ANALYSIS OF ULTRA LOW CYCLE FATIGUE PROBLEMS WITH THE BARCELONA PLASTIC DAMAGE MODEL

Transcription

1 XII International Conerence on Comutational Plasticity. Fundamentals and Alications COMPLAS XII E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) ANALYSIS OF ULTRA LOW CYCLE FATIGUE PROBLEMS WITH THE BARCELONA PLASTIC DAMAGE MODEL X. MARTINEZ *, S. OLLER *, L.G. BARBU * AND A.H. BARBAT * * International Center or Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña Camus Norte UPC, Barcelona, Sain x.martinez@uc.edu Deartament de Ciència i Enginyeria Nàutiques, UPC Deartament de Resistència de Materials i Estructures a l Enginyeria, UPC Key words: Ultra low cycle atigue, lastic damage, isotroic and kinematic hardening. Abstract. This aer resents a lastic ormulation based on the Barcelona lastic damage model ([1], []) caable o redicting the material ailure due to Ultra Low Cycle Fatigue. This is achieved taking into account the racture energy dissiated during the cyclic rocess. This aroach allows the simulation o ULCF in regular cyclic tests, but also in non-regular cases such as seismic loads. 1 INTRODUCTION Ultra Low Cycle Fatigue (ULCF) takes lace or a reduced number o cycles, usually less than 1000 and oten less than 100. Contrary to what occurs with high cycle atigue, in this ailure mechanism lasticity lays an imortant role. The most common rocedures used to simulate ULCF are those based on counting the number o cycles that can be alied to the material or a given lastic strain. Examles o those aroaches are the Coin-Manson rule [3], the Basquin rule [3], or the enhanced rule roosed by Xue in [4]. However, one o the main drawbacks o these ormulations is that they require o regular cycles to redict the material ailure, and oten this regularity does not exist. An examle o an ULCF ailure due to an irregular cyclic load is ound in the ailure o large diameter ies subjected to seismic loads, where the requency varies along time and each cycle may have dierent amlitude. Current work rooses the use o a lastic damage model to simulate Ultra Low Cycle Fatigue. The model develoed is based in the Barcelona model originally ormulated by Lubliner et al. [1]. Although this model was originally develoed or concrete, here is resented a new kinematic and isotroic hardening law seciically deined or the simulation o steel. One o the main characteristics o the model is that the hardening behaviour o the material is driven by the lastic energy dissiated. With this aroach, it is ossible to measure the energy required in each hysteresis cycle, as well as the available energy remaining on the material. Failure takes lace when all the lastic racture energy o the material has been dissiated. This aer shows that with the roosed model it is ossible to simulate an ULCF ailure

2 by calibrating the available lastic energy o the material. One o the main advantages o the roosed rocedure is that it is caable o redicting the material ailure indeendently o the number o cycles alied to the structure, as well as the amlitude and requency o those cycles. This is because the model is based on the actual loading history o the structure, being this loading history as random as needed. Thereore, it is ossible to simulate the material ailure roduced by ULCF in a seismic event, or on any other cyclic load scenario. Besides, the ormulation is caable o using any yield and otential suraces to characterize the material, which increases its alicability to dierent steel alloys. PLASTIC DAMAGE MODEL This work will not describe the comlete lastic damage model, as it can be obtained rom reerence [1]. Instead, it will ocus on the elements that dierentiate current model rom the Barcelona model roosed by Lubliner et al. The inelastic theory o lasticity can simulate the material behavior beyond the elastic range, taking into account the change in the strength o the material through the movement o the yield surace, isotroic and kinematic. It is assumed that each oint o the solid ollows a thermo-elasto-lastic constitutive law (stiness hardening/sotening) ([1], [5] and [6]) with the stress evolution deending on the ree strain variable and lastic internal variables. The yield surace is deined by a unction F that accounts or the residual strength o the material, which deends on the current stress state, the temerature and the lastic internal variables. This F unction has the ollowing orm, taking into account isotroic and kinematic lastic hardening (Bauschinger eect [7]), F ( S,, ) ( S ) K ( S,, ) 0 (1) where ( S ) is the uniaxial equivalent stress unctions deending o the current value o the stresses S, the kinematic lastic hardening internal variable, K ( S,, ) is the lastic strength threshold, is the lastic isotroic hardening internal variable, and is the temerature at current time t ([1], [5] and [6])..1 Kinematic Hardening Kinematic hardening accounts or a translation o the yield unction and allows the reresentation o the Bauschinger eect in the case o cyclic loading. A two dimensional reresentation o this movement in the lane S1 S is shown in the ollowing igure: Figure 1. Translation o the yield surace result o kinematic hardening

3 This translation is driven by the kinematic hardening internal variable which, in a general case, varies roortionally to the lastic strain o the material oint [7]. There are several laws that deine the evolution o this arameter. Current work uses a non-linear kinematic hardening law, which can be written as: P c E d () Where c k and d k are material constants, accumulative lastic strain, which can be comuted as: k k E is the lastic strain, and is the increment o /3 :. E Ekl. Isotroic Hardening Isotroic hardening rovides an exansion or a contraction o the yield surace. The exansion corresonds to hardening and the contraction to a sotening behavior. In the ollowing igure is deicted a two dimensional reresentation o this eect in the lane S : 1 S Figure. Exansion o the yield surace result o isotroic hardening The evolution o isotroic hardening is controlled by the evolution o the lastic hardening unction K, which is oten deined by an internal variable. The rate equation or these two unctions may be deined, resectively: K H k hk (3) H h k k : G h S where k denotes scalar and k states or a tensor unction. Deending on the unctions deined to characterize these two arameters dierent solid erormances can be obtained. 3 NEW ISOTROPIC HARDENING LAW Equation (3) allow the incororation o dierent hardening laws to describe the material erormance. In the Barcelona model deined in [1], the laws deined are driven by the racture energy o the material. This work resents a new law, secially develoed or steel materials, that has been designed to reroduce their hardening and sotening erormance k E 3

4 under monotonic and cyclic loading conditions. This law also deends on the racture energy o the material. 3.1 Fracture Energy Classical racture mechanics deines the racture energy o a material as the energy that has to be dissiated to oen a racture in a unitary area o the material. This energy is deined as: W G (4) A where W is the energy dissiated by the racture at the end o the rocess, and area o the surace ractured. The total racture energy dissiated, can be used to deine a racture energy by unit volume, mechanics ormulation: A is the W, in the racture rocess g, required in a continuum W G A g dv (5) V This last equation allows establishing the relation between the racture energy deined as a material roerty, G, and the maximum energy er unit volume: g W W G (6) V A l l Thus, the racture energy er unit volume is obtained as the racture energy o the material divided by the racture length. This racture length corresonds to the distance, erendicular to the racture area, in which this racture roagates. In a real section, this length tends to be ininitesimal. However, in a inite element simulation, in which continuum mechanics is alied to a discrete medium, this length corresonds to the smallest value in which the structure is discretized: the length reresented by a gauss oint. Threreore, in order to have a inite element ormulation consistent and mesh indeendent, it is necessary to deine the hardening law in unction o the racture energy er unit volume ([1], [6]). This value is obtained rom the racture energy o the material, G, and the size o the inite element in which the structure is discretized. 3. Hardening Function and Hardening Internal Variable The hardening unction deines the stress o the material when it is in the non-linear range. There are many ossible deinitions that can be used or this unction ulilling equation (3). Among them, here it is roosed to use a unction that describes the evolution o an equivalent uniaxial stress state, like the one shown in Figure 3. This equivalent stress state shown in Figure 3 has been deined to match the uniaxial stress evolution described by most metallic materials. This curve is divided in two dierent regions. The irst region is deined by curve itting rom a given set o equivalent stress-equivalent strain oints. The curve used to it the oints is a olynomial o any given order deined using 4

5 the least squares method. The data given to deine this region is exected to rovide an increasing unction, in order to obtain a good erormance o the ormulation when erorming cyclic analysis. Figure 3. Evolution o the equivalent lastic stress The second region is deined with an exonential unction to simulate sotening. The unction starts with a null sloe that becomes negative as the equivalent lastic strains increase. The exact geometry o this last region deends on the racture energy o the material. The hardening internal variable,, accounts or the evolution o the lastic hardening unction, K. In current ormulation is deined as a normalized scalar arameter that takes into account the amount o volumetric racture energy dissiated by the material in the actual strain-stress state. This is: t 1 S E dt g : (7) t0 In the ollowing igure is reresented, shaded in green, the volumetric racture energy required by a uniaxial material, or a given lastic strain E. The hardening internal variable deined in (7) is calculated normalizing this racture energy by the total racture energy o the system, g, which corresonds to the total area below the curve S eq ( E ), shaded with gray lines. Figure 4. Reresentation o the volumetric racture energy o a metallic material Using the deinition o the hardening internal variable deined in equation (7), it is ossible to deine the exression o the hardening unction as: 5

6 eq K S ( ) (8) It can be easily roven that the hardening unction and internal variable deined in equations (7) and (8) ulill the rate equations (3). And the h k and h k unctions deined in this exression become: eq S hk (9) S hk g t 3.3 Exressions o the hardening unction In this section are rovided the exact numerical exressions used to deine the new hardening law resented in this work. This law is characterized with two dierent unctions, each one deining the evolution o the equivalent stress in each region in which the equivalent stress erormance is divided (see Figure 3). Region 1: Curve itting with olynomial The irst region is characterized with a olynomial deined by curved itting rom a given exerimental data. Among the dierent available methods that can be used to deine this olynomial, here is roosed to use the least squares method due its simlicity, comutational cost, and good erormance rovided. The resultant relation between the stress and lastic strain in this region is: S N eq ( E ) a a i 0 i E (10) i1 with N the order o the olynomial. The volume racture energy that is dissiated in this region can be obtained calculating the eq area below the S E grah. This calculation rovides the ollowing value: N 1 i1 i i E E ai g t1 1 (11) i being E 1 and E the initial and inal lastic strain values, resectively, that delimit the olynomial unction region. Although the equivalent lastic stress should deend on the lastic internal variable, in a cyclic simulation with isotroic hardening this aroach will roduce hysteresis loos with increasing stress amlitude (or a ixed strain amlitude). For this reason, current ormulation calculates the equivalent lastic stress using the value o the equivalent lastic strain, which is calculated as: 6

7 S : E E (1) ( S) with (S) deined by the yield surace used to simulate the material, as it is shown in equation (1). Finally, the derivative o the hardening unction can be calculated with the ollowing exression: ds d ds de de d N i a eq eq i1 gt N 1 Exression (13) is valid or values o i1 a i1 i E E i i 1 1 (13) that are comrehended between 1 0 and gt 1 gt. The value o the uer limit o the internal variable shows that it is necessary deine a value or the volumetric racture energy o the material larger than g t1. I the value deined is lower, the material will not be able to reach its ultimate stress as this will imly having a racture internal variable larger than Region : Exonential sotening When the lastic internal variable reaches the volumetric lastic energy available in the irst region:. At this oint isotroic hardening is deined by region two, which unction is obtained with the ollowing arameters: 1. The initial equivalent stress value is deined by the equivalent stress reached in the eq irst region ( S ). This value can be the one deined in the material characterization or can be a lower value i there has been some lastic energy dissiation in a cyclic rocess. In this last case, the value stress value has to be obtained rom revious region.. The initial sloe o the unction is zero. 3. The volumetric racture energy dissiated in this region is the remaining energy in the material: gt gt gt1 With these considerations in mind, the resultant equation that relates the equivalent stress with the lastic strain is: eq eq be E be E S ( E ) S e e (14) eq 3 S where b gt The exression o the equivalent stress as a unction o the hardening variable is obtained combining equation (14) and (7), obtaining: eq eq S ) S (15) ( 7

8 ( ) b gt Being, 1 eq S And the derivative o the hardening unction is: eq ds d 1 b gt 1 (16) 4 PERFORMANCE OF THE FORMULATION In the ollowing are included the results obtained rom several simulations conducted to illustrate the erormance o the ormulation resented. These simulations can be divided in three dierent grous. The irst grou roves the ability o the ormulation to characterize the mechanical erormance o steel, under monotonic and cyclic loading conditions. The second set o simulations shows the erormance o the develoed ormulation when it is used to characterize Ultra Low Cycle Fatigue. Finally, the third set o simulations intends to demonstrate the advantages o the aroach roosed to simulate ULCF; this is done with the simulation o the steel resonse to a seismic-tye load. The main aim o all the simulations resented hereater is to show the resonse obtained with the constitutive equation develoed, and not to show the mechanical resonse o any articular structural element. Thereore, or the sake o simlicity, and to reduce the comutational cost o the simulations, all o them have been conducted on a single hexahedral inite element. The element is ixed in one o its aces and the load is alied to the oosite ace as an imosed dislacement. 4.1 Simulation o the mechanical erormance o steel under dierent loading conditions To rove the ability o the model to simulate monotonic and cyclic tests made on steel, in the ollowing are comared the results obtained rom the numerical model with the results obtained rom exerimental tests erormed by Universidade du Porto in the ramework o the ULCF roject. The tests were erormed to a X5 steel. The data used to deine the numerical model has been obtained adjusting the solution o the model to the results o the exerimental tests. To do so, it has been necessary to take into account that the eects o the kinematic and the isotroic hardening laws are couled. This imlies that the deinition o the irst region o isotroic hardening cannot be obtained rom the exerimental curve straightorward, as this curve does not take into account the dislacement o the yield surace due to the kinematic hardening law. The most relevant arameters o the model are described in the ollowing table. Table 1. Mechanical roerties o steel X5 Young Modulus MPa Poisson Modulus 0.30 Elastic Stress ( ) 70 MPa eq Y Plastic Strain Sotening ( E ) 7 % 8

9 C1 kinematic hardening MPa C kinematic hardening 450 Fracture Energy 8.0 MN m/m In the ollowing igures are included the stress-strain results obtained with the develoed ormulation or the material described. The green line corresonds to the numerical result and the red dots corresond to the exerimental values rovided by FEUP. Figure 5 shows the comarison made on a monotonic test. Figure 6 shows this comarison or two cyclic tests in which the samle is loaded with a ixed dislacement oscillating between a tensile and a comressive strain o 0.5% and 1.5%. Figure 5. Comarison o numerical vs. exerimental monotonic (a) and cyclic (b) test Figure 6. Comarison o numerical vs. exerimental cyclic tests or a strain o 0.5% (a) and 1.5% (b) Although the corresondence between the numerical and exerimental results obtained or the dierent simulations is not erect, it can be said that it is quite reasonable. Moreover i the discreancy between the exerimental tests is taken into account: while the numerical result is always the same, in Figure 5 and 6a the stress values seem over-redicted comared to the exerimental ones and, in Figure 6b they seem to be under-redicted. This indicates that the exerimental resonse obtained with the dierent secimens is not exactly the same, and that the numerical model rovides a solution that is ound between the limits o the exerimental tests. 4. Simulation o Ultra Low Cycle Fatigue Once having roved the ability o the model to characterize roerly a x5 steel, this 9

10 section shows the erormance o the ormulation i it is used to characterize an ULCF ailure. In the ollowing igures is shown the resonse rovided by the numerical model or a cyclic test. Figure 7a shows the stress-strain grah obtained or a simulation that is being loaded with 10 cycles. In this case all cycles ollow the same stress-strain ath. Figure 7b is reresented the stress-strain resonse o the material when the simulation is extended to 30 cycles. This igure shows a reduction in the stress rovided by the material or some cycles (the last ones). This stress reduction is consequence that the available hardening energy has been reached and that the material has started a sotening rocess. We can consider that ULCF starts when this sotening behavior starts. Figure 7. Resonse o the material during 10 cycles (a)-let, and during 30 cycles (b)-right With this aroach it is ossible to obtain the material erormance or dierent lastic strain amlitude cycles. The results obtained rom these simulations are shown in Figure 8. This igure shows that the resonse o the material to ULCF resents a logarithmic variation, as it is exected according to what is described in literature and reorted by several authors (such as Xue [1]). Figure 8. Number o cycles that can be alied to the material beore ULCF ailure 4.3 Advantages o the aroach roosed Previous results have shown that the roosed constitutive equation is caable o redicting material ailure ater alying several cycles to the material; it has shown also that the number o cycles deends on the lastic strain amlitude and, inally, that this is a logarithmic variation (as it was exected). However, these caabilities do not resent major advantages comared to other aroaches such as the Coin-Manson rule, or any other 10

11 analytical exression caable o deining the maximum number o cycles that can be alied or a given lastic strain. The main advantage o the roosed aroach is that the rediction o ULCF ailure does not deend on the lastic strain alied, but on the energy dissiated during the cyclic rocess. Thereore, it is ossible to vary the lastic strain in the cycles alied to the structure and the constitutive model will be still caable o redicting the material ailure. This is roved in the ollowing examle, where an irregular load, in requency and amlitude, is alied to the material (Figure 9a). This load will rovide the stress-strain resonse lotted Figure 9b. As it can be seen, the load alied roduces several loos, each one with a dierent lastic strain. Figure 9. Seismic-tye load alied (a) and material stress-strain resonse (b) The model is caable o caturing the energy dissiated in each one o these loos and, thereore, to evaluate the racture energy available in the material ater having alied the load. In the ollowing igure it is shown the resonse i a monotonic load is alied ater two reetitions o the load deicted in Figure 9a. This resonse is suerimosed with the resonse o a monotonic load. The result obtained shows that this two cycles have dissiated some energy and, thereore, the maximum strain reached by the model beore ailure is lower than the strain reached with the monotonic test. Figure 10. Resonse o the material ater alying two seismic-tye cycles It is also ossible to reeat several times the irregular load, shown in Figure 9a, to study the number o reetitions that are required to reach material ailure. Figure 11 shows the stress-strain resonse o the material ater twelve seismic-tye cycles. This grah shows that in the last six cycles the stress develoed by the material has been reduced, which allows to 11

12 conclude that the material can hold only six cycles o the load described. Figure 11. Resonse o the material ater twelve seismic-tye cycles 5 CONCLUSIONS This document has resented a constitutive model to characterize the mechanical erormance o steel, caable o taking into account the isotroic and kinematic hardening eects. The size o the yield surace is determined by the isotroic hardening law that deends on the amount o energy dissiated by the material. Preliminary results, made on the material, show that the simulation o steel with the roosed aroach is caable o caturing the eect o Ultra Low Cycle Fatigue in the material. Moreover, the model not only is caable o redicting the ailure or a seciic cyclic load, but it is also caable o redicting the material ailure or cyclic irregular loads. ACKNOWLEDGEMENTS This work has been suorted by the Research Fund or Coal and Steel through the ULCF roject (RFSR-CT ) and by the Euroean Research Council under the Advanced Grant: ERC-01-AdG COMP-DES-MAT "Advanced tools or comutational design o engineering materials" REFERENCES [1] J. Lubliner, J. Oliver, S. Oller and E. Oñate. A lastic-damage model or concrete. International Journal o Solids and Structures, 5(3): (1989) [] Lee, J. and Fenves, G. Plastic-Damage Model or Cyclic Loading o Concrete Structures. Journal o Engineering Mechanics, 14(8): (1998) [3] F.C. Cambell, Elements o Metallurgy and Engineering Alloys, ASM International, Ohio, USA (008) [4] L Xue, A uniied exression or low cycle atigue and extremely low cycle atigue and its imlication or monotonic loading, Int. J. Fatigue, 30, (008). [5] B. Luccioni, S. Oller and R. Danesi. Couled lastic-damage model. Comuter Methods in Alied Mechanics and Engineering, 19: (1996) [6] S. Oller. Fractura mecánica. Un enoque global. Centre Internacional de Mètodes Numèrics a l Enginyeria (CIMNE). Barcelona, Sain, 001. ISBN: [7] J. Lemaitre and J.-L. Chaboche. Mechanics o Solid Materials. Cambridge University Press. New York, USA,