Life Prediction Under Multiaxial Fatigue

Transcription

1 Lie Prediction Under Multiaxial Fatigue D. Ramesh and M.M. Mayuram Department o Mechanical Engineering Indian Institute o Technology, Madras Chennai (India) mayuram@iitm.ac.in ABSTRACT Traditionally stress lie characteristics o engineering materials are decided upon rotating bending atigue test. The test data thus obtained are subjected to dierent corrections during design or analysis, to account other eects such as size eect, surace texture eect, material quality and also nature o loading (service actor) etc. With increasing complexities in application and use o components, which experience biaxial or triaxial stressing with varying stress amplitudes, precise design or accurate analysis is not possible. Hence increased attention has been directed towards multiaxial atigue loading. In the present work experimental investigation was conducted on standard SAE 1040 steel, notched specimen samples. Experiments were conducted under combined bending and torsional cyclic loading on a modiied Rotating Bending Machine The stress and strains induced at notch root have been determined by modeling specimen geometry using commercial sotware. Using the estimated stress and strain values, lie prediction has been attempted through various mutliaxial atigue theories and correlated with the experimental results. 1. INTRODUCTION Fatigue is one o the important actors in design since majority o engineering components are subjected to variable loading. Notches or geometric discontinuities are common in geometries o most engineering components such as shats o automobiles, earth moving machinery etc. The induced stresses under such situations are complex in nature. Fatigue under such loading involving stresses along more than one axis is known as multiaxial atigue [1]. Despite the signiicant advance made in understanding and prediction o endurance under uniaxial loading, the criterion or multiaxial loading does not reach satisactory level. There has been limited research eort towards atigue under multiaxial loading though assessment o multiaxial atigue is an important design consideration or reliable operation and sae in-service use o many systems. 2. LITERATURE REVIEW The material atigue behavior is deined in the orm o a stress (or strain) parameter versus number o cycles to ailure. Such characterization is usually obtained or smooth specimen under uniaxial loading. Multiaxial atigue assessment is then carried out with the help o an appropriate criterion that reduces the complex multiaxial loading to

2 equivalent uniaxial loading. In general, the ailure mechanism may involve two stages, viz. crack initiation and their subsequent propagation. The research in multiaxial atigue can be divided into three main approaches namely stress based, strain based and energy based approaches. 2.1 Stress Based Approach: For stress based approach, the principal stress, or the maximum shear stress is used to compose the multiaxial atigue parameter. This approach is oten applied or high cycle atigue. Instead o the magnitude o stresses some theories like McDiarmid s [2], are based on the shear and normal stresses associated with a critical plane. The critical plane is the plane where the atigue crack usually initiates. This is supported by some experimental observations also. 2.2 The strain based approach: Strain can be measured and has been shown to be excellent parameter or correlating with atigue lie. The most common application o the strain based approach, is atigue o notch members. The local strains can be above the yield strain, and stresses are more diicult to estimate than strains. In notched specimen subjected to cyclic loads, the behavior o material at the notch root is best considered in terms o strain rather than stress. This local strain approach is now widely accepted as a tool or predicting crack initiation atigue lie in uniaxial loading situations. The strain lie method oten called local strain approach can be used or atigue lie predictions using material properties, geometry, stress strain analysis at the critical location, damage assessment and summation techniques. For strain-based approach involving multi axial atigue the octahedral shear strain has been used to ormulate the classical atigue ailure criterion. However as seen earlier atigue crack usually initiate in the plane o maximum shear strain. Thereore, some theoretical approaches suggested that the parameter governing atigue lives are related to the maximum shear strain or the combined eect shear and stress normal to the plane o maximum shear strain. The local strain approach has evolved to deal analytically with multi axial atigue crack initiation and assumes that when the luctuating state o strain at critical location in an engineering component is the same as the strain in uniaxial atigue test specimen, a crack o similar proportions will develop in approximately the same number o luctuations. Based on this it is oten considered that the shear strain and normal strain in the critical plane, to be governing multiaxial atigue parameter [3]. For the energy based approach, the stain energy density per cycle [4] has been used to orm the atigue parameter, which is related to atigue lie, and the concept o plastic work is introduced by Garud [5] considering strain energy density in the critical plane. In the present work experimental investigation was conducted on standard SAE 1040 steel, notched specimen samples. Experiments were conducted under combined bending and torsional cyclic loading on a modiied Rotating Bending Machine. The stress and strains induced at notch root have been determined by modeling specimen geometry using commercial sotware. Using the estimated stress and strain values, lie prediction has been attempted through various mutliaxial atigue theories and correlated with the experimental results.

3 3. EXPERIMENTAL WORK Experiments have been conducted under combined bending and torsional loading conditions. Rotating bending machine (RBM) is suitable to test the atigue properties at zero mean stresses. An extension to chuck, holding the atigue test specimen is made and a torsional exciter working on eddy current braking principle is added so as to produce input torque variations on the specimen under testing. 3.1 Test specimens The tests have been perormed on notched cylindrical, smooth specimens. The geometry o these specimen are given in Fig.1 Components are abricated rom SAE1040 steel adopting standard manuacturing procedure and V notch o angle o 30 0, 45 0 and 60 0 to a depth o 1mm with 0.2 mm notch root radius and was introduced at the center. The composition and mechanical properties o the material is given in Table 1 Fig. 1 N otched Test specim en N o tc h D e ta ils Table 1 Composition and room temperature material properties o type SAE1040 steel Material C Si Mn Ni Cr S P Cu SAE Fatigue Strength exponent b * Fatigue Ductility exponent c * Youngs Modulus E(MPa) 2.05*10 5 Poisson s Ratio ν * 0.3 Fatigue Strength coeicient σ * (MPa) 948 Yield Stress σ ys (MPa) 380 Fatigue Ductility coeicient ε * * Literature reported values. 3.2 Test Details Experiments are conducted on completely reversed bending mode with torsional excitation o constant amplitude at ive dierent stress level combinations. Test details are presented in Table 2

4 Table 2 Test details- Applied loadings and estimated stresses and strains Bending moment (M) (N-m) Torsional Moment (T)(N-m) Bending stress (σ b ) MPa Shear stress(τ) MPa Applied vonmises stress(σ von ) σ von = ( σ b τ 2 ) Equival ent strain (σ von ) rom Ansys Fatigue strength coeicient ( σ ) σ eq = σ (2N) b b= Fatigue strength coeicient (ε ) ε eq ε = (2N) c= c 4. ANALYTICAL WORK Notch geometry eects were assessed or all the test conditions. Specimen geometry is meshed with 20 noded hexahedral elements. The experimental loading conditions are simulated in commercial sotware, Ansys to evaluate the resulting stresses and strains or particular loading conditions. To evaluate lie o specimen under combined loading conditions, notch root vonmises stresses and strains are considered in the lie prediction models 5. LIFE PREDICTION MODELS 5.1 Stress and strain lie theory Generally the atigue behavior o a material is depicted by its S-N (amplitude stress versus lie) characteristics. Under multi axial loadings, instead o the amplitude stress an equivalent atigue lie parameter is evolved. Successul stress based multi axial atigue damage parameter have the general orm σ eq= τ + kσ n (1) The basic premise o the local stress-strain approach is that the local atigue response o the material at critical point is more dependant upon the strain than the stress. Manson- Coin s equation gives an empirical expression or the cyclic strain lie curve. Separating the cyclic strain amplitude ε / 2 into elastic and plastic components the inal expression yield to

5 ε = 2 ε 2 e ε p + 2 σ = (2N 2E ) b + ε (2N ) c (2) 5.2 The Lohr and Ellison Approach This approach takes two types o strain, termed type A and type B, The dierence between these shear strains is the direction in which they act in relation to the specimen surace and how the crack grow. Type A shear strain drive the crack along the surace o the specimen or component. Type B shear strain act into the depth o the specimen. Lohr and Ellison have proposed a critical plane theory [6], which attempts to resolve dierence between type A and B shear behavior. They argued only crack being driven into the specimen (i.e. type B shear strain) is critical. Their ormulation was presented in the orm γ + k ε n = C (3) A lie relation based on these parameter can be developed rom and the uniaxial case as γ ε σ n b + k = 1.44 (2N ) + 1.6ε (2N 2 2 E ) c (4) with k=0.4. As in the Brown and Miller [7] critical plane approach, the normal strain to the plane o shear is thought to have a modiying inluence. The normal strain is termed as εn ε + ε = (5) 6. RESULTS AND DISCUSSIONS One o the objectives o the present work is to compare the results predicted by using existing multiaxial atigue models with experimental data obtained under combined bending and torsional loading conditions or SAE 1040 steel. Notch geometry is simulated in commercial sotware; Ansys and resulting vonmises stress and strain or particular loading conditions are determined. Typical results rom the analysis are shown in Fig.2 and 3. Maximum stress and strain occur at the root o the notch and distribution o stresses conirm to the theory. A plot o the vonmises strain versus experimentally observed lie is shown in Fig. 4. The trend is similar to standard S-N characteristics. Assuming a strain lie equation o the nature ε eq = ε (2N) c the atigue ductility co-eicient ε and atigue ductility exponent c values are successively calculated. A plot o these calculated values versus lie is generated as shown in Fig 5 and a straight line it is assigned. The intercept o this line is the true stain at racture in uniaxial tensile testing. The resulting value is compared to the literature reported value o On similar lines, in the stress based

6 approach, assuming a stress lie equation o the nature σ eq = σ (2N) b the atigue strength co-eicient σ and exponent b values are successively determined under speciic experimental conditions. A plot o the σ value versus lie is generated as shown in Fig 6 and straight line it is generated. The intercept o this line at 1 cycle is the true stress at racture. A close match o this value was possible only i the equivalent (vonmises) stress is based on the basic theory i.e. σ eq = ( σ 2 b + 3 τ 2 ) and not on the notch root stresses determined rom the Ansys. This is understandable because the elastic strains are to be dominant and lie controlling aspect in this region. Now assuming an analytical expression on the lines o Manson-Coins equation o the nature ε ε ε p σ = e b c + = (2N ) + ε (2N ) and substituting the E experimental lie values, the resulting strain amplitudes are determined. This plot o the strain amplitude versus lie is also shown in Fig 4. or comparison. This expression predicts a higher magnitude o total strain value compared to the vonmises strain rom the Ansys analysis. Finally based on the Lohr-Ellipson approach, using the normal and shear strains determined rom the Ansys analysis and the material characteristics σ, ε expected lie ( 2N ) is predicted. A plot o the predicted lie is shown in Fig 5. The equivalent stress amplitude versus lie plot is also included here or comparison. A close match could be observed at higher strain magnitude or in the strain controlled region. 7. CONCLUSION Based on the above observations it can be stated that under multi axial atigue loading equivalent strain on the critical plane based on Lohr-Ellipson approach yielded a closer match o the predicted lie with experimentally observed lie. NOMENCLATURE b = Fatigue Strength exponent ε e = Elastic strain c = Fatigue Ductility exponent ε p = Plastic strain K = Constant ε n = Normal strain N = Number o cycles to ailure γ = Maximum shear stain σ eq = Equivalent stress ε 1 = Maximum principal strain τ = Torisnal shear stress σ ε 3 = Minimum principal strain n = Normal stress σ σ b = Bending stress = Fatigue Strength coeicient ε = Total strain ε = Fatigue Ductility coeicient

7 REFERENCES 1. Garud Y.S., Multiaxial atigue: A Survey o the State o the Art, The American Society or Testing and Materials,5, , McDiarmid. D.L., A Shear stress Based Critical-Plane Criterion o Multiaxial Fatigue or -Design and Lie prediction, Fatigue and Fracture o Engineering Materials and Structures, 17, , Socie D.F., A Critical Plane Approach or Multiaxial Fatigue Damage assessment. Advances in Multiaxial atigue, ASTM STP 1191, American Society or Testing and Materials, Philadelphia, 7-36, Morrow. J Cyclic Plastic strain Energy and Fatigue o Metals, International Friction, Damping and Cyclic Plasticity, ASTM STP 378, American Society or Testing and Materials, West Conshohocken, PA, 45-87, Garud Y.S., A New Approach to the Evaluation o Fatigue Under Multiaxial Loadings, Journal o Engineering Materials and Technolozy,103, , Ellyin.F., and Golos. K., Multiaxial atigue damage criterion, J. o Engineering Materials and Technology, 110, 63-68, Brown. M.W., and Miller. K.J, A Theory or Fatigue Under Multiaxial Stress- Strain Conditions, Proc. o the Institute o Mechanical Engineers, 187, ,1973 Fig. 2 vonmises Stress VFrom onmises Ans Strainys corresponding to partic 0. ular 004loading covnditions onmises Strain Fatigue Strain Strength amplitude coeicent Equivalent total strain (Manson- Coin s type Eqn.) Linear (Vonmises Strain) y = -6E-05x y = -5E-10x y = -4E-11x E E E E E E+06 Fatigue Ductility coeicient VonMisesStress in Mpa Fig. 3. vonmises Strain From Ansys corresponding to Particular Loading conditions Experimental Lie Predicted Lie y = 5E-08x E E E E E+06 Fig 6. Fatigue Stren 2N gth coeicient Vs Number o cycles to ailure Fig 4. Strain amplitude Vs 2 2N Fig 7. Comparison o Lie 2N Prediction based Fig on 5. the Fatigue Lohr and Ductility Ellison coeicient Theory *Lohr and Ellison Vs model 2 N Equivalent strain* N

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