晶体塑性

fcc lic si F. R nd U ed fo line 1 models such as [2–5] the flow rule is based on the Orowan equation and the hardening rule is based on the evolution of dislocation densities. More and more studies have shown, that it is important to optimize rate hardening behavior for metal forming applica- structures or subgrains. As the deformation increases especially for polyslip the microstructure of the single crystal can be characterized by cell blocks (CBs) sep- arated by dense dislocation walls (DDWs). Further- more these cell blocks consist of ordinary equiaxed dislocation cells [6]. At somewhat higher strain sections of the DDWs 3–361 * Corresponding author. Tel.: +49-211-6792-393; fax: +49-211-6792- The constitutive laws for single crystals are very important as they form the basis for the understand- ing of poly crystal deformation. For the flow rule of the single crystal the continuum slip theory has been well constructed and accepted commonly [1]. Although in the majority of FEM studies the flow rule is still described by a simple power law and the hardening behavior of commercial alloys is represented in terms of another power law of strain and strain rate em- pirically, there are more and more authors beginning to formulate constitutive relations based on disloca- tion density theory. For most of these more physical tions to forecast the occurrence of plastic flow local- ization and fracture phenomena [5]. Such optimized hardening laws should depend on microstructural state variables rather than macroscopic quantities such as accumulated plastic strain and strain rate. For materials with high stacking fault energy such as aluminium alloys, the plastic deformation is mainly caused by dislocation glide along well defined slip systems. When the deformation is very small the mi- crostructure of the single crystal consists of tangled dislocations. After a relatively small amount of plastic deformation the microstructure will change to cell A new dislocation density based constitutive model for fcc single crystals at elevated temperatures is developed. In addition to former composite models [Acta Mater 48 (2000) 4181; J Mater Proc Technol 123 (2002) 155; Acta Metall Mater 41 (1993) 589; Acta Metal 31 (1983) 1367] it distinguishes the individual slip systems and thus directly accounts for latent hardening. This distinction is an essential prerequisite for a later implementation into crystal plasticity finite element models. The model is applied to hot compression tests of aluminium single crystals with their h110i axis parallel to the compression axis. The predicted stress strain curves fit the experimental observations very well for a strain rate range from 1� 10�5 to 1� 10�1 s�1 and a temperature range from 623 to 723 K using a single set of parameters. � 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dislocation density; Internal variables; Constitutive equation; Aluminium; Single crystal; Slip system; Crystal plasticity FEM 1. Introduction the prediction of material strain hardening and strain A constitutive model for dislocation densities and its app of aluminium A. Ma, Max Planck Institut f€ur Eisenforschung, Mikrostrukturphysik u Received 25 March 2004; received in revis Available on Abstract Acta Materialia 52 (2004) 360 333. E-mail address: roters@mpie.de (F. Roters). 1359-6454/$30.00 � 2004 Acta Materialia Inc. Published by Elsevier Ltd. A doi:10.1016/j.actamat.2004.04.012 single crystals based on ation to uniaxial compression ngle crystals oters * mformtechnik, Max-Planck-Str. 1, 40237 D€usseldorf, Germany rm 14 April 2004; accepted 16 April 2004 0 May 2004 2 www.actamat-journals.com tend to degenerate to become first-generation micro- ll rights reserved. ateri bands (MB1s). The MB1s are three-dimensional walls consisting of small pancake-shaped subgrains. In Lef- fers’s opinion [7] it appears, that the basic wall character of a DDW is unaffected by its partial transformation to pieces of MB1s. Therefore, no distinction can be made between DDWs and MB1s, when one considers the microstructure for the single crystal. In many experi- ments the DDWs are found approximately parallel to the transverse direction (TD) of the rolled sheet, forming a certain angle with the rolling direction (RD). In Bay’s experiments [8] at higher strains a second generation of DDWs was observed intersecting the first ones in some grains. For Bay et al. [8] two intersecting families of DDWs/MB1s are the normal situation at higher strains and the small ordinary equiaxed dislocation cell groups are divided into big cell blocks. Meanwhile Leffers [7,9] only considers one predominant family of walls for the reason of simplicity and the cell blocks are replaced by band structures. From the experimental results a single crystal should be subdivided step by step if one wants to simulate its physical reactions and texture evolution. In fact for the FEM calculation this is not a wise choice because it will render the calculation more and more time consuming with increasing deformation. Starting from the contin- uum mechanics view proper internal variables have to be defined, which can reflect the microstructure evolution (the ordinary equiaxed dislocation cells, CBs, DDWs) collectively. The ordinary equiaxed dislocation cell is composed of cell walls with high dislocation density and enclosed cell interiors with low dislocation density. A group of these ordinary equiaxed dislocation cells, which have almost the same crystallographic orientation, constructs one cell block surrounded by those dense dislocation walls. In their experiments Zehetbauer and Seumer [10] find that the transformation from ordinary equiaxed dislo- cation walls to subgrain boundaries or DDWs is the result of the onset of dislocation climb. So for the reason of simplicity we assume the physical characteristics of cell walls and DDWs are the same. In the present work, we treat one cell block as a perfect single crystal and the microstructure has only two components: CBs and DDWs which therefore have a different physical mean- ing as compared with real structures observed in the experiment [8], the CBs do not contain any walls and the DDWs replace the experimental DDWs and cell walls altogether. According to experiments for copper the size of the subgrains will decrease as the strain increases at the beginning of deformation and saturate toward a value of about one micrometer after a not very large strain [11,12]. Analogous studies of the sizes of the CBs and DDWs for FCC metals show, that both shrink with increasing deformation, and furthermore the ratio of the 3604 A. Ma, F. Roters / Acta M two values can be regarded as almost constant during the deformation process [13]. Based on these results in this study we assume, that the volume fractions of CBs and DDWs are fixed. In this study, we will adopt Roters et al. [3] and Karhausen and Roters [4] idea, namely, that the im- mobile dislocations in the CBs are those which have been locked by other ones with nonparallel Burgers vectors, and the immobile ones in the DDWs contain additionally the dislocation dipoles with anti parallel Burgers vectors. A proper model should be able to predict the de- formation work hardening behavior in terms of in- ternal variables based on the special mechanisms of microstructure evolution. Similar as the concept of Roters et al. [3] the interaction of dislocations with the microstructure can be expressed as: dislocation sources inside the single crystal generate mobile dis- locations, at the same time certain parts of the mobile dislocations change to immobile ones and certain parts annihilate; those immobile dislocations are stored in the CBs and DDWs according to different manners: that is, the dislocation locks are stored in CBs and DDWs, while the dislocation dipoles are stored in the DDWs only; the applied stresses cause the mobile dislocations to ran across the CBs and DDWs to accommodate the plastic deformation. In the framework of this paper we only account for the externally applied stress and the dislocations as stress sources, i.e., we neglect all other possible stress sour- ces such as particles, precipitates, etc. Moreover we only treat the movement of single mobile dislocations and dislocation dipoles, where the latter does not contribute to the plastic deformation. We do not treat the cooperative motion of groups of dislocations, e.g., local tilt segments, as this only gives a minor contri- bution to the overall plastic strain. It is, however, important for the evolution of the dislocation cell structure, but, as stated above, we do not treat this evolution explicitly in the model. In this paper, we will use the following notation for the dislocation structure: • subscripts denote dislocation densities or slip systems: � M , the mobile dislocation density, � I , the immobile dislocation density, � F , the forest dislocation density, � P , the parallel dislocation density, � a; b, different slip systems, • superscripts denote areas of the cell structure: � B, the dislocation cell block, � W , the dense dislocation wall. Of the two regions of the microstructure of disloca- tions, the CBs contain the lower density of immobile dislocations for every slip system, and the DDWs con- tain the higher one. For the initial state without any deformation it is proper to assume all these immobile alia 52 (2004) 3603–3612 dislocation densities are identical for all slip systems. be expressed as density qPa and the forest dislocation density qFa for slip system a. qPa is the piercing density perpendicular to ena which means all the dislocation lines parallel to the slip plane, and qFa are the dislocation lines parallel to ena. A. Ma, F. Roters / Acta Materialia 52 (2004) 3603–3612 3605 L ¼ _eFeF�1 þ eFL0eF�1 ð5Þ with L0 ¼ �P�1 _P: ð6Þ In the case of crystal plasticity the following constitutive assumption is often used to connect the phenomeno- logical variable and the physical phenomena: L0 ¼ XN a¼1 _cafMa; ð7Þ here fMa ¼ eda � ena is the Schmid tensor for the slip system a where eda expresses the slip direction and ena the slip plane normal with respect to the undistorted con- figuration. N is the total number of slip systems, e.g., for fcc crystals there are 12 octahedral slip systems, f111gh110i, i.e., N ¼ 12. From the Orowan equation the plastic shear rate _ca of one slip system a is a function of the mobile dislocation density qMa on that slip system and the average slip velocity of the dislocation lines _ca ¼ qMabva: ð8Þ In order to determine the average dislocation slip ve- 2. The constitutive equations 2.1. The elastic law Based on the isomorphy assumption, the current elastic law of the single crystal can be represented by the referential elastic law through a plastic transformation P [14]. The elastic law for the current configuration is T2PK ¼ P~K 1 2 PTCP �� � I��PT; ð1Þ where T2PK is the second Piola–Kirchhoff stress tensor, C ¼ FTF is the right Cauchy–Green strain tensor, F the deformation gradient, I is the second order identity tensor, and the fourth order tensor eK is the elasticity tensor with respect to the undistorted configuration. From Eq. (1), the Cauchy stress T can be calculated as T ¼ 1eJ eF eK 12 eFTeF �h � I �ieF ð2Þ witheF ¼ FP; ð3Þ andeJ ¼ detðeFÞ: ð4Þ 2.2. The flow rule The velocity gradient tensor L ¼ _FF�1 is used to ex- press the loading process. Using Eq. (3) this tensor can locity va, we have to define the parallel dislocation Ignoring the mobile dislocations 1 and assuming the interactions of dislocations of different slip systems have the same interaction strength, we find qFa ¼ XN b¼1 qIb cosðena; enb��� � edbÞ���; ð9Þ qPa ¼ XN b¼1 qIb sinðena; enb��� � edbÞ���: ð10Þ Based on the framework of thermally activated dislo- cation motion the average velocity va then reads va ¼ kam0 exp � � Qslip KBh sinh jsaj � spass;a KBh Va � ð11Þ with sa ffi eK 12 eFTeF�h � I�i �fMa; ð12Þ and spass;a ¼ c1Gb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qPa þ qMa p ; ð13Þ where c1 is a constant, G the shear modulus, b the magnitude of the Burges vector, h the temperature, KB the Boltzman constant, m0 the attack frequency and Qslip the effective activation energy for dislocation slip. sa and sapass are the external driving force and the athermal re- sistance or the passing stress for the mobile dislocations. The jump width ka and the activation volume Va can be calculated as functions of qFa as ka ¼ c2ffiffiffiffiffiffiffiqFap ; ð14Þ and Va ¼ c3b2ka; ð15Þ where c2 and c3 are constants. In Eq. (11) the forward and backward jumps of dis- locations over the obstacle have been considered. For the reason of simplicity we will neglect the backward jumps in the following, so that the dislocation velocity changes to va ¼ 1 2 kam0 exp � � Qslip KBh exp jsaj � spass;a KBh Va � : ð16Þ 1 The mobile dislocation density is usually believed to be lower than the immobile one by at least one order of magnitude. 3. A scaling relation for the mobile dislocation density When dislocation densities are chosen as internal variables, their evolution laws are generally formulated as rate equations containing production terms and an- c1Gb 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip � KBh 2 ffiffiffiffiffiffiffi qFa p ¼ 0: ð20Þ 3606 A. Ma, F. Roters / Acta Materialia 52 (2004) 3603–3612 nihilation terms. The evolution laws for the immobile dislocation densities have been formulated based on several dislocation reactions such as immobilization of the mobile dislocations by the formation of dislocation locks and dipoles and dynamic recovery by dislocation climb [3]. However, for the mobile dislocation density there is no such simple model to describe its evolution. This difficulty arises from the fact, that both the number of the mobile dislocations and their average slip velocity effect the plastic deformation rate. One can accommo- date the external plastic deformation and its changes by fixing one and changing the other, but in general both quantities will change at the same time. This makes it very difficult to get a proper production term for the evolution law without any additional assumptions. An alternative idea has been presented in [15] in order to circumvent the above mentioned difficulties. In this concept it is assumed, that, although there are many ways for the dislocation structure to evolve, the proper one should satisfy, that the given external stress gener- ates the maximum plastic deformation or the given plastic deformation rate causes minimum external stress. 2 This is fulfilled if ðdsaÞ _ca ¼ osa oqMa dqMa þ osa oqPa dqPa þ osa oqFa dqFa ¼ 0: ð17Þ As the parallel dislocation density qPa and the forest dislocation density qFa have their independent evolution laws, 3 which are only a function of the loading history, and we should not give other constraints to them, we can only satisfy the criteria partially for one fixed time step by requesting osa oqMa � _ca;qPa;qFa ¼ 0: ð18Þ Eq. (18) can be used as constraint condition for the mobile dislocation density: First from (8), (13) and (16) we find the stress strain rate relation sa ¼ c1Gb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qPa þ qMa p þ Qslip c2c3b2 1 " þ KBh Qslip ln 2_ca bm0c2 ffiffiffiffiffiffiffi qFa p qMa !# ffiffiffiffiffiffiffi qFa p ; ð19Þ and then combining (18) and (19) we get the constraint equation for qMa 2 This assumption is equivalent to the minimization of global plastic work. 3 In fact these are functions of the immobile dislocation densities, see Eqs. (9) and (10). 2 qPa þ qMa c2c3b qMa Introducing A ¼ KBh c1c2c3Gb3 ; (20) can be reformulated as q2Ma � 4A2qFaqMa � 4A2qFaqPa ¼ 0: ð21Þ Solving (21) and knowing that qMa > 0 we get qMa ¼ 2A2qFa þ 2A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2q2Fa þ qPaqFa q : ð22Þ If we take the parameters KB ¼ 1:38� 10�23 J K�1, 0:16 c16 0:5, 16 c26 10, 16 c36 10 and material data (b;G) for aluminium, the parameter A is in the range of ½10�4; 10�1� for temperatures between room temperature and 723 K. So it is convenient to simplify Eq. (22) as qMa � 2A ffiffiffiffiffiffiffiffiffiffiffiffiffi qPaqFa p ; ð23Þ or qMa � Bh ffiffiffiffiffiffiffiffiffiffiffiffiffi qPaqFa p ð24Þ with another constant B ¼ 2KB c1c2c3Gb3 : This means the mobile dislocation density is propor- tional to the geometric mean of the parallel and the forest dislocation densities, and has a linear relationship with the temperature. Eq. (24) is an intrinsic constraint equation for the dislocation structure. One can easily see that the dislocation structure with two sets of indepen- dent internal variables ðqMa; qIa; a ¼ 1;NÞ has been re- duced to one with only one set of independent internal variable ðqIa; a ¼ 1;NÞ. 4 From (24) it can be seen that qMa qIa; qPa; qFa. It is interesting to realize at this point, that if one uses this assumption in (20), Eq. (24) can be directly derived. The same principle can be used for a composite structure. In this case we get sa ¼ fW c1Gb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qWPa þ qMa q( þ Qslip c2c3b2 1 " þ KBh Qslip ln 2 _ca bm0c2 ffiffiffiffiffiffiffi qWFa p qMa !# ffiffiffiffiffiffiffi qWFa q ) þ f B c1Gb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qBPa þ qMa q( þ Qslip c2c3b2 1 " þ KBh Qslip ln 2 _ca bm0c2 ffiffiffiffiffiffiffi qBFa p qMa !# ffiffiffiffiffiffiffi qBFa q ) ; ð25Þ 4 Remember that qF and qP are functions of qI (Eqs. (9) and (10)). of the immobile dislocation density inside the CBs. The A. Ma, F. Roters / Acta Materialia 52 (2004) 3603–3612 3607 first one is the immobilization of mobile dislocations, which for pure materials is mainly caused by the inter- action of the mobile dislocations with forest disloca- tions. We therefore assume, that the average distance LA traveled by each mobile dislocation is inversely pro- portional to the forest dislocation spacing: LA / 1ffiffiffiffiffiffiffi qBFa p : ð29Þ The frequency of immobilization Fimmob can then be calculated as Fimmob ¼ vaLA : Using the Orowan Eq. (8) the increasing rate for the immobile dislocations becomes _qBþIa / qMa � Fimmob; which can be changed to the final form _qBþIa ¼ c4 ffiffiffiffiffiffiffi qBFa q _ca ð30Þ with a constant c4. The second process is the interaction of mobile dis- locations of one slip system with the immobile disloca- and the derivative reads c1Gb 2 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qWPa þ qMa p þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qBPa þ qMa p ! � KBh c2c3b2 ffiffiffiffiffiffiffi qWFa p qMa þ ffiffiffiffiffiffiffi qBFa p qMa ! ¼ 0: ð26Þ Eq. (26) cannot be solved analytically, but if one uses again the approximation qMa qIa; qPa; qFa we get c1Gb 2 1ffiffiffiffiffiffiffi qWPa p þ 1ffiffiffiffiffiffiffi qBPa p ! � KBh c2c3b2 ffiffiffiffiffiffiffi qWFa p qMa þ ffiffiffiffiffiffiffi qBFa p qMa ! ; ð27Þ and the solution for qM for the composite structure reads qMa � Bh fW ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qWPaq W Faq B Pa p þ f B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qBPaq B Faq W Pa p fW ffiffiffiffiffiffiffi qBPa p þ f B ffiffiffiffiffiffiffi qWPa p : ð28Þ Therefore, it is again possible to remove the mobile dislocation density as independent internal variable. An important property of Eq. (28) is that for either fW ¼ 0 or f B ¼ 0 Eq. (24) is recovered. 4. The evolution of the dislocation densities 4.1. The immobile dislocation density inside CBs There are three processes contributing to the change tions on the same system. We assume this interaction mechanism will cause a decrease of the immobile dislo- cation density. This mechanism is especially important at low temperatures, where the probability of thermally activated cross slip of screw dislocations and climb of edge dislocations are very small. However, at the ele- vated temperatures treated in this paper, also annihila- tion by cross slip can be regarded as being included in this term. Calculating the meeting frequency of one mobile dislocation with another immobile one of the same slip system one can derive the nonthermal anni- hilation rate [3] _qB�Ia1 ¼ �c5qBIa _ca ð31Þ with a constant c5. The third o