9 Finite Elements for MicrostructureEvolution —Dierk Raabe 9.1 Fundamentals of Differential Equations 9.1.1 Introduction to Differential Equations Many of the laws encountered in materia
Trang 1The strain increments per simulation step∆tamount to
∆ε x1x1= ˙γ x1∆t ∆ε x2x2 = ˙γx2∆t ∆εx3x3 = ˙γx3∆t (8.186)and its shear components to
∆εx1x2= ∆2t ( ˙γx1+ ˙γx2) ∆εx1x3= ∆2t ( ˙γx1+ ˙γx3) ∆εx2x3= ∆2t ( ˙γx2+ ˙γx3)
(8.187)The rotation rate of the lattice affected, ˙ω xlatti x j, which results from the shears on the indi-vidual slip systems, can be computed from the rigid-body rotation rate, that is, from the skewsymmetric portion of the discretized crystallographic velocity gradient tensor, ˙ωspinx i x j, and fromthe antisymmetric part, ˙uantix i ,x j, of the externally imposed macroscopic velocity gradient tensor,
˙uextx i ,x j:
˙ωlattx i x j = ˙uantix i ,x j − ˙ω xspini x j = 12 ˙uextx i ,x j − ˙uextx j ,x i− 12˙γx i − ˙γ x j
(8.188)
8.6 Dislocation Reactions and Annihilation
In the preceding sections it was mainly long-range interactions between the dislocation ments that were addressed However, strain hardening and dynamic recovery are essentiallydetermined by short-range interactions, that is, by dislocation reactions and by annihilation,respectively
seg-Using a phenomenological approach that can be included in continuum-type simulations,one can differentiate between three major groups of short-range hardening dislocation reactions[FBZ80]: the strongest influence on strain hardening is exerted by sessile reaction productssuch as Lomer–Cottrell locks The second strongest interaction type is the formation of mobilejunctions The weakest influence is naturally found for the case in which junctions are formed.Two-dimensional dislocation dynamics simulations usually account for annihilation and theformation of sessile locks Mobile junctions and the Peach–Koehler interaction occur naturallyamong parallel dislocations The annihilation rule is straightforward If two dislocations onidentical glide systems but with opposite Burgers vectors approach more closely than a certain
Area of SpontaneousAnnihilation
FIGURE 8-8 Annihilation ellipse in 2D dislocation dynamics It is constructed by the values for the
spontaneous annihilation spacing of dislocations that approach by glide and by climb.
Introduction to Discrete Dislocation Statics and Dynamics 311
Trang 2minimum allowed spacing, they spontaneously annihilate and are removed from the simulation.Current 2D simulations [RRG96] use different minimum distances in the direction of glide(dg
ann≈ 20|b|) and climb (dc
ann≈ 5|b|), respectively [EM79] (Figure 8-8)
Lock formation takes place when two dislocations on different glide systems react to form
a new dislocation with a resulting Burgers vector which is no longer a translation vector of anactivated slip system In the 2D simulation this process can be realized by the immobilization ofdislocations on different glide systems when they approach each other too closely (Figure 8-9).The resulting stress fields of the sessile reaction products are usually approximated by a linearsuperposition of the displacement fields of the original dislocations before the reaction.Dislocation reactions and the resulting products can also be included in 3D simulations Due
to the larger number of possible reactions, two aspects require special consideration, namely,the magnitude and sign of the latent heat that is associated with a particular reaction, and thekinematic properties and the stress field of the reaction product
The first point addressed can be solved without using additional analytical equations Forinvestigating whether a particular reaction between two neighboring segments will take place
or not, one subtracts the total elastic and core energy of all initial segments that participate inthe reaction from that of the corresponding configuration after the reaction If the latent heat isnegative, the reaction takes place Otherwise, the segments pass each other without reaction Inface-centered cubic materials 2 dislocations can undergo24different types of reactions Fromthis number only12entail sessile reaction products Assuming simple configurations, that is,only a small number of reacting segments, the corresponding latent heat data can be included inthe form of a reference table
The short-range back-driving forces that arise from cutting processes are calculated from thecorresponding increase in line energy For either of the cutting defects, the increase in disloca-tion line amounts to the Burgers vector of the intersecting dislocation Although this short-rangeinteraction does not impose the same immediate hardening effect as a Lomer–Cottrell lock, itsubsequently gives rise to the so-called jog drag effect, which is of the utmost relevance to themobility of the dislocations affected
The treatment of annihilation is also straightforward If two segments have a spacing belowthe critical annihilation distance [EM79] the reaction takes place spontaneously However, thesubsequent reorganization of the dislocation segment vectors is not simple and must be treatedwith care
Area of SpontaneousReaction
FIGURE 8-9 Reaction ellipse in 2D dislocation dynamics It is constructed by the values for the
spontaneous reaction spacing of dislocations that approach by glide and by climb.
Trang 3The stress and mobility of glissile dislocation junctions can be simulated by using a simplesuperposition of the segments involved Unfortunately, this technique does not adequately reflectthe behavior of Lomer–Cottrell locks Such sessile junctions must therefore be artificially ren-dered immobile.
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Trang 79 Finite Elements for Microstructure
Evolution
—Dierk Raabe
9.1 Fundamentals of Differential Equations
9.1.1 Introduction to Differential Equations
Many of the laws encountered in materials science are most conveniently formulated in terms
of differential equations Deriving and solving differential equations are thus among the mostcommon tasks in modeling material systems
Differential equations are equations involving one or more scalar or tensorial dependent ables, independent variables, unknown functions of these variables, and their correspondingderivatives Equations which involve unknown functions that depend on only one independentvariable are referred to as ordinary differential equations If the equations involve unknownfunctions that depend on more than one independent variable they are referred to as partialdifferential equations The “order” of a differential equation is the highest order of any of thederivatives of the unknown functions in the equation Equations involving only the first deriva-tives are referred to as first-order differential equations Equations involving the second deriva-tives are referred to as second-order differential equations Second- and higher-order differentialequations such as
In these equationsuis the state variable that is a function of the independent time variablet,
vis the first time derivative ofu, andf a function ofuandv, respectively For instance, thefrequently occurring problem of (one-dimensional) motion of a particle or dislocation segment
of effective massmunder a force fieldf (x, t)in thex-direction is described by the second-order
differential equation
md2x (t)
317
Trang 8If one defines the particle momentum
dp(x, t)
Differential equations which contain only linear functions of the independent variables arecalled “linear differential equations.” For these equations the superposition principle applies.That means linear combinations of solutions which satisfy the boundary conditions are alsosolutions to the differential equation satisfying the same boundary conditions Differential equa-tions which involve nonlinear functions of the independent variables are denoted as “nonlineardifferential equations.” For such equations the superposition principle does not apply
Most problems in computational materials science lead in their mathematical formulation to
“partial differential equations”, which involve both space and time as independent variables.Usually, one is interested in particular solutions of partial differential equations, which aredefined within a certain range of the independent variables and which are in accord with certaininitial-value and boundary-value conditions In this context it is important to emphasize that
a problem that is in the form of a differential equation and boundary conditions must be wellposed That means only particular initial and boundary conditions transform a partial differentialequation into a solvable problem
Partial differential equations can be grouped according to the type of additional conditionsthat are required in formulating a well-posed problem This classification scheme will be out-lined in the following for the important group of linear second-order partial differential equa-tions with two independent variables, sayx1andx2 The general form of this equation is
F = F (x1, x2), andG = G(x1, x2)are given functions of the independent variablesx1and
x2 It is stipulated that the functionsA (x1, x2),B (x1, x2), andC (x1, x2)never be equal to zero
at the same point(x1, x2) In analogy to the classification of higher-order curves in analyticalgeometry that are described by
a x1 + b x1x2+ c x2 + d x1+ e x2+ f = 0 , a2+ b2+ c2= 0 (9.7)equation (9.6) can for given values ˆx1, ˆx2, of the variables x1 and x2 assume hyperbolic,parabolic, or elliptic character Roughly speaking, hyperbolic differential equations involvesecond-order derivatives of opposite sign when all terms are grouped on one side; parabolic
differential equations involve only a first-order derivative in one variable, but have second-orderderivatives in the remaining variables; and elliptic differential equations involve second orderderivatives in each of the independent variables, each of the derivatives having equal sign whengrouped on the same side of the equation
hyperbolic partial differential equation 4 A (ˆx1, ˆx2) C (ˆx1, ˆx2) < B2(ˆx1, ˆx2)parabolic partial differential equation 4 A (ˆx1, ˆx2) C (ˆx1, ˆx2) = B2(ˆx1, ˆx2)elliptic partial differential equation 4 A (ˆx1, ˆx2) C (ˆx1, ˆx2) > B2(ˆx1, ˆx2)
Trang 9In that context it must be considered that, sinceA (x1, x2),B (x1, x2), andC (x1, x2)depend onindependent variables, the character of the differential equation may vary from point to point.The approach to group differential equations according to the character of their discriminant(4 A C − B2) is due to its importance in substituting mixed derivatives by new independentvariables The fundamental classification scheme outlined here for second-order partial differ-ential equations can be extended to coupled sets of nonlinear higher-order partial differentialequations with more than two independent variables.
Classical examples of the three types of differential equations are the wave equation for thehyperbolic class, the heat or diffusion equation and the time-dependent Schr¨odinger equation forthe parabolic class, and the Laplace and time-independent Schr¨odinger equation for the ellipticclass In three dimensions and rectangular coordinates then can be written:
2u
∂x2+∂2u
∂x2+∂2u
∂x2 = 0wherex1,x2, and x3 are the spatial variables,t the temporal variable, uthe state variable,
Dthe diffusion coefficient, which is assumed to be positive and independent of the tion, andcthe propagation velocity of the wave The assumption that the diffusion coefficient
concentra-is independent of the concentration applies of course only for certain systems and very smallconcentrations In real materials the value of the diffusion coefficient is, first, a tensor quantityand, second, highly sensitive to the concentration
It is worth mentioning that for stationary processes where∂u/∂t = 0, the diffusion (heat)equation changes into the Laplace equation In cases where under stationary conditions sinks orsources appear in the volume being considered, the diffusion equation changes into the Poissonequation:
Helmholtz equation ∂2u
∂x2 +∂2u
∂x2+∂2u
∂x2 + α u = 0 α = const.
Using the more general Laplace operator∆ = ∇2instead of rectangular coordinates and ˙u
and¨ufor the first- and second-order time derivatives, respectively, the preceding equations can
be rewritten in a more compact notation:
Trang 10Hyperbolic and parabolic partial differential equations typically describe nonstationary, that
is, time-dependent problems This is indicated by the use of the independent variable t inthe corresponding equations For solving nonstationary problems one must define initial con-ditions These are values of the state variable and its derivative, which the solution shouldassume at a given starting time t0 These initial conditions could amount tou (x1, x2, x3, t0)and ˙u(x1, x2, x3, t0)for the wave equation andu (x1, x2, x3, t0)for the diffusion or heat equa-tion If no constraints are given to confine the solutions to particular spatial coordinates, that is,
−∞ < x1, x2, x3< +∞, the situation represents a pure initial-boundary problem
In cases where additional spatial conditions are required, such asu (x10, x20, x30, t)for thewave equation, and u (x10, x20, x30, t) or (∂u/∂x1) (x10, x20, x30, t), (∂u/∂x2) (x10, x20,
x30, t), and(∂u/∂x3) (x10, x20, x30, t)for the diffusion equation, or a combination of both,one speaks of a “boundary-initial-value problem.”
Models that are mathematically described in terms of elliptic partial differential equations
are typically independent of time, thus describing stationary situations The solutions of such
equations depend only on the boundary conditions, that is, they represent pure boundary-valueproblems Appropriate boundary conditions for the Laplace or stationary heat and diffusionequation, respectively, ∆u = 0, can be formulated as Dirichlet boundary conditions or asNeumann boundary conditions Dirichlet boundary conditions mean that solutions for the statevariableuare given along the spatial boundary of the system Neumann boundary conditionsmean that solutions for the first derivative∂u/∂x nare given normal to the spatial boundary ofthe system If both the function and its normal derivative on the boundary are known, the borderconditions are referred to as Cauchy boundary conditions
9.1.2 Solution of Partial Differential Equations
The solution of partial differential equations by use of analytical methods is only possible in alimited number of cases Thus, one usually has to resort to numerical methods [Coh62, AS64,BP83, EM88] In the following sections a number of techniques are presented that allow one toobtain approximate numerical solutions to initial- and boundary-value problems
Numerical methods to solve complicated initial-value and boundary-value problems have
in common the discretization of the independent variables (typically time and space) and thetransformation of the continuous derivative into its discontinuous counterpart, that is, its finite
difference quotient Using these discretization steps amounts to recasting the continuous lem expressed by differential equations with an infinite number of unknowns, that is, functionvalues, into a discrete algebraic one with a finite number of unknown parameters which can becalculated in an approximate fashion
prob-Numerical methods to solve differential equations which are essentially defined through
initial rather than boundary values, that is, which are concerned with time derivatives, are often
referred to as finite difference techniques Most of the finite difference simulations addressed inthis book are discrete not only in time but also in space Finite difference methods approximatethe derivatives that appear in differential equations by a transition to their finite differencecounterparts This applies for the time and the space derivatives Finite difference methods donot use polynomial expressions to approximate functions
Classical textbooks suggest a substantial variety of finite difference methods [Coh62,AS64, BP83, EM88] Since any simulation must balance optimum calculation speed andnumerical precision, it is not reasonable to generally favor one out of the many possiblefinite difference solution techniques for applications in computational materials science Forinstance, parabolic large-scale bulk diffusion or heat transport problems can be solved by
Trang 11using a simple central difference Euler method, while the solution of the equations of particlemotion in molecular dynamics is usually achieved by using the Verlet or the Gear predictor–corrector method In most cases, it is useful to select a discretization method with respect to theproperties of the underlying differential equations, particularly to the highest occurring order ofderivative.
A second group of numerical means of solving differential equations comprises the ous finite element methods These methods are designed to solve numerically both complexboundary-value and initial-value problems They have in common the spatial discretization ofthe area under consideration into a number of finite elements, the temporal discretization incases where time-dependent problems are encountered, and the approximation of the true spa-tial solutions in the elements by polynomial trial functions These features explain why they arereferred to as finite element techniques
vari-Although both the finite difference and the finite element techniques can handle space andtime derivatives, the latter approach is more sophisticated in that it uses trial functions and aminimization routine Thus, the finite difference techniques can be regarded as a subset of thevarious more general finite element approximations [ZM83, ZT89]
Many finite difference and particularly most finite element methods are sometimes intuitivelyassociated with the solution of large-scale problems Although this association is often true forfinite element methods which prevail at solving meso- and macroscale boundary-value prob-lems in computational materials science, it must be underlined that such general associationsare inadequate Finite difference and finite element methods represent mathematical approx-
imation techniques They are generally not intrinsically calibrated to any physical length or
timescale Scaling parameters are introduced by the physics of the problem addressed but not
by the numerical scheme employed to solve a differential equation
9.2 Introduction to the Finite Element Method
This section deals with the simulation of materials properties and microstructures at the scopic and macroscopic levels Of course, a strict subdivision of the numerous methods thatexist in computational materials science according to the length scales that they address is to acertain extent arbitrary and depends on the aspect under investigation
meso-The finite element method is a versatile numerical means of obtaining approximate tions to boundary and initial-value problems Its approach consists in subdividing the sample
solu-of interest into a number solu-of subdomains and by using polynomial functions to approximate thetrue course of a state function in a piecewise fashion over each subdomain [Cou43, ZM83]
Hence, finite element methods are not intrinsically calibrated to some specific physical length
or timescale The application of the finite element method in materials science, though, liesparticularly in the field of macro- and mesoscale simulations where averaging empirical or phe-nomenological constitutive laws can be incorporated that describe the actual material behavior
in a statistical fashion Emphasis of the finite element method is often placed on problems incomputational mechanics, particularly when the considered shapes are complicated, the materialresponse is nonlinear, or the applied forces are dynamic [RSHP91, NNH93] All three featuresare typically encountered in the calculation of large-scale structures and elastic–plastic defor-mation The extension of classical computational solid mechanics to microstructure mechanics
or computational micromechanics requires a scale-dependent physical formulation of theunderlying constitutive behavior that is admissible at the level addressed, and a detailed incor-poration of microstructure [GZ86, ABL+94, KK96].
Finite Elements for Microstructure Evolution 321
Trang 12This task reveals a certain resemblance to molecular dynamics, where the equations ofmotion are solved for a large number of interacting particles These calculations require someapproximate formulation of the interatomic potential It is clear that the accuracy of the under-lying potential determines the reliability of the predictions Similar arguments apply for the use
of computational solid mechanics in materials science The validity of the constitutive laws andthe level at which the microstructure is incorporated in the finite element grid determine thepredictive relevance of the simulation As a rule the accuracy of solid mechanics calculationscan be increased by decreasing the level at which the required microstructural and constitutivedata are incorporated Advanced finite element methods have more recently also been used on
a very fine microstructural scale
In addition to the previously mentioned field of solid-states mechanics and its treatmentwith finite element methods, many materials problems exist, which must be formulated asinitial-value rather than as boundary-value problems Such time-dependent simulations are char-acterized by the presence of time derivatives in the governing differential equations and theprescription of initial conditions at the time origin Methods to numerically integrate equa-tions which involve time derivatives are provided by the various finite difference approaches[FW60, RM67, Det69, Mar76, DeV94, BP95] Finite difference methods comprise a number
of general numerical means for solving initial-value problems Typical examples in materialsscience are the solution of the diffusion or heat equation, of the atomic equations of motion
in molecular dynamics, or of the equations of motion in dislocation dynamics As is apparentfrom these examples, the use of finite difference algorithms is also not confined to any particularscale The present section gives a discussion of the potential of the finite difference method inmeso- and macroscale materials science
Numerical methods to solve initial- and boundary-value problems have in common thediscretization of the independent variables, which are usually time and space, and the trans-formation of the continuous derivatives into finite difference quotients Performing these dis-cretizations amounts to recasting the continuous problem expressed by differential equationswith an infinite number of unknowns, that is, function values, into an algebraic one with a finitenumber of unknown parameters that can be calculated in an approximate fashion Although boththe finite difference and the finite element methods can essentially be used to solve boundary-and initial-value problems, the latter technique represents the more general approach since ituses polynomial shape functions and a minimalization procedure According to Zienkiewiczand Morgan [ZM83], the finite difference techniques can thus be regarded as a subset of thefinite element approximation
9.3 Finite Element Methods at the Meso- and Macroscale
9.3.1 Introduction and Fundamentals
This section is devoted to discussing particular applications of the method for the simulation ofmaterials problems at the meso- and macroscale with special emphasis on large-strain plasticity.The finite element technique is a numerical method for obtaining approximate solutions toboundary- and initial-value problems by using polynomial interpolation functions In contrast
to analytical techniques, finite elements are also applicable to complicated shapes The basiccharacteristic of the finite element method is the discretization of the domain of interest, whichmay have nearly arbitrary geometry, into an assembly of relatively simply shaped elements thatare connected