress

Seconds

Screen 1

Cylindrical notched specimen with axial sine loading

σ

τ

Tension

τ

σ

Compression

THE NEED FOR A SIGN

Comments on Equivalent Strain Methods

* Don’t account for the known fact that fatigue failure occurs in specifically oriented planes.

* These approaches “average” the stresses/strains to obtain a failure criterion with no regard to the direction of crack initiation.

* Tresca and von Mises are not sensitive to the hydrostatic stress or strain.

* They don’t account for mean stresses.

* They don’t handle out-of-phase stresses or strains.

ASME Pressure Vessel Code

* This method is based on the concept of relative von Mises Strain - equivalent to signed von Mises strain for proportional loadings.

* The ASME pressure vessel code uses the equivalent strain parameter:

* No path dependence.

* Non-conservative for non-proportional loading.

* No directionality.

* Not sensitive to hydrostatic stress.

( ) ( ) ( ) ( )Δ Δ Δ Δ Δ Δ Δ Δ Δ Δε ε ε ε ε ε ε ε ε εeq MAX wrt time= − + − + − + + +⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪( . )

23 611 22

222 33

233 11

2122

232

312

Simple Methods for Proportional Loadings

Stress Criterion

Absolute Maximum Principal



Strain Criterion


Any Tresca

-1<a<0 a~0 0<a<1

Notch Rules for Proportional Loading

* When the loading is no longer uniaxial, the uniaxial stress strain curve is no longer enough on its own

* Two methods which address this problem:

*  Klann, Tipton & Cordes

*  Hoffmann & Seeger

* Both these methods extend the use of the von Mises criterion to post yield behaviour

* Both methods assume fixed principal axes and fixed ratio of stresses or strains

εσ σ

qq q

n

= +⎛

⎝⎜

⎞

⎠⎟

Ε Κ '

'1

v vEe

q

q' =

⎛⎝⎜

⎞⎠⎟

12

12

σ

ε

First define cyclic stress-strain curve using the Ramberg-Osgood formula:

Digitize the cyclic stress-strain curve and for each point calculate Poisson’s ratio from the equation :

Calculate the biaxiality ratio from :

av

v=

+

+

εε

εε

2

1

2

11

'

'

The ratio ε2/ε1 of the principal strains is assumed to be constant in this case

KLANN-TIPTON-CORDES Method

It can be shown that the values of the principal strains and stresses can be calculated from:

Fit the following equation to the calculated modified parameters:

The modified modulus is calculated explicitly from:

σ σ 1 q 2=

+

1

1 a aε ε 1 q 2

=+

1

1

v a

a a

'

εσ σ

1 1 *

1 *

*

= +⎛⎝⎜

⎞⎠⎟

Ε Κ

1n

Ε *

e e

E1- a

=v

KLANN-TIPTON-CORDES Method, cont.

ε1

σ1 ae = 0

ae = -1

ae = 1

Modified Stress-Strain Curve Parameters

ε ε q,e 1,ee e

e e

a aa

=+1

1

2

v

σ ε ε q q q,eE= 2

HOFFMAN-SEEGER Method

Calculate von Mises equivalent strain from combined strain parameter e.g. from:

The Neuber correction is then carried out on this formulation:

The effective Poisson’s ratio is calculated as for the Klann-Tipton-Cordes Method, as are a, σ, ε1 and ε2/ ε1

ε εεε 2 1

2

1 =

⎛

⎝⎜

⎞

⎠⎟ ε ε 3 q 2

a)1- a + a

=+v' (1 σ σ 2 1a=

These can then be used to calculate any other combined parameter e.g. signed Tresca

The other required stresses and strains are calculated from:

HOFFMAN-SEEGER Method, cont.

Extending NEUBER to Non-Proportional Loadings

* This topic is important because it permits non-proportional multiaxial fatigue life predictions to be made based on elastic FEA. Still being researched and not working properly yet.

* The aim is to predict an average sort of elastic-plastic stress-strain response from a pseudo-elastic stress or strain history.

* It is necessary to combine a multiaxial plasticity model with an incremental formulation of a notch correction procedure and to make some other assumptions.

BUCZYNSKI-GLINKA Notch Method

* The Neuber method is only suitable for uniaxial or proportional loadings

* Where the loading is non-proportional and the stress-strain response is path dependent it must be replaced by an incremental version

σε σ ε= e e

σ ε ε σ σ ε ε σije

ije

ije

ije

ijN

ijN

ijN

ijNΔ Δ Δ Δ+ = +

* This rule has to be combined with a multiaxial plasticity model such as the Mroz-Garud model

* Additionally some assumptions are required, eg., that the ratios of the increments of strains, stresses or total strain energy in certain directions are the same for the elastic as the elastic-plastic case. Buczynski-Glinka uses total strain energy

* One of these assumptions is necessary to be able to reach a solution of the equations

BUCZYNSKI-GLINKA Notch Method, cont.

What to do When Loading is NOT Uniaial

* For proportional loadings a different cyclic stress-strain curve is required

* For non-proportional loadings, a 1-dimensional cyclic plasticity model is no longer sufficient

* Neuber’s rule does not work for non-proportional loadings

* Uniaxial rainflow counting does not work for non-proportional loadings

* Simple combined stress-strain parameters do not predict damage well

Directionality of Crack Growth

* When the biaxiality ratio is negative (type A), the maximum shear plane where cracks tend to initiate is oriented as shown in the diagram (on next page)

* In the early stages of initiation, type A cracks grow mainly along the surface in mode 2 (shear), before transitioning to Mode 1 normal to the maximum principal stress

* When the biaxiality is positive (type B), the cracks tend to be driven more through the thickness.

* These are therefore more damaging for the same levels of shear strain.

* Uniaxial loading is a special case of type B.

Directionality of Crack Growth, cont.

Crack Initiation & Multiaxial Fatigue

Crack Initiation demonstrated to be due to:

*  Slip occurring along planes of maximum shear, starting in grains most favorably oriented with respect to the maximum applied shear stress

*  Stage I (nucleation & early growth) is confined to shear planes. Here both shear and normal stress/strain control the crack growth rate.

*  Stage II crack growth occurs on planes oriented normal to the maximum principal stress. Here the magnitude of the maximum principal stress and strain dominates crack growth.

* Proportion of time spent in Stage I and II depends on:

•  Loading mode and amplitude

•  Material type (ductile vs brittle)

* Crack initiation life refers to the time taken to develop an engineering size crack and includes Stage I and II.

* Stage I or II may dominate life. In uniaxial case, the controlling parameters in both stages are directly related to the uniaxial stress or strain. But NOT so in multi-axial case.

Crack Initiation & Multiaxial Fatigue, cont.

* For non-proportional loading, the “critical planes” vary vary with time.

* Cracks growing on a particular plane may impede the progress of cracks growing on a different plane.

* Multi-axial fatigue theory for non-proportional loading, MUST attempt, to a greater or lesser extent, to incorporate some of the above observations, to have any chance of success in real situations.

Crack Initiation & Multiaxial Fatigue, cont.

Multiaxial Analysis in MSC Fatigue

* Shear Strain on the plane of maximum shear will extend the fatigue crack •  Progress will be opposed by the friction between the crack faces

* The separation of the cracked faces due to the presence of the normal strains in case b, will eliminate friction. Consequently the crack tip experiences all the applied shear load. Hence this case is more damaging.

(a) Torsion

γ

γ

(b) Tension

εσ1

εσ1

γ

*

Critical Plane Approach: *  Recognising that fatigue damage (cracking) is directional

*  Considers accumulation of damage on particular planes

*  Typically damage is considered at all possible planes say @ 15 degree intervals, and the worst (critical) plane selected.

*  Employs variations on the Brown-Miller Approach:

*  Equivalent fatigue life results for equivalent values of the

material constant, C

Δγ2

+ ΔΣn = C

Multiaxial Analysis in MSC Fatigue, cont.

*  Four Planar Approaches:

•  Normal Strain

•  Smith-Watson-Topper-Bannantine

•  Shear Strain

•  Fatemi-Socie

* Two complex Rainflow Counting Methods:

•  Wang-Brown

•  Wang-Brown with Mean Stress Correction

* Dang-Van Total Life Factor of Safety Method

Multiaxial Analysis in MSC Fatigue, cont.

Normal Strain Method

* Critical Plane Approach

•  Calculates the Normal Strain time history and damage on multiple planes

•  Fatigue results reported on the worst plane

•  Fatigue damage based on Normal Strain range

•  No mean stress correction

* Use with Type ‘A’ cracks

Shear Strain Method


•  Calculates the Shear Strain time history and damage on multiple planes


•  Fatigue damage based on Shear Strain range

•  No mean stress correction

* Use with Type ‘B’ cracks

SMITH-TOPPER-WATSON-BANNANTINE Method


•  Calculates the Normal Strain time history and damage on multiple planes


•  Fatigue damage based on Normal Strain range

•  Includes a mean stress correction based on Maximum Normal stress

*  Use with Type ‘A’ cracks

FATEMI-SOCIE Method


•  Calculates the Shear Strain time history and damage on multiple planes


•  Fatigue damage based on Shear Strain range

•  Includes a mean stress correction based on Maximum Normal stress

•  Requires a material constant ‘n’

* Use with Type ‘B’ cracks

Summary of Critical Plane Damage Modesl

* Normal Strain:

* SWT – Bannantine:

* Shear Strain:

* Fatemi-Socie:

( ) ( )Δε σεn f

fb

f fc

EN N

22 2=

′+ ′

( ) ( ) ( ) ( )Δγ ν σν ε

21

2 1 2=+ ′

+ + ′e ff

bp f f

c

EN N

( ) ( )Δεσ

σσ εn

nf

fb

f f fb c

EN N

22 2

22

⋅ =′

+ ′ ⋅ ′+

,max

( ) ( ) ( )

( ) ( ) ( ) ( )

Δγ σ

σν

σν σ

σ

ν εν ε σ

σ

21 1 2

12

2

1 21

22

22

+⎛

⎝⎜⎜

⎞

⎠⎟⎟ =

+′ +

+ ′

+ + ′ ++ ′ ′ +

nE

Nn

EN

Nn

N

n

y

ef f

b e f

yf

b

p f fc p f f

yf

b c

,max ( )

WANG-BROWN Method

* A complex recursive multi-axial rainflow counting method.

* A mean stress correction method is available.

* May be quite slow especially for long loading histories.

* Recommended for a variety of proportional and non-proportional loadings.

* Calculates a different critical plane for each rainflow reversal

* For each reversal the damage is calculated on the critical (maximum shear plane) whether case A or B

* Uses Normal Strain range, Maximum Shear strain

* Requires material parameter ‘S’

WANG-BROWN Method, cont.

( ) ( )εγ δεν ν

σ σε≡

+

+ ′ + − ′=

′ −+ ′max ,.

( ).S

S EN Nn f n mean

fb

f fc

1 12

2 2

Mean Stress Correction using mean Normal Stress:

WANG-BROWN Method, cont.

0

30

6090

120

150

180

210

240270

300

330

1E-91E-81E-71E-6

Polar Plot of Data : DEMO

Theta=90 Theta=45

Polar Plot of Type A and Type B damage for Wang-Brown Method

Example of Polar Damage Plot

Multiaxial Method Life (Repeats)

Normal Strain 106,000

STW-Bannantine 316,000

Shear Strain 18,500

Fatemi-Socie 27,000

Wang-Brown 30,500

Wang-Brown + Mean 26,000

Abs. Max. Principal Strain 97,300

Example: Non-Proportional Loading

Example: Steering Knuckle (Workshop 10)

At Node 1045:

Max. Stress Range = 508 MPa

Mean Biaxiality Ratio: -0.6

Most Popular Angle = -64 deg

Angle Spread = 90 deg

Multiaxial Method Life (Cycles)

Normal Strain 4.12E+07

STW-Bannantine 2.80E+04

Shear Strain 1.41E+05

Fatemi-Socie 1.70E+05

Wang-Brown 6.63E+06

Wang-Brown + Mean 8.55E+05

Abs. Max. Principal Strain 2.88E+07

Signed von Mises Strain 2.88E+07

Signed Tresca Strain 8.41E+06

Material: Manten

Axial Stress, σx = 25 ksi

Shear Stress, τxy = 14.4 ksi

Example: Out-of-Phase Loading

DANG-VAN Method

* High-cycle fatigue applications designed for infinite life

* Calculates factor-of-safety of the design

* Uses S-N total life method

* Applications:

•  Bearing design

•  Vibration induced fatigue

DANG-VAN Criterion

* The Dang Van criterion is a fatigue limit criterion

* It is based on the premise that there is plasticity on a microscopic level before shakedown

* After shakedown the important factors for fatigue are the amplitude of the microscopic shear stresses and the magnitude of the hydrostatic stress

* The method has a complicated way of estimating the microscopic residual stress

Fatigue damage occurs if:

τ ( ) ( )t a ph t b+ ⋅ − ≥ 0

where τ(t) and ph(t) are the maximum microscopic shear stress and the hydrostatic stress at time t in the stabilised state. They can be calculated from:

( ) ( ){ }τ ρt S tij ij= +12

Tresca dev * ( ) ( )( )ph t txx yy zz= + +13 σ σ σ

“a” and “b” are material properties

DANG-VAN Criterion, cont.

* The parameter “b” is the shear stress amplitude at the fatigue limit

* The parameter “a” is in effect the mean stress sensitivity, with the mean stress being represented by the hydrostatic stress

* dev rij* is the deviatoric part of the stabilised residual stress

DANG-VAN Criterion, cont.

τ(t)

ph(t)

τ + ⋅ − =a ph b 0

τ − ⋅ + =a ph b 0

Damage occurs here !!!

DANG-VAN Plot

Stabilized Residual Stresses

* The stabilised local residual stresses are calculated by means of an iteration in which convergence assumes a stabilised state (a state of elastic shakedown).

* As the loading sequence is repeated, the “yield surface” grows and moves with a combination of kinematic and isotropic hardening until it stabilises

* The stabilised yield surface is a 9-dimensional hypersphere that encompasses the loading history

( )ρ ρ* *devij

Summary of DANG-VAN Criterion

* Is a high-cycle fatigue criterion (infinite fatigue life).

* Can deal with three-dimensional loading.

* Can deal with general multiaxial loading.

* Works at the microstructural level, ie, the scale of one or two grains.

* Can identify the direction of crack initiation.

DANG-VAN Factor of Safety Plot

Summary of Multiaxial Approach

* Assume uniaxial and find critical locations

* Assess multiaxiality at critical locations by checking biaxiality ratio and angle of max. principal stress vs time

* If angle constant and constant ae < 0, use Hoffman-Seeger (or Klann-Tipton-Cordes) correction with Abs. Max. Principal stress

* If angle constant and constant ae > 0, use Hoffman-Seeger correction and signed Tresca stress

* If ae or angle varies greatly with time, need to use critical plane method

Example of Multiaxial Assessment

Perform crack initiation analysis of a knuckle.

Multiple (12) loading inputs.

Assess multiaxiality.

0 500 1000 1500-50.05

84.71Force(Newtons) LOAD03.PVX

point

Sample = 1Npts = 1610Max Y = 84.71Min Y = -50.05

0 500 1000 1500-7998

7720Force(Newtons) LOAD02.PVX

point

Sample = 1Npts = 1610Max Y = 7720Min Y = -7998

0 500 1000 1500-2654

3769Force(Newtons) LOAD01.PVX

point

Sample = 1Npts = 1610Max Y = 3769Min Y = -2654

Screen 1

12 loads associated with 12 FE results

Example of Multiaxial Assessment, cont.

Mean Biaxiality

Angle Spread


· a useful concept in multiaxial fatigue and especially in plasticity is that of deviatoric...

Documents