Incorporating strain gradient effects in a multi-scale constitutive framework for nickel-base superalloys

Transcription

1 Incorporating strain gradient effects in a ulti-scale constitutive fraework for nickel-base superalloys Tiedo Tinga, Marcel Brekelans, Marc Geers To cite this version: Tiedo Tinga, Marcel Brekelans, Marc Geers. Incorporating strain gradient effects in a ulti-scale constitutive fraework for nickel-base superalloys. Philosophical Magazine, Taylor Francis, 00, (0-), pp.-. <0.00/000>. <hal-000> HAL Id: hal Subitted on Sep 00 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Philosophical Magazine & Philosophical Magazine Letters Incorporating strain gradient effects in a ulti-scale constitutive fraework for nickel-base superalloys Journal: Manuscript ID: Journal Selection: Date Subitted by the Author: Philosophical Magazine & Philosophical Magazine Letters TPHM-0-Jan-000.R Philosophical Magazine -Apr-00 Coplete List of Authors: Tinga, Tiedo; Netherlands Defence Acadey, Military Platfor Systes Brekelans, Marcel; Eindhoven University of Technology, Mechanical Engineering Geers, Marc; Eindhoven University of Technology, Mechanical Engineering Keywords: Keywords (user supplied): dislocations, plasticity of crystals, superalloys strain gradients, size effects

3 Page of Philosophical Magazine & Philosophical Magazine Letters Incorporating strain gradient effects in a ulti-scale constitutive fraework for nickel-base superalloys T. TINGA#*, W.A.M. BREKELMANS and M.G.D. GEERS # Netherlands Defence Acadey, Military Platfor Systes section, P.O. Box 0000, 0 CA Den Helder, The Netherlands Eindhoven University of Technology, Departent of Mechanical Engineering, P.O. Box, 00 MB Eindhoven, The Netherlands An efficient ulti-scale constitutive fraework for nickel-base superalloys is proposed that enables the incorporation of strain gradient effects. Special interface regions in the unit cell contain the plastic strain gradients that govern the developent of internal stresses. The odel is shown to accurately siulate the experientally observed size effects in the coercial alloy CMSX-. The liited coplexity of the proposed unit cell and the icroechanical siplifications ake the fraework particularly efficient in a ulti-scale approach. This is deonstrated by applying the odel in a gas turbine blade finite eleent analysis. Keywords: Superalloys; Crystal plasticity; Dislocations; Strain gradients; Size effects. Introduction Strain gradient effects are only quite recently recognized as an iportant factor in echanical odelling at sall length scales. At these length scales, the aterial strength is observed [] to be size dependent, with an increase of strength at decreasing diensions, i.e. saller is stronger. The existence of a strain gradient dependent back stress and its relevance for crystal plasticity of sall coponents undergoing inhoogeneous plastic flow has been reported in several papers. The work of Gurtin and co-workers [-] is here ephasized in particular. Strain gradient effects are also relevant for single crystal nickel-base superalloys, which are widely used as gas turbine blade aterials because of their high resistance against high teperature inelastic deforation. The superior high teperature behaviour is attributed to the two-phase coposite icrostructure consisting of a -atrix containing a large volue fraction of '-particles (see Figure ). Cubic Ni Al (') precipitates are ore or less regularly distributed in a Ni-atrix (-phase). The typical precipitate size is 0. µ and the atrix channel width is typically 0 n. Since these very narrow atrix channels bear the ajority of the plastic deforation, considerable plastic strain gradients develop in the aterial. * Corresponding author. Eail: t.tinga@nlda.nl

4 Philosophical Magazine & Philosophical Magazine Letters Page of Therefore, strain gradient effects should be included in superalloy constitutive odels, as was done by Busso and co-workers [,] and Choi et al. []. They used a detailed unit cell FE odel of an elastic '- precipitate ebedded in an elasto-viscoplastic -atrix. Busso and co-workers [,] adopted a non-local gradient dependent crystal plasticity theory to describe the behaviour of the -atrix. The flow resistance and hardening of the atrix were based on the densities of statistically stored and geoetrically necessary dislocations. This enabled the prediction of a precipitate size dependence of the flow stress and allowed to capture the effect of orphological changes of the precipitate. Choi et al. [] extended this work using a ore phenoenological crystal plasticity forulation with no direct relation to dislocation densities. However, a strain gradient dependence was incorporated in the odel, which also resulted in the prediction of precipitate size effects and an influence of the icrostructure orphology. [Insert Fig. about here] The ability to perfor a reliable life tie assessent is crucial for both gas turbine coponent design and aintenance. Therefore, a vast aount of work has been done on odelling the echanical behaviour of superalloys. Initially the aterial was treated as a hoogeneous single phase aterial [0-0]. In all these approaches conventional crystal plasticity theory was used to odel the aterial response, which eans that constitutive laws were defined on the slip syste level. Since these solution ethods address the acroscopic level, they can easily be used as a constitutive description in a finite eleent (FE) analysis, which is nowadays the coon ethod used for coponent stress analysis and life tie assessent. However, to be able to quantify strain gradients in the aterial and to assess their effect on the acroscopic response, the two-phase nature of superalloys has to be odelled explicitly. In the resulting icrostructural odels the shape, diensions and properties of both phases are considered as odel paraeters. However, the length scale of the icrostructure, which is in the order of icroeters, is uch saller than the engineering length scale. Modelling a acroscopic coponent copletely, i.e. taking into account all icrostructural details, is therefore not feasible in the engineering practice. One way to bridge this gap in length scales is to use a ulti-scale approach in which an appropriate hoogenization ethod is applied to connect the icroscopic to the acroscopic level. A large nuber of ulti-scale fraeworks has been developed in the past decades and applied to different aterials. Exaples are Eshelby-type hoogenization ethods [] for aterials with (elastic) inclusions, variational bounding ethods

5 Page of Philosophical Magazine & Philosophical Magazine Letters [,] and asyptotic hoogenization ethods [,]. Soe ore recent exaples applicable to the class of unit cell ethods are the first order [,] and second order [] coputational hoogenization ethods and the crystal plasticity work by Evers [] that considered the effect of ultiple differently oriented grains in an FCC etal. Finally, Fedelich [0,] used a Fourier series hoogenization ethod to odel the echanical behaviour of Ni-base superalloys. Another way to overcoe the length scale proble is to use icrostructural odels that predict the aterial response in a closed-for set of equations on the level of the aterial point [-,-]. The icrostructural results are then used to develop constitutive descriptions that fit in traditional ethods at the acroscopic level. Clearly, the analyses on the icroscopic and the acroscopic level are copletely separated in this case. Svoboda and Lukas [,] developed an analytical unit cell odel consisting of a '-precipitate and three -channels. The deforation in the distinct regions was assued to be unifor and power law creep behaviour was used for the atrix aterial. The required copatibility at the /'-interfaces resulted in a relatively high overall stiffness. Kuttner and Wahi [] used a FE ethod to odel a unit cell representing the /'-icrostructure. A odified Norton s creep law was assued for both phases and threshold stresses for different deforation echaniss were included. The latter odels [-] as well as the odel by Fedelich [0,] adopted the Orowan stress as a threshold stress for plastic deforation. Since the Orowan stress is related to the spacing of the '-precipitates, a length scale dependence was explicitly introduced into the constitutive description. However, apart fro the odels by Busso and co-workers [,] and Choi et al. [], that were discussed before, none of these odels includes strain gradient effects, whereas the FE based unit cell odels are detailed but usually too coplex to be used efficiently in a ulti-scale analysis of structural coponents. Therefore, in this paper a new fraework is proposed that is particularly developed to incorporate strain gradient effects, and to be coputational efficient, thus enabling application in a ulti-scale approach (FE analyses on real coponents). A new unit cell approach is forwarded, in which the role of the /'-interfaces is included. More specifically, each phase in the aterial is represented by a cobination of a bulk aterial unit cell region and several interface regions. In the interface regions internal stresses will develop as a result of the lattice isfit between the two phases and the plastic strain gradients, represented by non-unifor distributions of geoetrically necessary dislocations (GNDs). Conditions requiring stress continuity and strain copatibility across the /'-interfaces are specified in these regions. Continuous dislocation densities and slip gradients, as typically used in a continuu forulation, are approxiated here by piece-wise constant fields.

6 Philosophical Magazine & Philosophical Magazine Letters Page of The coprehensible coplexity of the adopted unit cell and the icroechanical siplifications, which render a coposition of 0 piece-wise uniforly deforing regions, ake the fraework particularly efficient in a ulti-scale approach. The unit cell response is deterined nuerically on a aterial point level (integration point level) within a acroscopic FE code, which is coputationally uch ore efficient than a fully detailed FE-based unit cell. The aterial response is predicted accurately by using an extended version of an existing non-local strain gradient crystal plasticity odel [,] for the atrix aterial. The precipitate is treated as an elastic anisotropic solid. Finally, to reduce the odel coplexity, the echaniss of precipitate shearing and rafting are oitted, which liits the application range of the present odel soewhat. In the ajority of industrial gas turbines single crystal Ni-base superalloy blades operate at teperatures well below 0 o C and are not allowed to defor by ore than to %. Since the atrix phase of the aterial is known to accoodate the ajority of the deforation, it is justified, at these operating conditions, to assue that the '-precipitates reain elastic during deforation. This eans that the echanis of precipitate shearing by dislocations is not considered. Experiental work [,,-] has shown that precipitate shearing becoes iportant at teperatures above 0 o C and at larger strains (later stages of steady-state creep). At lower teperatures considerable stresses in the range of 00 to 00 MPa are required to initiate particle shearing. Moreover, the orphology of the icrostructure is assued to reain the sae during deforation, which also neglects the echanis of rafting. Again experiental work [] has shown that precipitate coarsening is copleted rapidly at teperatures above 0 o C and after proportionally longer ties at lower teperatures. Consequently, the assuption of an elastic precipitate and a fixed orphology liits the application region of the present odel to teperatures below 0 o C and strains saller than about % and also to relatively short loading ties. However, this liited application range is sufficient to deonstrate the iportance of strain gradient effects in nickel-base superalloys. Moreover, future extension of the odel with precipitate deforation echaniss and rafting kinetics can reove the present liitations. To suarize, the original aspect of the present odel is the incorporation of strain gradient effects, which are not included in the ajority of the existing odels, in an efficient ulti-scale fraework. Due to the icroechanical siplifications, the present odel is coputationally uch ore efficient than the strain gradient FE unit cell odels. In the next section the ulti-scale fraework is outlined, providing definitions of the unit cell and the interaction laws. Then the strain gradient effects are ipleented, both in the hardening law and through internal

7 Page of Philosophical Magazine & Philosophical Magazine Letters stresses: section describes the constitutive odels that are used, focusing ainly on the strain gradient crystal plasticity concepts, while section considers the internal stresses, describing the forulation of isfit and strain gradient induced back stresses. In section, the odel is applied to the Ni-base superalloy CMSX-. Siulated stress-strain curves and size effects are copared to experiental results, showing that the present fraework is able to describe the aterial response and size effects to a level of detail siilar to coplex FE unit cell odels, while being coputationally uch ore efficient. The coputational efficiency is deonstrated by applying the odel to a gas turbine blade finite eleent analysis. Finally, section forwards soe concluding rearks. Multi-scale odel description The strain gradient effects are incorporated in a newly developed ulti-scale odel for the prediction of the superalloy echanical behaviour. This odel covers several length scales, which is shown scheatically in Figure a. The acroscopic length scale characterises the engineering level on which a finite eleent (FE) odel is coonly used to solve the governing equilibriu proble. The esoscopic length scale represents the level of the icrostructure within a acroscopic aterial point. At this length scale the aterial is considered as a copound of two different phases: '-precipitates ebedded in a -atrix. Finally, the icroscopic length scale reflects the crystallographic response of the individual aterial phases. The constitutive behaviour is defined on this level using a strain gradient crystal plasticity fraework. [Insert Fig. about here] Considering the overall deforation level, a sall strain approxiation will be used in the odel. The intended application of the odel is the analysis of gas turbine coponents in which deforations are sall. Consequently, the initial and defored state are geoetrically nearly identical. Instead of a large deforation strain tensor, the linear strain tensor () will be used with the Cauchy stress tensor () as the appropriate stress easure. In this section the different aspects of the aterial point odel are described. Firstly the esoscopic unit cell is defined, after which the scale transitions and interaction laws are described.. Unit cell definition On the aterial point level the Ni-base superalloy icrostructure, consisting of '-precipitates in a -atrix, is represented by a unit cell containing regions (see Figure b):

8 Philosophical Magazine & Philosophical Magazine Letters Page of '-precipitate region -atrix channel regions ( j, j = ) with different orientations (noral to the [00], [00] and [00] direction) interface regions (I k and I p k, k = ) containing the /'-interfaces. A atrix and a precipitate region together for a bi-crystal, which is located on each face of the '-precipitate. The interface between the two different phases plays an iportant role in the echanical behaviour of the aterial, especially due to the large strain gradients that develop here. Therefore, special interface regions were included in the odel to take into account the processes that take place at the /'-interfaces. Consequently, each phase in the two-phase aterial, either a precipitate or a atrix channel, is represented by two types of regions in the unit cell. The first type represents the bulk aterial behaviour and in the second type all shortrange interface effects, including dislocation induced back stress and interaction with other phases, are incorporated. Inside each individual region quantities, like stresses and strains, are assued to be unifor, which leads to a particularly efficient fraework. The only relevant quantity that is not uniforly distributed inside a region is the GND density, as will be shown in section.. Thus, in the present fraework the behaviour of a specific phase in the real aterial, e.g. a atrix channel, is given by the (weighted) average behaviour of the bulk unit cell region and the appropriate sides of the interface regions. This also eans that the individual unit cell regions will not necessarily describe the real deforation behaviour on their own. Finally note that, throughout this paper, the word phase refers to a specific coponent in the real aterial, either atrix or precipitate, and the word region refers to a specific part of the odel unit cell. The constitutive behaviour of the atrix and precipitate fractions of the interface is identical to the behaviour of the bulk atrix and precipitate phases, respectively. However, additional interface conditions (section.) are specified and short-range internal stresses (back stress, see section ) are included in these regions, which distinguishes the fro the atrix and precipitate regions. Another internal stress, the lattice isfit stress, is a long-range stress field, which is consequently included in both bulk and interface regions. [Insert Fig. about here] The orphology of the icrostructure is defined by the values of the geoetrical paraeters L, w and h, as shown in Figure. The precipitate (') size, including the '-interface region, is given by the value of L in

9 Page of Philosophical Magazine & Philosophical Magazine Letters three directions (L, L and L ). These values also deterine two of the three diensions of the three atrix channel regions ( j ) and the six interface regions (I k ). The total channel width, including one channel and two - interface regions, is given by the paraeter h i for -channel i. Finally, the width of the interface regions is related to the values of L and h. The width of the atrix phase layer in the interface (I i ) is defined as 0% of the atrix channel width (w i = 0.0 h i ) and the width of the precipitate layer in the interface (I i p ) as % of the precipitate size (w i p = 0.0 L i ). The selection of these values will be otivated in section. The CMSX- icrostructure (Figure ) is rather regular, so for the present odel the precipitates are assued to be cubic with L = L = L = 00 n. The atrix channel width is taken as h = h = h = 0 n. These values yield a ' volue fraction of %. As will be shown later, the interface regions at opposite sides of the precipitate (e.g. I and I ) are assued to behave identically in ters of deforation, internal stress developent, etc. Therefore, to the benefit of coputational efficiency, only half of the interface regions need to be included in the equations in the next subsections, thereby effectively reducing the nuber of regions fro to 0. The opposite regions are then incorporated in the volue averaging by doubling the respective volue fractions.. Scale transitions and interaction law The relations between the different length scales of the odel are shown scheatically in Figure. Conventionally, a finite eleent ethod is used on the acroscopic level to solve the engineering proble with its boundary conditions. In the present ulti-scale approach the usual standard procedure to obtain the stress response for a given deforation (i.e. a local closed-for constitutive equation) is replaced by a esoscopic calculation at the unit cell level as indicated in Figure. [Insert Fig. about here] The deforation (total strain) for a certain acroscopic aterial point during a tie increent is provided by the acro scale and the stress response is returned after the coputations at the esoscopic level. The quantities used for this acro-eso scale transition are denoted as the esoscopic average strain ( tot ) and the esoscopic average stress ( tot ). The stress tensor tot is deterined fro the strain tensor tot based on the specified esoscopic configuration and the local constitutive equations of the different phases at the icro level.

10 Philosophical Magazine & Philosophical Magazine Letters Page of The esoscopic strain is obtained by averaging the icrostructural quantities in each of the regions, defined as i f i i tot = tot i = ',, p p p,i,i,i,i,i, I, where f i are the volue fractions and i tot the total strain tensors in the 0 different regions of the odel. The relation between the esoscopic and icroscopic level is provided by the constitutive odels, which relate the stress tensors to the individual strain tensors for all 0 regions i i constituti ve box i = ', p p p,,,i,i,i,i,i, I The constitutive odel at the icro level, for the atrix phase, is based on a strain gradient enhanced crystal plasticity theory and will be described in section. The precipitate phase is treated as an elastic ediu. Also, only at this point the internal stresses (isfit and back stress, see section ) play a role in the stress analysis. They are cobined with the externally applied stress, as obtained fro the equilibriu calculation, to for an effective stress that is used in the constitutive box. They are thus not part of the equilibriu calculation itself, as is also indicated in Figure. This separation of external and internal stress calculation is particularly possible in the context of the adopted Sachs approach (to be outlined in the following), as will be discussed in section.. Inside each of the different regions, both stress and deforation are assued to be unifor. To specify the coupling between the regions an interaction law has to be defined. Two frequently adopted liit cases can be distinguished: Taylor interaction: deforation is unifor across the regions, stresses ay vary; Sachs interaction: stresses are unifor across the regions, deforation ay vary. These two approaches for an upper and a lower bound for the stiffness, so the real echanical behaviour interediates between these cases. A Taylor-type interaction usually yields a response that is too stiff, thereby overestiating the resulting stresses for a given deforation, whereas a Sachs type interaction yields an overly weak response. A Taylor interaction odel is inappropriate for the present application, since the deforation is highly localized in the -atrix phase. A Sachs-type approach is actually a uch better approxiation, but it lacks the ability to incorporate kineatical copatibility conditions at the interface. Also, it would not correctly represent the stress redistribution between the two phases that occurs when the atrix starts to defor plastically. Therefore, a hybrid interaction law [] or a odified Taylor / Sachs approach [] is best suited here. () ()

11 Page of Philosophical Magazine & Philosophical Magazine Letters For the present odel a odified Sachs approach is used, in which the requireent of a unifor stress state is relaxed for the interface regions. In the - and '-regions the stresses are required to be equal to the esoscopic stress. In each pair of interface regions however, only the average stress is enforced to be equal to the esoscopic stress. This results in the following equations: - Sachs interaction between '- and -regions: ' = = = = tot - Modified Sachs interaction for the bi-crystal interfaces: f p k I p k k k p Ik Ik ( f + f ) tot =,, I I I + f = k where i are the stress tensors in the different regions, fractions of the respective regions. tot () () is the esoscopic stress tensor and f i are the volue The fact that each partition of the interface region ay respond differently to a echanical load enables the possibility (and necessity) to define additional conditions at the interfaces. Both stress continuity (across the interface) and kineatical copatibility (in the plane of the interface) are therefore added as additional requireents. This leads to the following suppleentary equations, where interface. - Copatibility between the atrix (I k ) and the precipitate side (I p k ) of the k th interface: p k I r k r k I r k r k k ( I n n ) = ( I n n ) k =,, - Stress continuity at the sae interface: p k I r n k I r k k = n k =,, where and are the stress and strain tensors in the different regions. k n r is the unit noral vector on the k th For the aterial point odel the esoscopic deforation (total strain) is provided by the acro scale analysis and the esoscopic stress ust be calculated. Since the odel consists of 0 distinct regions in which the deforation and stress are hoogeneous, and the syetric stress and strain tensors contain independent coponents, a total of 0 unknowns results. The systes () and ()-() represent a total nuber of 0 equations, while the constitutive odel () adds another 0 equations, which copletes the description. In suary, the stresses in the bulk aterial regions and the average stresses of the interface regions are coupled, whereas for the interface regions additional interface conditions in ters of stress and strain are specified. The assuption of unifor stress and strain inside the unit cell regions in cobination with the () ()

12 Philosophical Magazine & Philosophical Magazine Letters Page 0 of conditions proposed above copletely deterine the proble. Additional conditions are not required nor allowed, which eans that, for exaple, the absence of traction continuity between bulk aterial and interface is accepted for the sake of efficiency. Constitutive behaviour A strain gradient enhanced crystal plasticity approach is used to odel the constitutive behaviour of the atrix phase, whereas the precipitate is treated as an elastic anisotropic aterial. After a general introduction concerning the underlying crystal plasticity forulation, the atrix and precipitate constitutive odels will be described. Note that the atrix phase constitutive odel is applied to both bulk atrix unit cell regions and the atrix sides of the interface regions. The precipitate regions and the precipitate sides of the interface regions reain elastic.. Strain gradient crystal plasticity In a conventional crystal plasticity fraework, the plastic deforation of etals is a natural consequence of the process of crystallographic slip. For each type of crystal lattice a set of slip systes exists along which the slip process will take place. A slip syste is coonly characterised by its slip plane and its slip direction. For the considered superalloy, with a face-centred cubic (FCC) lattice, slip directions on each of the octahedral slip planes can be identified, resulting in slip systes. In addition to the plastic slip, elastic deforation is accoodated by distortion of the crystallographic lattice. In any superalloy crystal plasticity odels [,,] an additional set of cubic slip systes is incorporated to account for the cross slip echaniss that occur when the aterial is loaded in a direction other than <00>. The present odel here is only applied to the technologically iportant <00> loading direction, corresponding to the direction of centrifugal loading in turbine blades. However, a set of cubic slip systes can easily be incorporated when dealing with other orientations is required. Clearly, crystallographic slip is carried by the oveent of dislocations. Yet, also the hardening behaviour of etals is attributed to dislocations. Plastic deforation causes ultiplication of dislocations and their utual interaction ipedes the otion of gliding dislocations, which causes strengthening. The total dislocation population can be considered to consist of two parts: statistically stored dislocations (SSDs) geoetrically necessary dislocations (GNDs) [] 0

13 Page of Philosophical Magazine & Philosophical Magazine Letters The SSDs are randoly oriented and therefore do not have any directional effect and no net Burgers vector. They accuulate through a statistical process. On the other hand, when a gradient in the plastic deforation occurs in the aterial, a change of the GND density is required to aintain lattice copatibility. Individual dislocations cannot be distinguished as SSDs or GNDs. The GNDs are therefore the fraction of the total dislocation population with a non-zero net Burgers vector. Moreover, as will be shown later, a gradient in the GND density causes an internal stress which affects the plastic deforation. These strain gradient dependent influences give the odel a non-local character. They enable the prediction of size effects which cannot be captured by conventional crystal plasticity theories. In the present odel it is assued that all SSD densities are of the edge type, whereas for the GNDs both edge and screw dislocations are considered. This iplies that for an FCC etal edge SSD densities are taken into account, next to edge and screw GND densities [0]. A coplete overview of the dislocation densities, including their type and slip syste is given in Table. Each screw dislocation can ove on either of the two slip planes in which it can reside. This is indicated in Table by the two corresponding slip syste nubers, where a negative nuber eans that the defined slip direction should be reversed. The elastic aterial behaviour is odelled using a standard forulation for orthotropic aterials with cubic syetry. The three independent coponents of the elastic tensor C of both phases in CMSX- at 0 o C are given in Table []. [Insert Table about here] [Insert Table about here] The next subsection shows how the strain gradient based crystal plasticity fraework is used to elaborate the atrix phase constitutive odel.. Matrix constitutive odel The basic ingredient of the crystal plasticity fraework is the relation between the slip rates resolved shear stresses for all the slip systes. The following forulation is proposed here: & = & 0 s eff exp or eff n sign ( ) eff () & and the

14 Philosophical Magazine & Philosophical Magazine Letters Page of where or denotes the Orowan stress, s the actual slip resistance and, obtained fro the effective stress tensor eff by eff = : eff P where P is the syetric Schid tensor defined as r r r r P = s n + n s ( ) The unit length vectors n r and eff the effective shear stress on slip syste s r are the slip plane noral and slip direction, respectively. The effective stress tensor is defined as the cobination of the externally applied stress, the back stress and the isfit stress (see section ) according to eff = + isfit b The forulation in equation () is an extended version of the slip law used in the work of Evers et al. [,] for a single phase FCC aterial. For the present two-phase aterial an additional threshold ter is added to account for the Orowan stress, which is the stress required to bow a dislocation line into the channel between two precipitates. This stress is given by [] as or µ b d µ b = ln = d r 0 d where µ is the shear odulus, b the length of the Burgers vector, d the spacing between two precipitates (equal to the channel width) and r 0 the dislocation core radius (in the order of b). There is no generally accepted value for the constant. The used values range fro 0. to. for different aterials and conditions [,0-,,], where in soe cases the constant was used as an adjustable paraeter. A value of = 0. is taken here, as was done by Busso et al. []. If the effective stress exceeds this Orowan stress threshold, dislocation lines enter the atrix channel and the typical slip threshold (governed by s ) deterines whether or not they can ove any further. This is the case if the effective stress exceeds the slip resistance. The slip resistance is in a certain sense also an Orowan type stress related to the average spacing of obstacles inside the atrix phase, such as other dislocation segents. While both thresholds are a result of icroscopic phenoena, the Orowan threshold is related to a esoscopic length scale. Moreover, the slip resistance threshold ter deterines the actual slip rate value, whereas the Orowan threshold is essentially active or inactive (as the exponential ter is ranging fro 0 to ). As soon as the Orowan threshold or is exceeded by the effective stress or when the growth of the dislocation density triggers an increase of the slip resistance s to a value that exceeds the Orowan threshold, the slip resistance contribution becoes the active threshold that deterines the slip rate. () () (0) ()

15 Page of Philosophical Magazine & Philosophical Magazine Letters Slip resistance Generally speaking, slip resistance or dislocation drag is caused by several obstacles such as solute atos, precipitates (e.g. carbides, interetallics) and other dislocations, each having a contribution to the overall slip resistance. The physical echanis associated with an increasing slip rate at increasing teperature is the decrease of dislocation drag (related to the slip resistance). The teperature dependence of all these contributions is assued to be identical, resulting in a classical expression for the teperature dependent slip resistance where s = s 0 Q exp kt s 0 is the atheral slip resistance, Q is an activation energy for overcoing the barriers, k =. x 0 J K - is the Boltzann constant and T the absolute teperature. The aount and spacing of solute atos and precipitates (other than ') in the atrix phase is assued to be constant, which eans that the isotheral lattice slip resistance due to these obstacles is constant as well. The second contribution to the total slip resistance depends on the dislocation densities in the aterial. This contribution is related to the resistance of sessile / forest dislocations and therefore depends on the total dislocation density, coposed of the SSDs and the GNDs. The relation between the slip resistance and the dislocation density is defined according to s = c + disl µ b! SSD! GND Evers et al. [] used an interaction atrix containing experientally deterined entries to define the interactions between dislocations on different slip systes. These values are not available for Ni-base superalloys, so only interactions with dislocations on the sae slip syste (self hardening) will be taken into account, as was done by Busso et al. []. Interactions with dislocations on other slip systes (cross hardening) are neglected. Also the contribution to the slip resistance of the screw-type GND densities ( = ) whose slip plane is abiguous (see Table ) is neglected. The exploitation of equation () requires the knowledge of all dislocation densities ( edge dislocation densities for the SSDs and edge and screw dislocation densities for the GNDs). The GND densities can be obtained fro the plastic deforation gradients in the aterial as will be explained in section. dealing with the back stresses. The SSD densities are calculated on the basis of an appropriate evolution () () equation [], starting fro their initial value SSD,0 :

16 Philosophical Magazine & Philosophical Magazine Letters Page of &! SSD = yc! SSD & b L,! SSD ( t = 0) =! SSD, 0 which is the net effect of dislocation accuulation (left ter) and annihilation (right ter). The paraeter y c represents the critical annihilation length, i.e. the average distance below which two dislocations of opposite sign annihilate spontaneously. The accuulation rate is linked to the average dislocation segent length of obile dislocations on syste, which is deterined by the current dislocation state through L = K! SSD +! GND where K is a aterial constant. Further, the experiental tensile curves in section of this paper show that after soe aount of yielding strain softening occurs in the aterial. This phenoenon is typical for superalloys and has been the subject of several studies. Busso and co-workers [,,] and Choi et al. [] perfored unit cell finite eleent analyses and concluded that the softening ight be attributed to lattice rotations around the corners of the precipitates. These rotations induce activation of additional slip systes and result in a fast increase of plastic slip. In these analyses, the precipitate was assued to behave elastically. On the other hand, Fedelich [] states that the softening is related to the onset of precipitate shearing, a phenoenon which was not accounted for in the FE unit cell analyses entioned above. The present odel is specifically developed to be efficient in a ulti-scale approach. The consequential choice for sall strain kineatics and unifor stress and strain in the unit cell regions ean that local lattice rotations cannot be predicted. Also the echanis of precipitate shearing is not included. Therefore, in the present fraework the softening effect is incorporated in a phenoenological way by adding a softening ter p SSD s soft C! = soft ~ ()! SSD to the slip resistance, where C soft and p are constants and ~! SSD () () is the equilibriu value of the SSD density. This equilibriu value follows fro equation () by requiring that the creation and annihilation ters are equal. Rather than a real slip resistance, the contribution s soft should be considered as a reflection of the lack of dislocation obility. It represents, in a phenoenological way, the increase of dislocation obility, and consequential decrease of s, associated with either local lattice rotations or precipitate shearing. In forthcoing work, a precipitate constitutive odel will be proposed to properly capture precipitate shearing. The need for a phenoenological ter that accounts for softening will then be re-assessed.

17 Page of Philosophical Magazine & Philosophical Magazine Letters Finally, it is assued that the atheral lattice slip resistance is caused by an initial SSD density SSD,0, which eans that its effect on the total slip resistance is incorporated in the dislocation slip resistance as given by equation (). Therefore, cobination of equations () and () yields the total atheral slip resistance to be used in equation (): s = s disl + s soft 0 (). Precipitate constitutive odel In the present approach, the precipitate in the superalloy is assued to be elastic, which iplies that both the unit cell precipitate region and the precipitate sides of the interface regions are treated as anisotropic elastic edia. As was entioned in the introduction, this assuption is only acceptable under certain conditions. The precipitate ay defor inelastically when it is sheared by a dislocation or bypassed by dislocation clib. However, these processes have considerable thresholds in ters of stress and teperature. Therefore, at teperatures below 0 o C and oderate stress levels the siplification of an elastically deforing precipitate is justified. These conditions are, nevertheless, sufficient to deonstrate the iportance of strain gradient effects, which is the ai of the present paper. The developent of an enhanced constitutive odel that includes crystal plasticity in the precipitate for ore extree conditions, is the subject of future work. Internal stresses The interface between the two different phases plays an iportant role in the echanical behaviour of the ultiphase aterial, because of the developent of significant internal stresses that interact with the externally applied stress, see equation (0). In the present odel the following internal stresses are incorporated: isfit stress: stress that originates fro the lattice isfit between the and '-phases at the level of the coherent interface that is fored. This is a long-range stress field that spans the coplete unit cell. back stress: stress that originates fro deforation-induced plastic strain gradients inducing a gradient in the GND density at the interfaces. This is a short-range stress field that only acts in the interface regions. Apart fro these two explicitly defined internal stress fields, an internal redistribution of stresses occurs due to differences in plastic deforation between both phases.. Lattice isfit The and '-phases both have an FCC lattice structure with a slightly different lattice (diension) paraeter. They for a coherent interface, which eans that the crystal lattice planes are continuous across the interface,

18 Philosophical Magazine & Philosophical Magazine Letters Page of but a isfit strain exists to accoodate the difference in lattice paraeter. For ost superalloys the isfit is called negative, which eans that the lattice paraeter of the precipitate is saller than the atrix lattice paraeter. To bridge the isfit, both the precipitate and atrix are strained, causing copressive isfit stresses in the atrix (parallel to the interface) and tensile stresses in the precipitate. The aount of straining of the atrix and precipitate is dependent on the agnitude of the isfit, the elastic oduli of both aterials and their relative sizes []. The unconstrained isfit is defined as a ' a " = a with a and a the lattice paraeters of the ' and -phases respectively. If the coefficient of theral expansion is not equal for both phases, the isfit is teperature dependent, since the difference in lattice paraeter changes with teperature. The isfit is assued to be accoodated equally by both phases, leading to a isfit strain # isfit = ( a a ) ' a in the atrix (in the two directions in the plane of the interface) and the sae strain with opposite sign in the i precipitate. Using the noral vector of the interface n r, the coponents of the isfit strain tensor are defined as i isfit isfit ri ri i ri ri ( n n ) n n = # I + # n with isfit given by () for the atrix regions and with # the isfit strain in noral direction resulting fro the requireent that the associated stress coponent vanishes. This isfit strain tensor represents an initial elastic strain (also called eigenstrain), triggering an initial stress in each of the regions. Since the isfit is accoodated elastically in both phases, the isfit stress and strain tensor are directly related to each other by a odified (plane stress) elastic stiffness tensor the interface are non-zero i isfit i isfit = B : i isfit i B isfit i n () () (0), which ensures that only the stress coponents parallel to Usually, the isfit strain is used as an initial strain in the equilibriu calculation of the local stresses. However, in an approach that is based on the Sachs interaction law this is not straightforward, since the different regions are coupled by their stresses, which affects the stress redistribution due to the isfit strains. At the sae tie, the use of the Sachs interaction law akes it possible to superpose a separately calculated internal stress (e.g. isfit stress, equation ()) to the calculated local stress to constitute an effective stress tensor. The effective stress tensor is then used in the constitutive law, equation (), to calculate the plastic strains. ()

19 Page of Philosophical Magazine & Philosophical Magazine Letters The isfit between the two phases can be partially relaxed by plastic deforation of one or both phases. Plastic deforation generates isfit dislocations at the interface resulting in a loss of coherency between the phases and a corresponding relaxation of the isfit. When the total isfit strain would be copletely accoodated by plastic slip, the effective stresses in both phases would be siilar and the isfit would effectively vanish. For the interface regions this is autoatically ensured by the copatibility requireents, according to equation (). Plastic deforation in one region causes a local stress redistribution across the two phases and a corresponding decrease of the isfit. For the bulk regions ( precipitate and atrix regions), which are not subject to copatibility requireents, the absolute values of the isfit strain coponents (equation (0)) are reduced by the absolute value of the plastic strain difference ( $ ) between the two phases, until the isfit copletely vanishes: i i i isfit, ij isfit, ij pl, ij $ () This siulates the loss of coherency due to plastic deforation in one or both phases. When a tensile stress is applied to the aterial, the effective stress in the atrix channels parallel to the loading direction will be lowered by the copressive isfit stress. This is not the case for the channels perpendicular to the loading axis, and it was observed that the deforation is initially localized in these atrix channels.. Strain gradient induced back stress The back stress on a slip syste originates fro the spatial distribution of dislocations and is therefore only related to the GND density. For SSDs, which usually have a rando orientation, the back stress contribution will be negligible. The value of the back stress tensor is calculated by suation of the internal stress fields caused by the individual edge and screw dislocation densities. int int ( ) b = e + s For a field of edge dislocations the stress field in a point is approxiated by suation of the contributions of all dislocation systes in a region with radius R around that point [], resulting in int e µ br r = '! ( & ) % = % GND i pl r % r % ( n s s s s n s n s + n n n + & n p p ) where the vectors s r and n r are in the direction of the Burgers vector and slip plane noral respectively and p r is r r r defined as p = s n, i.e. the dislocation line vector for an edge dislocation. For the field of screw dislocations the stress field is given by () ()

20 Philosophical Magazine & Philosophical Magazine Letters Page of int s µ br r = '! % = % GND r % r % r % r % ( n s p n p s + p s n + p n s ) r r r where p = s n is now perpendicular to the dislocation line direction (since the Burgers vector is parallel to the dislocation line). Note that only a non-zero gradient of the GND densities causes a non-vanishing contribution. To calculate the back stress, it is necessary to know the distribution of the dislocation densities for all individual slip systes. These densities can be obtained fro the slip gradients in the aterial. Since the two phases for a coherent interface this can be done on the slip syste level [,]. Slip gradients in the direction of the slip will be accoodated by edge dislocations while slip gradients perpendicular to the slip direction will be accoodated by screw dislocations. For the edge dislocations ( = ) the GND densities are obtained fro the slip gradients by % % % %! GND =! GND 0 ' s () r r, () b and for the screw dislocations ( = ) by r r r r (' p + ' ) % %! GND =! GND,0 + p b The screw dislocation densities are the result of the cobined effect of the slip gradients on the two available slip planes and, as given in Table. The initial values of the GND densities, for pre-deforation effects, if necessary. %! GND, 0 (), can be used to account Since the real deforation distribution in the unit cell is siplified by assuing unifor deforation inside each region, gradients in slip are captured through discrete steps in between regions only. This is illustrated in Figure, where the solid curve represents the expected distribution of plastic slip and the set of horizontal solid lines the piecewise unifor approxiation. The GND density distribution corresponding to the real deforation is approxiated by the dashed line. The gradients in the dislocation density and slip, as used in the equations () to (), are replaced by their piece-wise discrete analogons. For exaple, when defined relative to a x,y,z-coordinate syste, the gradient in GND density can be written as r % % $! GND '! GND = l r n x for the interface regions with their noral in the x-direction. In this relation GND is the difference in GND density between both sides of the region and l is the width of the region. No gradient in y- or z-direction is present in these regions. ()

21 Page of Philosophical Magazine & Philosophical Magazine Letters [Insert Fig. about here] Further, the slip gradient is assued to be accoodated by the interface regions only, which eans that the total slip difference between the atrix and a precipitate ' is distributed over the two interface regions in between both bulk regions. Moreover, it is assued that the GND densities increase (or decrease) linearly fro zero in the - and '-regions to a axiu (or iniu) value at the boundary between the constituents of the interface regions (see Figure ). According to the equations () and () the GND density is proportional to the gradient in plastic slip. This gradient is based on the slip difference between the bulk - and '-regions. Due to the assuption of a linear variation of GND density inside a region, the GND density is the only relevant quantity whose distribution is not unifor inside a region. This assuption is necessary since only a gradient in GND density induces a back stress, but also physically ore sound than a unifor GND density. Finally, Figure also shows that the two interface regions on either side of a atrix or precipitate region behave identically, both in ters of plastic deforation and in ters of GND density gradients (which deterine the back stress). This otivates the reduction of the nuber of interface regions in the odel that was entioned in section... Model suary The coplete odel as described in sections, and is suarized in

22 Philosophical Magazine & Philosophical Magazine Letters Page 0 of Table. [Insert Table about here] Application The fraework described in the previous sections has been applied to the single crystal Ni-base superalloy CMSX- to deonstrate the effect of strain gradients on the echanical response. First, the deterination of the odel paraeters is discussed and siulated tensile and creep curves are copared to experiental results. Then, the contributions of the Orowan threshold and the strain gradient induced back stresses to the observed size effects are deonstrated and the siulated size effects due to a change in the icrostructural diensions are copared to experiental results. Finally, a real ulti-scale analysis is perfored, showing the effect of a change in icrostructural diensions on the creep strain accuulation in a gas turbine blade. The lattice constants of the and '-phase at 0 o C are 0.0 n and 0. n respectively, which leads to an unconstrained isfit of -. x 0 -, see equation (). The odel paraeters used for CMSX- are given in 0

23 Page of Philosophical Magazine & Philosophical Magazine Letters Table. The paraeters, k, b and µ are physical quantities with a fixed value, obtained fro [0,]. The atrix phase paraeters & and in the slip law, equation (), and c, y 0 c, SSD,0, K, C soft, Q and p in relations () to () for the slip resistance and the SSD density evolution, deterine the echanical behaviour for a fixed icrostructure. Their values were obtained by calibrating the odel to the experiental results [] shown in Figure, using a least-squares fitting ethod. The paraeters n and R and the relative width of the atrix phase interface layer deterine the aterial size dependence. The value of n deterines the relative strength of the Orowan threshold and the radius of the dislocation influence region R quantifies the agnitude of the back stress. The values for R and n were obtained by fitting the odel to the two endpoints (only two points) of the experientally deterined size dependence curve in Figure, using a least-squares fitting ethod. The obtained value for the radius of the dislocation influence region R is significantly saller than the rather large value of. µ that Evers et al. [] obtained by fitting their odel to results on pure copper. The value used here is ore realistic for the present application, since it is in the sae order as the diensions of the atrix phase. Moreover, several studies [-] recently showed that R should be in the order of the dislocation spacing, which is equivalent to the inverse square root of the dislocation density. In our odel, dislocation densities develop fro an initial value of x 0 to values up to 0 -, which corresponds to R-values ranging fro 0 to 0 n. This is in the sae order of agnitude as the resulting value for R. The width of the precipitate part of the interface layer does not affect the size dependence, because no plastic deforation occurs in the precipitate. Therefore this width can be chosen freely in between soe liits, e.g. a physically acceptable fraction of the precipitate size. Increasing the width of the atrix interface layer decreases the slip gradients and the resulting back stress and therefore diinishes the size dependence. But on the other hand it increases the volue fraction of the interface regions, which results in stronger size effects. An interfacial width of 0 % of the atrix phase width proved to yield the best coproise between these two counteracting phenoena. This eans that the interface effects are acting in a boundary zone with a characteristic size of % of the total channel width, located on each side of the channel. This is very close to the value of % presented by Busso et al. [] for the noralized channel width that contains the strain gradients in their FE unit cell analysis. [Insert Table about here]

24 Philosophical Magazine & Philosophical Magazine Letters Page of Siulation results In this subsection the odel capabilities are deonstrated by coparing siulated tensile and creep curves to experiental results. The odel has been ipleented in a finite eleent (FE) code. Tensile tests at 00 and 0 o C at strain rates of 0 -, 0 - and 0 - s - and creep tests at 0 o C and, and MPa are siulated by using an FE odel with only a single eleent. The results are shown in Figure, together with experiental results for CMSX- []. [Insert Fig. about here] The results in Figure deonstrate that the present fraework is able to siulate the real aterial response adequately. For the tensile curves, especially the steady-state stress levels correspond well to the experiental values, whereas the deviations are soewhat larger at the initial yielding stage of the curves. This is due to the use of the phenoenological description of the softening behaviour. The siulated creep curves describe the aterial behaviour quite well for the priary and secondary creep regie. The tertiary regie is associated with precipitate shearing and icrostructural degradation. Since these echaniss are not included in the present fraework, the odel is not able to accurately siulate the aterial response for this part of the creep curve. To deonstrate the contributions of the individual unit cell regions to the acroscopic response, the evolution of the effective stresses and plastic strains on the icro-level during one of the tensile tests in Figure is plotted in Figure and Figure. [Insert Fig. about here] Figure a shows that the regions have different starting values, which is caused by the isfit stress. In the precipitate (both bulk and interface) regions the tensile isfit stress increases the effective stress, while the copressive isfit stress in the atrix regions parallel to the applied load decreases the effective stress. As the isfit stresses only occur in the plane of the / -interface, they do not affect the stresses in the load direction in the regions that are noral to the applied load. The figure also shows that the isfit stresses quickly disappear as soon as the plastic deforation starts. After that, the stresses in all bulk regions equal the acroscopic stress, as is required by the Sachs interaction law. Also, the average value of the atrix and precipitate interface regions

25 Page of Philosophical Magazine & Philosophical Magazine Letters equals the acroscopic stress, but the strain gradient related back stresses cause a large difference between the two sides (atrix and precipitate) of the interface regions, especially for the regions parallel to the applied load. Finally, the results for the bulk and interface unit cell regions in Figure a are used to calculate the volue averaged values for the different phases (atrix and precipitate) in the aterial, as is shown in Figure b. [Insert Fig. about here] Figure shows the evolution of plastic strain in the different regions. There is a difference in plastic deforation rate between the atrix bulk regions parallel and noral to the applied load. This is caused by the different (initial) stress levels and the resulting differences in evolution rate of the slip resistance. The plastic flow in the interface regions is liited due to the developent of strain gradient related back stresses that reduce the effective stress. The precipitate bulk and interface regions are absent in this figure, since they only defor elastically. Finally, the evolution of the dislocation densities is illustrated in Figure, which copares the SSD and GND densities on a specific slip syste in both bulk and interface regions at three stages during a tensile test. This shows that in the bulk regions, where no GNDs are present, the SSD density increases with a factor two during the test. In the interface regions strain gradients develop, which are accoodated by a rapidly increasing GND density. The resulting back stresses reduce the effective stress and therefore lower the slip rate. Consequently, the SSD density hardly increases in these regions. [Insert Fig. about here]. Strain gradient effects and size dependence Nickel-base superalloys show a clear size dependence, which eans that the echanical behaviour changes when proportionally increasing or decreasing the icrostructural diensions while keeping all volue fractions constant. The present fraework is able to siulate these icrostructural size effects. There are two essential contributions that ake the odel response size dependent. Firstly, the Orowan threshold stress is size dependent, since it is inversely proportional to the -channel width h. Secondly, the GND density is size dependent, because it is related to strain gradients. Reducing the icrostructural diensions will increase the strain gradients and consequently the GND densities. GND densities in their turn contribute to the

26 Philosophical Magazine & Philosophical Magazine Letters Page of slip resistance and, through their gradients, govern the back stress. The Orowan threshold is incorporated in several existing superalloy odels [0-], whereas only strain gradient effects are present in the two FE unit cell approaches [-] discussed before. This section will deonstrate the necessity and possibility of including both ingredients in superalloy constitutive odels. The odel paraeters fro

27 Page of Philosophical Magazine & Philosophical Magazine Letters Table lead to an initial slip resistance (s = MPa) which is lower than the Orowan stress ( or = MPa). This eans that the Orowan threshold is the decisive threshold in this case. Downsizing the coplete unit cell by a factor or increases the Orowan threshold by the sae factor. The GND densities in the interface regions are proportional to the gradients in the plastic slip. Changing the interfacial width by increasing or decreasing the unit cell diensions affects the gradients and hence the GND densities. Consequently, the aterial response changes since the GNDs contribute to the slip resistance and constitute the source of back stresses. The unit cell diensions were varied to quantify these size effects. Figure 0 shows the stress-strain curves at a rate of 0 - s - for the reference case, for which the icrostructural diensions used are L = 00 n and h = 0 n, and for three other cases with all unit cell diensions ultiplied by a factor 0., 0. and. In these siulations both the Orowan effect and the GND effects are included. [Insert Fig. 0 about here] To separate the effects of GNDs and the Orowan threshold, the internal stresses were reoved fro the odel by setting all GND densities to zero. The resulting stress-strain curves are shown in Figure for the reference case and for three other unit cell sizes, where the observed effects are now due to the change of the Orowan threshold only. This figure shows that the shape of the curves for the different unit cell sizes reains the sae, while the axiu stress again shifts to a higher level with a decreasing atrix channel width. [Insert Fig. about here] Coparison of the curves in Figure 0 and Figure shows that their shape is identical for sall strains (<. %), but quite different for larger strains. Also, the yield points for the curves in Figure 0 are higher than those in Figure. The increased slip resistance and back stresses that develop due to increasing internal strain gradients reduce the slip rates. Hence the aterial stress response is increasing. This effect is stronger when the icrostructural diensions decrease below the values of the reference case, i.e. saller is stronger. Finally, the siulated size effects were copared to experiental results (Figure ). Duhl [] easured the change of the steady-state flow stress at different precipitate sizes for PWA0, a siilar nickelbase superalloy with a high precipitate volue fraction. The steady-state flow stresses were noralised by the values for the reference cases to enable a direct coparison in Figure. Although the experiental results

28 Philosophical Magazine & Philosophical Magazine Letters Page of were deterined at 0 o C and the present odel was calibrated for 00 o C, the observed size effect could be siulated quite well. In the siulations, the steady-state flow stress was defined as the stress level at the end of the curve (at % total strain). During the deforation, the slip resistance and back stress evolve until the plastic strain rate equals the externally applied strain rate. Fro that point on the stress has attained a steady-state value. In the ajority of the siulations this point was reached before a strain level of %. Figure also shows the siulated size dependence for the odel without strain gradient effects, where the Orowan threshold is the only cause of size effects. For large icrostructural diensions, the curve is alost identical to the strain gradient curve, but for sall diensions there is a significant difference. At these diensions the plastic strain gradients play an iportant role and the aterial size dependence cannot be described by the Orowan effect only. Busso et al. [] perfored the sae siulations with their gradient-dependent unit cell finite eleent odel, consisting of about 00 finite eleents. The present odel with only 0 unit cell regions predicts the size effects at least as good as the odel by Busso et al. [], but with a considerably lower coputational effort. [Insert Fig. about here]. Coputational efficiency and ulti-scale analysis The present fraework is developed specifically to be used in a ulti-scale analysis on real gas turbine coponents. Therefore, it is essential that the odel is coputationally efficient. This efficiency is largely deterined by the odel s level of detail and the associated nuber of internal variables. Even though a direct coparison between different odelling approaches in ters of coputational efficiency is difficult (in view of the any sall or large differences in odel capabilities and assuptions), an attept is ade in Table. The left-hand side of the table provides inforation about the odel characteristics and capabilities, also classified by one or two + or signs. In the right-hand side of the table the coputational efficiency is quantified. The first colun provides the nuber of degrees of freedo (d.o.f.) to be solved to obtain the local stress distribution for a given plastic strain distribution. For the analytical odels this is the nuber of equations, for the FE odels three (nodal d.o.f.) ties the nuber of nodes, both for one aterial point. Then the nuber of internal variables per aterial point is specified and the tie integration ethod is entioned. For the present odel, the tie integration is fully explicit, both on the local (unit cell) and global (acro) level. Finally, the overall coputational efficiency is classified by one or two + or signs.

29 Page of Philosophical Magazine & Philosophical Magazine Letters [Insert Table about here] Table shows that the different approaches can be classified in three groups: traditional single phase odels, analytical unit cell odels and finite eleent unit cell odels, requiring sall, ediu and very large coputational effort, respectively. Within the groups the coputational effort is coparable, but the level of detail and odel capabilities differ considerably. The present odelling approach can be considered as the ost extensive ethod aongst the analytical unit cell odels, where it is the only odel including strain gradient effects. To clearly ephasize the coputational efficiency of the present fraework and to deonstrate the ulti-scale capabilities, the proposed ulti-scale odel was applied to a finite eleent odel of an aero-engine low pressure turbine blade discretized with hexagonal (-noded) eleents and 0 nodes. The proposed unit cell odel was ipleented in a user-subroutine of the coercial finite eleent code MSC.Marc. During the analysis, it is solved for each integration point in the turbine blade odel. A short-ter creep analysis, divided into tie steps, was perfored in only half an hour on a desktop PC. The teperature and stress distribution used in the creep analysis are shown in Figure a and b respectively. Then the creep strain accuulation in a coponent with a reference icrostructure (L = 00 n, h = 0 n) can be copared to the accuulation in a blade with a coarsened icrostructure (L = 000 n, h = 0 n). The results are shown in Figure c and d. The evolution of the creep strain in tie is plotted in Figure, which shows that the creep strain rate in the coponent with the coarsened icrostructure is considerably higher than in the blade with the reference icrostructure. [Insert Fig. about here] [Insert Fig. about here] This analysis clearly deonstrates that the proposed fraework can be used efficiently in a ulti-scale approach. And it also reveals the relevance, for the engineering practice, of including strain-gradient effects in superalloy echanical odels.

30 Philosophical Magazine & Philosophical Magazine Letters Page of Conclusions Strain gradient effects are incorporated in a newly developed efficient crystal plasticity fraework for nickelbase superalloys with the following innovative characteristics: The proposed unit cell contains special interface regions, in which the plastic strain gradients are located. In these interface regions strain gradient induced back stresses will develop as well as stresses due to the lattice isfit between the two phases. The liited size of the unit cell and the icroechanical siplifications, which condense the governing equations to 0 uniforly deforing regions, ake the fraework particularly efficient in a ulti-scale approach. The unit cell response is deterined nuerically on a aterial point level within a acroscopic FE code, which is coputationally uch ore efficient than a detailed FE-based unit cell discretization. The aterial constitutive behaviour is siulated by using an extension of an existing non-local strain gradient crystal plasticity odel for the atrix aterial. In this odel, non-unifor distributions of geoetrically necessary dislocations (GNDs), induced by strain gradients in the interface regions, affect the hardening behaviour. Further, the odel has been odified here for the present two-phase aterial by adding a threshold ter related to the Orowan stress to the hardening law. Continuous dislocation densities and slip gradients, as typically used in the FE forulation, are approxiated here by piecewise unifor fields for application in an efficient unit cell approach. The odel was applied to the Ni-base superalloy CMSX-, where it proved to be capable of accurately siulating the experientally observed change of the steady-state flow stress with varying icrostructural diensions. Further, the ulti-scale capability was deonstrated by applying the odel in a gas turbine blade finite eleent analysis. Therefore, it can be concluded that the proposed fraework is able to describe the aterial response and size effects to a level of detail siilar to coplex FE unit cell odels, while being coputationally uch ore efficient. Acknowledgeents The authors want to acknowledge the National Aerospace Laboratory NLR for facilitating this research and the Ministry of Defence for funding part of this research under contract NTP N0/. References [] J.W. Hutchinson, Int. J. Sol. Struct. (000).

31 Page of Philosophical Magazine & Philosophical Magazine Letters [] M.E. Gurtin, J. Mech. Phys. Sol. 0 (00). [] P. Cerelli and M.E. Gurtin, Int. J. Sol. Struct. (00). [] M.E. Gurtin and L. Anand, Int. J. Plast. (00). [] M.E. Gurtin, L. Anand and S.P. Lele, J. Mech. Phys. Sol. (00). [] E.P. Busso, F.T. Meissonnier and N.P. O'Dowd, J. Mech. Phys. Sol. (000). [] F.T. Meissonnier, E.P. Busso and N.P. O'Dowd, Int. J. Plast. 0 (00). [] Y.S. Choi, T.A. Parthasarathy and D.M. Diiduk, Mat. Sci. Eng. A (00). [] S.J. Moss, G.A. Webster and E. Fleury, Metall. Mat. Trans. A A (). [0] R.N. Ghosh, R.V. Curtis and M. McLean, Acta Metall. Mat. (0). [] F. Hanriot, G. Cailletaud and L. Rey, in High Teperature Constitutive odeling - Theory and application, edited by (ASME, ). [] E.H. Jordan, S. Shi and K.P. Walker, Int. J. Plast. (). [] Z.F. Yue, Z.Z. Lu and C.Q. Zheng, Theor. Appl. Fract. Mech. (). [] L.-M. Pan, B.A. Shollock and M. McLean, Proc. R. Soc. Lond. A (). [] Z. Yue and Z. Lu, J. Mat. Sci. Techn. (). [] E.P. Busso, N.P. O'Dowd and R.J. Dennis, in Proc. of th IUTAM syposiu on creep in structures, edited by Murakai, S. and Ohno, N. (Kluwer Acadeic, Japan, 000). [] M.A. Rist, A.S. Oddy and R.C. Reed, Scr. Mat. (000). [] R. Daniel, T. Tinga and M.B. Henderson, in Proc. of Materials for Advanced Power Engineering, edited by Lecote-Beckers, J., Carton, M., Schubert, F., and Ennis, P. J. (Forschungszentru Jülich, Jülich, Gerany, 00). [] D.W. MacLachlan, G.S.K. Gunturi and D.M. Knowles, Cop. Mat. Sci. (00). [0] G. Cailletaud, J.L. Chaboche, S. Forest and L. Rey, Rev. Metall. 00 (00). [] J.D. Eshelby, Proc. R. Soc. Lond. A (). [] J.R. Willis, Adv. Appl. Mech. (). [] P. Ponte Casteñeda and P. Suquet, Advances in Applied Mechanics (). [] A. Bensoussan, J.L. Lionis and G. Papanicolaou, Asyptotic analysis for periodic structures (North- Holland, Asterda, ). [] E. Sanchez-Palencia, Non-hoogeneous edia and vibration theory, Lecture notes in physics (Springer, Berlin, 0).

32 Philosophical Magazine & Philosophical Magazine Letters Page 0 of [] C. Miehe, J. Schröder and J. Schotte, Coputer Methods in Applied Mechanics and Engineering (). [] J.C. Michel, H. Moulinec and P. Suquet, Cop. Meth. Appl. Mech. Eng. 0 (). [] V.G. Kouznetsova, M.G.D. Geers and W.A.M. Brekelans, Int. J. Nu. Meth. Eng. (00). [] L.P. Evers, D.M. Parks, W.A.M. Brekelans and M.G.D. Geers, J. Mech. Phys. Sol. 0 0 (00). [0] B. Fedelich, Cop. Mat. Sci. (). [] B. Fedelich, Int. J. Plast. (00). [] E.P. Busso and F.A. McClintock, Int. J. Plast. (). [] J. Svoboda and P. Lukas, Acta Mat. (). [] T. Kuttner and R.P. Wahi, Mat. Sci. Eng. A (). [] J. Svoboda and P. Lukas, Acta Mat. (000). [] L.P. Evers, W.A.M. Brekelans and M.G.D. Geers, J. Mech. Phys. Sol. (00). [] L.P. Evers, W.A.M. Brekelans and M.G.D. Geers, Int. J. Sol. Struct. 0 (00). [] T.M. Pollock and A.S. Argon, Acta Metall. Mat. (). [] V. Sass and M. Feller-Kniepeier, Mat. Sci. Eng. A (). [0] S.S.K. Gunturi, D.W. MacLachlan and D.M. Knowles, Mat. Sci. Eng. A (000). [] T. Link, A. Epishin, U. Brückner and P. Portella, Acta Mat. (000). [] N. Miura, Y. Kondo and N. Ohi, in Proc. of Superalloys 000, edited by Pollock, T. M., Kissinger, R. D., and Bowan, R. R. (The Minerals, Metals & Materials Society, 000). [] R. Srinivasan, G. Eggeler and M.J. Mills, Acta Mat. (000). [] D.W. MacLachlan, L.W. Wright, G.S.K. Gunturi and D.M. Knowles, Int. J. Plast. (00). [] C.M.F. Rae, N. Matan and R.C. Reed, Mat. Sci. Eng. A 00 (00). [] Q.Z. Chen and D.M. Knowles, Mat. Sci. Eng. A (00). [] S.C. Prasad, I.J. Rao and K.R. Rajagopal, Acta Mat. (00). [] J.A.W.v. Doelen, D.M. Parks, M.C. Boyce, W.A.M. Brekelans and F.P.T. Baaijens, J. Mech. Phys. Sol. (00). [] M.F. Ashby, Phil. Mag. (0). [0] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis and Y. Brechet, Sol. St. Phen. & (). [] C. Yuan, J.T. Guo and H.C. Yang, Scr. Mat. (). 0

33 Page of Philosophical Magazine & Philosophical Magazine Letters [] E.P. Busso, F.T. Meissonnier, N.P. O'Dowd and D. Nouailhas, J. Phys. IV France (). [] D.A. Porter and K.E. Easterling, Phase transforations in etals and alloys (Chapan & Hall, London, UK, ). [] C.J. Bayley, W.A.M. Brekelans and M.G.D. Geers, Int. J. Sol. Struct. (00). [] J. Lecote-Beckers, M. Carton, F. Schubert and P.J. Ennis, Materials for advanced power engineering 00 (Forschungszentru Jülich, Jülich, Gerany, 00). [] I. Groa, F.F. Csikor and M. Zaiser, Acta Mat. (00). [] M.G.D. Geers, W.A.M. Brekelans and C.J. Bayley, Mod. Si. Mat. Sci. Eng. S (00). [] A. Roy, R.H.J. Peerlings, M.G.D. Geers and Y. Kasyanyuk, Mat. Sci. Eng. A in press (00). [] D.N. Duhl, in Superalloys II: High teperature aterials for aerospace and industrial power, edited by C.T. Sis, N.S. Stoloff and W.C. Hagel (John Wiley and Sons Ltd, New York, ).

34 Philosophical Magazine & Philosophical Magazine Letters Page of Tables Table List of indices and vectors for dislocation densities and slip systes in an FCC etal. Dislocation density Slip syste Slip direction type s r Slip plane noral n r edge ½ [I0] [III] edge ½ [I0] [III] edge ½ [0I] [III] edge ½ [0] [I] edge ½ [I0I] [I] edge ½ [0I] [I] edge ½ [II0] [I] edge ½ [0I] [I] edge ½ [0] [I] 0 edge 0 ½ [I0] [I] edge ½ [0] [I] edge ½ [0II] [I] screw - or ½ [II0] [I] or [I] screw or - ½ [I0I] [I] or [I] screw - or ½ [0II] [I] or [I] screw or -0 ½ [I0] [III] or [I] screw or - ½ [I0] [III] or [I] screw or - ½ [0I] [III] or [I] Table C elastic tensor coponents for CMSX- at 0 o C []. -atrix '-precipitate C (GPa) 0.. C (GPa).. C (GPa)

35 Page of Philosophical Magazine & Philosophical Magazine Letters Table Overview of odel equations unit cell region(s) Equilibriu i i Strain averaging all f = () Sachs interaction ',,, Modified Sachs interaction Strain copatibility Traction continuity Constitutive odel Internal stresses Misfit Back stress i tot tot ' = = = = p p I,I,I, p p p Ik Ik Ik Ik Ik Ik f f = ( f + f ) p tot I,I,I p p I,I,I, p I,I,I p p I,I,I, p I,I,I all Matrix constitutive odel, Effective stress I,I all p I,I,I, I,I,I p,i p,, p k I p tot eq.nr. () + () r k r k I r k r k k ( I n n ) = ( I n n ) r = r Ik k Ik k n n () i i isfit () i constituti ve box () = B : () i isfit int int ( ) b = e + s int e int s eff i isfit µ br r = '! % n s s s s n r r r r r r r % r % n s p n p s r % r % + p s n + p n s ( ) GND % % % % % % % % % & % = s n s + n n n + & n p p µ br = ' r! % = isfit % GND b r r r () () () = + (0) n eff eff & = & 0 exp sign ( ) or eff () Slip rate,,, I,I,I s Slip resistance p,,, = +! + SSD Q s cµ b! SSD! GND Csoft I,I,I ~ exp! kt SSD Precipitate constitutive odel p p p ',I,I, no plastic slip, purely elastic - Dislocation densities GND densities SSD evolution I p p I,I,I, I,I,I I,I p,i,,, r r b r % % r r r! GND =! GND,0 + (' p + ' p ) (screw disl.) b % % % %! GND =! GND, 0 ' s (edge dislocations) b L &!! & SSD = y c SSD () () () () () ()

36 Philosophical Magazine & Philosophical Magazine Letters Page of Table Model paraeters for the atrix phase of CMSX-. Model paraeter Sybol Value Unit Used in equation Reference slip rate & 0. x 0-0 s - () Rate sensitivity exponent Orowan threshold n () Rate sensitivity exponent slip resistance. () Reference activation energy Q. x 0-0 J () Strength paraeter c 0.0 () Shear odulus µ 00. GPa () Burgers vector length b 0. n (),(),() Paraeter in Orowan stress 0. () Critical annihilation length y c. n () Initial SSD density SSD,0.0 x 0 - () Material constant K () Softening constant C soft 0 MPa () Softening exponent p 0. () Radius of dislocation influence region R. n (),() Table Coparison of different odelling approaches. Author(s) Characteristics / level of detail / capabilities Coputational efficiency # d.o.f. # int. vars tie integr. Traditional odels - single phase odel no icrostructural details Svoboda and - regions unit cell - not specified + Lukas [] - analytic stress calculation - dislocation based creep law in / - for <00> only - isotropic elastic properties Fedelich [] - regions unit cell +, using Runge-Kutta + - Fourier series approach for stress calculation - dislocation based plasticity in / structural tensor ( KJ ) coponents Tinga et al. - 0 regions unit cell ++ 0 fully explicit + (present odel) - analytic stress calculation - dislocation based strain gradient plasticity in / elastic precipitate Kuttner and Wahi - el FE odel + ~ not specified - [] - Norton creep law for / Busso et al. [] - el FE odel ++ ~ 000 ~ 0 fully iplicit -- - dislocation based strain gradient plasticity in / elastic precipitate Choi et al. [] - el FE odel - phenoenological strain gradient plasticity in / elastic precipitate ++ ~ 000 ~ 0 cobined explicit / iplicit --

37 Page of Philosophical Magazine & Philosophical Magazine Letters Figure captions Figure Micrograph of a superalloy icrostructure showing the cube-like '-precipitates in a -atrix []. Figure Scheatic overview of the odel, showing (a) the ulti-scale character and (b) the ulti-phase unit cell, consisting of one precipitate ( ), three atrix ( i ) and six double interface(i i ) regions. Figure Definition of the unit cell diensions in the y-z plane cross section. Figure Overview of the interaction between the different levels of the ulti-scale odel. In the acroscopic FE analysis, the usual standard procedure to obtain the stress response for a given deforation is replaced by a esoscopic calculation at the unit cell level. A crystal plasticity (CP) odel yields the relation between the local stress and plastic strain rate for the atrix regions. Figure Overview of gradients in slip and GND density. The solid curved line represents the continuous plastic slip distribution that is expected in the real aterial and the series of straight solid lines the piecewise unifor approxiation. The dashed line represents the GND density distribution. Figure Siulated curves (solid lines) copared to experiental results (arkers) for CMSX-; a) stressstrain curves at strain rates of 0 - s - at 0 o C and 0 - and 0 - s - at 00 o C; b) creep curves at, and MPa at 0 o C. Figure Variation of icro-level effective stresses during a tensile test at 00 o C and a strain rate of 0 - s -. a) stresses in individual unit cell regions; b) stresses in aterial phases, obtained fro cobining the results in a), copared to the acroscopic (unit cell averaged) stress. In all cases the stress coponent in the direction of the applied load is plotted. The sybols and // denote the regions oriented noral and parallel to the applied load, respectively. Figure Plastic strain evolution in the unit cell regions during a tensile test at 00 o C and a strain rate of 0 - s -. The strain coponent in the direction of the applied load is plotted. The acroscopic (unit cell averaged) stress and strain are plotted as a reference. Figure Evolution of the SSD and GND densities during a tensile test at 00 o C and a strain rate of 0 - s -. Values are given for a bulk atrix region and an interface region at three stages of the tensile test. Figure 0 Effect of changing icrostructural diensions on the stress-strain curve at a strain rate of 0 - s -, including the effect of Geoetrically Necessary Dislocations and resulting back stresses. The reference case is L = 00 n and h = 0 n. Figure Effect of changing icrostructural diensions on the stress-strain curve, using the odel without Geoetrically Necessary Dislocations. The reference case is L = 00 n and h = 0 n. Figure Siulated effect of changing icrostructural diensions on the steady-state flow stress for CMSX- at 00 o C. Results for siulations with Orowan effect only and with Orowan + strain gradient effects. Experiental data for PWA0 at 0 o C obtained fro []. Figure Finite eleent analysis results on a gas turbine blade. a) teperature distribution, b) equivalent stress distribution, c) accuulated creep strain for reference icrostructure, d) accuulated creep strain for coarsened icrostructure. Figure Creep strain accuulation in tie for a reference and a coarsened icrostructure.

38 Philosophical Magazine & Philosophical Magazine Letters Page of Figures Figure Micrograph of a superalloy icrostructure showing the cube-like '-precipitates in a -atrix []. a) b) Figure Scheatic overview of the odel, showing (a) the ulti-scale character and (b) the ulti-phase unit cell, consisting of one precipitate ( ), three atrix ( i ) and six double interface(i i) regions. z ½h y I I p w p w ' L I p p I I I L ½h p I I Figure Definition of the unit cell diensions in the y-z plane cross section.

39 Page of Philosophical Magazine & Philosophical Magazine Letters Figure Overview of the interaction between the different levels of the ulti-scale odel. In the acroscopic FE analysis, the usual standard procedure to obtain the stress response for a given deforation is replaced by a esoscopic calculation at the unit cell level. A crystal plasticity (CP) odel yields the relation between the local stress and plastic strain rate for the atrix regions.

40 Philosophical Magazine & Philosophical Magazine Letters Page of Figure Overview of gradients in slip and GND density. The solid curved line represents the continuous plastic slip distribution that is expected in the real aterial and the series of straight solid lines the piecewise unifor approxiation. The dashed line represents the GND density distribution. Stress (MPa) C / 0- s-s C / 0- s-s 0 C / 0- s-s Strain (%) Tie (hrs) a) b) Strain (%) C / MPa 0 C / MPa 0 C / MPa Figure Siulated curves (solid lines) copared to experiental results (arkers) for CMSX-; a) stress-strain curves at strain rates of 0 - s - at 0 o C and 0 - and 0 - s - at 00 o C; b) creep curves at, and MPa at 0 o C.

41 Page of Philosophical Magazine & Philosophical Magazine Letters Stress (MPa) precipitate atrix atrix // interface precipitate // interface precipitate & atrix interface atrix // Strain (%) Stress (MPa) precipitate Strain (%) Figure Variation of icro-level effective stresses during a tensile test at 00 o C and a strain rate of 0 - s -. a) stresses in individual unit cell regions; b) stresses in aterial phases, obtained fro cobining the results in a), copared to the acroscopic (unit cell averaged) stress. In all cases the stress coponent in the direction of the applied load is plotted. The sybols and // denote the regions oriented noral and parallel to the applied load, respectively. Stress (Mpa) acro stress atrix // acro atrix interface atrix // interface atrix acro atrix atrix // Tie (s) Figure Plastic strain evolution in the unit cell regions during a tensile test at 00 o C and a strain rate of 0 - s -. The strain coponent in the direction of the applied load is plotted. The acroscopic (unit cell averaged) stress and strain are plotted as a reference. Plastic strain

42 Philosophical Magazine & Philosophical Magazine Letters Page 0 of Dislocation density ( - ).00E+.00E+.00E+.00E+.00E+.00E+ 0.00E+00 SSD bulk region SSD interface region GND interface region Total strain (%) Figure Evolution of the SSD and GND densities during a tensile test at 00 o C and a strain rate of 0 - s -. Values are given for a bulk atrix region and an interface region at three stages of the tensile test. Stress (MPa) Ref. 0. Ref. Reference Ref. 0 0 Strain (%) Figure 0 Effect of changing icrostructural diensions on the stress-strain curve at a strain rate of 0 - s -, including the effect of Geoetrically Necessary Dislocations and resulting back stresses. The reference case is L = 00 n and h = 0 n. 0

43 Page of Philosophical Magazine & Philosophical Magazine Letters Stress (MPa) Ref. 0. Ref. Reference Ref. 0 0 Strain (%) Figure Effect of changing icrostructural diensions on the stress-strain curve, using the odel without Geoetrically Necessary Dislocations. The reference case is L = 00 n and h = 0 n. Noralised flow stress Experient Orowan + gradients Orowan Precipitate size (µ) Figure Siulated effect of changing icrostructural diensions on the steady-state flow stress for CMSX- at 00 o C. Results for siulations with Orowan effect only and with Orowan + strain gradient effects. Experiental data for PWA0 at 0 o C obtained fro [].

44 Philosophical Magazine & Philosophical Magazine Letters Page of a) b) c) d) Figure Finite eleent analysis results on a gas turbine blade. a) teperature distribution, b) equivalent stress distribution, c) accuulated creep strain for reference icrostructure, d) accuulated creep strain for coarsened icrostructure. Creep strain Tie Reference icrostructure Coarsened icrostructure Figure Creep strain accuulation in tie for a reference and a coarsened icrostructure.