S18-1 PAT318, Section 18,March 2005 SECTION 18 MULTIAXIAL FATIGUE

S18-2 PAT318, Section 18,March 2005

S18-3 PAT318, Section 18,March 2005 WHY DO MULTIAXIAL FATIGUE CALCULATIONS? n Fatigue analysis is an increasingly important part of the design and development process n Many components have multiaxial loads, and some of those have multiaxial loading in critical locations n Uniaxial methods may give poor answers needing bigger safety factors

S18-4 PAT318, Section 18,March 2005 measured strains stress and strain components LIFE plasticity modelling damage model constitutive model and notch rule elastic strains from FEA THE LIFE PREDICTION PROCESS E - N APPROACH

S18-5 PAT318, Section 18,March 2005 x y xy yx xy yx xx yy xx 2-D STRESS STATE

S18-6 PAT318, Section 18,March 2005 z x y xx yy zz yy xy xz yz yx zx zy 3-D STRESS STATE

S18-7 PAT318, Section 18,March 2005 xxxyxz yxyyyz zxzyzz TENSOR REPRESENTATION OF STRESS STATE n Stresses can be represented as tensor n Diagonal terms are direct stresses n Other terms are shear stresses n For equilibrium purposes it must be symmetric n On free surface (z is surface normal) all terms with z disappear. Can be written ij

S18-8 PAT318, Section 18,March 2005 xxxyxz yxyyyz zxzyzz STRAIN TENSOR n Strains can also be represented by tensors n Diagonal terms are the direct strains and the other terms are shear strains n For equilibrium the matrix is symmetric Shear strains, e.g. xy are half the engineering shear strain xy Can be written ij

S18-9 PAT318, Section 18,March 2005 X Y Z Y Z X TRANSFORMATION OF CO-ORDINATES

S18-10 PAT318, Section 18,March 2005 STRESS TENSOR ROTATION n Stress or strain tensors can be rotated to a different coordinate system by a transformation matrix. n The matrix contains the direction cosines of the new co-ordinate axes in the old system n The tensor is pre-multiplied by the matrix and post-multiplied by its transpose l 11, l 12, l 13 are the direction cosines of the X axis in the original system and so on. T lll lll lll

S18-11 PAT318, Section 18,March 2005 PRINCIPAL STRESSES (AND STRAINS) n The principal stress axes are the set in which the diagonal terms disappear. In these directions the direct stresses reach their extreme values n The maximum shear strains occur at 45 degrees to the principal axes. n The principal stresses can be calculated from:

S18-12 PAT318, Section 18,March 2005 x y xy 1 2 max 2 MOHRS CIRCLE FOR STRESS (2D)

S18-13 PAT318, Section 18,March max 3 MOHRS CIRCLES FOR TRIAXIAL STRESS

S18-14 PAT318, Section 18,March 2005 GENERALIZED HOOKES LAW FOR 3-D

S18-15 PAT318, Section 18,March 2005 GENERALIZED HOOKES LAW FOR 3-D

S18-16 PAT318, Section 18,March 2005 x y z Stress state on free surface is biaxial - principal stresses 1 and 2 ( where | 1 |>| 2 |) lie in the x-y plane FREE SURFACE STRESSES

S18-17 PAT318, Section 18,March 2005 MULTIAXIAL ASSESSMENT n Ratio of Principals or Biaxiality Ratio: u Stress state can be characterised by ratio of principal stresses and their orientation (angle) u If orientation and ratio are fixed, loading is proportional. u Otherwise loading is non-proportional u Biaxiality analysis: l ae = -1:Pure Shear l ae = +1:Equi-Biaxial l ae = 0: Uni-axial

S18-18 PAT318, Section 18,March 2005 EXAMPLE: NEAR PROPORTIONAL LOADING

S18-19 PAT318, Section 18,March 2005 Biaxiality ratio vs. 1 EXAMPLE: NEAR PROPORTIONAL LOADING Orientation of 1 vs. 1

S18-20 PAT318, Section 18,March 2005 EXAMPLE: NEAR PROPORTIONAL LOADING n 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. n The right hand plot shows that the orientation of the principal stresses is more or less fixed. n This is effectively a proportional loading (these calculation assume elasticity)

S18-21 PAT318, Section 18,March 2005 EXAMPLE: NON-PROPORTIONAL LOADING

S18-22 PAT318, Section 18,March 2005 Both the ratio and orientation of 1 and 2 vary considerably: non-proportional loading. EXAMPLE: NON-PROPORTIONAL LOADING

S18-23 PAT318, Section 18,March 2005 Uniaxial Proportional Multiaxial Non-Proportional Multiaxial Increasing Difficulty (and Rarity) Decreasing Confidence OK Need a Tricky p a p constant p may vary a = 0 -1 < a < +1 a may vary EFFECT OF MULTIAXIALITY ON PLASTICITY, NOTCH MODELLING AND DAMAGE MODELLING

S18-24 PAT318, Section 18,March 2005 DEVIATORIC STRESSES The shear stresses are unchanged The deviatoric stresses S x,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. The hydrostatic stress P h is an invariant:

S18-25 PAT318, Section 18,March 2005 or 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, i.e. 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: The Tresca Criterion can be expressed:

S18-26 PAT318, Section 18,March 2005 hydrostatic stress von Mises yield surface THE VON MISES YIELD CRITERION

S18-27 PAT318, Section 18,March 2005 von Mises Tresca S1S1 S2S2 S3S3 VON MISES AND TRESCA IN DEVIATORIC STRESS SPACE

S18-28 PAT318, Section 18,March von Mises Tresca VON MISES AND TRESCA IN PRINCIPALS

S18-29 PAT318, Section 18,March 2005 EQUIVALENT STRESS AND STRAIN METHODS Extension of the use of yield criteria to fatigue under combined stresses

S18-30 PAT318, Section 18,March 2005 EQUIVALENT STRESS AND STRAIN METHODS n They dont account for the known fact that fatigue failure occurs in specifically oriented planes. Rather these approaches average the stresses/strains to obtain a failure criterion with no regard to the direction of crack initiation. n Tresca and von Mises are not sensitive to the hydrostatic stress or strain n They dont account for mean stresses n They dont really handle out-of-phase stresses or strains

S18-31 PAT318, Section 18,March 2005 SOME EQUIVALENT STRESS/STRAIN CRITERIA n Maximum Principal Stress n Maximum Principal Strain n Maximum Shear Stress (Tresca Criterion) n Shear Strain (Tresca) n von Mises stress n von Mises strain n Note that n can be found from

S18-32 PAT318, Section 18,March 2005 S-N WITH EQUIVALENT STRESS n Basquin equation for uniaxial n Using (Abs) Max Principal n Using Max Shear n Using von Mises

S18-33 PAT318, Section 18,March 2005 E-N WITH EQUIVALENT STRAIN n Coffin-Manson-Basquin equation for uniaxial n Using (Abs) Max Principal n Adapted for Torsion n But if we assume the principal stress/strain criterion

S18-34 PAT318, Section 18,March 2005 E-N WITH EQUIVALENT STRAIN (CONT.) n Similarly, based on the Tresca criterion…… n …and based on the von Mises Criterion n which is the same as...

S18-35 PAT318, Section 18,March 2005 Cylindrical notched specimen with axial sine loading Tension Compression THE NEED FOR A SIGN

S18-36 PAT318, Section 18,March 2005 Torsional Strain-Life curve Coefficient VonMises Stress/Strain Max. Principal Stress/Strain Max Shear Stress Strain TORSION STRAIN-LIFE COEFFICIENTS PREDICTED BY 3 EQUIVALENT STRAIN THEORIES (IN TERMS OF UNIAXIAL FATIGUE CONSTANTS) If you compare the results of these methods for axial and torsion there can be differences of up to a factor of 2 on stress and strain

S18-37 PAT318, Section 18,March 2005 COMMENTS ON EQUIVALENT STRAIN METHODS n They dont account for the known fact that fatigue failure occurs in specifically oriented planes. Rather these approaches average the stresses/strains to obtain a failure criterion with no regard to the direction of crack initiation. n Tresca and von Mises are not sensitive to the hydrostatic stress or strain n They dont account for mean stresses n They dont really handle out-of-phase stresses or strains

S18-38 PAT318, Section 18,March 2005 FAILURE OF EQUIVALENT STRESS METHOD FOR OUT-OF-PHASE (AXIAL AND TORSIONAL) LOADING - AN EXAMPLE LOAD: n Axial Stress: n Shear Stress: n Von Mises Stress: n Signed Von Mises will predict damage, but it will underestimate the damage (non-conservative) (No Alternating Stress) = No Fatigue Damage?

S18-39 PAT318, Section 18,March 2005 ASME PRESSURE VESSEL CODE n This method is based on the concept of relative von Mises Strain - equivalent to signed von Mises strain for proportional loadings n The ASME pressure vessel code uses the equivalent strain parameter: n No path dependence n Non-conservative for non-proportional loading n No directionality n Not sensitive to hydrostatic stress

S18-40 PAT318, Section 18,March

S18-41 PAT318, Section 18,March 2005 Uniaxial Proportional Multiaxial Non-Proportional Multiaxial p a p constant p may vary a = 0 -1 < a < +1 a may vary Increasing Difficulty (and Rarity) OK Need a Tricky Decreasing Confidence EFFECT OF MULTIAXIALITY ON PLASTICITY, NOTCH MODELLING AND DAMAGE MODELLING

S18-42 PAT318, Section 18,March 2005 NOTCH RULES FOR PROPORTIONAL LOADING n When the loading is no longer uniaxial, the uniaxial stress strain curve is no longer enough on its own n Two methods which address this problem are the parameter modification method due to Klann, Tipton and Cordes, and the Hoffmann-Seeger method. n Both these methods extend the use of the von Mises criterion to post yield behaviour n Both methods assume fixed principal axes and fixed ratio of stresses or strains

S18-43 PAT318, Section 18,March 2005 q qq n ' ' 1 vv E e q q ' First define cyclic stress-strain curve using the Ramberg- Osgood formula : Digitise the cyclic stress-strain curve and for each point calculate Poissons ratio from the equation : Calculate the biaxiality ratio from : a v v ' ' The ratio 2 / 1 of the principal strains is assumed to be constant in this case PARAMETER MODIFICATION METHOD (KLANN-TIPTON-CORDES)

S18-44 PAT318, Section 18,March 2005 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: 1 q 2 1 q va aa aa ' 1 1 * 1 * * 1 n * ee E 1-a v PARAMETER MODIFICATION METHOD (KLANN-TIPTON-CORDES) CONT.

S18-45 PAT318, Section 18,March 2005 a e = 0 a e = -1 a e = 1 MODIFIED STRESS-STRAIN CURVE PARAMETERS

S18-46 PAT318, Section 18,March 2005 q,e 1,e ee ee aa a v q q q,e E 2 HOFFMAN AND 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 Poissons ratio is calculated as for the Parameter Modification Method, as are: a, and /

S18-47 PAT318, Section 18,March = 3 q 2 a) 1-a+a v'(1 2 1 a HOFFMAN AND SEEGER METHOD CONT. These can then be used to calculate any other combined parameter e.g. signed Tresca The other required stresses and strains are calculated from:

S18-48 PAT318, Section 18,March 2005 EXTENDING NEUBER TO NON-PROPORTIONAL LOADINGS n This topic is important because it permits non-proportional multiaxial fatigue life predictions to be made based on elastic FE n The aim is to predict an average sort of elastic-plastic stress-strain response from a pseudo-elastic stress or strain history n It is necessary to combine a multiaxial plasticity model with an incremental formulation of a notch correction procedure and to make some other assumptions

S18-49 PAT318, Section 18,March 2005 BUCZYNSKI-GLINKA NOTCH METHOD n The Neuber method is only suitable for uniaxial or proportional loadings n Where the loading is non-proportional and the stress-strain response is path dependent it must be replaced by an incremental version

S18-50 PAT318, Section 18,March 2005 BUCZYNSKI-GLINKA METHOD n This rule has to be combined with a multiaxial plasticity model such as the Mroz-Garud model n Additionally some assumptions are required, e.g. 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. Glinka-Buczynski use total strain energy n One of these assumptions is necessary to be able to reach a solution of the equations

S18-51 PAT318, Section 18,March 2005 WHAT HAPPENS WHEN THE LOADING IS NOT UNIAXIAL? n For proportional loadings a different cyclic stress - strain curve is required n For non-proportional loadings, a 1 dimensional cyclic plasticity model is no longer sufficient n Neubers rule does not work for non-proportional loadings n Uniaxial rainflow counting doesnt work for non-proportional loadings n Simple combined stress-strain parameters do not predict damage well

S18-52 PAT318, Section 18,March 2005 DIRECTION OF CRACK GROWTH

S18-53 PAT318, Section 18,March 2005 DIRECTION OF CRACK GROWTH n When the biaxiality ratio is negative (type A), the maximum shear plane where cracks tend to initiate is oriented as shown in the diagram. u In the early stages of initiation the type A cracks grow mainly along the surface in mode 2 (shear), before transitioning to Mode 1 normal to the maximum principal stress. n When the biaxiality is positive (type B), however, the cracks tend to be driven more through the thickness. u These are therefore more damaging for the same levels of shear strain. n Uniaxial loading is a special case.

S18-54 PAT318, Section 18,March 2005 MULTI-AXIAL FATIGUE THEORY n Crack Initiation demonstrated to be due to: u Slip occuring along planes of Maximum Shear, starting in grains most favorably oriented w.r.t. the Maximum Applied Shear stress u Stage I (Nucleation & Early growth) is confined to Shear Planes. Here both Shear and Normal Stress/ Strain control the crack growth rate. u Stage II crack growth occurs on planes oriented normal to the Max. Principal Stress. Here the magnitude of the Max. Principal stress and strain dominates crack growth.

S18-55 PAT318, Section 18,March 2005 MULTI-AXIAL FATIGUE THEORY (CONT.) n Proportion of Life spent in Stage I, and II depend on: u Loading Mode and Amplitude u Material Type (Ductile Vs. Brittle) n Crack Initiation Life Refers to the time taken to develop Engineering Size Crack, and Includes Stage I and Stage II. n Stage I or Stage II may dominate Life. In uniaxial, n the Controlling Parameters in both Stages are directly n related to the uniaxial stress or strain. But n NOT so in Multi-axial case.

S18-56 PAT318, Section 18,March 2005 MULTI-AXIAL FATIGUE THEORY (CONT.) n For non-proportional loading, the Critical Planes vary vary with time. n Cracks growing on a particular Plane may impede the progress of cracks growing on a different plane. n 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.

S18-57 PAT318, Section 18,March 2005 MSC.FATIGUE MULTI AXIAL ANALYSIS n Shear Strain on the plane of maximum shear will extend the fatigue crack u Amongst other things, progress will be opposed by the friction between the crack faces n 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 (b) Tension

S18-58 PAT318, Section 18,March 2005 MSC.FATIGUE MULTI AXIAL ANALYSIS n Critical Plane Approach: u (Recognizing that fatigue damage (crack) is directional) considers accumulation of damage on particular planes u Typically damage is considered at all possible planes 15 deg. interval, and the worst (critical) plane selected. u Employs variations on the Brown-Miller Approach: u Equivalent fatigue life results for equivalent values of the material constant, C

S18-59 PAT318, Section 18,March 2005 MSC.FATIGUE MULTI AXIAL ANALYSIS n Four Planar Approaches: u Normal Strain u Smith-Watson-Topper-Bannantine u Shear Strain u Fatemi-Socie n Two complex Rainflow Counting Methods: u Wang-Brown u Wang-Brown with Mean Stress Correction n Dang-Van Total Life Factor of Safety Method

S18-60 PAT318, Section 18,March 2005 NORMAL STRAIN METHOD n A Critical Plane Approach u Calculates the Normal Strain Time History and Damage on 18 multiple planes, Fatigue Results reported on the worst plane u Fatigue Damage Based on Normal Strain Range u No mean Stress Correction n Used with Type A Cracks:

S18-61 PAT318, Section 18,March 2005 SHEAR STRAIN METHOD n A Critical Plane Approach u Calculates the Shear Strain Time History and Damage on 36 multiple planes, Fatigue Results reported on the worst plane u Fatigue Damage Based on Shear Strain Range u No mean Stress Correction n Used with Type B Cracks:

S18-62 PAT318, Section 18,March 2005 SMITH-TOPPER-WATSON-BANNANTINE METHOD n A Critical Plane Approach u Calculates the Normal Strain Time History and Damage on 18 multiple planes, Fatigue Results reported on the worst plane u Fatigue Damage Based on Normal Strain Range u Uses a Mean Stress Correction based on Maximum Normal Stress n Used with Type A Cracks

S18-63 PAT318, Section 18,March 2005 FATEMI-SOCIE METHOD n A Critical Plane Approach u Calculates the Shear Strain Time History and Damage on 36 multiple planes, Fatigue Results reported on the worst plane u Fatigue Damage Based on Shear Strain Range u Uses a Mean Stress Correction based on Maximum Normal Stress u Uses Material Constant n n Used with Type B Cracks

S18-64 PAT318, Section 18,March 2005 CRITICAL PLANE DAMAGE MODELS n Normal Strain n SWT - Bannantine n Shear Strain n Fatemi-Socie

S18-65 PAT318, Section 18,March 2005 WANG-BROWN METHOD n A complex recursive multi-axial Rainflow Counting Method n A Mean Stress Correction Method is available n May be quite slow especially for long loading histories n Recommended for a variety of proportional and non-proportional loadings

S18-66 PAT318, Section 18,March 2005 WANG-BROWN METHOD n Calculates a different Critical Plane for each rainflow reversal n For each reversal the damage is calculated on the critical i.e. maximum shear plane, whether case A or B n Uses Normal Strain Range, Maximum Shear Strain n Material Parameter S

S18-67 PAT318, Section 18,March 2005 WANG-BROWN METHOD Mean Stress Correction using Mean, Normal Stress

S18-68 PAT318, Section 18,March 2005 TYPICAL POLAR DAMAGE PLOT

S18-69 PAT318, Section 18,March 2005 I. Multiaxial MethodLife (Repeats) Normal Strain106,000 SWT-Brannantine316,000 Shear Strain18,500 Fatemi-Socie27,000 Wang-Brown30,500 Wang-Brown + Mean26,000 II. Equivalent Strain Method Abs. Max. Principal Strain97,300 MULTI-AXIAL LIFE CALCULATION METHODS: NONPROPORTIONAL LOADING Example: Knuckle, Chapter 11 (QSG) At Node 1045: Max Stress Range = 508 Mpa Mean Biaxiality Ratio: -0.6 Biaxiality S.D. = 0.18 Most Popular Angle = -64 deg Angle Spread = 90 deg

S18-70 PAT318, Section 18,March 2005 I. Multiaxial MethodLife (Cycles) Normal Strain4.12E+07 SWT-Brannantine2.80E+04 Shear Strain1.41E+05 Fatemi-Socie1.70E+05 Wang-Brown6.63E+06 Wang-Brown + Mean8.55E+05 II. Eqivalent Strain Method Abs. Max. Principal Strain2.88E+07 Signed Von Mises Strain2.88E+07 Signed Tresca Strain8.41E+06 MULTI-AXIAL LIFE CALCULATION METHODS: 90 DEG OUT OF PHASE LOADING Material: Manten Axial Stress, Sx = 25,000 psi Shear Stress, Sxy = 14,434 psi

S18-71 PAT318, Section 18,March 2005 DANG-VAN METHOD n High-Cycle Fatigue applications designed for infinite life n Calculates Factor-of-Safety of the design n Uses S-N total life method n Ideal applications: Bearing Design, Vibration induced fatigue

S18-72 PAT318, Section 18,March 2005 THE DANG VAN CRITERION n The Dang Van criterion is a fatigue limit criterion n It is based on the premise that there is plasticity on a microscopic level, leading to shakedown n After shakedown, the important factors for fatigue are the amplitude of the microscopic shear stresses and the magnitude of the hydrostatic stress n The method has a complicated way of estimating the microscopic residual stress

S18-73 PAT318, Section 18,March 2005 THE DANG VAN CRITERION Fatigue damage occurs if: where (t) and ph(t) are the maximum microscopic shear stress and the hydrostatic stress at time t in the stabilized state. They can be calculated from: a and b are material properties

S18-74 PAT318, Section 18,March 2005 THE DANG VAN CRITERION n The parameter b is the shear stress amplitude at the fatigue limit n The parameter a is in effect the mean stress sensitivity, with the mean stress being represented by the hydrostatic stress dev ij * is the deviatoric part of the stabilised residual stress

S18-75 PAT318, Section 18,March 2005 (t) ph(t) Damage occurs here !!! DANG-VAN PLOT

S18-76 PAT318, Section 18,March 2005 STABILIZED RESIDUAL STRESSES: n The stabilized local residual stresses are calculated by means of an iteration in which convergence assumes a stabilized state, a state of elastic shakedown. n As the loading sequence is repeated the yield surface grows and moves with a combination of kinematic and isotropic hardening until it stabilises n The stabilised yield surface is a 9 dimensional hypersphere that encompasses the loading history

S18-77 PAT318, Section 18,March 2005 DANG-VAN CRITERION SUMMARY n Is a High-Cycle fatigue criterion (infinite fatigue life) n Can deal with three-dimensional loading n Can deal with general multiaxial loading n Is constructed on the basis of microscopic level: the scale of one or a few grains n Can identify the direction of crack initiation

S18-78 PAT318, Section 18,March 2005 DANG VAN FACTOR OF SAFETY PLOT

S18-79 PAT318, Section 18,March 2005 SUMMARY OF APPROACH n Assume uniaxial and find critical locations n Assess multiaxiality at critical locations by checking biaxiality ratio and angle of principal If angle constant and constant e < 0, use Hoffman-Seeger (or Parameter Modification) biaxiality correction and abs max principal If angle constant and constant e > 0, use Hoffman-Seeger biaxiality correction and signed Tresca n If angle varies greatly with time, needs multiaxial If e varies greatly with time, needs multiaxial

S18-80 PAT318, Section 18,March 2005 A MULTIAXIAL ASSESSMENT Perform crack initiation analysis of a knuckle. Multiple (12) loading inputs. Assess multiaxiality.

S18-81 PAT318, Section 18,March 2005 LOADING INFO SETUP 12 loads associated with 12 FE results

S18-82 PAT318, Section 18,March 2005 LOG-LIFE CONTOUR PLOT (IN REPEATS)

S18-83 PAT318, Section 18,March 2005 Mean Biaxiality Angle Spread EXAMPLE MULTIAXIALITY INDICATORS

S18-84 PAT318, Section 18,March 2005 EXERCISE n Perform Quick Start Guide Chapter 10 Lesson, A Multiaxial Assessment. Part 1: Sections 10.1 through 10.4, where you will assess if the stress state is multiaxial, and Part 2: Section 10.5, where you will compute, and compare life estimates using multi-axial methods at a single location. n Be sure to ask for help if theres anything you dont understand