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
Life Prediction Process: E-N Approach MEASURED STRAINS Plasticity Modelling STRESS & STRAIN COMPONENTS Damage Model LIFE ELASTIC STRAINS FROM FEA Constitutive Model and Notch Rule
-D Stress State σ yy τ yx τ xy σ xx σ xx τ xy y τ yx x σ yy
3-D Stress State σ yy σ zz σ xx τ yx τ yz τ xz τ zx τ τzy xy σ xx y σ zz x z σ yy
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. σ τ τ xx xy xz τ σ τ τ τ σ yx yy yz zx zy zz * Can be written s ij
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. e xy are half the engineering shear strain g xy ε ε ε ε ε ε ε ε ε xx xy xz yx yy yz zx zy zz * Can be written e ij
Transformation of Stress / Strain Z Z Y Y X X
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. T = = ~ l l l l l l l l l 11 1 13 1 3 31 3 33 l 11, l 1, l 13 are the direction cosines of the X axis in the original system and so on.
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: σ 1 3 3 1 where I I I I σ = σ = σ = σ x x x + σ σ σ y y + I y z + σ σ σ I + σ y σ z z + τ 3 xy C = 0 + σ τ xz y σ τ z yz τ σ xy x τ τ yz xz σ τ y τ yz xz σ z τ xy
MOHR S Circle for -D Stress τ xy σ σ y θ σ 1 σ σ x τ xy τ max τ
MOHR S Circle for 3-D Stress σ 3 σ σ 1 σ τ max τ
Generalized HOOKE S LAW for 3-D (Strains) ε ε ε γ x y z xy σx = E σ y = E σz = E τxy =, G v E v E v E γ ( σ + σ ) ( σ + σ ) ( σ + σ ) yz y z x = τ yz G z x y, γ zx = τ G zx where G = E ( 1+ v)
Generalized HOOKE S LAW for 3-D (Stress) σ σ σ xx yy zz = = = ve v ( )( ) { ε } xx + ε yy + ε zz 1+ 1 v ve v ( )( ) { ε } xx + ε yy + ε zz 1+ 1 v ve v ε + ε + ε E + ε 1+ v E + ε 1+ v ( )( ) { } xx yy zz zz 1+ 1 v 1+ v + E ε xx yy
Free Surface Stresses z x y Stress state on free surface is biaxial - principal stresses σ 1 and σ (where σ 1 > σ ) lie in the x-y plane
Multiaxial Assessment Ratio of Principals or Biaxiality Ratio: a e = σ σ 1 * 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: a e = -1 Pure Shear a e = +1 Equi-Biaxial a e = 0 Uniaxial
Example of Near Proportional Loading Strain(UE) S131A.DAC 1301-39.3 0 4 6 8 10 1 Seconds Strain(UE) S131B.DAC 11.1-84.3 0 4 6 8 10 1 Seconds Strain(UE) S131C.DAC 663-98.7 0 4 6 8 10 1 Seconds Sample = 409.6 Npts = 9446 Max Y = 1301 Min Y = -39.3 Sample = 409.6 Npts = 9446 Max Y = 11.1 Min Y = -84.3 Sample = 409.6 Npts = 9446 Max Y = 663 Min Y = -98.7 Screen 1
Example of Near Proportional Loading, cont. S131.ABS Strain UE S131.ABS Strain UE Time range : 0 secs to 3.06 secs Time range : 0 secs to 3.06 secs 5000 5000 4000 4000 3000 3000 000 000 1000 1000 0 0-1000 -1-0.5 0 0.5 1-1000 -50 0 50 Biaxiality Ratio (No units) Screen 1 Angle (Degrees) Screen 1 Biaxiality Ratio vs. σ 1 Orientation of σ 1 vs. σ 1
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 Non-Proportional Loading 161.4 GAGE 1X( us) GAGE103.DAC Sample = 00 Npts = 3.67E4 Max Y = 161.4 Min Y = -81.3-81.3 0 50 100 150 559.5 GAGE 1Z( us) GAGE10.DAC Sample = 00 Npts = 3.67E4 Max Y = 559.5 Min Y = -74.6-74.6 0 50 100 150 716. GAGE 1Y( us) GAGE101.DAC Sample = 00 Npts = 3.67E4 Max Y = 716. Min Y = -651-651 0 50 100 150 Screen 1
Example of Non-Proportional Loading, cont. GAGE1.ABS Stress MPa GAGE1.ABS Stress MPa 00 Time range : 0 secs to 183.6 secs 00 Time range : 0 secs to 183.6 secs 100 100 0 0-100 -100-00 -1-0.5 0 0.5 1 Biaxiality Ratio (No units) Screen 1-00 -50 0 50 Angle (Degrees) Screen 1 Both the ratio and orientation of σ 1 and σ vary considerably: non-proportional loading.
Effect of Multiaxiality on Plasticity, Notch Modelling and Damage φ p a e Increasing Difficulty (and Rarity) Uniaxial φ p constant a e = 0 OK Proportional Multiaxial φ p constant -1 < a e < +1 Need a e Non-Proportional Multiaxial φ p may vary a e may vary Tricky Decreasing Confidence
Deviatoric Stresses 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. The hydrostatic stress P h is an invariant: P h = 3 1 ( σ + σ + σ ) x y z The deviatoric stresses S x,y,z are given by: S S S = σ P = σ P = σ P x x h y y h z z h The shear stresses are unchanged.
( S S S ) 1 3 3 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: 1 + + = σ or ( ) ( ) ( ) 1 3 3 1 σ σ + σ σ + σ σ = σ y An alternative, the Tresca Criterion can be expressed as: σ σ σ σ σ σ σ 1 3 3 1 τ max = max,, = y
VON MISES & TRESCA in Deviatoric Space S 1 Tresca von Mises 3 σ y S 3 S
VON MISES & TRESCA in Principals σ Tresca von Mises σ 1
* 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 σ ε σ 1 = eq ε 1 = eq σ σ σ 1 3 eq * Maximum Shear Stress (Tresca Criterion) = τeq = ε ε γ ( νε 1 1+ 3 max ) * Shear Strain (Tresca) = = * von Mises stress * von Mises strain 1 1 ( 1+ ν) ( ) ( ) ( ) 1 + 3 + 3 1 = eq σ σ σ σ σ σ σ eq ( ) ( ) ( ) ε1 ε + ε ε3 + ε3 ε1 = ε eq Note that ν can be found from: ν = νε ε + ν ε e e p p e + ε p
S-N Methods with Equivalent Stress * Basquin equation for uniaxial * Using (Abs) Max Principal Δσ = σ Δ σ 1 = σ ( ) b f N f ( ) b f N f * Using Max Shear Δτ σ max f = ( N ) b f * Using von Mises Δσ VM = σ f ( N ) b f
S-N Methods with Equivalent Strain * Coffin-Manson-Basquin equation for uniaxial ' Δε σ f = + E ( ) b ( ) ' N f ε f N f c * Using (Abs) Max Principal Δε 1 ' σ f = + E ( ) b ( ) ' N f ε f N f c * Adapted for Torsion Δγ ' τ f = + G N N ( ) b ( ) ' f γ f f c * But if we assume the principal stress/strain criterion: γ Δγ σ f σ1 = τ and ε1 =, so = + G N N ' ( ) b ( ) ' f ε f f c
S-N Methods with Equivalent Strain, cont. * Tresca criterion Δγ = σ ' f G ( ) b ' N + (1 + ν ) ε ( N ) c f p f f * von Mises Criterion Δγ = (1+ ν E e ) σ ' f b ' ( N ) + (1+ ν ) ε ( N ) c f p f f * which is the same as... Δγ 1 ( + νe) σ f = 3 E ' ( ) b ( ) ' N f + 3ε f N f c
THE NEED FOR A SIGN Cylindrical notched specimen with axial sine loading 50 Stress(MPa) maximum principal -50 0 1 3 Seconds 50 Stress(MPa) minimum principal τ Tension σ Compression σ τ -50 0 1 3 Seconds 50 Stress(MPa) absolute maximum principal -50 0 1 3 Seconds 50 Stress(MPa) von Mises stress -50 0 1 3 Seconds 50 Stress(MPa) maximum shear stress -50 0 1 3 Seconds Screen 1
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: Δεeq = MAX ( wrt. time) Δε11 Δε + Δε Δε33 + Δε33 Δε11 + Δε1 + Δε3 + Δε 3 ( ) ( ) ( ) 6( ) 31 * No path dependence. * Non-conservative for non-proportional loading. * No directionality. * Not sensitive to hydrostatic stress.
Simple Methods for Proportional Loadings -1<a<0 a~0 0<a<1 Stress Criterion Absolute Maximum Principal Absolute Maximum Principal Absolute Maximum Principal Strain Criterion Absolute Maximum Principal Any Tresca
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
KLANN-TIPTON-CORDES Method First define cyclic stress-strain curve using the Ramberg-Osgood formula: The ratio ε /ε 1 of the principal strains is assumed to be constant in this case σ σ ε q q n q = + 1 ' Ε Κ ' Calculate the biaxiality ratio from : Digitize the cyclic stress-strain curve and for each point calculate Poisson s ratio from the equation : v ε + ε v ' a = 1 ε 1 + v ' ε 1 q ' = 1 1 σ ve E ε q
KLANN-TIPTON-CORDES Method, cont. It can be shown that the values of the principal strains and stresses can be calculated from: 1 v a ' ε 1 = ε q 1 a + a Fit the following equation to the calculated modified parameters: σ σ * ε 1 1 1 = * + 1 n * Ε Κ The modified modulus is calculated explicitly from: Ε * E = 1 - v e a e 1 σ 1 = σ q 1 a + a
Modified Stress-Strain Curve Parameters σ 1 a e = 1 a e = 0 a e = -1 ε 1
HOFFMAN-SEEGER Method Calculate von Mises equivalent strain from combined strain parameter e.g. from: ε = ε q, e 1, e 1 a e + a e 1 a e v e The Neuber correction is then carried out on this formulation: σ q ε q = E ε q, e The effective Poisson s ratio is calculated as for the Klann-Tipton-Cordes Method, as are a, σ, ε 1 and ε / ε 1
HOFFMAN-SEEGER Method, cont. The other required stresses and strains are calculated from: ε ε 1 ε v ' ( 1 + a) = ε 3 = ε q ε 1 1 - a + a σ = a σ 1 These can then be used to calculate any other combined parameter e.g. signed Tresca
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 e ε e e ij e ij e ij e ij N ij N ij N ij σ Δε + ε Δσ = σ Δε + ε Δσ N ij
BUCZYNSKI-GLINKA Notch Method, cont. * 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
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 (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.
Crack Initiation & Multiaxial Fatigue, cont. * 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.
Multiaxial Analysis in MSC Fatigue γ (a) Torsion * (b) Tension σ 1 ε γ * 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. ε σ 1 γ
Multiaxial Analysis in MSC Fatigue, cont. 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: Δγ + ΔΣn = C * Equivalent fatigue life results for equivalent values of the material constant, 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
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 * Critical Plane Approach Calculates the Shear Strain time history and damage on multiple planes Fatigue results reported on the worst plane Fatigue damage based on Shear Strain range No mean stress correction * Use with Type B cracks
SMITH-TOPPER-WATSON-BANNANTINE 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 Includes a mean stress correction based on Maximum Normal stress * Use with Type A cracks
FATEMI-SOCIE Method * Critical Plane Approach Calculates the Shear Strain time history and damage on multiple planes Fatigue results reported on the worst plane 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: Δεn σ f b = + E ( N ) ε ( ) f f N f c * SWT Bannantine: Δεn σ f σn =,max + E b ( N f ) σ f εf ( N f ) b+ c * Shear Strain: Δγ = ( 1 ) + ν σ e E f b ( N f ) + ( 1+ νp) ε f ( N f ) c * Fatemi-Socie: ( 1 νp) εf ( N f ) ( 1 ) Δγ σ ν ν σ σ ( ) 1 n,max 1 + σ = ( + n + e ) b e f n f N f + y E Eσ y + + + n ( 1 ν ) ( N f ) + ε σ + σ c p f f y b ( N f ) b c
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.
WANG-BROWN Method, cont. * 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. Mean Stress Correction using mean Normal Stress: ε γ + S. δε 1+ ν + S( 1 ν ) σ σ =. b + E max n f n, mean ( N ) ε ( N ) f f f c
Example of Polar Damage Plot Theta=90 Polar Plot of Data : DEMO Theta=45 10 90 60 150 30 180 1E-91E-81E-71E-6 0 10 330 40 70 300 Polar Plot of Type A and Type B damage for Wang-Brown Method
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 (Repeats) Normal Strain 106,000 STW-Bannantine 316,000 Shear Strain 18,500 Fatemi-Socie 7,000 Wang-Brown 30,500 Wang-Brown + Mean 6,000 Abs. Max. Principal Strain 97,300
Example: Out-of-Phase Loading Material: Manten Axial Stress, σ x = 5 ksi Shear Stress, τ xy = 14.4 ksi Multiaxial Method Normal Strain STW-Bannantine Shear Strain Fatemi-Socie Wang-Brown Wang-Brown + Mean Abs. Max. Principal Strain Signed von Mises Strain Signed Tresca Strain Life (Cycles) 4.1E+07.80E+04 1.41E+05 1.70E+05 6.63E+06 8.55E+05.88E+07.88E+07 8.41E+06
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
DANG-VAN Criterion, cont. 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: 1 { } τ() t = S () ij t + ρij Tresca dev * () ( ph t = xx + yy + zz )() t 3 σ σ σ 1 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 r ij * is the deviatoric part of the stabilised residual stress
DANG-VAN Plot τ(t) Damage occurs here!!! τ + a ph b= 0 τ a ph+ b= 0 ph(t)
Stabilized Residual Stresses ρ ( ) devρ ij * * * 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
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 a e < 0, use Hoffman-Seeger (or Klann- Tipton-Cordes) correction with Abs. Max. Principal stress * If angle constant and constant a e > 0, use Hoffman-Seeger correction and signed Tresca stress * If a e or angle varies greatly with time, need to use critical plane method
Example of Multiaxial Assessment Perform crack initiation analysis of a knuckle. Multiple (1) loading inputs. Assess multiaxiality.
Example of Multiaxial Assessment, cont. 1 loads associated with 1 FE results Force(Newtons) LOAD03.PVX 84.71-50.05 0 500 1000 1500 point Force(Newtons) LOAD0.PVX 770-7998 0 500 1000 1500 point Force(Newtons) 3769 LOAD01.PVX -654 0 500 1000 1500 point Sample = 1 Npts = 1610 Max Y = 84.71 Min Y = -50.05 Sample = 1 Npts = 1610 Max Y = 770 Min Y = -7998 Sample = 1 Npts = 1610 Max Y = 3769 Min Y = -654 Screen 1
Example of Multiaxial Assessment, cont.
Example of Multiaxial Assessment, cont. Angle Spread Mean Biaxiality