European Comsol Conference

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1 universiy of ferrara Presened a he COMSOL Conference 2009 Milan European Comsol Conference Tile: Faigue Damage Evaluaion on Mechanical Componens Under Muliaxial Loadings Auor name: Ing. Simone Capea simone.capea@unife.i i Prof. Robero Tovo robero.ovo@unife.iovo@unife i Deparmen of Engineering, Universiy of Ferrara (Ialy)

2 universiy of ferrara CONTENTS. Aim of he work 2. Inroducion o he non-local model 3. Sress-invarian based muliaxial crierion 4. Experimenal analysis 5. Procedure and calculaion ools 6. Numerical resuls 7. Conclusions

3 . AIM OF THE WORK AIM. Faigue srengh h assessmen of mechanical componens. 2. Taking ino accoun he effec on he faigue srengh due o he presence of complex hreedimensional (3D) noches (gradien effec). 3. Considering he muliaxial effec caused by exernal loadings as well as by muliaxial sress fields due o severe sress raisers (muliaxial effec). 4. Developing a numerical ool in conjuncion wih hree-dimensional modelling ools o be used by indusrial engineers. 5. Comparing heoreical faigue esimaions o experimenal daa in order o validae he developed approach. Slide 3/5

4 2. INTRODUCTION TO THE NON-LOCAL MODEL Non-local model y x V x x,y y dy V Local sress ensor in y r y k 2 xy 2 2 l x, y e Weigh funcion r V x x,y dy V x Reference volume Non-local sress ensor in x x x V F a F a r Slide 4/5

5 2. INTRODUCTION TO THE NON-LOCAL MODEL From inegral equaion o parial differenial equaion x x,y y dy V x r V () I is imporan o observe ha any scalar sress field defined in volume V as a funcion of sress ensor can be used as equivalen scalar in equaion (). x x,yydy V r x (2) V Inegral definiion of non-local scalar quaniies (2) can be approximaed by a second order parial differenial equaion (3): 2 2 x c x x (3) The equaion (2) urns ou o be equal an implici differenial formulaion (3), i.e. socalled implici gradien approach. n 0 n c Neumann s boundary condiions, where denoes he normal o he surface of V Diffusive lengh relaed o he relevan maerial properies Slide 5/5

6 Non-local model in faigue 2. INTRODUCTION TO THE NON-LOCAL MODEL 2 2 x c x x Scalars and can be differen in order o consider differen problems:, ij Elaso-plasic model, low cycle faigue,,,... High-cycle faigue damage evaluaions a eq 2 2 a,eq x c a,eq x a,eq x Under non-proporional loading he principal sress direcions are no consan. To correcly define he equivalen deviaoric componen, i could be necessary o apply amuliaxial i l crierion i ogeherh wih he sress gradien approach under mixed-mode loadings. Slide 6/5

7 Time variabiliy of deviaoric componen 3. STRESS-INVARIANT BASED MULTIAXIAL CRITERION As an exernal load is applied, he ip of he vecor s describes a curve called deviaoric sress componen loading pah s 3 s s In-phase loading s 3 s Ou-of-phase loading Usually he ampliude is calculaed as he maximum difference of vecor Slide 7/5 s J s s 2 2,a 2 a wo differen ime insans s s 3 s s Variable ampliude loading

8 PbP approach 3. STRESS-INVARIANT BASED MULTIAXIAL CRITERION A.Crisofori,L.Susmel,R.Tovo,Asressinvarian based crierion o esimae faigue damage under muliaxial i l loading. Inernaional i Journal of Faigue 30 (2008), Fundamenal hypohesis: The faigue damage due o he applied loading pah can be esimaed from he projecion p,i of he loading pah on a convenien reference frame. Procedure: s 3 n * 3 Projeced pahs Assessmen of he projecion reference frame n* Definiion of equivalen ampliude of he deviaoric sress componen 2 d,a d,a i i p,3 Formulaion of he muliaxial high-cycle crierion d,a d,a 0 FL d,a 0 d,a FL FL FL FL H,max FL d,a FL 3 s p, s d,a d,a FL Slide 8/5

9 ,7 00 4,5 6,98 4,5 4,50 4. EXPERIMENTAL ANALYSIS 70 ess were carried ou by esing cylindrical 3D-noched specimens having gross diameer equal o 8mm and made of a commercial cold-rolled low-carbon seel (En3B). 3,97 5 Ø2,95 23,48 2 Ø ,97 A 6,54 R 0,03 DETTAGLIO A SCALA 20: The specimens are characerized by severe sress raisers wih roo radius equal o 0.03mm. All ess were carried ou in he Deparmen of Mechanical Engineering, Triniy College of Dublin, under he supervision of prof. David Taylor. Slide 9/5

10 Definiions: a a universiy of ferrara 4. EXPERIMENTAL ANALYSIS, Tensile and orsional sress ampliude refered o he gross secion Biaxial load raio defined as a a R Phase angle beween load componens Nominal load raio Experimenal deails: One series of ess under pure ension faigue loading, R=-. One series of ess under pure orsion faigue loading, R=-. Two series of ess under combined ension orsion faigue loading, consan biaxial load raio 3, R=-, phase angle =0 or 90 Two series of ess under combined ension orsion faigue loading, consan biaxial load raio, R=-, phase angle =0 or 90 Slide 0/5

11 Crack pahs universiy of ferrara Torsional Muliaxial loading 4. EXPERIMENTAL ANALYSIS Torsional Muliaxial loading Torsional Muliaxial loading Torsional Muliaxial loading Slide /5

12 The procedure framework: COMSOL Srucural Mechanics 5. PROCEDURE AND TOOLS OF CALCULATIONS COMSOL PDE Modes MATLAB Sep Sep 2 LINEAR ELASTIC STRESS ANALYSIS HAVE TO BE CARRIED OUT FOR ANY EXTERNAL LOADING APPLIED EXCTRACTION S ROUTINE OF NODAL SOLUTIONS FOR ANY APPLAIED EXTERNAL LOAD ROUTINE OF CALCULATION OF PROJECTIONS REFERENCE FRAME Sep 3 Sep 4 ASSESSMENT OF NON-LOCAL VALUES d,a AND H,max BY USING THE IMPLICIT GRADIENT METHOD ROUTINE TO CALCULATE THE LOCAL VALUES OF DEVIATORIC AND HYDROSTATIC COMPONENT ROUTINE TO IMPORT THE LOCAL SCALAR FIELDS INTO THE FEM MODEL Sep 5 Slide 2/5 FATIGUE LIFE ESTIMATION IS PERFORMED AT EACH NODAL POINT BY COMPARING d,a TO LOCAL FATIGUE LIMIT. d,a FL

13 Damage index universiy of ferrara 77.5 MPa NUMERICAL RESULTS 67. MPa 7 83 Uniaxial loading a Torsional loading a σ A =77.5 MPa τ A =67.5 MPa MPa 376 MPa a 50.8 MPa a 53.4 MPa Muliaxial loading a a a σ A =50.8 MPa In-phase MPa Muliaxial loading a a a σ A =54.3 MPa Ou-of-phase MPa 66.2 MPa 66.2 MPa 7 7 Muliaxial loading 3 a a 3 a a σ A =66.3 MPa In-phase Muliaxial loading a a 3 a a σ A =66.3 MPa Ou-of-phase MPa 424 MPa Slide 3/5

14 Damage index universiy of ferrara 6. NUMERICAL RESULTS GEOMETRY σa [Mpa] LOAD CONDITIONS τa [Mpa] φ R EXPERIMENTAL RESULTS σ A [Mpa] τ A [Mpa] FATIGUE STRENGTH PREDICTION σ A [Mpa] τ A [Mpa] ERROR INDEX E(%) CRITICAL POINT LOCATION x [mm] y [mm] z [mm] % % % % % % Slide 4/5

15 7. CONCLUSIONS The devised approach is seen o be highly accurae in esimaing high-cycle faigue damage in mechanical componens wihou he need for assuming a priori he posiion of he criical poin. This approach is capable of efficienly aking ino accoun he presence of boh muliaxial l loading and non zero ou-of-phase angles. The implici gradien mehod applied in conjuncion wih PbP approach proved o be a powerful engineering ool capable of efficienly designing complex, i.e. hree-dimensional (3D), sress concenraions agains muliaxial faigue. The faigue life esimaion echnique proposed in he presen work is suiable for being used in siuaions of pracical ineres by direcly pos- processing simple linear-elasic FE models. Even if he resuls obained so far are very saisfacory, more work needs o be done for a complee validaion of his mehod. Slide 5/5

16 Thank you for your aenion Slide 6/5

17 3. STRESS-INVARIANT BASED MULTIAXIAL CRITERION General form for muliaxial faigue limi crieria f T, N Maerial parameer ha characerizes he faigue limi condiion Shear sress componen Normal sress componen Criical plane Ampliude of he shear Mean or maximum normal approach sress on he criical plane sress perpendicular o criical plane Sress-invarian based approach Ampliude of he square roo of he second invarian of he sress deviaor Mean or maximum hydrosaic sress Slide 7/5

18 3. STRESS-INVARIANT BASED MULTIAXIAL CRITERION Definiion of he sress quaniies relaed o sress invarian approach Deviaoric and hydrosaic componens of Cauchy s sress ensor x xy xz yx y yz H I d zx zy z H r 3 2x y z xy xz 3 2y x z d yx yz 3 2 z x y zx zy 3 Deviaoric componen can be Square roo of he second represened as a 5-elemen vecor invarian of he sress deviaor 3 s 2 s2 s s3 2 s 4 s5 xyz J2 s s i y z s i xy xz yz How o find he ampliude? Slide 8/5

19 The projecion reference frame 3. STRESS-INVARIANT BASED MULTIAXIAL CRITERION Cenroid of he pah sm,i si d T T i,...,5 s 3 Projecion reference frame n * 3 Recangular momen of ineria of he pah ij i m,i j m, j T C s s s s d ij i,j,...,5 n* Eigenvecors are invarian for coordinae ransformaion and heir calculaion is numerically efficien. The eigenvecor reference frame coincides wih he maximum variance reference frame. sm s Faigue damage is srongly relaed o he variance of he signals. For insance, under ampliude consan load he maximum damage reference frame coincides wih he maximum variance reference frame. Slide 9/5

20 Calibraion of he faigue srengh crierion 5. PROCEDURE AND TOOLS OF CALCULATIONS EN3B Serie R ρ FL d,a FL a 0 6 cicli [MPa] d,a FL a cicli [MPa] Monoassiale Monoassiale Torsione pura da d,a CURVA High-cycle DI CALIBRAZIONE faigue srengh DEL MODELLO versus muliaxial i l load raio Relaion beween faigue srengh a 2 million cycle and FL 0.77 FL 0.2 d,a FL [MPa] 30e FL γ min CURVA DI TARATURA DI C Faigue limi condiion for a crack 2.0 KI ynom a Kh.5.0 High-cycle faigue safey facor d,a FL d,a 0.5 C=0.055 mm C [mm] Slide 20/5

21 The procedure framework: 5. PROCEDURE AND TOOLS OF CALCULATIONS. A usual linear elasic sress analysis have o carry ou for each ype of exernal load applied o he specimen. 2. Local projecion reference frame is evaluaed a every mesh nodal poin. 3. Local values of he ampliude of deviaoric componen and maximum hydrosaic componen are calculaed a every node. d,a H,max H,max 4. Non-local values and are calculaed a each nodal poin by using he implici gradien mehod: d,a d,a d,a 2 2 H,max c H,max Hmax H,max H,max 2 2 c n d,a 0 n H,max 0 H,max 5. Muliaxial load raio can be calculaed as: FL 3 FL d,a 6. Faigue life esimaion is finally performed a each nodal poin comparing wih local faigue limi calculaed as funcion of FL. d,a FL d,a d,a Slide 2/5

22 Faigue ess Tes muliassiali EXPERIMENTAL ANALYSIS Saisical analysis Tes muliassiali i li orsionale normalizzaa Rigidezza o Serie Nr. provini Cicli Failure crierion R δ φ k σ a, τ a [MPa] σ A,50% a 0 6 cicli [MPa] OF 8OF 9OF 200 8OF 75 9OF 50 20OF 2OF Roura Run-ou Rea di regressione k = 5.36 Banda di dispersione (95%) Banda di confidenza (95%) Inercea a cicli = 60.2 MPa σ A,50% Inercea a σ A,50% cicli = 52.9 MPa Inercea a cicli = 44.5 MPa a cicli 25 [MPa] a cicli [MPa] Monoassiale Torsione pura Muliassiale Muliassiale Muliassiale Muliassiale δ=, φ=90 Synhesis of he experimenal resuls generaed under uniaxial and Numero cicli muliaxial i l loads Slide 22/5

23 Definiion of he sress quaniies 5. PROCEDURE AND TOOLS OF CALCULATIONS Firs load condiion ( ) Second load condiion ( ) F a k: index of node : phase angle beween F a and M M x, k xy, k xz, k, k xy, k 23 y, k yz, k xz, yz, z, k k k 2 x,m x,asin x,m x,asin y,m y,asin 23 sin sin sin s 2 x y z y,m y,a z,m z,a z,m z,a x, k xy, k xz, k, k xy, k y, k yz, k, k, k, k s 3 xy xy,m xy,asin xy,m xy,asin xz yz z s sin sin 4 xz xz,m xz,a xz,m xz,a s s s 2 s 3 s 4 s 5 s s s 2 s 3 s 4 s 5 s5 yz yz,m yz,asin yz,myz,asin s2 y z 2 y,m y,asin y,m y,asins s s s s2 s3 s4 s5 2 z,m z,asin z,m z,asin Slide 23/5

24 The maximum variance reference frame Variance and covariance erms of C s s s s d ij i i,m j j,m T s 5. PROCEDURE AND TOOLS OF CALCULATIONS i, j,...,5 Covariance marix is a symmeric square marix of order 5 C k C C C C, k j, k i, k ij, k i, j,...,5 k,...,n The covariance marix have o be diagonalised in order o find he direcion of maximum variance Projecion reference frame n 2 O s * n 2 * ni * O s ;...;s,m 5,m n * n T C NN k 2 C s s d, k,m T N n n * * 5 k x,a 4 x,a y,a y,a z,a z,a 8 x,ax,a cos 4 x,ay,a 4x,ay,a 4x,ay,acos 4x,ay,acos 4x,a z,a 244 x,az,acos4 x,az,acos4 x,a z,a 2 y,a y,a cos 2 y,a z,a 2 y,a z,acos2 y,a z,acos2 y,az,a 2 z,az,a cos Slide 24/5 n i T C C s s s s d 2, k 2, k,m 2 2,m T z,a z,a y,a y,a 2 x,a y,a2 x,ay,acos 83 2x,az,a 2x,az,acos2x,ay,acos2x,a y,a 2 cos 2 2 x,az,a x,az,a y,ay,acos2 z,az,acos

25 5. PROCEDURE AND TOOLS OF CALCULATIONS Analyical formulaions of deviaoric and hydrosaic sress componens s s s s s sin s s sin s * i n* * i n i n,n * * i i m a m a s,n i,...,5 * n2 Projeced pah sin sin H H,m H,a H,m H,a Calculaion of maximum values of s and H funcions ds n * i d 0 i,...,5 d s,n * a i arcan n* i s,n * sin an a i H,a arcan H H,a sin an H d i,...,5 0 n* * i ni H H * ni s n *i *2 s n * O s s n * * n Analyical soluions s * * * * * s a,n i sin s a,n i sin n i n i ni a i,...,5 Slide 25/5 5 2 s d,a d,a i n* i i i 2 a H,max H,a H H,a sin sin H