COMPUTATIONAL MATERIALS ENGINEERING Episode 12 doc

interface; subsequently, it rapidly solidifies. Nucleation of the opposite phase within the concave portion of the interfaces stabilizes the lamellae and regular lamellar growth to continue at half the original lamellae spacing. This suggests that nucleation ahead of the eutectic front may provide a method of stabilizing the lamellae. Morphological instabilities with a regular oscillatory structure of lamellar eutectics are reported in experiments with a transparent organic alloy and also in numerical studies by Karma and Sarkission [KS96]. In three dimensions, an analogous type of oscillation can be observed for eutectic microstructure formations, Figure 7-32. Performing an alternating topological change, α solid rods are embedded in a β matrix followed by the opposite situation of β crystals embed- ded in an α matrix. As a next example, we apply the phase-field model to simulations of solidification processes in ternary A–B–C alloys. Simulation results of ternary dendritic and ternary eutectic growth are exemplarily illustrated. In particular, the ternary Ni 60 Cu 40−x Cr x alloy system is consi- dered as a prototype system to investigate the influence of interplaying solute fields on the interface stability, on the growth velocity, and on the characteristic type of morphology. The Ni–Cu–Cr system serves as an extension of the binary Ni–Cu system which has been explored by phase-field modeling (e.g., ref. [WB94]) and by molecular dynamics simulations in several papers (e.g., refs. [HSAF99, HAK01] ). Hence, physical parameters of Ni–Cu are relatively well established. A series of numerical computations for different alloy compositions varying from Ni 60 Cu 28 Cr 12 to Ni 60 Cu 12 Cr 28 has been carried out in Figure 7-33. The concentration of Ni was kept constant at 60 at.%, and the initial undercooling was fixed at 20 K measured from the equilibrium liquidus line in the phase diagram at a given composition of the melt. A mor- phological transition from dendritic to globular growth occurs at a melt composition of about Ni 60 Cu 20 Cr 20 . FIGURE 7-32 Topological change of the microstructure with α rods embedded in a β matrix phase and vice versa. The structure formation results from regular 3D oscillations along the solid–solid interface. Phase-Field Modeling 261 Cr Ni Cu Ni 60 Cu 24 Cr 16 Ni 60 Cu 16 Cr 24 (a) (b) Ni 60 Cu 28 Cr 12 Ni 60 Cu 12 Cr 28 FIGURE 7-33 (a) Schematic drawing of an isothermal cut through the ternary Ni–Cu–Cr phase diagram. The arrow marks the path where the simulations of the morphological changes were per- formed. (b) Dendritic to globular morphological transition for different alloy compositions. The atomic percents of Cu and Cr are exchanged while keeping Ni fixed at 60at.%. The gray region corresponds to the solid phase, and the solid lines represent the isolines of average concentration of Ni in the solid phase. The left side of Figure 7-33(b) shows the dendritic morphologies observed for Cr concen- trations less than 20at.%. The right side of Figure 7-33(b) displays globular morphologies for Cr concentrations crossing this threshold. The velocity of the dendritic–globular tip increases linearly from 1.19 cm/s to 3.24 cm/s with increasing the concentration of Cr. The morpholog- ical transition is related to the transition from a two-phase region (above the solidus line) to a one-phase region (below the solidus line) in the phase diagram. In a ternary eutectic system, we considered two regions within a completely symmetric model phase diagram, namely, (i) the region of four phase equilibrium at the ternary eutectic composition and temperature and (ii) a region where one component has a minor contribution of the amount of a ternary impurity. At the ternary composition c =0. 3, three solid phases α, β, and γ grow into an undercooled melt L with equal volume fractions by a ternary eutectic reaction L → α+ β +γ.Different permutations of lamellar structures αβγ and αβαγ . . . are possible. Phase-field simulations can be used to investigate which phase sequence is favored to grow at certain solidification conditions. An example for an αβγ configuration is displayed in Figure 7-34(a) showing in addition the concentration of one of the components ahead of the growing eutectic front. In three dimensions, a regular hexagonal shape of the three solid phases occurs for isotropic growing phases at three different time steps [Figure 7-34(b)]. The hexag- onal symmetry breaks if anisotropy is included and if, hence, strong facets form in preferred growth directions. In a eutectic phase system with ternary impurity, it can be observed that the impurity is pushed ahead of the solidifying lamellae and builds up. For the simulation in Figure 7-35, we have set an initial composition vector of (c A ,c B ,c C )=(0.47, 0.47, 0.06) so that c C is the concentration component of minor amount. The main components c A and c B are incorporated in the growing α–β solid front whereas the impurity is rejected by both growing solid phases. To further observe the effect of eutectic cell/colony formation, computations in larger domains including noise as well as nucleation have to be performed. 262 COMPUTATIONAL MATERIALS ENGINEERING (a) (b) FIGURE 7-34 (a) Phase transformations in a ternary eutectic system with three solid phases growing into an undercooled melt. The concentration of component A is shown for different time steps ahead of a regular αβγ configuration. (b) Ternary eutectic growth in three dimensions forming a steadily propagating hexagonal structure. (a) (b) FIGURE 7-35 (a) Concentration profile of the main component c A in the melt. (b) The ternary impurity c C is pushed ahead of the growing eutectic front so that the concentration enriched zones of component c C can be observed at the solid–liquid interface. Another important field of application for phase-field modeling is the computation of grain structure evolution and anisotropic curvature flow in a polycrystalline material. In this case, the phase-field variables φ α ,α =1, ,N represent the state of crystals with different crystallo- graphic orientations. Figure 7-36 shows the effect of grain boundary motion on the growth selec- tion in comparison with an experimental microstructure. As initial configuration, a distribution of small grains was posed along a thin layer at the upper wall of the simulation box. The grains started to grow toward the bottom of the domain. Certain grains with their crystal orientation in the direction of the shear movement of the lower wall grew faster than the neighboring grains, which finally ceased to grow. Phase-Field Modeling 263 (a) (b) FIGURE 7-36 Grain selection process as a result of grain boundary motion of differently oriented cr ystals, (a) experimental microstructure observed in geological material, [Hil] and, (b) phase-field simulation. Grains with their growth direction in alignment with the shear movement of the lower wall dominate the structure formation. (a) (b) FIGURE 7-37 Dendritic growth of 10 Ni-Cu nuclei with different crystal orientation; (a) illustration of the Ni concentration in the crystallized solid dendrites and in the surrounding melt, (b) view of the sharp crystal boundaries showing the different crystal orientations in gray shades. The final example in Figure 7-37 shows a distribution of differently oriented nuclei growing into an isothermally undercooled Ni–Cu melt. To reach the state of complete crystallization, the system is quenched. The individual dendrites match and form grain boundaries of a poly- crystalline grain structure. After the solidification is finalized, the grain boundaries continue to migrate as a result of curvature minimization. The phase-field model recovers the generic features of grain growth such as grain bound- ary motion, crystalline curvature flow, the force balance known as Young’s law at triple junc- tions, the (in)stability of quadruple junctions, wetting phenomena, and the symmetry behavior of neighboring triple junctions in microstructures of polycrystalline thin films. 264 COMPUTATIONAL MATERIALS ENGINEERING 7.5 Acknowledgments The author of this chapter thanks her co-workers Frank Wendler and Denis Danilov for provid- ing images of the simulation results. Bibliography [Alm99] R. F. Almgren. SIAM J. Appl. Math., 59:2086–2107, 1999. [AMW98] D. M. Anderson, G. B. McFadden, and A. A. Wheeler. Annu. Rev. Fluid Mech., 30:139–165, 1998. [BGJ82] J. Q. Broughton, G. H. Gilmer, and K. A. Jackson. Crystallization rates of a Lennard–Jones liquid. Phys. Rev. Lett., 49:1496–1500, 1982. 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[Sto96] B. Stoth. Eur. J. Appl. Math., 7:603–634, 1996. [STSS97] M. Seesselberg, J. Tiaden, G. J. Schmitz, and I. Steinbach. In J. Beek and H. Jones (B. B. R. Sheffield), eds., Proceedings of the Fourth Decennial International Conference on Solidification Processing, Sheffield, UK, pp. 426–430, 1997. [TA98] T. T ¨ onhardt and G. Amberg. J. Cryst. Growth, 194:406, 1998. [Vis96] A. Visintin. 1996. [WB94] J. A. Warren and W. J. Boettinger. Acta. Metall. Mater., 43:689, 1994. [WBM92] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phys. Rev. A, 45:7424, 1992. [WBM93] A. A. Wheeler, W. J. Boettinger, and G. B. McFadden. Phys. Rev. E, 47:1893, 1993. [WM96] A. A. Wheeler and G. B. McFadden. Eur. J. Appl. Math., 7:369, 1996. [WM97] A. A. Wheeler and G. B. McFadden. Proc. R. Soc. London, Ser. A, 453:1611, 1997. [WMS93] A. A. Wheeler, B. T. Murray, and R. J. Schaefer. Physica D, 66:243, 1993. [Woo73] D. P. Woodruff. The Solid–Liquid Interface. Cambridge, 1973. [WS96] S L. Wang and R. F. Sekerka. J. Comp. Phys., 127:110, 1996. [WSW + 93] S L. Wang, R. F. Sekerka, A. A. Wheeler, S. R. Coriell, B. T. Murray, R. J. Braun, and G. B. McFadden. Physica D, 69:189, 1993. 266 COMPUTATIONAL MATERIALS ENGINEERING 8 Introduction to Discrete Dislocation Statics and Dynamics —Dierk Raabe 8.1 Basics of Discrete Plasticity Models This chapter deals with the simulation of plasticity of metals at the microscopic and mesoscopic scale using space- and time-discretized dislocation statics and dynamics. The complexity of discrete dislocation models is due to the fact that the mechanical interaction of ensembles of such defects is of an elastic nature and, therefore, involves long-range interactions. Space-discretized dislocation simulations idealize dislocations outside the dislocation cores (few atomic positions in the center of the dislocation) as linear defects which are embedded within an otherwise homogeneous, isotropic or anisotropic, linear elastic medium. Therefore, the elastic material outside of the dislocation cores can be described in terms of the Hooke law of elasticity. This applies for both straight infinite dislocations (2D discretization) and dislocation segments (3D discretization). The simulation work in this field can be grouped into simulations which describe the dislo- cations in two dimensions and in three dimensions. 2D calculations are conducted either with dislocations that cannot leave their actual glide plane (view into the glide plane) [FM66, BKS73, R ¨ on87, Moh96], or with nonflexible infinite straight dislocations which may leave their glide plane but cannot bow out (view along the line vector of the dislocation) [LK87, GA89, GSLL89, Amo90, GLSL90, GH92, LBN93, WL95]. 3D simulations which decompose each dislocation into a “spaghetti-type” sequence of piecewise straight segments with a scaling length much below the length of the complete dislocation line are independent of such geometrical con- straints [KCC + 92, DHZ92, DC92, DK94, RHZ94, Raa96a, Raa96b, Dev96, FGC96, Raa98]. The motion of the dislocations or of the dislocation segments in their respective glide planes are usually described by assuming simple phenomenological viscous flow laws. “Viscous flow” means that the dislocation is in an overdamped state of motion so that its velocity is linearly proportional to the local net force which acts in the dislocation glide plane. Viscous motion phenomenologically describes strain rate sensitive flow. A more detailed formulation of the dynamics of dislocations can be obtained by solving Newton’s second law of motion for each dislocation or dislocation segment, respectively. This formulation which takes into account the effective mass of the dislocation (which is a measure for the reluctance of the dislocation against acceleration) is of relevance only for very small and 267 for very large dislocation velocities. The solution of the temporal evolution of the dislocation positions is as a rule obtained by simple finite difference algorithms. It is not the aim of this chapter to provide an exhaustive review of the large number of analytical statistical, phenomenological models that dominate the field of mesoscopic non- space-discretized materials modeling, but to concentrate on those simulations that are discrete in both space and time, and explicitly incorporate the properties of individual lattice defects in a continuum formulation. The philosophy behind this is twofold. First, the various classi- cal phenomenological mesoscopical modeling approaches, which are discrete in time but not in space, have already been the subject of numerous thorough studies in the past, particu- larly in the fields of crystal plasticity, recrystallization phenomena, and phase transformation [Koc66, Arg75, KAA75, Mug80, MK81, Mug83, EM84, PA84, EK86, GA87, Koc87, EM91]. These models usually provide statistical rather than discrete solutions and can often be solved without employing time-consuming numerical methods. This is why they often serve as a phys- ical basis for deriving phenomenological constitutive equations that can be incorporated in advanced larger-scale finite element, self-consistent, or Taylor-type simulations [Arg75, GZ86, ABH + 87, Koc87, NNH93, KK96]. However, since such constitutive descriptions only pro- vide an averaged picture of the material response to changes in the external constraints, they are confined to statistical predictions and do not mimic details of the microstructural evolu- tion. Hence, they are beyond the scope of this chapter. Second, physically sound micro- and mesoscale material models that are discrete in both space and time must incorporate the stat- ics and kinetics of individual lattice defects. This makes them superior to the more descriptive statistical models in that they allow simulations that are more precise in their microscopical pre- dictions due to the smaller number of phenomenological assumptions involved. An overview of statistical analytical dislocation models can be found in the more recent overview volume of Raabe et al. [RBC04]. 8.2 Linear Elasticity Theory for Plasticity 8.2.1 Introduction This section reviews the theoretical framework of linear continuum elasticity theory as required for the formulation of basic dislocation mechanics. The mechanical interaction between dif- ferent dislocations (2D case) or different dislocation segments (3D case) is transmitted by the constituent elements (atoms) of the material. In this approach the material is described as an isotropic or anisotropic linear elastic unbounded homogeneous continuum in which the dislo- cations are embedded as elementary carriers of displacement and stress. This statement already implies some essentials associated with the mathematical treatment of dislocations, namely, that they are outside their cores simulated as line defects in the framework of linear elasticity. Large strains occurring close to the dislocation cores are naturally excluded from the elastic treatment. For this purpose an inner cutoff radius in the order of the magnitude of the Burgers vector is used. The dislocations are generally treated as stationary defects, that is, their displacement field does not depend on time. This implies that for all derivations the time-independent Green’s function may be used. While in the pioneering studies [DC92, Kub93, Raa96a] the field equations for the isotropic elastic case were used for 3D simulations, this chapter presents the general anisotropic field approach [Raa96b, Raa98]. For this reason the following sections recapitulate the elementary concepts of isotropic and anisotropic linear elastic theory. On this basis the field equations for both infinite dislocations (2D) and finite dislocation segments (3D) will be developed. 268 COMPUTATIONAL MATERIALS ENGINEERING In what follows, the notation x 1 ,x 2 ,x 3 will be used in place of x, y, z for the Cartesian coordinate system. This notation has the particular advantage that in combination with Einstein’s summation convention it permits general results to be expressed and manipulated in a concise manner. The summation convention states that any term in which the same Latin suffix occurs twice stands for the sum of all the terms obtained by giving this suffix each of its possible values. For instance, the trace of the strain tensor can be written as ε ii and interpreted as ε ii ≡ 3  i=1 ε ii = ε 11 + ε 22 + ε 33 (8.1) For the trace of the displacement gradient tensor the same applies: ∂u i ∂x i ≡ 3  i=1 ∂u i ∂x i = ∂u 1 ∂x 1 + ∂u 2 ∂x 2 + ∂u 3 ∂x 3 (8.2) On the other hand, certain results are more conveniently expressed in vector notation (bold symbols, e.g., u) or as components (suffixes 1, 2, 3, e.g., u 1 ,u 2 ,u 3 ). In lengthy terms, partial spatial derivatives will be expressed by the abbreviation v 1,2 instead of ∂v 1 /∂x 2 . 8.2.2 Fundamentals of Elasticity Theory The Displacement Field In a solid unstrained body the position of each infinitesimal volume element 1 can be described by three Cartesian coordinates, x 1 ,x 2 , and x 3 . In a strained condition the position of the volume element considered will shift to a new site described by x 1 + u 1 , x 2 + u 2 , x 3 + u 3 , where the triple u 1 , u 2 , u 3 is referred to as displacement parallel to the x 1 -, x 2 -, and x 3 -axis, respectively. The displacement field corresponds to the values of u 1 , u 2 , and u 3 at every coordinate x 1 , x 2 , x 3 within the material. In general, the displacement is a vector field which depends on all three spatial variables. It maps every point of the body from its position in the undeformed to its position in the deformed state. For instance, translations represent trivial examples of displacement, namely, that of a rigid-body motion where u 1 , u 2 , and u 3 are constants. The Strain Field Let the corners of a volume element, which is much larger than the atomic volume, be given by the coordinates (x 1 ,x 2 ,x 3 ), (x 1 +∆x 1 ,x 2 ,x 3 ), (x 1 ,x 2 +∆x 2 ,x 3 ) and so on. During straining, the displacement of the corner with the coordinates (x 1 ,x 2 ,x 3 ) will amount to (u 1 ,u 2 ,u 3 ). Since the displacement is a function of space it can be different for each corner. Using a Taylor expansion the displacements can be described by         u 1 + ∂u 1 ∂x 1 δx 1 + ∂u 1 ∂x 2 δx 2 + ∂u 1 ∂x 3 δx 3 u 2 + ∂u 2 ∂x 1 δx 1 + ∂u 2 ∂x 2 δx 2 + ∂u 2 ∂x 3 δx 3 u 3 + ∂u 3 ∂x 1 δx 1 + ∂u 3 ∂x 2 δx 2 + ∂u 3 ∂x 3 δx 3         (8.3) 1 Since for small displacements the elastic bulk modulus is proportional to the spatial derivative of the interatomic forces, any cluster of lattice atoms can be chosen as an infinitesimal volume element. Introduction to Discrete Dislocation Statics and Dynamics 269 Using concise suffix notation, equation (8.3) can be rewritten ( u 1 + u 1,j δx j ,u 2 + u 2,j δx j ,u 3 + u 3,j δx j ) (8.4) where the summation convention is implied. The abbreviation u 1,2 refers to the spatial deriva- tive ∂u 1 /∂x 2 . These partial derivatives represent the components of the displacement gradient tensor ∂u i /∂x j = u i,j . In linear elasticity theory only situations in which the derivatives ∂u i /∂x j are small compared with 1 are treated. If the extension of the considered volume element ∆x 1 , ∆x 2 , ∆x 3 is sufficiently small, the displacement described by equation ( 8.3) can be written         u 1 + ∂u 1 ∂x 1 ∆x 1 + ∂u 1 ∂x 2 ∆x 2 + ∂u 1 ∂x 3 ∆x 3 u 2 + ∂u 2 ∂x 1 ∆x 1 + ∂u 2 ∂x 2 ∆x 2 + ∂u 2 ∂x 3 ∆x 3 u 3 + ∂u 3 ∂x 1 ∆x 1 + ∂u 3 ∂x 2 ∆x 2 + ∂u 3 ∂x 3 ∆x 3         (8.5) Using suffix notation equation (8.5) is found to be ( u 1 + u 1,j ∆x j ,u 2 + u 2,j ∆x j ,u 3 + u 3,j ∆x j ) (8.6) For the corner of the volume element with the coordinates (x 1 ,x 2 +∆x 2 ,x 3 +∆x 3 ) equation (8.5) reduces to  u 1 + ∂u 1 ∂x 2 ∆x 2 + ∂u 1 ∂x 3 ∆x 3 ,u 2 + ∂u 2 ∂x 2 ∆x 2 + ∂u 2 ∂x 3 ∆x 3 , u 3 + ∂u 3 ∂x 2 ∆x 2 + ∂u 3 ∂x 3 ∆x 3  (8.7) Similar displacement expressions can be obtained for the other corners of the volume element being considered. For situations where all of the derivatives except those denoted by ∂u 1 /∂x 1 , ∂u 2 /∂x 2 , and ∂u 3 /∂x 3 are equal to zero, it is straightforward to see that a rectangular volume element pre- serves its shape. In such a case the considered portion of material merely undergoes positive or negative elongation parallel to its edges. For the x 1 -direction the elongation amounts to (∂u 1 /∂x 1 )∆x 1 . Hence, the elongation per unit length amounts to (∂u 1 /∂x 1 ) ·(∆x 1 )/(∆x 1 )= ∂u 1 /∂x 1 . This expression is referred to as strain in the x 1 -direction and is indicated by ε 11 . Pos- itive values are defined as tensile strains and negative ones as compressive strains. The sum of these strains parallel to x 1 , x 2 , and x 3 defines the dilatation, which equals the change in volume per unit volume associated with a given strain field, that is, ε ii = ε 11 + ε 22 + ε 33 = div u, where div is the operator ∂/∂x 1 + ∂/∂x 2 + ∂/∂x 3 . In case of a nonzero dilatation the strain components describe the change in both shape and size. The situation is different when each of the derivatives denoted by ∂u 1 /∂x 1 , ∂u 2 /∂x 2 , ∂u 3 /∂x 3 is zero, but the others are not. In such cases the considered initial rectangular volume element is no longer preserved but can both rotate and assume a rhombic shape. A single component of the displacement gradient ten- sor, for instance, ∂u 2 /∂x 1 , denotes the angle by which a line originally in the x 1 -direction rotates towards the x 2 -axis during deformation. However, the rotation of an arbitrary boundary line of a small volume element does not necessarily imply that the material is deformed. One could rotate the boundary line simply by rotating the other boundaries accordingly. Such an 270 COMPUTATIONAL MATERIALS ENGINEERING [...]... 0 0 C1111 C 1122 C 1122 C 1122 C1111 C 1122 0 0 0    C 1122 C 1122 C1111 0 0 0    = 0 0  0 0 C2323  0   0 0  0 0 0 C2323 0 0 0 0 0 C2323 276 COMPUTATIONAL MATERIALS ENGINEERING (8.37) In matrix notation this can be rewritten  cub Cmn  0 0 C11 C12 C12 0 C12 C11 C12 0 0 0    C12 C12 C11 0 0 0    = 0  0 0 C44 0  0   0 0 0 0 C44 0  0 0 0 0 0 C44 (8.38) For hexagonal materials the... σ22 + σ33 = σ1 + σ2 + σ3 1 2 2 2 σii σjj − σij σij = σ11 σ22 − 12 + σ22 σ33 − σ23 + σ11 σ33 − σ13 I2 = 2 σ13 12 σ23 σ σ σ = σ1 σ2 + σ1 σ3 + σ2 σ3 = 11 + 11 + 22 σ23 σ33 σ13 σ33 12 σ22 2 2 2 I3 = det σij = σ11 σ22 σ33 + 2 12 σ13 σ23 − σ11 σ23 − σ22 σ13 − σ33 12 σ11 12 σ13 = σ1 σ2 σ3 = 12 σ22 σ23 σ13 σ23 σ33 272 COMPUTATIONAL MATERIALS ENGINEERING (8.16) Since conservative dislocation motion is practically... independent elastic constants remain, that is, C11 , C12 , and C44 , in the hexagonal lattice five constants C11 , C12 , C13 , C33 , and C44 must be considered In the case of cubic symmetry, the inversion of the stiffnesses to the compliances and vice versa leads to the relations C11 = (S11 + S12 ) (S11 − S12 ) (S11 + 2 S12 ) C12 = − S12 (S11 − S12 ) (S11 + 2 S12 ) C44 = 1 S44 (8.41) Introduction to Discrete... C12 ) (C11 − C12 ) (C11 + 2 C12 ) S12 = −C12 (C11 − C12 ) (C11 + 2 C12 ) S44 = 1 C44 (8.42) Before using general anisotropic elastic field equations in simulations, it is useful to test the predictions in the isotropic limit A material is defined as elastically isotropic if the elastic properties are independent of direction In the case of cubic crystal symmetry, this is realized when C44 = (C11 − C12... − C12 (8.43) Indeed, most metals deviate considerably from the isotropic limit Tungsten has the smallest deviation with a Zener ratio of Az = 1 and lithium the largest one with a ratio of Az = 9.39 (Table 8-1) The two elastic constants of isotropic materials are often expressed in terms of µ and ν , which are defined by µ = C44 = C2323 = 1 (C11 − C12 ) 2 (8.44) and ν= C12 C 1122 S S = = − 12 = − 1122 ... hex Cijkl 0 0 C1111 C 1122 C1133 C 0 0  1122 C1111 C1133 C 0 0  1133 C1133 C3333 = 0 0 0 0 C2323    0 0 0 0 C2323  0 0 0 0 0  0 0 0 0 0          1 (C1111 − C 1122 ) 2 (8.39) In matrix notation it can be rewritten  hex Cmn 0 C11 C12 C13 0 C 0  12 C11 C13 0 C 0  13 C13 C33 0 = 0 0 0 C44 0    0 0 0 0 C44  0 0 0 0 0 0 0 0 0 0           1 (C11 − C12 ) 2 (8.40) While in... invariants of the stress deviator are J1 = Sii = 0 1 2 1 2 2 2 2 2 Sij Sij = S + S22 + S33 + 12 + σ23 + σ13 2 2 11 1 1 2 2 2 2 2 2 S S S = (σ11 − σ22 )2 + (σ22 − σ33 )2 + (σ33 − σ11 )2 + 12 + σ23 + σ13 = 2 1 2 3 6 J2 = J3 = det Sij = S11 S12 S13 S11 12 σ13 1 (Sij Sjk Ski ) = S1 + S2 + S3 = S12 S22 S23 = 12 S22 σ23 3 S13 S23 S33 σ13 σ23 S33 (8.20) where Sij are the components and S1 , S2 , and S3... Stiffness Constants for Some Cubic Metals Structure C1111 (GPa) C 1122 (GPa) C2323 (GPa) Az ν fcc fcc fcc bcc fcc bcc bcc 12. 40 10.82 18.60 35.00 16.84 24.20 1.48 9.34 6.13 15.70 5.78 12. 14 14.65 1.25 4.61 2.85 4.20 10.10 7.54 11.20 1.08 3.013 1.215 2.987 0.691 3.209 2.346 9.391 0.43 0.36 0.46 0.14 0.42 0.38 0.46 278 COMPUTATIONAL MATERIALS ENGINEERING In addition to these constants the Lam´ constants µ... contribution is given by σh = 1 1 1 σii = ( σ11 + σ22 + σ33 ) = I1 3 3 3 (8.17) The deviatoric stress tensor can then be written  d σij = σij  σ11 − σ h 12 σ13   − δij σ h =  σ21 σ22 − σ h σ23  = Sij σ31 σ32 σ33 − σ h     S11 S12 S13 S11 12 σ13 = S21 S22 S23  = σ21 S22 σ23  S31 S32 S33 σ31 σ32 S33 (8.18) The operation of adding or subtracting hydrostatic contributions to or from the deviator... Dislocation Statics and Dynamics 273 for the variation of stress as a function of position and Newton’s second law gives the dynamic equations of equilibrium: m ∂ 2 u1 = ∂t2 σ11 + ∂σ11 ∂ 12 δx1 − σ11 δx2 δx3 + 12 + δx2 − 12 δx1 δx3 ∂x1 ∂x2 + σ13 + m ∂ 2 u2 = ∂t2 σ22 + ∂σ22 ∂σ21 δx2 − σ22 δx1 δx3 + σ21 + δx1 − σ21 δx2 δx3 ∂x2 ∂x1 + σ23 + m ∂ 2 u3 = ∂t2 σ33 + ∂σ13 δx3 − σ13 δx1 δx2 + P1 δx1 δx2 δx3 ∂x3 . form: C cub ijkl =         C 1111 C 1122 C 1122 000 C 1122 C 1111 C 1122 000 C 1122 C 1122 C 1111 000 000C 2323 00 0000C 2323 0 00000C 2323         (8.37) 276 COMPUTATIONAL MATERIALS ENGINEERING In matrix. relations C 11 = (S 11 + S 12 ) (S 11 − S 12 )(S 11 +2S 12 ) C 12 = −S 12 (S 11 − S 12 )(S 11 +2S 12 ) C 44 = 1 S 44 (8.41) Introduction to Discrete Dislocation Statics and Dynamics 277 S 11 = (C 11 + C 12 ) (C 11 −. isotropic materials are often expressed in terms of µ and ν, which are defined by µ = C 44 = C 2323 = 1 2 (C 11 − C 12 ) (8.44) and ν = C 12 C 11 + C 12 = C 1122 C 1111 + C 1122 = − S 12 S 11 = − S 1122 S 1111 (8.45) where µ