Characterization of Material Parameters

Transcription

1 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Characterization of Material Parameters S. M. Humayun Kabir, Tae-In Yeo, Sang-Ho Kim Abstract The resent work is concerned with the characterization of hardening arameters for an elasto-lastic continuum model, taking into account the memory effect of lastic strain amlitude, in order to redict the hysteretic resonses of 429EM steel. This elasto-lastic three-dimensional model is based on the internal thermodynamic variables which comosed of the nonlinear kinematic hardening and isotroic hardening with the lastic strain memorization. The emhasis is ut on the determination of strain memory arameters along with other material arameters of the roosed model in order to better simulate the behavior of the material at different strain range. The material arameters are calibrated with the exerimental stabilized loos of stress-strain curves available in the literature. The redicted stabilized loos from the simulation with the determined arameters show good agreement with the exerimental results signifying the validity of the considered model. Index Terms Elasto-lastic continuum model, Material arameters, Nonlinear hardening law, Plastic strain memory, Stabilized hysteresis loos. I. INTRODUCTION Under cyclic loading, the structural materials show comlicated mechanical resonses involved with the lastic deformation at isothermal and anisothermal conditions. In the framework of elasto-lasticity, many constitutive models were established to describe these cyclic inelastic resonses of the materials. The concets, based on the internal thermodynamic variables for time-indeendent lasticity, have been studied under many different ways in order to generalize the classical isotroic and kinematic theories [1]. Based on the yield surface, Mroz [2] roosed the multiyield surface model and Dafalias & Poov [3] roosed a model with two surfaces only. Armstrong and Frederick [4] roosed the nonlinear kinematic rule in terms of differential equation which was develoed further by Chaboche [1] and Ohno and Wang [5].Various hardening rules including multi-surfaces, two surfaces with the stationary limit surface and non-linear surface were reviewed by Chaboche [1]. Some significant modifications on kinematic hardening were done by Chaboche [6] and Dafalias et al.[7] concerned with the time indeendent lasticity theories in the range of cyclic loading. Furthermore, Valanis [8] roosed the lasticity Manuscrit received November 24, 28. S. M. H. Kabir is with the School of Mechanical and Automotive Engg., University of Ulsan, P.O Box18 Ulsan , Reublic of Korea, (hone: ; fax: ; dalimuou@yahoo.com). Tae-In Yeo is with the School of Mechanical and Automotive Engg., University of Ulsan, P.O Box18 Ulsan , Reublic of Korea. ( ytn@mail.ulsan.ac.kr) S. Kim is with the School of Mechanical and Automotive Engg., University of Ulsan, P.O Box18 Ulsan , Reublic of Korea. ( nannom18@sjku.co.kr) theory without the concet of yield surface based on endochronic theory. Iwan [9] and Besseling [1] roosed the overlay model based on an aroach which views the system as consisting of a series of ideal elasto-lastic element. From the subject oint of view i.e., at describing the cyclic elasto-lastic behavior of materials, all these models are said to be meaningful and reresentative examles. Various alloy steels are facilitated in a variety of engineering structural alications such as automotive structure, ressure vessels, and so on. It is ossible that the structural comonents made from these alloy steels are subjected to cyclic loading. The 429EM ferritic stainless steel is generally a good selection in the exhaust systems of the automobile engines as well as many high temerature-structures due to its excellent corrosion resistance and enhanced thermal fatigue resistance. So the material arameters and mechanical roerties of 429EM steel in elasto-lastic cyclic behavior have been the object of many studies during recent years on life rediction of high-temerature structures. Many real materials usually exhibit cyclic hardening or softening which deends, in general, not only on the number of cycles but also on strain amlitude. It has been observed that some alloy steels resent a significant strain range-deendent cyclic hardening under strain-controlled cyclic loading in different exerimental studies [1], [11]-[13]. Chaboche et al. [14] roosed first the strain amlitude deendence of cyclic hardening in the constitutive model to describe the cyclic hardening behavior of SS316 stainless steel under varied strain amlitude. Then, Ohno [15] extended this concet by introducing a cyclic non-hardening region, inside which the cyclic hardening does not takes lace, to describe the deendence of cyclic hardening on the strain amlitude. It consists of an index surface in the sace of lastic strain with a hardening variable that memorize the maximum lastic strain amlitude exerienced by the material. Therefore, this strain amlitude deendency of cyclic hardening should be considered by the constitutive model used for analyzing structural comonents subjected to cyclic loading. Exerimental studies of the 429EM steel in [12] and [13] revealed a comlex behavior under elasto-lastic cyclic loading. In addition to the classical Bauschinger effect and cyclic hardening, a memory effect of the lastic strain amlitude was observed. Yoon et al. [12] and Yoon [13] have studied the low cycle fatigue tests of 429EM steel at different temeratures and roosed a model, based on the overlay model, that has the ability of describing the change of the stress amlitude and the strain range deendence in hysteresis loos. And they determined the set of arameters for their roosed constitutive model. The objective of this work is to roose a ISBN: WCE 29

2 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. three-dimensional elasto-lastic model that can describe the lastic resonse of 429EM steel under cyclic loading conditions. The emhasis is ut on identifying the comlete material arameters including the arameters of the lastic strain memorization in order to better simulate the stabilized elasto-lastic cyclic behavior of the 429EM steel at different strain range. This elasto-lastic law is based on the internal thermodynamic variables and takes into account the combined nonlinear kinematic hardening and isotroic hardening with lastic strain memory effect of Chaboche [1].The material arameters in terms of this Chaboche model are calibrated utilizing the available exerimental stress-strain curves obtained from [12] and [13]. On the basis of the exerimental results, a cyclic hardening is observed. The calibration of material arameters in this aer has been carried out under strain-controlled cyclic loading. The aer is organized as follows. Section II concisely describes the adoted constitutive equations including memory effect of lastic strain amlitude. Section III contains a resentation of integration rocedure in brief. Section IV is devoted to the strategy of material arameters determination concerning the elasto-lastic cyclic behavior of 429EM steel. Section V describes the simulation of the adoted constitutive model utilizing the determined arameters and comarison of analysis results with those of the exeriment. Finally the closing remarks are resented in Section VI. II. CONSTITUTIVE EQUATIONS The constitutive equations dealt with the time-indeendent elasto-lastic behavior of structures subjected to cyclic loading must take into account the comlex henomena of Bauschinger effect, cyclic hardening and strain memory effect. The constitutive equations adoted here is commonly called Chaboche model which combines the nonlinear kinematic and isotroic hardenings with memory effect [1].The general exression of this model is the following: (every italicized bold variable indicates a tensor, for examle, σ reresents stress tensor. This is the convention to be adoted throughout the aer.) A. Decomosition of Strain e ε ε ε (1) where, ε is the total strain tensor, ε is the elastic strain tensor and ε is the lastic strain tensor. B. Associated Flow Rule With Yield Criterion e the consistency condition f df when the lastic flow occurs, ( f / σ) gives the direction of the increment of lastic strain tensor, x is the kinematic hardening tensor called back stress tensor, R is the isotroic internal stress or drag stress, k is a temerature deendent material arameter which reresents the initial size of the elastic domain, and J ( σ -x) is defined by Von Mises criterion as follows, J ( σ -x) (3/ 2)( σ -x ) : ( σ-x) (4) where, σ and x are the deviatoric art of the stress tensor and back stress tensor resectively. The tensorial oeration : on two second order tensors A and B imlies the following, A B (5) : Aij Bij C. The Non-linear Kinematic Hardening The evolution of the back stress ( x ) in kinematic hardening is based on Prager s linear hardening law and a recall term which can be written in its simlest form as, dx (3/ 2) Cdε xd (6) where, is the accumulated lastic strain. C and are material arameters describing the kinematic hardening. This modification of Prager s rule initially roosed by Armstrong and Frederick [4] to take into account the non-linear evolution of the back stress. D. Plastic Strain Memory and Isotroic Hardening To exress the deendence between the saturated value of the isotroic internal stress and the maximum lastic strain amlitude, a model has been roosed first in [14]. The general formulation consists of having an non-hardening index surface in the sace of lastic strain. The evolution of this enveloing surface is described by, F (2/3) J ( ε ξ) q where, q and ξ are the radius and the center of this non-hardening surface. The change in the memory state takes lace only if F and ( F / ε ): dε. The evolution rule for q and ξ can be defined by the following two equations, (7) dε d ( f / σ) with Von Mises yield criteria, (2) dq H ( F ) n: n d (8) dξ (1 ) H( F) n: n 3 / 2 n d (9) f J( σ -x) Rk (3) where, f is the yield function, d is the lastic multilier which is derived from the hardening rule through where, H ( F ) is a Heaviside function and is a material arameter regarding the lastic strain memory, introduced by Ohno [15] in order to allow gradual effect into memorization. In (8) and (9), n and n are the unit outward normals to the ISBN: WCE 29

3 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. load surface ( f ) and to the memory surface ( F ) resectively which are defined as follows, n 3 ( σ-x) 3 ( ε - ξ) and n 2 ( -x J σ ) 2 J ( ε - ξ) The isotroic hardening law is then modified to take into account the evolution of q. The evolution of the drag stress R in (3) can be written as, dr b( Q R)d (11) where, Q is the asymtotic value of the isotroic hardening variable R and b is the material arameter which describes the raidity of the isotroic hardening. The following relation reveals the deendence between the asymtotic value Q of the isotroic hardening variable and the size q of the non-hardening memory surface, dq 2 ( Qs Q) dq In the case of uniaxial (tension-comression) loading with the constant lastic strain amlitude, q leads to the amlitude of the lastic strain, i.e., max /2 max (1) (12) q (13) Then integrating (12), the saturation value of the isotroic hardening becomes, max /2) ( Qq ( ) Q( Qs ( Q Qs) e where, reresents the maximum lastic strain range and Q s, Q, and are the four material arameters regarding memorization of the lastic strain. There are nine material arameters E, k, C,, b,, Q s, Q, and which need to be determined using the roosed model for 429EM steel. We determine these arameters in section IV in such a way so that they minimize the error between the exerimental curves and numerical curves deduced from the model, which is the rime objective of this aer. max III. INTEGRATION OF THE CONSTITUTIVE EQUATIONS In this section, numerical integration rocedure for the Chaboche model is resented. In contrast to linear elastic roblems in which there exists a unique relationshi between stress and elastic strain, no such uniqueness holds for lasticity roblems due to non-linear nature. An incremental aroach is therefore almost always necessary to solve the cyclic lasticity equations numerically for caturing the history deendence inelastic behavior of material. The imlicit Backward Euler algorithm is favored by many researchers, such as Ortiz and Poov [16], Chaboche and Cailletaud [17], for large increments because of its stability ) (14) and accuracy characteristics. An imlicit algorithm for the combined non-linear kinematic/isotroic hardening has also been roosed by Doghri [18] and modified by Mahnken [19], whereby discretized rate equation reduced to one-dimensional roblem. In this imlicit scheme only a lastic multilier (equivalently, accumulated lastic strain for von-mises material) aears to be unknown. We aly the imlicit integration scheme in a strain-driven aroach to the roosed model in a similar fashion described in [19]. We emloy this integration scheme due to two reasons, a). the resulting relations for linearization of the constitutive equations are obtained in a straightforward manner which avoids the inversion of second order tensors, and b). the resulting roblem is reduced to one-dimensional roblem which gives the oortunity to combine the Newton-Rhason method with different one dimensional solution scheme, such as the Bisection method or the Pegasus method, for raid convergence. We decomose the external loading in iterative rocedure in order to check the yield criteria at each ste and to follow correctly the hardening rule. The resulting discretized equations for the constitutive equations mentioned in section II are summarized as follows, n1 n1 n1 n f J( σtr x) n1 n1 with (3 G C) Rk n 1 n e n e tr G K Kb n1 (15) σ 2 ( ε ε) ( ) I( ε ε): I (16) 1/(1 ) (17) where, the index n 1 reresents the current time ste and the symbol stands for the increment, for examle, defines the increment of the accumulated lastic strain. G and K are the lame constants. K b is the elastic bulk modulus and is the unknown variable at the current time ste which is solved in the iterative rocedure. We combine the Bisection method with the Newton-Rhason iteration for better convergence in the iterative rocedure. IV. ON THE PARAMETERS IDENTIFICATION The resent section is concerned with the determination of arameters in terms of the Chaboche model with the lastic strain memorization for modeling the cyclic behavior of 429EM steel described in [12]. The strengths of an advanced lasticity model might be undermined if the model arameters are not calibrated well for the exerimental resonses of the material. Therefore, in this aer we emhasize on the calibration of the arameters that can simulate the actual (exerimental) hysteresis curve firmly well at different strain amlitude. On the calibration of material arameters, the cyclic test data are obtained from [12] and [13] where a series of the strain-controlled low cycle fatigue tests had been erformed on 429EM steel for several strain amlitude.3% ( / 2).7%. In this aer, the calibration of the arameters is undertaken at elevated temerature of 2 C. In order to identify the material ISBN: WCE 29

4 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. arameters in terms of the roosed model, we adot the following rocedure. Considering the fact that low cycle fatigue failure occurs usually after several hundreds of load cycles, the arameters are calibrated using the stabilized loos. Fig.1 shows the exerimental stabilized hysteresis loo ( ) for different strain amlitude obtained from [12]. The Young s modulus E is derived directly from the linear art of the stabilized hysteresis loo. Fig.2 shows the exerimental stabilized hysteresis loos in transosed stress versus transosed strain lot obtained from Fig.1 where each of the hysteresis loos translated to the lower eak. From Fig.2 we can stated that the suerosition of the stabilized stress-strain loos (tensile branch) is imossible and the cyclic curve is different from that redicted by the Masing s rule. Fig.3 describes the translation of hysteresis loos for suerosing the uer branches of the all stabilized hysteresis loos. Isotroic hardening is a henomenon in the rogressive behavior of cycle by cycle, but for a single cycle it can be considered constant. Therefore, for the stabilized loo isotroic hardening will be taken constant. Taking this fact into account, the differences in the translational values in Fig.3 rovide us the differences in the asymtotic values of twice the isotroic hardening variable R. After measuring the differences in the saturated values ( Q ), we define the function Q( max ) of (14) and its coefficients Q s, Q, and. In Fig.4, the exerimental data are comared to the comuted ones after identifying the arameters of the lastic strain memorization. After estimating the saturated values Q of the isotroic hardening variable R for the strain amlitude ointed out in Fig.3, we deduct these values from the stress range which leads to the ure kinematic effect. Choosing the initial size of the elastic domain, k is evaluated [2], and then the exerimental kinematic hardening ( x ) is easily extracted from the lastic resonse of the stabilized loo. Utilizing this stabilized hysteresis data with the built-in calibration rocedure of the ABAQUS code [21], C and are determined rimarily. And the values of C and are further calibrated to fit well with the hysteresis loos for the uniaxial cyclic loading. Stress, (MPa) Strain, Fig. 2. Hysteresis loos adjusted to the lower eak. Stress, MPa Strain, (%) Fig. 3. Translation of hysteresis loos defining the asymtotic value of isotroic hardening variable. Q (MPa) Model curve, [Q=Q s +(Q -Q s )ex(- )] Exerimental data Stress, (MPa) Strain, (%) Fig. 1. Exerimental stabilized hysteresis loos obtained from [12] Plastic strain amlitude, Fig. 4. Identification of arameters for lastic strain memorization. Then the coefficient b of (11), the ace of the isotroic hardening, is calculated from the evolution of isotroic hardening variable R. Fig.5 illustrates the relationshi between the exerimental stress amlitude and accumulated lastic strain ( ( /2)- ) for different strain amlitude obtained from [13]. The evolution of R in (11) is related to the stress amlitude during cyclic test as follows [1], ISBN: WCE 29

5 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. ( ) /( s ) R/ Q 1ex( b) (18) where, s and are the stress ranges for the stabilize cycle and the first cycle resectively and is the stress range in between. Utilizing Fig.5, we lot the total history data of (18) which is shown in Fig.6. From the history lot in Fig.6, it reveals that the ace of isotroic hardening ( b ) deends only on the accumulated lastic strain, indeendent of secific strain amlitude. In Fig.5 and Fig.6, we show only the strain amlitude of /2.3% and /2.7% but the above observation is also true for other strain amlitudes in between. Table. I gives the different material arameters which are determined. The unit of material arameters k, C, Q s, Q is MPa, the unit of E is GPa, and all others are dimensionless. Stress,MPa) Strain, Symbols : Exerimental data Solid lines : Simulation results Fig.7. Exerimental and uniaxial simulation resonses (stabilized loo). V. SIMULATION OF THE CONSTITUTIVE EQUATIONS AND COMPARISON OF RESULTS The model figured out from the uniaxial simulation is adequate for a reroduction of the real three dimensional behavior of the material. In addition to the simlicity of analysis, uniaxial simulation allows determining and calibrating the model arameters straightforwardly Stress,MPa) Stress amlitude, (MPa) Accumulated lastic strain, Fig. 5. Variation of stress amlitude with accumulated lastic strain obtained from [13]. (- )/( s - ).8.4 (- )/( s - ) = 1-e Strain, (%) Exeriment FE simulation Fig 8. Stabilized loos - a comarison between exerimental results and FE simulation at /2.7%. with minimum memory required for analysis. Fig.7 shows, to comarison urose, the exerimental and comuted stabilized resonses when the whole loading history is considered in the uniaxial simulation. This comarison evidences that the simulation results with the identified material arameters redict well the exerimental stabilized loos for all the strain amlitudes excet at the strain amlitude of.4% but the shae of the loo remains unchanged. This haens because the identified asymtotic value Q for the strain amlitude of.4% overestimate the exerimental ones as shown in Fig.4. However, the analysis result for the strain amlitude of.4% at isothermal conditions would be accetable from the viewoint of safety in design. In the uniaxial simulation, we assign the value.1 to take into account the gradual effect into the memorization. The higher the value of, the higher the rate of raidity in stabilization of stress Accumulated lastic strain, Fig. 6. Variation of hardening with accumulated lastic strain. Table I. Material arameters for 429EM steel at 2 C. E k C b Q s Q ISBN: WCE 29

6 Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. This three-dimensional model is introduced into a finite element (FE) rogram ABAQUS through user-defined material subroutine called UMAT [21] with the determined arameters. For numerical simulation, an axisymmetric version of the cylindrical samle (the cylindrical samle that described in [12]) is emloyed. In FE simulation, only one finite element, in the middle of the samle, is submitted to an imosed strain / 2.7% and subsequently analysis is carried out. The stabilized cyclic resonse of the material is calculated by emloying a 3-D 8-noded isoarametric brick element with full-integrated formulation (C3D8 element in ABAQUS). Fig.8 gives a comarison of stabilized loos ( ) between exerimental data and FE simulation for the strain range / 2.7%. For all the discussed simulation, very good correlation is obtained between the resonses simulated using the determined arameters and the exerimental observations. Comarisons reveal that the obtained arameters of the roosed model for describing the inelastic behavior of 429EM steel aroach as well as can be exected, those in the exerimental curves. These arameters are said to be stationary because a small variation of arameters does not have significant influence on the stabilized resonse of the material. VI. CONCLUSION The roosed model with lastic strain memorization for describing the stabilized cyclic resonse of 429EM steel is verified by using the available test results. The model is tested through the uniaxial simulation and FE simulation utilizing the determined material arameters. The comuted resonses agree reasonably well with the exerimental results. The aim of the study is the characterization of hardening arameters for an elasto-lastic continuum model, taking into account the memorization effect of lastic strain amlitude. The use of continuum mechanics constitutive models into engineering alication encounter the difficulties to find references about the material arameters obtained by exerimental data. Therefore, unveiling the material arameters to find the stabilized hysteresis resonse of the steel in the case of elasto-lastic cyclic loading, is an imerative ste regarding fatigue life studies. The determined material arameters are the elemental to extend the results into the continuum damage model. Therefore, couling the adoted model with a damage law to redict the life of the selected material is what we shall do in our future work. nonisothermal conditions, Int. J. Plasticity, vol. 7, 1991, [6] J. L. Chaboche, On some modifications of kinematic hardening to imrove the descrition of ratcheting effects, Int. J. 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Chaboche, K. Dang Van and G. Cordier, Modelization of the Strain Memory effect on the cyclic Hardening of 316 Stainless Steel, Transactions of SMIRT 5, vol. L, 1979, Paer No. L11/3. [15] N. Ohno, A constitutive model of cyclic lasticity with a non-hardening strain region, J. Al. Mech., vol. 49, 1982, [16] M. Ortiz and E. P. Poov, Accuracy and stability of integration algorithm for elastolastic constitutive equations, Int. J. Numer. Meth. Eng., vol. 21, 1985, [17] J. L. Chaboche and G. Cailletaud, Integration methods for comlex lastic constitutive equations, Com. Meth. Al. Mech. Engng., vol. 133, 1996, [18] I. Doghri, Fully imlicit integration and consistent tangent modulus in elasto-lasticity, Int. J. Numer. Meth. Eng. vol. 36, 1993, [19] R. Mahnken, Imroved imlementation of an algorithm for nonlinear isotroic kinematic hardening, Commun. Numer. Meth. Engng., vol. 15, 1999, [2] S. M. H. Kabir, S. Kim and T. Yeo, Unified material arameters for 429EM stainless steel at Elevated temerature in terms of Chaboche model and Modified Overlay model, Proceedings of the 4th BSME-ASME Int. Conf. Thermal Engineering, 28, [21] Hibbitt, Karlsson, Sorensen, ABAQUS version 6.6, 26. REFERENCES [1] J. L. Chaboche, Time-indeendent constitutive theories for cyclic lasticity, Int. J. Plasticity, vol. 2(2), 1986, [2] Z. Mroz, On the descrition of anisotroic work hardening, J. Mech. Phys. Solids, vol 15, 1967, [3] Y. F. Dafalias, and E.P. Poov, Plastic internal variables formalism of cyclic lasticity, J. Al. Mech., vol. 43, 1976, [4] P. J. Armstrong and C.O. Frederick, A mathematical reresentation of the multiaxial Bauschinger effect, G.E.G.B. Reort RD/B/N, 1966, [5] N. Ohno and J. D. Wang, Transformation of a nonlinear kinematic hardening rule to a multi-surface form under isothermal and ISBN: WCE 29