In a ternary eutectic system, we considered two regions within a completely symmetricmodel phase diagram, namely, i the region of four phase equilibrium at the ternary eutecticcompositio
Trang 1interface; subsequently, it rapidly solidifies Nucleation of the opposite phase within the concaveportion of the interfaces stabilizes the lamellae and regular lamellar growth to continue at halfthe original lamellae spacing This suggests that nucleation ahead of the eutectic front mayprovide a method of stabilizing the lamellae.
Morphological instabilities with a regular oscillatory structure of lamellar eutectics arereported in experiments with a transparent organic alloy and also in numerical studies by Karmaand Sarkission [KS96] In three dimensions, an analogous type of oscillation can be observed foreutectic microstructure formations, Figure 7-32 Performing an alternating topological change,
αsolid rods are embedded in aβmatrix followed by the opposite situation ofβcrystals ded in anαmatrix
embed-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 growthare exemplarily illustrated In particular, the ternary Ni60Cu40−xCrx alloy system is consi-dered as a prototype system to investigate the influence of interplaying solute fields on theinterface stability, on the growth velocity, and on the characteristic type of morphology TheNi–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 severalpapers (e.g., refs [HSAF99, HAK01]) Hence, physical parameters of Ni–Cu are relatively wellestablished
A series of numerical computations for different alloy compositions varying from
Ni60Cu28Cr12to Ni60Cu12Cr28has been carried out in Figure 7-33 The concentration of Niwas kept constant at 60 at.%, and the initial undercooling was fixed at 20K measured fromthe 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
Ni60Cu20Cr20
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
Trang 2Ni60Cu28Cr12 Ni60Cu12Cr28
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 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
per-Ni in the solid phase.
The left side of Figure 7-33(b) shows the dendritic morphologies observed for Cr trations less than 20at.% The right side of Figure 7-33(b) displays globular morphologies for
concen-Cr concentrations crossing this threshold The velocity of the dendritic–globular tip increaseslinearly 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 aone-phase region (below the solidus line) in the phase diagram
In a ternary eutectic system, we considered two regions within a completely symmetricmodel phase diagram, namely, (i) the region of four phase equilibrium at the ternary eutecticcomposition 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, andγgrow into an undercooled meltLwith equal volume fractions by a ternary eutecticreactionL → α + β + γ Different permutations of lamellar structuresαβγ andαβαγ arepossible Phase-field simulations can be used to investigate which phase sequence is favored togrow at certain solidification conditions An example for anαβγ configuration is displayed
in Figure 7-34(a) showing in addition the concentration of one of the components ahead of thegrowing eutectic front In three dimensions, a regular hexagonal shape of the three solid phasesoccurs 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 preferredgrowth directions
In a eutectic phase system with ternary impurity, it can be observed that the impurity ispushed ahead of the solidifying lamellae and builds up For the simulation in Figure 7-35, wehave set an initial composition vector of (c A , c B , c C ) = (0.47, 0.47, 0.06) so thatc C is the
concentration component of minor amount The main componentsc Aandc Bare incorporated
in the growingα–βsolid front whereas the impurity is rejected by both growing solid phases
To further observe the effect of eutectic cell/colony formation, computations in larger domainsincluding noise as well as nucleation have to be performed
Trang 3(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 grainstructure evolution and anisotropic curvature flow in a polycrystalline material In this case, thephase-field variablesφ 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 grainsstarted to grow toward the bottom of the domain Certain grains with their crystal orientation inthe direction of the shear movement of the lower wall grew faster than the neighboring grains,which finally ceased to grow
Phase-Field Modeling 263
Trang 4(a) (b)
FIGURE 7-36 Grain selection process as a result of grain boundary motion of differently oriented
crystals, (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 growinginto 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 tomigrate as a result of curvature minimization
The phase-field model recovers the generic features of grain growth such as grain 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
bound-of neighboring triple junctions in microstructures bound-of polycrystalline thin films
Trang 57.5 Acknowledgments
The author of this chapter thanks her co-workers Frank Wendler and Denis Danilov for ing images of the simulation results
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Trang 78 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 mesoscopicscale using space- and time-discretized dislocation statics and dynamics The complexity ofdiscrete dislocation models is due to the fact that the mechanical interaction of ensembles ofsuch 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 embeddedwithin 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 ofelasticity This applies for both straight infinite dislocations (2D discretization) and dislocationsegments (3D discretization)
The simulation work in this field can be grouped into simulations which describe the cations in two dimensions and in three dimensions 2D calculations are conducted either withdislocations 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 glideplane 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 dislocationinto a “spaghetti-type” sequence of piecewise straight segments with a scaling length muchbelow 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 planesare 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 linearlyproportional to the local net force which acts in the dislocation glide plane Viscous motionphenomenologically describes strain rate sensitive flow
dislo-A more detailed formulation of the dynamics of dislocations can be obtained by solvingNewton’s second law of motion for each dislocation or dislocation segment, respectively Thisformulation which takes into account the effective mass of the dislocation (which is a measurefor the reluctance of the dislocation against acceleration) is of relevance only for very small and
267
Trang 8for very large dislocation velocities The solution of the temporal evolution of the dislocationpositions 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 ofanalytical 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 cal phenomenological mesoscopical modeling approaches, which are discrete in time but not
classi-in space, have already been the subject of numerous thorough studies classi-in the past, 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 solvedwithout 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 inadvanced larger-scale finite element, self-consistent, or Taylor-type simulations [Arg75, GZ86,ABH+87, Koc87, NNH93, KK96] However, since such constitutive descriptions only pro-
particu-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 tion Hence, they are beyond the scope of this chapter Second, physically sound micro- andmesoscale material models that are discrete in both space and time must incorporate the stat-
evolu-ics and kinetevolu-ics 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 dictions due to the smaller number of phenomenological assumptions involved An overview ofstatistical analytical dislocation models can be found in the more recent overview volume of
The dislocations are generally treated as stationary defects, that is, their displacement fielddoes not depend on time This implies that for all derivations the time-independent Green’sfunction may be used
While in the pioneering studies [DC92, Kub93, Raa96a] the field equations for the isotropicelastic case were used for 3D simulations, this chapter presents the general anisotropic fieldapproach [Raa96b, Raa98] For this reason the following sections recapitulate the elementaryconcepts of isotropic and anisotropic linear elastic theory On this basis the field equations forboth infinite dislocations (2D) and finite dislocation segments (3D) will be developed
Trang 9In what follows, the notation x1, x2, x3 will be used in place of x, y, z for the Cartesiancoordinate system This notation has the particular advantage that in combination with Einstein’ssummation convention it permits general results to be expressed and manipulated in a concisemanner The summation convention states that any term in which the same Latin suffix occurstwice 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 asε iiand interpreted as
ε ii ≡3i=1
For the trace of the displacement gradient tensor the same applies:
∂u i
∂x i ≡3i=1
8.2.2 Fundamentals of Elasticity Theory
The Displacement Field
In a solid unstrained body the position of each infinitesimal volume element1can be described
by three Cartesian coordinates,x1, x2, andx3 In a strained condition the position of the volumeelement considered will shift to a new site described byx1+ u1,x2+ u2,x3+ u3, where thetripleu1,u2,u3is referred to as displacement parallel to thex1-,x2-, andx3-axis, respectively.The displacement field corresponds to the values ofu1,u2,andu3at every coordinatex1,
x2,x3within the material In general, the displacement is a vector field which depends on allthree 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 ofdisplacement, namely, that of a rigid-body motion whereu1,u2, andu3are constants
The Strain Field
Let the corners of a volume element, which is much larger than the atomic volume, be given bythe coordinates(x1, x2, x3),(x1+∆x1, x2, x3),(x1, x2+∆x2, x3)and so on During straining,the displacement of the corner with the coordinates (x1, x2, x3)will amount to(u1, u2, u3).Since the displacement is a function of space it can be different for each corner Using a Taylorexpansion the displacements can be described by
1Since 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
Trang 10Using concise suffix notation, equation (8.3) can be rewritten
( u1+ u 1,j δx j , u2+ u 2,j δx j , u3+ u 3,j δx j) (8.4)where the summation convention is implied The abbreviationu 1,2refers to the spatial deriva-tive∂u1/∂x2 These partial derivatives represent the components of the displacement gradienttensor ∂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 volumeelement∆x1,∆x2,∆x3is sufficiently small, the displacement described by equation (8.3) can
For situations where all of the derivatives except those denoted by∂u1/∂x1,∂u2/∂x2, and
∂u3/∂x3are equal to zero, it is straightforward to see that a rectangular volume element serves its shape In such a case the considered portion of material merely undergoes positive
pre-or negative elongation parallel to its edges Fpre-or the x1-direction the elongation amounts to
(∂u1/∂x1) ∆x1 Hence, the elongation per unit length amounts to(∂u1/∂x1) · (∆x1)/(∆x1) =
∂u1/∂x1 This expression is referred to as strain in thex1-direction and is indicated byε11 itive values are defined as tensile strains and negative ones as compressive strains The sum ofthese strains parallel tox1,x2, andx3defines the dilatation, which equals the change in volumeper unit volume associated with a given strain field, that is,ε ii = ε11+ ε22+ ε33 = div u,where div is the operator∂/∂x1 + ∂/∂x2+ ∂/∂x3 In case of a nonzero dilatation the straincomponents describe the change in both shape and size The situation is different when each
Pos-of the derivatives denoted by∂u1/∂x1,∂u2/∂x2,∂u3/∂x3is zero, but the others are not Insuch cases the considered initial rectangular volume element is no longer preserved but canboth rotate and assume a rhombic shape A single component of the displacement gradient ten-sor, for instance,∂u2/∂x1, denotes the angle by which a line originally in thex1-directionrotates towards thex2-axis during deformation However, the rotation of an arbitrary boundaryline of a small volume element does not necessarily imply that the material is deformed Onecould rotate the boundary line simply by rotating the other boundaries accordingly Such an
Trang 11operation would leave the body undeformed and is therefore referred to as rigid-body rotation.However, if different boundary lines rotate by different angles the volume element undergoesboth deformation and rigid-body rotation By subtracting the rotation components from the dis-placement gradients, one obtains the elements which describe the shape changes These com-ponents are denoted as shear strains 1
From the preceding, it becomes clear that a deformation state is entirely characterized bythe displacement vector field u i However, this quantity is an inconvenient representation ofdeformation since it does not naturally separate shape changes from rigid-body rotations or vol-ume changes For this purpose the displacement gradient tensoru i,j, which in the general case
still contains the strain tensor ε ij (tensile/compressive and shear components) and the
rigid-body rotationω ij, seems more appropriate if adequately dismantled Simple geometrical siderations show that the former portion corresponds to the symmetric part of the displacementgradient tensor and the latter one to its antisymmetric (skew symmetric) part:
equiva-Since the trace elements of the antisymmetric part of the displacement gradient tensor are
by definition equal to zero, only three independent components remain These represent smallpositive rotations about the axes perpendicular to the displacements from which they are derived,that is, ω23 = ω x1 denotes a rotation about thex1-axis,ω13 = ω x2 about thex2-axis, and
ω12= ω x3 about thex3-axis By using the totally antisymmetric Levi–Civita operator ijk, the
components of which are defined to be1if the suffixes are in cyclic order,−1if they are inreverse cyclic order, and0if any two suffixes are the same, the three rotations can be compactlywritten as
ω x k= 12 ijk ∂x ∂u i
j
(8.10)Summarizing the three rotation components as a vector, one obtains
! = 12curl u = 12r × u (8.11)
Introduction to Discrete Dislocation Statics and Dynamics 271
Trang 12whererdenotes the operator ∂
The Stress Field
The introduction of the traction vector serves as a starting point for deriving the stress tensor.The traction is defined by
whereT denotes the traction vector,Athe area, andF the externally imposed force vector In
suffix notation the traction can be written
Since the traction vector depends on the inclination of the area element considered, it is pertinent
to look for a more general form to describe the effect of external forces on the material Such
a description can be found by considering the traction vectors of three orthogonal sections, theunit normal vectors of which are denoted byn1, n2, n3:
Thus, by definition of equation (8.14), the stressσ ij is a tensor field which connectsT tonatany point within the material Equation (8.14) is referred to as the Cauchy stress formula.The components σ ij with i = j give the respective force component along the positive
x j-axis acting through the area element having its normal along the same direction They arereferred to as normal stresses The components σ ij withi = j give the corresponding twoorthogonal force components acting in the same area element along the two respective positive
x j-axes, wherei = j They are referred to as shear stresses Considering momentum equilibriumunder static conditions one obtainsσ ij = σ ji By solving the eigenvalue problem
| σ ij − δ ij σ |= σ3− I1σ2+ I2σ − I3= 0 (8.15)whereδ ij is the Kronecker symbol, andI1, I2, I3the invariants of the stress state, one obtainsthe principal stressesσ = σ1, σ2, σ3 The principal axes are the eigenvectors associated withthis stress tensor The invariants amount to
σ23 σ33
+ σ11 σ13
σ13 σ33
σ11 σ12 σ13
σ12 σ22 σ23
σ13 σ23 σ33
... σ33
σ11 σ12< /small> σ13
σ12< /small> σ22 σ23
σ13