precipitate strengthening in nickel- based superalloys
precipitate strengthening in nickel- based superalloys
precipitate strengthening in nickel- based superalloys
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3768 A. Takahashi et al.Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012coherency stra<strong>in</strong>s, effects of elastic modulus mismatch between the matrix and<strong>precipitate</strong>, and the <strong>in</strong>fluence of <strong>precipitate</strong> crystal structure <strong>in</strong> relation to the matrix.The primary objective of the theory is to provide an accurate evaluation of thecritical resolved shear stress (CRSS), and its dependence on the <strong>in</strong>dividualmechanisms.Hirsch and Kelly proposed an analytical equation for the dependence of theCRSS on the stack<strong>in</strong>g-fault energy, and discussed the <strong>in</strong>fluence of the differencebetween the stack<strong>in</strong>g-fault energy of the matrix and that of the <strong>precipitate</strong> [2].They used an extended dislocation model, where the dislocation was represented bytwo partial dislocations and a stack<strong>in</strong>g-fault <strong>in</strong>-between. Nembach modified thissolution, and developed another equation for accurate evaluation of the CRSS [3].On the other hand, Gerold and Haberkorn pursued an analytical study of coherencystra<strong>in</strong> <strong>strengthen<strong>in</strong>g</strong>, and proposed an equation for its contribution to the CRSS [4].They assumed a spherical <strong>precipitate</strong> with a coherency stra<strong>in</strong>, where the elastic fieldimpedes dislocation motion, either <strong>in</strong>side the <strong>precipitate</strong> or <strong>in</strong> the matrix. All theseanalytical approaches capture the qualitative behavior of the CRSS. However,quantitative evaluation of the CRSS as a result of <strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong>necessitates that the assumptions <strong>in</strong>volved <strong>in</strong> analytical estimates should be removed.In analytical approximations, the dislocation is assumed to be straight, and thedislocation core structure is ignored. Therefore, to provide quantitative evaluationsof <strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong>, we must exam<strong>in</strong>e the <strong>in</strong>fluence of the dislocation l<strong>in</strong>eflexibility, and the role that the dislocation core structure plays dur<strong>in</strong>g its <strong>in</strong>teractionwith <strong>precipitate</strong>s. Analytical equations can then be revised on the basis of detailednumerical calculations, when necessary.One of the <strong>in</strong>terest<strong>in</strong>g applications is the <strong>strengthen<strong>in</strong>g</strong> caused by precipitation <strong>in</strong><strong>nickel</strong>-<strong>based</strong> <strong>superalloys</strong>. These alloys have been extensively <strong>in</strong>vestigated, becauseL1 2 -type long-range ordered <strong>in</strong>termetallic compounds, such as Ni 3 Al, Ni 3 Si andCo 3 Ti, are considered for high-temperature structural applications. Miller observedextensive formation of spherical -<strong>precipitate</strong>s <strong>in</strong> the 0 matrix phase of a <strong>nickel</strong><strong>based</strong>superalloy us<strong>in</strong>g 3-D atom probe measurements [5]. Nemoto et al. measuredthe <strong>in</strong>crease <strong>in</strong> the yield strength of the alloy result<strong>in</strong>g from the formation of-<strong>precipitate</strong>s, and provided a clear picture of the <strong>in</strong>fluence of -<strong>precipitate</strong><strong>strengthen<strong>in</strong>g</strong> on the <strong>in</strong>verse temperature dependence of the yield strength ofthe alloy [6]. Thus, understand<strong>in</strong>g the role of -<strong>precipitate</strong>s on alloy strength, and thedetailed mechanisms <strong>in</strong>volved dur<strong>in</strong>g the <strong>in</strong>teraction between a dislocation and-<strong>precipitate</strong>s is fundamental to the control and improvement of the mechanicalproperties.Recently, computer simulation techniques have provided an opportunity to<strong>in</strong>vestigate the detailed mechanisms of material deformation without many of theassumptions that must be <strong>in</strong>voked <strong>in</strong> order to develop analytical approaches.The dislocation dynamics (DD) computer simulation technique has been developedto study metal deformation from a fundamental perspective, as the method allowssimulation of the collective behavior of dislocation ensembles, and to derive themacroscopic deformation response on the basis of dislocation mechanisms [7].However, the orig<strong>in</strong>al development of the DD technique was <strong>based</strong> on elasticitysolutions for dislocations <strong>in</strong> an <strong>in</strong>f<strong>in</strong>ite medium, without the <strong>in</strong>fluence of external or<strong>in</strong>ternal surfaces. Subsequently, Van der Giessen and Needleman utilized the
Philosophical Magaz<strong>in</strong>e 3769Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012superposition pr<strong>in</strong>ciple to account for the geometry of the material <strong>in</strong> twodimensionalDD simulations [8], employ<strong>in</strong>g the f<strong>in</strong>ite element method (FEM) tosolve the correction problem aris<strong>in</strong>g from the application of the superpositionpr<strong>in</strong>ciple. Weygand et al. extended the method to three-dimensional DDsimulations [9]. More recently, Takahashi and Ghoniem extended the superpositionpr<strong>in</strong>ciple to general three-dimensional dislocation–<strong>precipitate</strong> <strong>in</strong>teraction problems[10]. They derived a set of boundary-volume <strong>in</strong>tegral equations for the correctionproblem, and solved the equations us<strong>in</strong>g the boundary element method (BEM) witha volume <strong>in</strong>tegral term. The method allows us to deal with flexible dislocation l<strong>in</strong>es,and can thus give important <strong>in</strong>formation on the <strong>in</strong>fluence of dislocation shapechange on its <strong>in</strong>teraction with <strong>precipitate</strong>s. Xiang et al. developed a level set methodto simulate the dynamics of dislocations. They simulated the complexdislocation behavior <strong>in</strong> the vic<strong>in</strong>ity of spherical <strong>in</strong>clusions <strong>in</strong>clud<strong>in</strong>g the cross-slipof screw parts of the dislocation [11]. Devicre et al. developed a discrete cont<strong>in</strong>uousmodel (DCM), where the dislocations are represented as eigenstra<strong>in</strong>s <strong>in</strong> thecont<strong>in</strong>uum calculation us<strong>in</strong>g the FEM, and the dynamics of dislocations iscalculated <strong>based</strong> on the FEM result of the stress calculation. Us<strong>in</strong>g the DCMmethod, they calculated and discussed the critical stress for dislocations to move the 0 // 0 channel [12]. Rao et al. <strong>in</strong>vestigated the <strong>in</strong>teraction between dislocations andboth - and 0 -<strong>precipitate</strong>s us<strong>in</strong>g the DD method [13]. They studied the dependenceof the CRSS on the shape and volume fraction of the -<strong>precipitate</strong>s. However, thesestudies did not deal with an extended dislocation core structure, and hence it isclearly necessary to implement a new idea that can <strong>in</strong>corporate the dislocationcore structure <strong>in</strong>to DD simulations. Banerjee et al. developed a generalizedPeierls–Nabarro (GPN) model to determ<strong>in</strong>e the dislocation core structure, tak<strong>in</strong>g<strong>in</strong>to account crystal lattice resistance to slip through ab <strong>in</strong>itio calculations, and alsogeneraliz<strong>in</strong>g the dislocation shape as a 3-D space curve [14]. The ma<strong>in</strong> idea is todescritize the elastic <strong>in</strong>teraction part of the equation with a number of fractionaldislocations hav<strong>in</strong>g small Burgers vectors. The lattice restor<strong>in</strong>g part of the forceequilibrium equation is evaluated tak<strong>in</strong>g the derivative of the -surface energyobta<strong>in</strong>ed by separate ab <strong>in</strong>itio calculations. They applied the method to staticdislocation problems, and <strong>in</strong>vestigated the core structure of dislocation loops <strong>in</strong>f.c.c. metals.The present model for the <strong>in</strong>teraction between dislocations and <strong>precipitate</strong>s hasseveral advantages and a few limitations. First, <strong>in</strong> MD simulations, we cannot clearlyseparate the <strong>in</strong>fluence of the specific mechanisms on precipitation <strong>strengthen<strong>in</strong>g</strong>(e.g. the specific effects of stack<strong>in</strong>g-fault energy, coherency stra<strong>in</strong>, etc.). On the otherhand, <strong>in</strong> the present model, it is very easy to add and delete the <strong>strengthen<strong>in</strong>g</strong>mechanisms, which provides an opportunity to understand the relative magnitude of<strong>strengthen<strong>in</strong>g</strong> mechanisms, as compared to classical precipitation theory. Second, thepresent model is not dependent on the accuracy of the <strong>in</strong>teratomic potential used <strong>in</strong>MD simulations, because lattice resistance to dislocation motion is obta<strong>in</strong>ed fromab <strong>in</strong>itio <strong>in</strong>formation. In particular, when we deal with new types of <strong>precipitate</strong>s, wedo not need to develop a valid <strong>in</strong>teratomic potential for the matrix, <strong>precipitate</strong>, andtheir <strong>in</strong>teraction zone, which is a significant task for each type of <strong>precipitate</strong>. Andf<strong>in</strong>ally, the present model is not limited by the size of the MD simulation box, nor theboundary conditions used, which are usual limitations <strong>in</strong> many MD simulations.
3770 A. Takahashi et al.Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012S<strong>in</strong>ce the – 0 <strong>in</strong>terface is coherent, restructur<strong>in</strong>g of the dislocation core along thismatrix–<strong>precipitate</strong> <strong>in</strong>terface may not be significant. However, the present modelmust be extended to account for weak <strong>in</strong>terfaces between <strong>precipitate</strong>s and the matrixas discussed <strong>in</strong> the work of Shehadeh et al. [15]. To extend the present model toclimb<strong>in</strong>g dislocations (e.g. at high-temperature and under irradiation), we need toconsider coupl<strong>in</strong>g between thermal/irradiation osmotic po<strong>in</strong>t defect forces and glideforces, which will move the dislocation out of the glide plane. However, the speed ofdislocation glide is much faster than climb as a general rule, and thus we wouldexpect that the effects of climb are small. This topic requires future modifications ofthe present model us<strong>in</strong>g k<strong>in</strong>etic <strong>in</strong>formation on defect fluxes generated by thermalor irradiation effects.The objective of this paper is to develop and apply a computational methodthat can simulate the dislocation–<strong>precipitate</strong> <strong>in</strong>teraction process, <strong>in</strong>clud<strong>in</strong>g thedynamics of the dislocation core itself. The development of this new computationalprocedure will be <strong>based</strong> on a comb<strong>in</strong>ation of the Parametric Dislocation Dynamics(PDD) [7], and the GPN model [14]. Us<strong>in</strong>g the computational method, we will thensimulate the <strong>in</strong>teraction between a super-dislocation and a spherical -<strong>precipitate</strong>embedded <strong>in</strong> a 0 matrix of a <strong>nickel</strong>-<strong>based</strong> superalloy. The focus here will be an<strong>in</strong>vestigation of two ma<strong>in</strong> effects: (1) the <strong>in</strong>fluence of the dislocation l<strong>in</strong>e flexibility,and (2) the effects of the dislocation core structure on precipitation <strong>strengthen<strong>in</strong>g</strong>.We will exam<strong>in</strong>e, separately and <strong>in</strong> comb<strong>in</strong>ation, the relative effects of changes <strong>in</strong>the stack<strong>in</strong>g-fault energy between the <strong>precipitate</strong> and matrix, and the <strong>in</strong>fluence ofcoherency stra<strong>in</strong>s on the CRSS. On the basis of computer simulation results, wewill revise the classical theoretical equations for coherency <strong>strengthen<strong>in</strong>g</strong> to accountfor the <strong>in</strong>fluence of the dislocation core structure, and we will show thatclassical analytical results extremely over-estimate the CRSS. F<strong>in</strong>ally, we presentresults for <strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong>, when stack<strong>in</strong>g-fault and coherency<strong>strengthen<strong>in</strong>g</strong> are simultaneously operat<strong>in</strong>g together. In the next section, we presentdetails of the computational method, followed by a description of the <strong>precipitate</strong>dislocationmodel <strong>in</strong> Section 3. The results for stack<strong>in</strong>g-fault and coherencystra<strong>in</strong> <strong>strengthen<strong>in</strong>g</strong> are then presented <strong>in</strong> Section 4, while conclusions are f<strong>in</strong>allydrawn <strong>in</strong> Section 5.2. Computational methodTo determ<strong>in</strong>e the elastic <strong>in</strong>teraction between <strong>precipitate</strong>s and dislocations of an<strong>in</strong>f<strong>in</strong>itesimal-size core, Takahashi and Ghoniem proposed a computational method,which is <strong>based</strong> on a comb<strong>in</strong>ation of the orig<strong>in</strong>al PDD method, and the BEM method[10]. On the other hand, the development of a GPN model by Banerjee et al. enablesexplicit representation of the core structure of dislocations with<strong>in</strong> the PDDframework [14]. Thus, coupl<strong>in</strong>g between these two methods should open the doorto <strong>in</strong>vestigations of the roles played by the dislocation l<strong>in</strong>e flexibility and its corestructure on precipitation <strong>strengthen<strong>in</strong>g</strong>. In the follow<strong>in</strong>g, we develop basic conceptsfor the elastic <strong>in</strong>teraction between dislocations and <strong>precipitate</strong>s. We also show howthe PDD and the GPN methodologies can be comb<strong>in</strong>ed.
Philosophical Magaz<strong>in</strong>e 37712.1. Solution to dislocation–<strong>precipitate</strong> elastic <strong>in</strong>teraction problemConsider an <strong>in</strong>f<strong>in</strong>ite elastic body D, with elastic constants C ijkl , conta<strong>in</strong><strong>in</strong>g N p<strong>precipitate</strong>s and N d dislocations, and subjected to an external applied stress ij 0.The mth <strong>precipitate</strong>, m , has elastic constants Cijkl m , and <strong>in</strong>duces a coherencystra<strong>in</strong> kl m . Follow<strong>in</strong>g Mura [16], the stress <strong>in</strong> an <strong>in</strong>f<strong>in</strong>ite elastic body can be written as(ij 0 þ ij ¼C ijkl " 0 kl þ " kl <strong>in</strong> D Cijkl m " 0 kl þ " kl klm <strong>in</strong> m ,ð1ÞDownloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012where " kl ¼ C 1klij 0 ij , and is the total volume occupied by N p <strong>precipitate</strong>s. S<strong>in</strong>ce it isdifficult to solve Equation (1) directly, the superposition pr<strong>in</strong>ciple will be utilized<strong>in</strong> its solution: 0 ij þ ij ¼ ^ ij þ ~ ij ,where ^ ij is the stress <strong>in</strong> the elastic problem conta<strong>in</strong><strong>in</strong>g dislocations with an externalapplied stress, which is given asð2Þ^ ij ¼ C ijkl " 0 kl þ ^" kl: ð3ÞThe last equation can be solved us<strong>in</strong>g the PDD method. On the other hand, ~ ij is acorrection stress to the <strong>in</strong>f<strong>in</strong>ite medium solution, which should have an exact solutionpresented by Equation (1), when comb<strong>in</strong>ed with Equation (3). Therefore, ~ ij can bedef<strong>in</strong>ed as(~ ij ¼ C ijkl ~" kl <strong>in</strong> D C m " kl kl m þ CmC ijkl " 0 kl þ ~" kl <strong>in</strong> m ð4Þ:ijkl~ijklAccord<strong>in</strong>g to Equation (4), the correction field can be obta<strong>in</strong>ed from the solution ofan <strong>in</strong>homogeneous <strong>in</strong>clusion problem, with an <strong>in</strong>itial stress ðCijklm C ijkl Þð" 0 kl þ ~" klÞ <strong>in</strong>the <strong>precipitate</strong>s. Takahashi and Ghoniem derived boundary and volume <strong>in</strong>tegralequations for such an <strong>in</strong>homogeneity problem (Equation (4)), and solved theboundary and volume <strong>in</strong>tegral equations us<strong>in</strong>g the BEM method, with a volume<strong>in</strong>tegral term. In the <strong>nickel</strong>-<strong>based</strong> superalloy -<strong>precipitate</strong> case, the elastic constantsof both 0 and -phase are nearly the same, and thus the <strong>in</strong>itial stress can be assumedto be zero. Then Equation (4) can be converted from the <strong>in</strong>homogeneous <strong>in</strong>clusionproblem to an <strong>in</strong>clusion problem. Additionally, if the shape of the <strong>precipitate</strong> isassumed to be an ellipsoid, the equation can be solved us<strong>in</strong>g the Eshelbytensor <strong>in</strong>stead of the BEM method, which will drastically decrease the computationaleffort [17].2.2. Parametric dislocation dynamics methodIn the PDD method [7], flexible dislocation ensembles are discretized <strong>in</strong>to a numberof curved dislocation segments. The segments have two edge nodes, and the nodeshave a generalized coord<strong>in</strong>ate vector, which is a comb<strong>in</strong>ation of position and tangentvectors. The curved shape of segments is represented by a cubic spl<strong>in</strong>e function that
3772 A. Takahashi et al.Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012can preserve high-order cont<strong>in</strong>uity conditions. The equation of motion of eachdislocation ensembles is given byZ dQ F k B k r k jdsj ¼0,ð5Þdtwhere F k is the force per unit length of the dislocation, B k is the resistivity matrixand Q k is the generalized coord<strong>in</strong>ate for dislocations, which conta<strong>in</strong>s both positionand tangent vectors at a given node of a dislocation. In this work, F k is def<strong>in</strong>ed asF k ¼ F extkþ F <strong>in</strong>tkþ F selfkþ F pptk, ð6Þwhere F extkis the force produced by the external stress, F <strong>in</strong>tkis the <strong>in</strong>teractionforce with other dislocation loops, F selfkis the self-force, and F pptkis the force due tothe elastic field generated by -<strong>precipitate</strong>s. Integrat<strong>in</strong>g Equation (5), we canobta<strong>in</strong> the matrix form of the equation of motion for dislocation ensembles asK ijdQ jdt ¼ F i:The dislocation velocity can be calculated by solv<strong>in</strong>g the above equation, and its time<strong>in</strong>tegration simulates the dynamical behavior of dislocations <strong>in</strong> response to externaland <strong>in</strong>ternal stress fields. Details of the method can be found elsewhere [7].2.3. Elastic field of <strong>in</strong>coherent spherical c-<strong>precipitate</strong>sS<strong>in</strong>ce it is experimentally observed that most -<strong>precipitate</strong>s <strong>in</strong> Ni-<strong>based</strong> <strong>superalloys</strong>have spherical or cuboidal shapes, we thus assume that the shape of the -<strong>precipitate</strong>is spherical, and utilize an analytical solution to the elastic field generated by the-<strong>precipitate</strong> [17]. The lattice constant of the -<strong>precipitate</strong> a 0¼ 0:352 nm differs fromthat of the 0 -phase a 00¼ 0:357 nm. This difference provides a coherency stra<strong>in</strong> 2<strong>in</strong> the -<strong>precipitate</strong>, which is def<strong>in</strong>ed by =a 02¼ a 0a 00ð7Þ0 : ð8ÞAs a result of Equation (8), the coherency stra<strong>in</strong> is 0.0143. When the position of<strong>in</strong>terest is <strong>in</strong>side the <strong>in</strong>clusion, the analytical solution to an elastic problem of aspherical <strong>in</strong>clusion with a coherency stra<strong>in</strong> 2 is given by r ¼ t ¼1 þ 2, ð9Þ3ð1 Þwhere is Poisson’s ratio, and r and t are the stra<strong>in</strong> <strong>in</strong> the radial and tangentialdirections, respectively. On the other hand, when the po<strong>in</strong>t of <strong>in</strong>terest is outside the<strong>in</strong>clusion, the stra<strong>in</strong>s are given by r ¼ 2 1 þ 3 1 r 3 2 t ¼ 1 1 þ a 33 1 r 3 2,a 3ð10Þ
Philosophical Magaz<strong>in</strong>e 3773where a is the radius of the spherical <strong>in</strong>clusion, and r is the distance between thecenter of <strong>in</strong>clusion and the po<strong>in</strong>t of <strong>in</strong>terest. The stress tensor at the po<strong>in</strong>t can thenbe easily calculated us<strong>in</strong>g Hooke’s law, once the stra<strong>in</strong> tensor is available.Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 20122.4. Dislocation core structureIn the classical dislocation dynamics method, the character of dislocation l<strong>in</strong>es isdef<strong>in</strong>ed only by the l<strong>in</strong>e sense vector and the Burgers vector. However, thedislocation core structure is atomistic <strong>in</strong> nature, and hence cannot be accounted for<strong>in</strong> the classical dislocation dynamics method. To overcome this difficulty and to<strong>in</strong>corporate atomistic <strong>in</strong>formation <strong>in</strong>to dislocation dynamic, the generalization of theorig<strong>in</strong>al Peierls–Nabarro (P-N) model [18,19] by Banerjee et al. will be used [14].In the P-N model, the dislocation core structure is described with a distribution ofdisplacements with<strong>in</strong> the core, and is determ<strong>in</strong>ed by balanc<strong>in</strong>g the elastic <strong>in</strong>teractionenergy between the displacements <strong>in</strong> the dislocation core and the lattice restor<strong>in</strong>genergy. Orig<strong>in</strong>ally, the force equilibrium equation of the P-N model is solved forsimple straight dislocations <strong>in</strong> an <strong>in</strong>f<strong>in</strong>ite medium. However, the complicated shapeof three-dimensional dislocations makes a direct analytical solution rather difficult,and one must resort to numerical methods. The GPN model of Banerjee et al. isdesigned to determ<strong>in</strong>e dislocation core structures of complex shapes of dislocations.The basic idea <strong>in</strong> the model is to discretize the cont<strong>in</strong>uous distribution ofdisplacement <strong>in</strong> the dislocation core by a number of fractional dislocations withfractional Burgers vectors. Thus, the distribution of fractional dislocationscorresponds to equi-displacement contours with<strong>in</strong> the dislocation core. Then theforce equilibrium equation of the P-N model can be written asp i ¼ð1b X N fÞ 2N fj6¼i1x j x i, ð11Þwhere p i is the lattice restor<strong>in</strong>g stress on a fractional dislocation i, is the elasticshear modulus, b is the Burgers vector, N f is the number of fractional dislocations,and x i is the position of the i-th fractional dislocation. From Equation (11), theelastic <strong>in</strong>teraction term <strong>in</strong> the P-N model is replaced with the elastic <strong>in</strong>teractionbetween fractional dislocations. Therefore, the implementation of the P-N modelwith<strong>in</strong> the dislocation dynamics framework is straightforward. In the dislocationdynamics method, calculation of the elastic <strong>in</strong>teraction between dislocations is anessential part, which can account for the elastic <strong>in</strong>teraction between fractionaldislocations. The lattice restor<strong>in</strong>g stress can be calculated by tak<strong>in</strong>g the derivative ofthe -surface energy obta<strong>in</strong>ed by separate atomistic calculations, such as ab <strong>in</strong>itio, oralternatively, reliable <strong>in</strong>teratomic potentials, and the force is simply added toEquation (6).2.5. Rigid dislocation dynamicsTo clarify the <strong>in</strong>fluence of dislocation l<strong>in</strong>e flexibility on its <strong>in</strong>teraction with<strong>precipitate</strong>s, we need to modify the equations of motion so as to constra<strong>in</strong>
3774 A. Takahashi et al.dislocations not to bend dur<strong>in</strong>g the <strong>in</strong>teraction process. In other words, constra<strong>in</strong>eddislocation dynamics will allow dislocations to move, with additional constra<strong>in</strong>ts ontheir shapes. Let us consider a dislocation ly<strong>in</strong>g on an x–z slip plane, and glides <strong>in</strong> thex-direction. To <strong>in</strong>sure that the dislocation l<strong>in</strong>e rema<strong>in</strong>s straight, additionalconditions are applied to Equation (5):Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012dP ydt ¼ dP zdt ¼ dT xdt ¼ dT ydt ¼ dT zdt ¼ 0:By the application of the above constra<strong>in</strong>t, the velocity of the dislocation <strong>in</strong> bothy- and z-directions are zero, and the shape is constra<strong>in</strong>ed to be rigidly straight.Therefore, we need to consider only the velocity of the dislocation <strong>in</strong> the x-direction.Moreover, the velocity of the dislocation <strong>in</strong> the x-direction is identical along theentire dislocation l<strong>in</strong>e, which reduces the degrees of freedom of the dislocationproblem to only one. As a consequence, Equation (5) can be simplified to thefollow<strong>in</strong>g s<strong>in</strong>gle degree of freedom:X X dP xK ijdt ¼ X i jiwhere the <strong>in</strong>dexes i and j are taken only for the rate of the dislocation translation <strong>in</strong>the x-direction. The implementation of rigid dislocation dynamics is an easy andstraightforward implementation with<strong>in</strong> the classical dislocation dynamics.3. Dislocation-<strong>precipitate</strong> <strong>in</strong>teraction3.1. Geometric modelFigure 1 shows the simulation volume used <strong>in</strong> this work. The matrix material of thesimulation volume is the 0 -phase, and the x-, y- and z-axes are along the crystalorientations ½ 101Š, [111], ½ 12 1Š, respectively. A straight edge super-dislocation isLF i ,g -PrecipitateEigen stra<strong>in</strong> : –0.0143ð12Þð13Þy:[111]z:[121]x:[101]Edge super-dislocation(Fractional dislocations : 20 l<strong>in</strong>es)g ′ MatrixFigure 1. Schematic of simulation model for PDD simulations.
Philosophical Magaz<strong>in</strong>e 3775Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012<strong>in</strong>troduced <strong>in</strong>to the simulation volume. The super-dislocation has a Burgers vector ofa 00 ½ 101Š, and is on the (111) slip plane, where a 0 is the lattice constant of the L1 2ordered <strong>nickel</strong>-alum<strong>in</strong>um lattice (0.3571 nm) [20]. The core structure of the superdislocationis represented with 20 fractional dislocations. The -surfaces of the 0 -phase and of the -<strong>precipitate</strong> are calculated us<strong>in</strong>g an <strong>in</strong>teratomic potentialdeveloped for the <strong>nickel</strong>-alum<strong>in</strong>um b<strong>in</strong>ary system [20]. The lattice restor<strong>in</strong>g stressesare then calculated by tak<strong>in</strong>g the derivative of the -surface, and are used as afunction of position of the fractional dislocations. When part of a fractionaldislocation is <strong>in</strong> the 0 -phase, the lattice restor<strong>in</strong>g stress for the 0 -phase is given tothat part of the fractional dislocation, whereas the lattice restor<strong>in</strong>g stress for the-<strong>precipitate</strong> is used for the part of the fractional dislocation located <strong>in</strong>side the-<strong>precipitate</strong>. In reality, the -surface at the <strong>in</strong>terface between the - and 0 -phase isgenerally different from both the -surface of the - and 0 -phases. However, forsimplicity, the unique shape of the -surface at the <strong>in</strong>terface is ignored. A spherical-<strong>precipitate</strong> with a coherency stra<strong>in</strong> of 2¼ 0.0143 is placed at the front of thesuper-dislocation. Periodic boundary conditions are applied <strong>in</strong> the z-direction,assum<strong>in</strong>g that the dislocation is <strong>in</strong>f<strong>in</strong>itely long, and the <strong>precipitate</strong> makes a 1-Dperiodic array <strong>in</strong> the z-direction. To move the dislocation, an external shear stress of xy is applied to the volume, and is controlled to measure the CRSS for the<strong>in</strong>teraction. The size of the simulation volume <strong>in</strong> the z-direction and the diameter ofthe <strong>precipitate</strong> are denoted with L and D, respectively, which are importantparameters controll<strong>in</strong>g the CRSS.3.2. c-surface energiesBefore start<strong>in</strong>g the simulation of the super-dislocation and -<strong>precipitate</strong> <strong>in</strong>teraction,the -surface energies of the - and 0 phases must be determ<strong>in</strong>ed, and <strong>in</strong> this work,are calculated us<strong>in</strong>g an <strong>in</strong>teratomic potential for the <strong>nickel</strong> and alum<strong>in</strong>um b<strong>in</strong>aryalloy [20]. In these calculations, we used a volume with a size of 2.5 12.4 4.4 nm.The x-, y- and z-axes are along the crystal orientations ½ 101Š, [111], ½ 12 1Š. Periodicboundary conditions are applied on all sides of the simulation volume. In order tomake a stack<strong>in</strong>g-fault <strong>in</strong> the simulation volume, a displacement of Du ¼ a 0 =6=20½ 1 12Šis given only to the upper half of the simulation volume. Then, the motion of atoms<strong>in</strong> the x- and z-directions are constra<strong>in</strong>ed, and that <strong>in</strong> the y-direction is relaxed us<strong>in</strong>ga numerical quench<strong>in</strong>g technique. The process is repeated until the total displacementgiven to the upper half of the simulation volume reaches u ¼ 20Du. Figure 2shows the results of these calculations. Brandl et al. calculated the -surface energyus<strong>in</strong>g the ab <strong>in</strong>itio method [21]. In order to check the accuracy, the -surface is alsoplotted <strong>in</strong> the figure. In Figure 2a, the -surface for <strong>nickel</strong> calculated us<strong>in</strong>g the<strong>in</strong>teratomic potential is <strong>in</strong> excellent agreement with that obta<strong>in</strong>ed us<strong>in</strong>g the ab <strong>in</strong>itiomethod. The unstable stack<strong>in</strong>g fault energy is at u ¼ 0.27b, whereas the <strong>in</strong>tr<strong>in</strong>sicstack<strong>in</strong>g-fault energy is the -energy at u ¼ 0.5b. The dislocation core tends to havelower displacement density <strong>in</strong> the region around the unstable stack<strong>in</strong>g-fault energy,and to have higher displacement density <strong>in</strong> the region around the <strong>in</strong>tr<strong>in</strong>sic stack<strong>in</strong>gfaultenergy, result<strong>in</strong>g <strong>in</strong> the formation of an extended dislocation, which is acomb<strong>in</strong>ation of two partial dislocations and a stack<strong>in</strong>g-fault.
3776 A. Takahashi et al.(a)300(b)600GSFE [mJ/m 2 ]250200150100GSFE [mJ/m 2 ]500400300200Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 201250MDab-<strong>in</strong>itio00.0 0.2 0.4 0.6 0.8 1.0Displacement, u [b]Figure 2b shows the calculated -surface energy for the 0 -phase, where ananti-phase boundary (APB) forms beh<strong>in</strong>d a lead<strong>in</strong>g dislocation, and a trail<strong>in</strong>gdislocation term<strong>in</strong>ates the APB. Thus, dislocations must move <strong>in</strong> pairs <strong>in</strong> asuper-dislocation configuration. In the present calculation of the -surface energy forthe 0 -phase, the maximum displacement is 40Du, which is twice that given to theatomic volume <strong>in</strong> the -surface calculation <strong>in</strong> the -phase. In the figure, there are twolocal maxima <strong>in</strong> the -surface energy, result<strong>in</strong>g <strong>in</strong> the formation of an extendeddislocation <strong>in</strong> the super-dislocation core. Moreover, there are two local m<strong>in</strong>ima atu ¼ 0.42b and 1.58b, which correspond to the complex stack<strong>in</strong>g-fault between twopartial dislocations of the extended dislocation, and the APB between twosuper-partials of the super-dislocation. The complex stack<strong>in</strong>g-fault and APB energiesare 202 mJ/m 2 and 252 mJ/m 2 [20], whereas the experimental results of the energiesare 235 mJ/m 2 and 175 mJ/m 2 [22], respectively.4. -<strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong>00.0 0.5 1.0 1.5 2.0Displacement, u [b]Figure 2. Generalized stack<strong>in</strong>g-fault energies for the -(top) and 0 -(bottom) phases.-<strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong> <strong>in</strong> <strong>nickel</strong>-<strong>based</strong> <strong>superalloys</strong> is a result of severalmechanisms that operate concurrently. Follow<strong>in</strong>g Ardell [1], chemical <strong>strengthen<strong>in</strong>g</strong>and modulus harden<strong>in</strong>g do not substantially affect the overall strength of the alloy.Therefore, <strong>in</strong> this work, we focus only on the <strong>in</strong>fluence of the stack<strong>in</strong>g-fault andcoherency stra<strong>in</strong> mechanisms, and study the <strong>strengthen<strong>in</strong>g</strong> effect of each oneseparately. The ma<strong>in</strong> focus will thus be on the <strong>in</strong>fluence of the dislocation l<strong>in</strong>eflexibility, and the details of its core structure on the overall strength. F<strong>in</strong>ally, themechanism of -<strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong>, which can be def<strong>in</strong>ed as the mixed<strong>strengthen<strong>in</strong>g</strong> of the stack<strong>in</strong>g-fault and coherency stra<strong>in</strong>, will be fully explored.1004.1. Coherency <strong>strengthen<strong>in</strong>g</strong>Gerold and Haberkorn developed an analytical solution to the <strong>in</strong>crease <strong>in</strong> the CRSSas a result of coherency <strong>in</strong>teraction between a dislocation and a <strong>precipitate</strong> [4]. Toobta<strong>in</strong> a closed form solution, they assumed that the dislocation shape is straight,and that the dislocation core is <strong>in</strong>f<strong>in</strong>itely narrow (i.e. the displacement field has
3778 A. Takahashi et al.(a)Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012(b)Figure 4. Snapshots of the dislocation core structure dur<strong>in</strong>g the <strong>in</strong>teraction between asuper-dislocation and a -<strong>precipitate</strong> account<strong>in</strong>g only for <strong>precipitate</strong> coherency stra<strong>in</strong>. (a) Slipplane position: 3 nm above the mid-plane; (b) slip plane position: 3 nm below the mid-plane.Upon <strong>in</strong>creas<strong>in</strong>g the applied shear stress, the lead<strong>in</strong>g super-partial starts to cutthrough the <strong>precipitate</strong>. Once the dislocation moves <strong>in</strong>side the <strong>precipitate</strong>, it isstrongly pushed to the outside of the <strong>precipitate</strong> by the <strong>in</strong>teraction. After break<strong>in</strong>gaway from the <strong>precipitate</strong>, the super-dislocation is still <strong>in</strong>fluenced by a repulsive<strong>in</strong>teraction force that enhances its glide. On the other hand, as shown <strong>in</strong> Figure 4b,the central part of the super-dislocation spontaneously dissociates <strong>in</strong>side the<strong>precipitate</strong>, because most of the <strong>precipitate</strong> is on the tension side of the dislocationand the <strong>precipitate</strong> has an attractive <strong>in</strong>teraction force with the super-dislocation.Thus, the super-dislocation tends to stay <strong>in</strong>side the <strong>precipitate</strong> so that the maximum<strong>in</strong>teraction appears at the center of the <strong>precipitate</strong>. When the applied shear stressreaches the CRSS, the dislocation can f<strong>in</strong>ally cut through the <strong>precipitate</strong>.Figure 5 shows the <strong>in</strong>crease <strong>in</strong> the CRSS due to the <strong>precipitate</strong> coherency stra<strong>in</strong>.In the figure, the CRSS is calculated us<strong>in</strong>g four different computational methods: (1)PDD, (2) PDD with the GPN model, (3) rigid PDD, and (4) rigid PDD with theGPN model, and the results are all plotted <strong>in</strong> the same figure. In order to comparethe simulation results to classical analytical solutions [4], the CRSS calculated byEquation (15) is also plotted. It is observed that when the dislocation is placed at thecenter of the -<strong>precipitate</strong>, the CRSS does not <strong>in</strong>crease, while the maximum CRSSoccurs when the dislocation slip plane is at z/R ¼0.75. A shear stress must beapplied to the dislocation to overcome the coherency stra<strong>in</strong> field, even when thedislocation does not directly cut the <strong>precipitate</strong>, because of the long-range nature ofthe stra<strong>in</strong> field, as can be seen <strong>in</strong> the figure. The results of calculations us<strong>in</strong>g the fullPDD method, the rigid PDD method, and those of Equation (15) show excellentagreement. However, the results of the (rigid) PDD with the GPN model differ
Philosophical Magaz<strong>in</strong>e 3779t crss / t 00.300.250.200.150.10Rigid PDDPDDRigid PDD-GPNPDD-GPNGerold (1966)0.05Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012significantly, particularly when the dislocation is on the slip plane at z/R ¼0.75.S<strong>in</strong>ce Equation (15) is derived with the assumptions that dislocation is perfectlystraight, and that the core structure is ignored, the agreement between the rigid PDDmethod and Equation (15) is reasonable. S<strong>in</strong>ce the results of the full PDD simulationare also identical to those of the rigid PDD method, we conclude that the <strong>in</strong>fluence ofthe dislocation l<strong>in</strong>e flexibility on the coherency <strong>strengthen<strong>in</strong>g</strong> mechanism must benegligibly small. Thus, the significant difference between the results of Equation (15)and those of the (rigid) PDD with the GPN model must be attributed to the <strong>in</strong>fluenceof the core structure of the dislocation.The present simulation results will now be utilized to extend the applicability ofEquation (15) by <strong>in</strong>corporat<strong>in</strong>g the <strong>in</strong>fluence of the core structure of dislocations asthey <strong>in</strong>teract with <strong>precipitate</strong>s. Referr<strong>in</strong>g to the coord<strong>in</strong>ate system of Figure 3, wedecompose the super-dislocation <strong>in</strong>to four partial dislocations D i , where thepositions of each partial dislocation x i are assumed to be x 2 ¼ x 1 þ w c , x 3 ¼ x 2 þ w aand x 4 ¼ x 3 þ w c , where w c and w a are the widths of the complex stack<strong>in</strong>g-faultbetween two partial dislocations of a super-partial dislocation, and the anti-phaseboundary between the super-partial dislocations. For simplicity, we consider an<strong>in</strong>f<strong>in</strong>itely long straight edge super-dislocation <strong>in</strong> an <strong>in</strong>f<strong>in</strong>ite isotropic elastic body,and <strong>in</strong>clude the elastic <strong>in</strong>teraction between the partial dislocations as well as theenergy of the complex stack<strong>in</strong>g-fault and anti-phase boundaries. The widths w c andw a can be calculated by solv<strong>in</strong>g the follow<strong>in</strong>g equilibrium equations:Gb e2 1 1 1þ þ2ð1Þw c w c þ w a 2w c þ w aGb e22ð10.00–5 –4 –3 –2 –1 0 1 2 3 4 5þ Gbs22Þþ Gbs22Slip plane position (y/R)Figure 5. The <strong>in</strong>crease <strong>in</strong> the CRSS as a function of the slip plane position as a resultof coherency <strong>strengthen<strong>in</strong>g</strong> by the <strong>precipitate</strong> stra<strong>in</strong> field.1 1 1þ¼ cð16Þw c w c þ w a 2w c þ w a1þ 1 1þw c w a w c þ w a1 1 1þ ¼ a c : ð17Þw c w a w c þ w a
3780 A. Takahashi et al.t crss / t 00.300.250.200.150.100.05PDD-GPNGerold (1966)Analytical modelDownloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012Similar to Equation (15), tak<strong>in</strong>g the summation of the K(x i ), and f<strong>in</strong>d<strong>in</strong>g themaximum, the CRSS can be calculated asP 4i Kðx i Þj crss ¼max: ð18ÞbLFigure 6 shows the results of the PDD with the GPN model simulation, andcompared to calculations <strong>based</strong> on Equation (15) as well as Equation (18). TheCRSS <strong>in</strong>crease calculated by Equation (18) is considerably smaller than thosecalculated by Equation (15), which is <strong>in</strong> excellent agreement with the numericalsimulation results. Thus, it is clear that the <strong>in</strong>fluence of the core structure ofdislocations on coherency <strong>strengthen<strong>in</strong>g</strong> is very significant (the maximum is loweredby a factor of almost 3), and that the <strong>in</strong>fluence can be accurately accounted for us<strong>in</strong>gthe proposed Equation (18).4.2. Stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong>0.00–5 –4 –3 –2 –1 0 1 2 3 4 5Slip plane position (y/R)Figure 6. The <strong>in</strong>crease <strong>in</strong> the CRSS as a function of the slip plane position as a resultof coherency <strong>strengthen<strong>in</strong>g</strong> by the <strong>precipitate</strong> stra<strong>in</strong> field, calculated by direct numericalsimulation, and compared to the results of Gerold and the proposed analytical model given byEquation (18).If the energy of the stack<strong>in</strong>g-fault between two partial dislocations is larger <strong>in</strong>side the<strong>precipitate</strong> than <strong>in</strong> the matrix, an additional shear stress must be applied on thedislocation to overcome that energy difference, D, and allow the dislocation to cutthrough the <strong>precipitate</strong>. Nembach derived an analytical expression to calculate therequired shear stress due to stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong> [3]. An extended dislocation,composed of two partial dislocations and a stack<strong>in</strong>g-fault <strong>in</strong>-between, wasconsidered, and the length of each partial dislocation <strong>in</strong>side the <strong>precipitate</strong>, L(x),was used to f<strong>in</strong>d the <strong>in</strong>teraction force between the <strong>precipitate</strong> and the dislocation K:KðxÞ ¼ðLðxÞ Lðx þ wÞÞD, ð19Þwhere w is the distance between the lead<strong>in</strong>g and trail<strong>in</strong>g partial dislocations, L(x)and L(x þ w) are the lengths of the lead<strong>in</strong>g and trail<strong>in</strong>g partial dislocations, at x
Philosophical Magaz<strong>in</strong>e 3781Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012Figure 7. Snapshots of the <strong>in</strong>teraction between a super-dislocation and a -<strong>precipitate</strong> as aresult of differences <strong>in</strong> the stack<strong>in</strong>g-fault between the matrix and <strong>precipitate</strong> (stack<strong>in</strong>g-fault<strong>strengthen<strong>in</strong>g</strong>).and x þ w, <strong>in</strong>side the <strong>precipitate</strong>, respectively. The CRSS can be calculated by f<strong>in</strong>d<strong>in</strong>gthe maximum <strong>in</strong>teraction force, and is given by [3] crss ¼ KðxÞj max: ð20ÞbLTo determ<strong>in</strong>e the accuracy of the analytical solution of Nembach, and anypossible effects of the exact core structure on stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong>, weperform numerical simulations where the <strong>in</strong>fluence of the coherency stra<strong>in</strong> isremoved by sett<strong>in</strong>g 2¼ 0, and that of the stack<strong>in</strong>g-fault is accounted for by giv<strong>in</strong>g the-surface <strong>in</strong>formation of both the - and 0 -phases. When part of the fractionaldislocation is located <strong>in</strong> the -phase, the lattice restor<strong>in</strong>g stress of -phase, which canbe calculated from the -surface of -phase, is given to that part, whereas the latticerestor<strong>in</strong>g stress of the 0 -phase is given to the part of the fractional dislocation <strong>in</strong>sidethe 0 -phase. The simulation volume used here is identical to that used <strong>in</strong> theprevious section, and the diameter of the <strong>precipitate</strong> is changed <strong>in</strong> the range from 2to 30 nm. The slip plane position of the super-dislocation is fixed at the center of the<strong>precipitate</strong>. Figure 7 shows snapshots of the dislocation configuration as it <strong>in</strong>teractswith the <strong>precipitate</strong> (here the <strong>precipitate</strong> diameter is 16 nm). It is clear that thesuper-dislocation and the <strong>precipitate</strong> are attracted to one another, and that thesuper-dislocation is first absorbed by the <strong>precipitate</strong>. When the dislocation entersthe -<strong>precipitate</strong>, the anti-phase boundary disappears, and the complex stack<strong>in</strong>gfault is changed to an <strong>in</strong>tr<strong>in</strong>sic stack<strong>in</strong>g-fault, which has a lower energy than thecomplex stack<strong>in</strong>g-fault. Thus, the super-dislocation tends to be attracted <strong>in</strong>sidethe <strong>precipitate</strong> to reduce the <strong>in</strong>teraction energy. Increas<strong>in</strong>g the externally appliedshear stress, the dislocation gradually starts to bow-out, and f<strong>in</strong>ally breaks awayfrom the <strong>precipitate</strong> when the applied shear stress reaches its critical value(the CRSS).Figure 8 shows the results of the numerical simulations (both the full PDD-GPNmodel, and the rigid PDD-GPN model). Also, the results of Equation (20) areplotted for comparison. As the <strong>precipitate</strong> diameter <strong>in</strong>creases, the CRSS also<strong>in</strong>creases, which is a consequence of an <strong>in</strong>crease <strong>in</strong> the stack<strong>in</strong>g-fault area betweenpartials when they enter <strong>in</strong>to the <strong>precipitate</strong>. Additionally, the results of all numericalsimulations with both methods, and those of Equation (20), are with<strong>in</strong> a few percent.
3782 A. Takahashi et al.t crss / t 00.300.250.200.150.10Rigid PDD-GPNPDD-GPNNembach (1986)0.05Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012Therefore, Equation (20) reasonably accounts for stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong>, aslong as the dislocation core can be described by two isolated peaks of displacementdistributions (i.e. well-isolated partials). The core structure of dislocations is then animportant factor <strong>in</strong> determ<strong>in</strong><strong>in</strong>g the CRSS, and Equation (20) can be used <strong>in</strong> caseswhere the two partial dislocations are well-isolated.4.3. Mixed <strong>strengthen<strong>in</strong>g</strong>0.000 10 20 30 40 50 60Diameter (D/2b)Figure 8. Dependence of the CRSS on <strong>precipitate</strong> diameter for the stack<strong>in</strong>g-fault<strong>strengthen<strong>in</strong>g</strong> mechanism alone.Precipitate <strong>strengthen<strong>in</strong>g</strong> is a result of several factors, such as the stack<strong>in</strong>g-fault,coherency, and modulus <strong>strengthen<strong>in</strong>g</strong> mechanisms. However, <strong>in</strong> the present caseof <strong>nickel</strong>-<strong>based</strong> <strong>superalloys</strong>, <strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong> is primarily due to thestack<strong>in</strong>g-fault and the coherency <strong>strengthen<strong>in</strong>g</strong> mechanisms, because the difference<strong>in</strong> the elastic moduli between the matrix and <strong>precipitate</strong> is very small. Therefore, wedescribe here -<strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong> as a comb<strong>in</strong>ation of the stack<strong>in</strong>g-fault andcoherency <strong>strengthen<strong>in</strong>g</strong> effects, and propose a method to evaluate the overall CRSS<strong>in</strong>crease when the two mechanisms are operat<strong>in</strong>g simultaneously. The simulationvolume and boundary conditions used <strong>in</strong> this section are the same as those used<strong>in</strong> previous simulations, while the diameter of the -<strong>precipitate</strong> is fixed to 8 nm.Figure 9 shows the dependence of the CRSS on the location of the slip plane. As canbe seen, <strong>in</strong> the stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong> mechanism, the distribution of the CRSSis symmetric about the center of the <strong>precipitate</strong>, and shows a maximum at the center.When the position of the slip plane is outside the <strong>precipitate</strong>, there is no <strong>in</strong>crease<strong>in</strong> the CRSS <strong>in</strong>crease, as expected.It is clear that the CRSS outside the <strong>precipitate</strong> is identical to that due to thecoherency stra<strong>in</strong> mechanism, while <strong>in</strong>side the <strong>precipitate</strong> both the coherency andstack<strong>in</strong>g-fault mechanisms operate simultaneously. In fact, as the slip plane getscloser to the <strong>precipitate</strong> mid-plane, the <strong>in</strong>fluence of coherency stra<strong>in</strong> dim<strong>in</strong>ishes,while stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong> reaches its maximum. However, the distributionof the CRSS when the two mechanisms operate simultaneously (mixed <strong>strengthen<strong>in</strong>g</strong>)has a complex structure as a function of the slip plane position, and its
Philosophical Magaz<strong>in</strong>e 3783Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012t crss / t 00.180.160.140.120.100.080.060.040.020.00–5 –4 –3 –2 –1 0 1 2 3 4 5Slip plane position (y/R)g-<strong>precipitate</strong>t crsscoh.t crsssf.t crsst coh. sf.crss+ t crssFigure 9. The CRSS as a function of the slip plane position for coherency (triangles)and stack<strong>in</strong>g-fault (open circles), simple summation (dashed l<strong>in</strong>e), and full calculation(open squares).maximum cannot be simply the sum of the two <strong>in</strong>dependent maxima for eachmechanism separately, a procedure that is common <strong>in</strong> precipitation harden<strong>in</strong>gestimates. To further demonstrate this po<strong>in</strong>t, we also used a simple summation rulefor the CRSS result<strong>in</strong>g from the stack<strong>in</strong>g-fault and the coherency <strong>strengthen<strong>in</strong>g</strong>mechanisms, and plot the results <strong>in</strong> Figure 9 (dashed l<strong>in</strong>e). Here, the simple sum doesnot represent overall <strong>precipitate</strong> harden<strong>in</strong>g very well. When the dislocation is on thecompression side of the <strong>precipitate</strong>, the stack<strong>in</strong>g-fault shr<strong>in</strong>ks, while it widens when itis on the tension side, giv<strong>in</strong>g rise to the asymmetric distribution of the CRSS as afunction of the slip plane position when the two mechanisms operate simultaneously,a result that cannot be predicted by simple addition of the two effects.The <strong>in</strong>teraction forces between the super-dislocation and the <strong>precipitate</strong>, us<strong>in</strong>gEquations (18) and (20), are calculated and plotted <strong>in</strong> Figure 10 as functions of thedislocation position from the <strong>precipitate</strong> center, when the slip plane is at 2 nm fromthe mid-plane. It is observed that the peaks of the <strong>in</strong>teraction force due to thestack<strong>in</strong>g-fault and the coherency mechanisms appear at different positions,suggest<strong>in</strong>g that the summation of the CRSS (which is equivalent to summ<strong>in</strong>gthe maximum forces) is not a good way to represent the true spatial dependence ofthe CRSS for mixed <strong>strengthen<strong>in</strong>g</strong>. The figure also shows the total <strong>in</strong>teraction forcedue to the stack<strong>in</strong>g-fault and coherency <strong>strengthen<strong>in</strong>g</strong> mechanisms as a function ofdislocation position. S<strong>in</strong>ce the dislocation has to overcome the maximum total<strong>in</strong>teraction force, we calculated the CRSS at different slip planes, us<strong>in</strong>g onlythe maximum <strong>in</strong>teraction force, and we show the results <strong>in</strong> Figure 11. We comparehere the CRSS as a function of the slip plane position, us<strong>in</strong>g the full PDD-GPNmodel, and compar<strong>in</strong>g it to the more simplified model of us<strong>in</strong>g only the maximumvalue of the calculated shear stress at each slip plane. These two methods showexcellent agreement, <strong>in</strong>dicat<strong>in</strong>g that, <strong>in</strong> order to calculate the CRSS for mixed<strong>strengthen<strong>in</strong>g</strong>, the maximum <strong>in</strong>teraction force for a given slip plane can be firstcomputed and stored, and then used <strong>in</strong> subsequent DD simulations without hav<strong>in</strong>gto perform detailed simulations.
3784 A. Takahashi et al.Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 20125. Conclusionst crss / t 0Interaction force : k (x10 –09 N)0.140.120.100.080.060.040.024.03.02.01.00.0–1.0–2.0–3.0t coh.t sft coh. +t sf–4.0–8 –6 –4 –2 0 2 4 6 8Figure 10. Dependence of the coherency and stack<strong>in</strong>g-fault <strong>in</strong>teraction forces on thedislocation position, when it lies on a slip plane at 2 nm, which is half of the <strong>precipitate</strong>radius, from the mid-plane.0.00–5 –4 –3 –2 –1 0 1 2 3 4 5Slip plane position (y/R)PDD-GPNAnalytical modelFigure 11. Comparison of the dependence of the CRSS on the slip plane position.Numerical simulations with the full PDD-GPN model are compared with the analyticalmodel. The results of the analytical model are given as the maximum total <strong>in</strong>teraction forceof the stack<strong>in</strong>g-fault and the coherency <strong>strengthen<strong>in</strong>g</strong>.A computational method for <strong>in</strong>vestigations of the role of dislocation flexibility andthe dislocation core structure on the dislocation–<strong>precipitate</strong> <strong>in</strong>teraction has beendeveloped. The method is a comb<strong>in</strong>ation of the PDD, with lattice resistance todislocation motion obta<strong>in</strong>ed from the GPN model. To quantitatively evaluate the<strong>in</strong>fluence of dislocation flexibility on its <strong>in</strong>teraction with <strong>precipitate</strong>s, we added aconstra<strong>in</strong>t condition on the dislocation shape <strong>in</strong> PDD simulations. The methodallows us to simulate the dynamics of rigid dislocations <strong>in</strong> <strong>in</strong>ternal and external stressfields. Us<strong>in</strong>g the computational method, we studied -<strong>precipitate</strong> <strong>strengthen<strong>in</strong>g</strong> <strong>in</strong><strong>nickel</strong>-<strong>based</strong> <strong>superalloys</strong>, and determ<strong>in</strong>ed the roles played by the dislocation l<strong>in</strong>eflexibility and its core structure. Conclusions of the present study are summarized <strong>in</strong>the follow<strong>in</strong>g:x/R
Philosophical Magaz<strong>in</strong>e 3785Downloaded by [University of California, Los Angeles (UCLA)] at 09:30 20 February 2012(1) Us<strong>in</strong>g four different <strong>in</strong>teraction models (PDD, rigid PDD, PDD with GPN,and rigid PDD with GPN), it is found that the <strong>in</strong>fluence of dislocation l<strong>in</strong>eflexibility is negligibly small, whereas the dislocation core structure isfound to play a major role <strong>in</strong> determ<strong>in</strong><strong>in</strong>g the CRSS.(2) An analytical equation for coherency <strong>strengthen<strong>in</strong>g</strong> is revised to <strong>in</strong>corporatethe dislocation core <strong>in</strong>formation <strong>in</strong>to the evaluation of the CRSS. The revisedequation reproduces very well the results of detailed PDD simulations withthe GPN model.(3) Aga<strong>in</strong>, the <strong>in</strong>fluence of the dislocation l<strong>in</strong>e flexibility on the stack<strong>in</strong>g-fault<strong>strengthen<strong>in</strong>g</strong> is found to be negligible.(4) An analytical equation, which is derived with a dislocation model composedof two partial dislocations and a stack<strong>in</strong>g-fault <strong>in</strong>-between, preciselyreproduces the CRSS of the stack<strong>in</strong>g-fault <strong>strengthen<strong>in</strong>g</strong> mechanism.(5) The summation of the CRSSs of the stack<strong>in</strong>g-fault and coherency <strong>strengthen<strong>in</strong>g</strong>mechanisms is found to give unacceptable values of the CRSS.(6) Tak<strong>in</strong>g the summation of the shear stresses of the stack<strong>in</strong>g-fault andcoherency <strong>strengthen<strong>in</strong>g</strong> mechanism, and f<strong>in</strong>d<strong>in</strong>g the maximum of the total<strong>in</strong>teraction force reproduces very well the CRSS for mixed <strong>strengthen<strong>in</strong>g</strong>.The developed method is a promis<strong>in</strong>g approach <strong>in</strong> determ<strong>in</strong><strong>in</strong>g the CRSS ofmixed types of precipitation <strong>strengthen<strong>in</strong>g</strong> mechanisms, without the detailedsimulations.AcknowledgementThis research is supported by the US Air Force Office for Scientific Research (AFOSR), GrantNo. FA9550-07-1-0396 at UCLA.References[1] A.J. Ardell, Metall. Trans. A 16 (1985) p.2131.[2] P.B. Hirsch and A. Kelly, Phil. Mag. 12 (1965) p.881.[3] E. Nembach, Scripta Metall. 20 (1986) p.763.[4] V. Gerold and H. Haberkorn, Physica Status Solidi 16 (1966) p.675.[5] M.K. Miller, Micron 32 (2001) p.757.[6] M. Nemoto, W.H. Tian, K. Harada, C.S. Han and T. Sano, Mater. Sci. Eng. A 152 (1992)p.247.[7] N.M. Ghoniem, S.-H. Tong and L.Z. Sun, Phys. Rev. B 61 (2000) p.913.[8] E. van der Giessen and A. Needleman, Model. Simul. Mater. Sci. Eng. 3 (1995) p.689.[9] E. Weygand, L.H. Friedman, E. Van der Giessen and A. Needleman, Model. Simul.Mater. Sci. Eng. 10 (2002) p.437.[10] A. Takahashi and N.M. Ghoniem, J. Mech. Phys. Solids 56 (2008) p.1534.[11] Y. Xiang, D.J. Srolovitz, L.-T. Cheng and E. We<strong>in</strong>an, Acta Mater. 52 (2004) p.1745.[12] B. Dev<strong>in</strong>cre, L.P. Kub<strong>in</strong>, C. Lemarchand and R. Madec, Mater. Sci. Eng. A 309–310(2001) p.211.[13] S.I. Rao, T.A. Parthasarathy, D.M. Dimiduk and P.M. Hazzled<strong>in</strong>e, Phil. Mag. 86 (2004)p.3195.[14] S. Banerjee, N.M. Ghoniem, G. Lu and N. Kioussis, Phil. Mag. 87 (2007) p.4131.
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