Materials and Design 31 (2010) 296–305 Contents lists available at ScienceDirect Materials and Design journal homepage: www.elsevier.com/locate/matdes Inﬂuence of continuum damage mechanics on the Bree’s diagram of a closed end tube A. Nayebi * Department of Mechanical Engineering, Shiraz University, Shiraz, Iran a r t i c l e i n f o Article history: Received 30 April 2009 Accepted 11 June 2009 Available online 13 June 2009 Keywords: Continuum damage mechanics Nonlinear kinematic hardening Bree’s diagram Shakedown Ratcheting Return mapping algorithm a b s t r a c t This paper extends the Bree’s cylinder behaviors, which is subjected to the constant internal pressure and cyclic temperature gradient loadings, with considering continuum damage mechanics coupled with nonlinear kinematic hardening model. The Bree’s biaxial stress model is modiﬁed using the uniﬁed damage and the Armstrong–Frederick nonlinear kinematic hardening models. With the help of the return mapping algorithm, the incremental plastic strain in axial and tangential directions is obtained. Continuum damage mechanics approach can be used to extend the Bree’s diagram to the damaging structures and reduce the plastic shakedown domain. Kinematic hardening behavior was considered in the material model which shifts the ratcheting zone. The role of the material constants in the Bree’s diagram is also discussed. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The need for a suitable constitutive model to predict the shakedown or cyclic failure (ratcheting) of structures under cyclic loading is increasing in many industries, and in order to better obtain the structures behaviors, many researchers tried to develop improved constitutive models . Although, the proposed models simulate well the uniaxial ratcheting responses, there are other factors that inﬂuence the biaxial stress cyclic loading. Shakedown loads and different behaviors of structures were also studied by many authors with the plasticity and cyclic plasticity models . In particular, the two-bar problem and the Bree’s cylinder were studied under different loading conditions and materials behaviors. Parkes  studied thermal ratcheting in an aircraft wing resulting from the cyclic thermal stresses superimposed on the normal wing loads. Miller  showed that the material strain hardening reduces considerably the strains due to ratcheting in the two-bar structure. Jiang and Leckie  presented a method for direct determination of the steady solutions in shakedown analysis with application to the two-bar problem. Bree  analyzed the elastic–plastic behavior of a thin cylindrical tube subjected to constant internal pressure and cyclic temperature gradient across the tube thickness. A simple one-dimensional model, a linear temperature drop distribution across the cylinder thickness and an elastic–perfectly-plastic material model were assumed in his analysis. Later, he used a biaxial stress model and obtained a more * Tel.: +98 711 6133029. E-mail address: [email protected] 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.06.014 complete interaction diagram (Bree’s diagram) for a closed tube . In that study the material was assumed to be perfect plastic. The Poisson’s effect was also neglected. The interaction diagram proposed by Bree received adequate investigation by a number of researchers [8,9]. His one-dimensional diagram is a part of ASME boiler and pressure vessel code  (Fig. 1). Mulcahy  improved the Bree’s analysis with incorporating a linear kinematic hardening model in the analysis of a beam element. Classical shakedown theory of Melan–Koiter for elastic– perfectly-plastic bodies has been well established in the literature . Melan procedure for the shakedown theorem can also be extended to encompass the linear kinematic hardening, but one encounters mathematical difﬁculty in treating more general cases, and the procedure does not apply straightforwardly for the case of Prager’s linear kinematic hardening. It seems that some additional assumptions as well as further mathematical tricks are needed to deal with the general kinematic hardening. Recently, Abdalla et al.  proposed a simple shakedown method with perfect plasticity material behavior to study the Bree’s cylinder problem and the 90° pipes bending. Very limited work has been done on damage related to the interaction diagram. The shakedown theory has been extended to include hardening and damage by Hachemi and Weichert  and Druyanov and Roman . However, the extensions should be made often at the expense of losing certain ﬁne features of the classical plasticity theory and shakedown theorems. Without the theorems in Melan–Koiter sense, which are valid only with certain restrictions, generally in practice one has to implement numerical incremental analysis to check for shakedown of a A. Nayebi / Materials and Design 31 (2010) 296–305 297 Nomenclature A ~ A C Dc E ~ E ez eh f FK1 K2 nij P p_ Dp q Q r Rt Sz Sh t T DT virgin surface resistant effective surface NLKH model’s constant interatomic decohesion damage parameter elastic modulus effective elastic modulus nondimensional axial strain nondimensional tangential strain yield function dissipative potential function mean tangential strain mean axial strain outward normal to the yield surface internal pressure equivalent plastic strain rate equivalent plastic strain increment material damage constant material damage constant mean radius triaxiality function nondimensional axial stress nondimensional tangential stress thin wall thickness temperature temperature difference between the inner and outer surfaces DTx x X X’ Y temperature difference in each radius coordinate back stress tensor deviatoric back stress tensor associate thermodynamics damage variable tensor expansion coefﬁcient NLKH model’s constant incremental plastic multiplier nondimensional thickness plastic strain tensor thermal strain tensor plastic damage threshold strain stress tensor effective stress tensor deviatoric stress tensor equivalent von-Mises stress hydrostatic stress nondimensional mechanical stress parameter yield stress nondimensional temperature difference nondimensional temperature difference increment Poisson’s ratio damage parameter a c dk g ep eT epD r ~ r r0 req rH rP ry s Ds t - structure under speciﬁc loading histories. Recently, Nayebi and El Abdi  and Kang et al.  used continuum damage mechanics (CDM) to predict the material behavior, including ratcheting and shakedown in 1-D analysis. In this research, the effect of continuum damage mechanics on the Bree’s diagram of a thin cylinder structure under speciﬁc loading histories is studied. Nonlinear kinematic hardening (NLKH) theory is coupled with continuum damage mechanics in order to model the behavior of the Bree’s cylinder. A uniﬁed damage mechanics model, which is also appropriate for low cyclic loading, is used. During each loading, the damage analysis is performed. An iterative method is used to analyze the cylinder under the cyclic thermal and constant mechanical loads. The model extends Bree’s 2-D diagram to incorporate damage effects. The proposed method can also be applied to other structures subjected to cyclic thermal and constant mechanical loadings. The results illustrate the inﬂuence of material damage on the behaviors of structures under cyclic loading in comparison with the conﬁrmed results on undamaged structures. material can then be represented by the constitutive equations of the virgin material where the usual stress tensor, r, is replaced by ~ deﬁned by the effective stress r 2. Constitutive behavior relations where k_ is calculated from the constitutive equations of plasticity coupled with the damage deduced from the dissipative potential _ , and function, F-. Y is the associate variable of the damage rate, epD is the plastic damage threshold strain. Also, many experimental results indicated that Fx must be a nonlinear function of Y  2.1. Continuum damage mechanics According to the applied theory of damage mechanics, microscopic change in a material element of surface A develops into macroscopic defect as a result of loading. In the damaged state, ~ (Fig. 2), from which the isotropic the new area is denoted by A damage variable - is deﬁned as  ~ AA -¼ ; A ð1Þ where - may be considered as an internal state variable characterizing the irreversible deterioration of a material in the thermodynamic sense. Following this theory, the behavior of a damaged ~¼ r r ; 1- ð2Þ where the value - = 0 corresponds to the undamaged state, 2 (0, Dc) corresponds to a partly damaged state, and - = Dc deﬁnes the element state rupture by interatomic decohesion (Dc 2 [0, 1]). In the sequel, superposed tilde indicates quantities related to the damaged state of the material. From a physical point of view, the material degradation involves the initiation, growth and coalescence of micro-cracks or microvoids generally induced by large plastic strains. This phenomenon is called ductile plastic damage and leads to plastic (ductile) fracture. Many observations and experiments indicated that the damage is also governed by the plastic strain which is introduced into _ as  the model through the plastic multiplier k, -_ ¼ k_ F- ¼ @F @Y if ep epD ; qþ1 Q Y ðq þ 1Þð1 -Þ Q ð3Þ ð4Þ from which Eq. (3) reduces to -_ ¼ q Y p_ Q ð5Þ where Q and q are material parameters and p_ is the equivalent plastic strain rate. According to Lemaitre and Desmorat  298 A. Nayebi / Materials and Design 31 (2010) 296–305 Fig. 1. Bree’s diagram for a 1-D tube model . 8 ~2 > < Y ¼ req2ERt where p_ is the accumulated plastic strain rate given by 2 ; > : Rt ¼ 2 ð1 þ tÞ þ 3ð1 2tÞ rH 3 req ð6Þ p_ ¼ 0 0 1/2 where Rt is the triaxiality function of stress, req = (3/2r :r ) is the von-Mises equivalent stress (r0ij ¼ rij rH dij and: indicates the inner product of two tensors), rH = 1/3tr(r) denotes the hydrostatic stress (dij is the Kronecker unit tensor), E is Young modulus, and t is the Poisson’s ratio. Using Eqs. (2) and (6), one can reduce the damage law (Eq. (5)) to the uniﬁed damage law for low cycle fatigue as  -_ ¼ r 2 eq Rt 2EQð1 -Þ2 !q 1 2 p p 2 e_ : e_ 3 So, the damage parameter is related to the equivalent plastic strain and is coupled with the plasticity. Assuming the von-Mises yield criteria, the yield surface can be rewritten by replacing the stress ~, as with the effective stress r f ð~ r; XÞ ¼ f _ p; ð7Þ ð8Þ r ;X ; 1- where X is the back stress. ð9Þ 299 A. Nayebi / Materials and Design 31 (2010) 296–305 where C and c are material parameters and Surface without voids and cracks dk ¼ ð1 -Þdp ð15Þ 3. Closed-tube biaxial-stress model based on CDM A Fig. 2. Deﬁnition of the surface (A) and the effective resistant surface (Ã). The studied pressure vessel was assumed to be a thin cylinder with the mean radius r, and the wall thickness, t. It is closed at both ends. The thin cylinder is subjected to an internal pressure P and a heat ﬂux through its internal surface (Fig. 3). The temperature difference decreases linearly across the wall thickness and is changed cyclically between DT and zero. It is assumed that the cylindrical shell is very long and the end effects and the curvature can be neglected. The axial and hoop strains, ez and eh, are spatially constant . This is because of the bending prevention; however they change during each cycle. The axial and hoop stresses, rz and rh, vary across the thickness and are only dependent on the coordinate x shown in the Fig. 3a. Using the equilibrium conditions, it is required that Z t 2 2.2. Nonlinear kinematic hardening Nonlinear kinematic hardening is introduced using the differential form of the governing equations for the kinematic variables. Based on the von-Mises yield criteria, the equation of the yield surface is written as f ¼ J 2 ð~ r XÞ ry ¼ 12 3 0 r0 X 0 Þ ry ¼ 0; ð~ r X 0 Þ : ð~ 2 ð10Þ where X is the back stress deﬁning the position of the yield surface and ry characterizes the size of the surface. The plastic ﬂow follows the normality rule dep ¼ dk @f 3 dk ð~ r0 X 0Þ : ¼ 1 - @r 2 1 - ð~ r0 X 0 Þeq ð11Þ The plastic multiplier dk is derived from the consistency condition, f = df = 0, if plastic ﬂow occurs. Different kinematic hardening models are available for the plastic analysis of structures. Sehitoglu et al.  pointed out that the material model was critical for the stress analysis of a damaged component. Many cyclic plasticity models were tested under different cyclic loadings [18–21]. A number of loading responses may be predicted by these models, while they fail to predict other types of cyclic loading conditions. Accordingly, it is difﬁcult to understand which case of loading should be simulated with which model. In this paper, the Armstrong–Frederick nonlinear kinematic hardening model  coupled with continuum damage mechanics is used to determine the structure behavior under cyclic loadings. If the plastic strain ðepij Þ and the back stress tensor (Xij) are assumed as the internal variables, the evolution equations are t 2 t 2 Z t 2 rh dx ¼ Pr; rz dx ¼ Pr 2 x DT x ¼ DT; t ð18Þ where DT is the temperature difference between inner and outer surface of the tube. The mean temperature in each cycle was assumed sufﬁciently low so that creep effects could be ignored. Fig. 3 demonstrates the loading steps. R ΔTout x t ΔTin (a) Thermal loading Mechanical loading depij ¼ dknij ; r~ 0ij X 0ij 3 1 ~ ij X ij k 2 1- kr where nij ¼ ¼ is the outward normal to the yield surface and dk is the incremental plastic multiplier calculated from the consistency condition @f @f @f drij þ dX ij þ d- ¼ 0; @ rij @X ij @- ð13Þ (b) (b) Nonlinear kinematic hardening model 2 Cð1 -Þdepij þ cX ij dk; 3 Loading ð12Þ @f @ rij dX ij ¼ ð17Þ Consequently, every element of the tube is subjected to the mean in axial direction and Prt in hoop direction. The temperature stress Pr 2t difference in start up half cycle is assumed to vary linearly with respect to x. It is zero in the thin cylinder mid-wall and at the shutdown second half cycle (a) Flow rule: df ¼ ð16Þ ð14Þ Time Fig. 3. (a) Tube geometry and applied temperature gradient across the tube wall thickness, t, (b) constant mechanical and cyclic temperature gradient loadings history. 300 A. Nayebi / Materials and Design 31 (2010) 296–305 Since bending is prevented in both hoop and axial direction, eh and ez are constants across the tube thickness. Therefore, the total strain in both directions and for every loading is constant Table 1 Materials models constants . E t ry c C Q q epD Dc eh ¼ K 1 ez ¼ K 2 134 GPa 0.3 85 MPa 250 5500 MPa 0.6 MPa 2 0.2 ey 0.2 ð19Þ ð20Þ Using the strain partition principle, the strains in hoop and axial direction have three parts (in small strain hypothesis): elastic, thermal and plastic strains. The total strains in two directions are: e h p h T h r~ h eh ¼ e þ e þ e ¼ ez ¼ eez þ eTz þ epz t r~ z p h þ aDT x þ e E E r~ z r~ h þ aDT x þ epz ¼ t E E ð21Þ ð22Þ where ee, eT and ep are elastic, thermal and plastic strains, respec~ z are effective tangential and longitudinal stresses ~ h and r tively. r according to the continuum damage mechanics. They are deﬁned in the Section 2.1. Eqs. (18)–(22) were solved to obtain hoop and axial stresses as E r~ h ¼ ðK 1 þ tK 2 ð1 þ tÞaDT x eph tepz Þ; 1 t2 E r~ z ¼ ðK 2 þ tK 1 ð1 þ tÞaDT x epz teph Þ 1 t2 ð23Þ where ( rp ¼ tPrry DT x s ¼ eyað1 tÞ ð32Þ Plastic strains can be determined from the constitutive relations that include the yield criterion, the normality rule, and the back stress model. The yield function for the biaxial stress using vonMises criterion is ~ h X h Þ2 þ ðr ~ z X z Þ2 ðr ~ h X h Þðr ~ z X z Þ r2y ¼ 0 f ¼ ðr ð33Þ By the normality rule, Eq. (11), the relation between hoop and axial plastic strain increments can be obtained as ð24Þ Substituting Eqs. (19) and (20) into Eqs. (21) and (22) and using Eqs. (16) and (17), we can determine the constants K1 and K2 as Rt 1 2t PrE þ t2 ð1 -ÞðaDT x þ eph Þdx 2 K1 ¼ ; R 2t t ð1 -Þdx 2 1 Rt t PrE þ t2 ð1 -ÞðaDT x þ epz Þdx 2 2 K2 ¼ : R 2t t ð1 -Þdx ð25Þ ð26Þ 2 In this part the following dimensionless parameters are presented: 8 ep ep > eph ¼ ehy ; epz ¼ ezy ; g ¼ xt ; > > < Sh ¼ rryh ; Sz ¼ rryz ; rp ¼ tEPrey > > > : k ¼ K 1 ; k ¼ K 2 ; e ¼ ry 1 2 y ey ey E ð27Þ Using these dimensionless parameters, we can simplify Eqs. (25) and (26). k1 ¼ 1 2t rp þ R 12 1 2 ð1 -Þ aDT x =ey þ eph dg R 12 ð1 -Þdg 1 R 12 t rp þ 1 ð1 -Þ aDT x =ey þ epz dg 2 2 k2 ¼ R 12 1 ð1 -Þdg ð28Þ 1 2 ð29Þ 2 Substituting the above relations for k1 and k2 into Eqs. (23) and (24), we can obtain the nondimensional stresses as a function of plastic strains as 0 r þ 1 B p ð1t2 Þ Sh ¼ ð1 -Þ@ hR 1 i ð1 -Þðeph þ tepz þ sÞdg R 12 1 ð1 -Þdg 2 2 1 2 1 ðep þ tepz Þ s ð1 t2 Þ h 0 hR 1 i p p 1 1 2 r p þ ð1t2 Þ 1 ð1 -Þðez þ teh þ sÞdg 2 B 2 Sz ¼ ð1 -Þ@ R 12 1 ð1 -Þdg 2 1 ðep þ teph Þ s ð1 t2 Þ z ð30Þ ð31Þ Fig. 4. Elastic shakedown behavior of the thin cylinder for smax = 1.976 and rp = 0.5, material models constants are given in Table 1. (a) variation of the maximum nondimensional tangential plastic strain, eph , at the outer surface of the thin cylinder as a function of the number of thermal loading cycles and (b) normalized equivalent stress versus normalized equivalent plastic strain at the outer surface. A. Nayebi / Materials and Design 31 (2010) 296–305 ~z r ~ h 2X z þ X h depz 2r ¼ ~h r ~ z 2X h þ X z deph 2r ð34Þ In order to obtain the variations of the plastic strain and stresses, Eqs. (7), (14), (30), (31), (33), and (34) are to be solved. 4. Numerical procedure In order to solve Eqs. (30), (31), (33), and (34), the return mapping algorithm RMA [23,24] was used. This method represents a well established integration scheme to integrate the rate constitutive equations. This method consists of an elastic trial and plastic corrector step. When the yield function is convex, i.e. fntrial > fn at time step n, the elastic trial step is employed to characterize the plastic loading/unloading state of the material using the algorithmic Kuhn–Tucker conditions fn 0; Dkn 0; Dkn f n ¼ 0: ð35Þ where Dkn is the increment of the plastic multiplier. Fig. 5. Plastic shakedown behavior of the thin cylinder for smax = 2.196 and rp = 0.5, material models constants and thin cylinder geometry are given in Table 1. (a) Variation of the maximum nondimensional tangential plastic strain, eph at the outer surface of the thin cylinder as a function of the number of thermal loading cycles and (b) normalized equivalent stress versus normalized equivalent plastic strain at the outer surface. 301 At each time step, the yield function is evaluated at the trial elastic step, in order to determine whether the yield occurs or not. If the trial yield function is less than zero, then the material is assumed to be elastic or plastic but under elastic unloading. Otherwise, the material is subjected to the plastic loading. The normality rule is used to deﬁne the plastic strain increment in the radial RMA. The yield condition is used to determine the incremental value of the plastic multiplier Dcn at the current time step n. Since a nonlinear kinematic hardening is assumed, an iterative procedure is required to determine the plastic strain at the current time step. Having determined Dcn, one can update the plastic strains and the hardening parameter, and the stress at the current time step is calculated using updated parameters. The increment of plastic damage parameter, D-n+1, is updated with the help of plastic increment computations. Total damage parameter can be obtained as: -n + D-n+1 = -Zn+1. The new value of the damage parameter is used to obtain the studied parameters for the new increment of loading. In order to obtain the trial elastic solution of the model, Eqs. (30) and (31) were modiﬁed to Fig. 6. Ratcheting behavior of the thin cylinder for smax = 2.928 and rp = 0.5, material models constants and thin cylinder geometry are given in Table 1. (a) Variation of the maximum nondimensional tangential plastic strain, eph , at the outer surface of the thin cylinder as a function of the number of thermal loading cycles and (b) normalized equivalent stress versus normalized equivalent plastic strain at the outer surface. 302 A. Nayebi / Materials and Design 31 (2010) 296–305 8 Bree’s biaxial results  Coupled damage – NLKH results 7 6 τ max 5 4 Ratcheting Domain 3 Plastic Shakedown 2 Shakedown in in and out surfaces Shakedown in out surface & Elastic in inner surafce 1 Elastic Domain 0 0 0.1 0.2 0.3 0.4 0.5 0.6 σp 0.7 0.8 0.9 1 1.1 1.2 Fig. 7. Interaction diagram for thermal load parameter, smax, versus internal pressure loading parameter, rp, solid line (—) shows the two-dimensional Bree’s diagram and the dashed line () presents the new results based on the continuum damage mechanics. 0 r þ 1 B p ð1t2 Þ STrial;nþ1 ¼ ð1 -n Þ@ h hR 1 2 1 2 i p;n ð1 -n Þðep;n h þ tez þ sn Þdg R 12 1 ð1 -n Þdg 2 1 1 ð1 -n Þ C ðep;n þ tep;n z Þ sn A þ R 1 ð1 t2 Þ h 2 ð1 - Þdg 1 2 Z 1 2 1 2 n ð1 -n ÞDsdg ð1 -n ÞDs 0 B ¼ ð1 -n Þ@ STrial;nþ1 z rp þ ð11t2 Þ hR 1 2 1 2 ð36Þ i p;n ð1 -n Þðep;n z þ teh þ sn Þdg 1 R2 1 ð1 -n Þdg 2 1 1 ð1 -n Þ C ðep;n þ tep;n h Þ sn A þ R 1 ð1 t2 Þ z 2 ð1 - Þdg 1 2 Z 1 2 1 2 ð1 -n ÞDsdg ð1 -n ÞDs With the help of the incremental form of Eqs. (14), (15), (30), (31), and (34) as below and using Newton–Raphson method for Eq. (33), the plastic strains increment can be determined p enþ1 ¼ eTrial nþ1 þ Denþ1 p p enþ1 ¼ epn þ Denþ1 ð38Þ 2 X nþ1 ¼ X n þ ð1 -nþ1 Þ C Depnþ1 cX nþ1 Dpnþ1 3 s Y nþ1 -nþ1 ¼ -n þ Dpnþ1 S 12 2 2 2 Dpnþ1 ¼ pﬃﬃﬃ Dezp;nþ1 þ Dehp;nþ1 þ Dezp;nþ1 Dehp;nþ1 3 ~ nþ1 ~ nþ1 Dep;nþ1 2r r 2X z þ X h z h z ¼ p;nþ1 nþ1 ~ ~ nþ1 2rh r 2X h þ X z Deh z ð39Þ ð40Þ ð41Þ ð42Þ ð43Þ n ð37Þ Ds is the increment of nondimensional temperature difference across the thickness. It is assumed that the maximum pressure is not greater than the yield pressure and elastic solution is only needed for pressure loading. The loading associated with the thermal gradient is cyclic and can be considered with the constant pressure loading. With these trial stresses, the yield function (Eq. (33)) is veriﬁed. If the yield criterion is violated, plastic solution is used. Table 2 Material models constants and applied loadings. I II III IV V VI c Q (MPa) q smax rp Fig. 250 250 250 250 50 50 0.06 0.6 0.06 0.6 0.6 0.6 2 0.5 2 0.5 2 2 2.928 2.928 2.196 2.196 2.196 2.928 0.5 0.5 0.5 0.5 0.5 0.5 8a 8b 8c 8d 9a 9b A. Nayebi / Materials and Design 31 (2010) 296–305 0 hR 1 0 hR 1 i p;nþ1 1 2 r þ tep;nþ1 þ snþ1 dg p þ ð1t2 Þ z 1 ð1 -nþ1 Þ eh B 2 Snþ1 ¼ ð1 -nþ1 Þ@ h R 12 1 ð1 -nþ1 Þdg 2 1 1 C snþ1 A ð44Þ ep;nþ1 þ tep;nþ1 z ð1 t2 Þ h r þ 1 B p ð1t2 Þ Snþ1 ¼ ð1 -nþ1 Þ@ z 2 1 2 i ð1 -nþ1 Þ ep;nþ1 þ tep;nþ1 þ snþ1 dg z h R 12 1 ð1 -nþ1 Þdg 2 1 1 C snþ1 A þ tep;nþ1 ep;nþ1 z h 2 ð1 t Þ ð45Þ 5. Results and discussion The constitutive model parameters for 2 14 CrMo steel at 580 °C were given by Lemaitre and Desmorat  and are shown in Table 1. Model constants are temperature dependent but the average values were chosen and it was assumed that the mean temperature is constant during loading and unloading. The mean radius, R, and thickness, t, are 100 mm and 10 mm, respectively. Constant mechanical and cyclic thermal loading were applied. Maximum inside pressure (mechanical loading) was not allowed to exceed the yield pressure. Starting from zero, an incremental thermal loading with a linearly varying temperature difference distribution across the wall thickness was applied. When the linear temperature gradient attained its maximum, it was reduced incre- 303 mentally to zero. At this point, a full thermal stress cycle is completed. The number of loading increments varies in different load cases in the test matrix, as each increment applies less than 1 °C in temperature gradient. Up to 100 load increments per cycle and up to 1000 cycles are applied in the most severe temperature gradients. In some cases, a steady cyclic state stress–strain plot is attained after the ﬁrst cycle. As an example, the amplitude of the nondimensional cyclic thermal gradient across the thin wall of the cylinder, smax, and the nondimensional constant mechanical stress, rp, were assumed to be 1.976 and 0.5, respectively. Fig. 4a shows the variation of the tangential plastic strain at the outer surface of the thin cylinder as a function of the number of cycles. Plastic strain stays constant during cyclic loading. Fig. 4b shows the variation of the equivalent stress as a function of equivalent plastic strain at the outer surface. The thin cylinder behavior is elastic after the ﬁrst cycle and elastic shakedown was obtained in the ﬁrst cycle. The amplitude of the nondimensional temperature gradient was increased to smax = 2.196 and nondimensional mechanical stress was not changed (rp = 0.5). Plastic shakedown was obtained (Fig. 5a and b). Fig. 5a shows that the nondimensional tangential plastic strain at the outer surface stays constant after 62nd cycle and the equivalent stress-equivalent plastic strain loop does not evolve (Fig. 5b). Finally, as it was shown in Fig. 6a, when the cyclic and constant loadings are smax = 2.928 and rp = 0.5, respectively, the plastic strain at the outer surface increases with a constant slope in each loading cycle. Fig. 6b shows that the stress–strain loop evolves and ratcheting phenomenon was resulted. Fig. 8. Variation of the maximum nondimensional tangential plastic strain, eph at the outer surface as a function of the number of thermal loading cycles for smax = 2.928 and rp = 0.5: (a) Q = 0.06 MPa (Table 2, case I), (b) q = 0.5 (Table 2, case II), and smax = 2.196 and rp = 0.5: (c) Q = 0.06 MPa (Table 2, case III) and (d) q = 0.5 (Table 2, case IV) (for each case, other material constants are given in Table 1). 304 A. Nayebi / Materials and Design 31 (2010) 296–305 Different results were obtained for the inner and outer surfaces of the thin cylinder, with changing the applied mechanical and thermal loading. In order to identify different regions in the Bree’s diagram, more than 100 combinations of mechanical and thermal loadings were simulated. The Bree’s diagram is divided into different regions as a function of the ﬁrst yield situation and the inner and outer surface of the thin cylinder behavior. These regions are shown in Fig. 7 which is known as the Bree’s diagram. In order to obtain the interaction diagram for Bree’s results , the inner pressure and the temperature gradient were normalized according to Eqs. (32). It should be noted that the Bree’s model was obtained using the Tresca’s criterion. In the present study, the von-Mises criterion was used. In order to be able to compare the results, the inner pressure in this study was normalized using the following relation: pﬃﬃ rp ¼ 23 tPrry . Region A in Fig. 7 corresponds to the elastic answer. The outer surface of the thin cylinder yields and elastic shakedown begins with increasing the temperature difference (region B). The inner surface remains elastic. In the third region C, the inner surface also yields and both surfaces are in elastic shakedown. When maximum thermal stress reaches 2ry, the plastic strain loops are formed and plastic shakedown is obtained (region D). For the thermal and mechanical stresses in the region D, the inner and outer surfaces have the same behavior and both are in plastic shakedown state. Finally, the ratcheting was obtained for two directions in the thin walled cylinder, in the region E. Continuum damage mechanics, which is dependent on the accumulated plastic strain, limits the plastic shakedown zone. The new boundaries were obtained by applying constant internal pressure while the temperature gradient was increased gradually to obtain each boundary between elastic, elastic shakedown, plastic shakedown, and ratcheting. As it was shown in the Bree’s diagram (Fig. 7), the two dimensional model of Bree can not predict the ratcheting because of the damage progress due to the accumulated plastic strain. But the 2ry limit of the shakedown behavior is independent of the models and the material behavior. Taking into consideration the nonlinear kinematic hardening behavior into the model, leads to the shift of the boundary between the plastic shakedown and ratcheting for greater internal pressures. It should be noted that other simulations have been conducted using different values of the damage model constants, Q and q. Although, the rate of ratcheting and the number of cycles before shakedown were changed, the same boundaries between different material behaviors were obtained. In the following simulations, one of the parameters, Q, q and c were changed in each example, and other material models parameters were as the same values in Table 1. Table 2 gives the used material models constants in these simulations and Figs. 8 and 9 show the variation of the nondimensional plastic strain as a function of the number of thermal loading cycles. For the ﬁst simulation, material damage model constant, Q, was decreased to 0.06 with respect to the simulations of Fig. 7. The cyclic thermal and constant mechanical loadings are smax = 2.928 and rp = 0.5, respectively (case I, Table 2). For the same loading, ratcheting behavior had been resulted for Q = 0.6 (see Fig. 6) and the same behavior was obtained for Q = 0.06 (Fig. 8a). In order to show the effect of the other material damage constant, q, on the thin cylinder behavior, the loadings were not changed and q was decreased to 0.5 (case II, Table 2). For this case, Fig. 8b shows that the behavior of the thin cylinder under theses loadings was not changed. Two other examples were considered for shakedown behavior. The loadings were changed to smax = 2.196 and rp = 0.5, and Q parameter was set to 0.06 (case III, Table 2). For the next simulation, the loadings were not changed and q parameter was diminished to 0.5 (case IV, Table 2). In both cases, plastic shakedown behavior was obtained (Fig. 8c and d). It was shown in Fig. 5 that the behavior was also plastic shakedown for the same loadings. The effect of the, c, constant of the kinematic hardening model was studied with changing it to 50. For two different above loadings (case V and VI, Table 2), the behavior of the thin cylinder was unchanged. As it was shown in Fig. 9a, number of cycles needed for plastic shakedown, was increased from 62 (Fig. 5a) to 300, and the rate of ratcheting was increased (Fig. 9b). 6. Conclusions Fig. 9. Variation of the maximum nondimensional tangential plastic strain, eph at the outer surface as a function of the thermal loading cycles when nonlinear kinematic hardening constant, c, was decreased to 50 and the loadings are (a) smax = 2.196 and rp = 0.5 (Table 2, case V), (b) smax = 2.928 and rp = 0.5 (Table 2, case VI) (for each case, other material constants are given in Table 1). The results of Kang et al.  showed that the coupling continuum damage mechanics and cyclic constitutive models, can improve the prediction of materials behavior in cyclic loading. Motivated by their results, the behavior of the Bree’s cylinder was studied with combining the uniﬁed continuum damage law and the nonlinear kinematic hardening model. The behavior of thin cylinder subjected to the constant internal pressure and the cyclic temperature gradient was studied. The effect of damage was considerable and the plastic shakedown domain obtained for low pri- A. Nayebi / Materials and Design 31 (2010) 296–305 mary stresses and cyclic temperature by Bree  was modiﬁed. Taking into consideration the hardening effect, which was neglected in Bree’s analysis, shifts the plastic shakedown boundary for higher internal pressure. The role of the material constants was also studied. It was shown that the variation of these constants did not change the behavior of materials, however, the rate of the accumulated plastic strain and cycles needed to obtain plastic shakedown were affected. It should be noted that because of the creep strains, the continuum damage mechanics of the creep mechanisms must also be considered in order to obtain more realistic boundaries in the interaction diagram. This will be the subject of a future research. Acknowledgments Discussions and advice of Professor M. Mahzoon (Shiraz University) and Dr. M. Dadfarnia (University of Illinois at Urbana Champaign) are appreciated. References  Hassan T, Lakhdar T, Krishna S. 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