COMPUTATIONAL MATERIALS ENGINEERING Episode 14 pptx

The strain increments per simulation step ∆t amount to ∆ε x 1 x 1 =˙γ x 1 ∆t ∆ε x 2 x 2 =˙γ x 2 ∆t ∆ε x 3 x 3 =˙γ x 3 ∆t (8.186) and its shear components to ∆ε x 1 x 2 = ∆t 2 (˙γ x 1 +˙γ x 2 )∆ε x 1 x 3 = ∆t 2 (˙γ x 1 +˙γ x 3 )∆ε x 2 x 3 = ∆t 2 (˙γ x 2 +˙γ x 3 ) (8.187) The rotation rate of the lattice affected, ˙ω latt x i 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 skew symmetric portion of the discretized crystallographic velocity gradient tensor, ˙ω spin x i x j , and from the antisymmetric part, ˙u anti x i ,x j , of the externally imposed macroscopic velocity gradient tensor, ˙u ext x i ,x j : ˙ω latt x i x j =˙u anti x i ,x j − ˙ω spin x i x j = 1 2  ˙u ext x i ,x j − ˙u ext x j ,x i  − 1 2  ˙γ x 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 seg- ments that were addressed. However, strain hardening and dynamic recovery are essentially determined by short-range interactions, that is, by dislocation reactions and by annihilation, respectively. 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 products such as Lomer–Cottrell locks. The second strongest interaction type is the formation of mobile junctions. The weakest influence is naturally found for the case in which junctions are formed. Two-dimensional dislocation dynamics simulations usually account for annihilation and the formation of sessile locks. Mobile junctions and the Peach–Koehler interaction occur naturally among parallel dislocations. The annihilation rule is straightforward. If two dislocations on identical glide systems but with opposite Burgers vectors approach more closely than a certain Area of Spontaneous Annihilation climb glide Glide Climb ann ann d d 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 minimum allowed spacing, they spontaneously annihilate and are removed from the simulation. Current 2D simulations [RRG96] use different minimum distances in the direction of glide ( d g ann ≈ 20|b|) and climb (d c 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 an activated slip system. In the 2D simulation this process can be realized by the immobilization of dislocations 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 linear superposition 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 the kinematic properties and the stress field of the reaction product. The first point addressed can be solved without using additional analytical equations. For investigating 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 in the reaction from that of the corresponding configuration after the reaction. If the latent heat is negative, the reaction takes place. Otherwise, the segments pass each other without reaction. In face-centered cubic materials 2 dislocations can undergo 24 different types of reactions. From this number only 12 entail sessile reaction products. Assuming simple configurations, that is, only a small number of reacting segments, the corresponding latent heat data can be included in the form of a reference table. The short-range back-driving forces that arise from cutting processes are calculated from the corresponding 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-range interaction does not impose the same immediate hardening effect as a Lomer–Cottrell lock, it subsequently gives rise to the so-called jog drag effect, which is of the utmost relevance to the mobility of the dislocations affected. The treatment of annihilation is also straightforward. If two segments have a spacing below the critical annihilation distance [EM79] the reaction takes place spontaneously. However, the subsequent reorganization of the dislocation segment vectors is not simple and must be treated with care. Area of Spontaneous Reaction climb glide Glide Climb react react d d 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. 312 COMPUTATIONAL MATERIALS ENGINEERING The stress and mobility of glissile dislocation junctions can be simulated by using a simple superposition of the segments involved. 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[Yof60] E. H. Yoffe. Phil. Mag., 5:161, 1960. [ZSL97] N. Zacharopoulos, D. J. Srolovitz, and R. LeSar. Acta Mater., 45:3745, 1997. 316 COMPUTATIONAL MATERIALS ENGINEERING 9 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 most common tasks in modeling material systems. Differential equations are equations involving one or more scalar or tensorial dependent vari- ables, independent variables, unknown functions of these variables, and their corresponding derivatives. Equations which involve unknown functions that depend on only one independent variable are referred to as ordinary differential equations. If the equations involve unknown functions that depend on more than one independent variable they are referred to as partial differential equations. The “order” of a differential equation is the highest order of any of the derivatives 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 differential equations such as d 2 u(t) dt 2 = f (u, t) (9.1) can be transformed into a coupled set of lower-order equations by substitution: dv(t) dt = f (u, t) v = du(t) dt (9.2) In these equations u is the state variable that is a function of the independent time variable t, v is the first time derivative of u, and f a function of u and v, respectively. For instance, the frequently occurring problem of (one-dimensional) motion of a particle or dislocation segment of effective mass m under a force field f(x, t) in the x-direction is described by the second-order differential equation m d 2 x(t) dt 2 = f (x, t) (9.3) 317 If one defines the particle momentum p(x, t)=m dx(t) dt (9.4) equation (9.3) becomes the two coupled first-order equations (Hamilton’s equations) dx(t) dt = p m dp(x, t) dt = f (x, t) (9.5) Differential equations which contain only linear functions of the independent variables are called “linear differential equations.” For these equations the superposition principle applies. That means linear combinations of solutions which satisfy the boundary conditions are also solutions to the differential equation satisfying the same boundary conditions. Differential equa- tions which involve nonlinear functions of the independent variables are denoted as “nonlinear differential 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 di fferential equations, which are defined within a certain range of the independent variables and which are in accord with certain initial-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 well posed. That means only particular initial and boundary conditions transform a partial differential equation into a solvable problem. Partial differential equations can be grouped according to the type of additional conditions that 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, say x 1 and x 2 . The general form of this equation is A ∂ 2 u ∂ 2 x 1 + B ∂ 2 u ∂x 1 ∂x 2 + C ∂ 2 u ∂ 2 x 2 + D ∂u ∂x 1 + E ∂u ∂x 2 + Fu+ G =0 (9.6) where A = A(x 1 ,x 2 ), B = B(x 1 ,x 2 ), C = C(x 1 ,x 2 ), D = D(x 1 ,x 2 ), E = E(x 1 ,x 2 ), F = F (x 1 ,x 2 ), and G = G(x 1 ,x 2 ) are given functions of the independent variables x 1 and x 2 . It is stipulated that the functions A(x 1 ,x 2 ), B(x 1 ,x 2 ), and C(x 1 ,x 2 ) never be equal to zero at the same point (x 1 ,x 2 ). In analogy to the classification of higher-order curves in analytical geometry that are described by ax 1 2 + bx 1 x 2 + cx 2 2 + dx 1 + ex 2 + f =0,a 2 + b 2 + c 2 =0 (9.7) equation (9.6) can for given values ˆx 1 , ˆx 2 , of the variables x 1 and x 2 assume hyperbolic, parabolic, or elliptic character. Roughly speaking, hyperbolic differential equations involve second-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-order derivatives in the remaining variables; and elliptic differential equations involve second order derivatives in each of the independent variables, each of the derivatives having equal sign when grouped on the same side of the equation. hyperbolic partial differential equation 4 A (ˆx 1 , ˆx 2 ) C (ˆx 1 , ˆx 2 ) <B 2 (ˆx 1 , ˆx 2 ) parabolic partial differential equation 4 A (ˆx 1 , ˆx 2 ) C (ˆx 1 , ˆx 2 )=B 2 (ˆx 1 , ˆx 2 ) elliptic partial differential equation 4 A (ˆx 1 , ˆx 2 ) C (ˆx 1 , ˆx 2 ) >B 2 (ˆx 1 , ˆx 2 ) 318 COMPUTATIONAL MATERIALS ENGINEERING In that context it must be considered that, since A(x 1 ,x 2 ), B(x 1 ,x 2 ), and C(x 1 ,x 2 ) depend on independent 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 AC − B 2 ) is due to its importance in substituting mixed derivatives by new independent variables. The fundamental classification scheme outlined here for second-order partial differ- ential equations can be extended to coupled sets of nonlinear higher-order partial differential equations with more than two independent variables. Classical examples of the three types of differential equations are the wave equation for the hyperbolic class, the heat or diffusion equation and the time-dependent Schr ¨ odinger equation for the parabolic class, and the Laplace and time-independent Schr ¨ odinger equation for the elliptic class. In three dimensions and rectangular coordinates then can be written: wave equation ∂ 2 u ∂t 2 − c 2  ∂ 2 u ∂x 2 1 + ∂ 2 u ∂x 2 2 + ∂ 2 u ∂x 2 3  =0 diffusion equation ∂u ∂t − D  ∂ 2 u ∂x 2 1 + ∂ 2 u ∂x 2 2 + ∂ 2 u ∂x 2 3  =0 Laplace equation ∂ 2 u ∂x 2 1 + ∂ 2 u ∂x 2 2 + ∂ 2 u ∂x 2 3 =0 where x 1 , x 2 , and x 3 are the spatial variables, t the temporal variable, u the state variable, D the diffusion coefficient, which is assumed to be positive and independent of the concentra- tion, and c the propagation velocity of the wave. The assumption that the diffusion coefficient is independent of the concentration applies of course only for certain systems and very small concentrations. In real materials the value of the diffusion coefficient is, first, a tensor quantity and, 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 or sources appear in the volume being considered, the diffusion equation changes into the Poisson equation: Poisson equation ∂ 2 u ∂x 2 1 + ∂ 2 u ∂x 2 2 + ∂ 2 u ∂x 2 3 − f(x 1 ,x 2 ,x 3 )=0 which in two dimensions is identical to the differential equation for the description of the trans- verse displacement of a membrane. An important differential equation similar to the Poisson equation is the Helmholtz equation, which contains both the dependent function itself and its second spatial derivative: Helmholtz equation ∂ 2 u ∂x 2 1 + ∂ 2 u ∂x 2 2 + ∂ 2 u ∂x 2 3 + αu=0 α = const. Using the more general Laplace operator ∆=∇ 2 instead of rectangular coordinates and ˙u and ¨u for the first- and second-order time derivatives, respectively, the preceding equations can be rewritten in a more compact notation: ¨u −c 2 ∆u =0 ˙u −D ∆u =0 ∆u =0 ∆u −f =0 ∆u + αu=0 Finite Elements for Microstructure Evolution 319 Hyperbolic and parabolic partial differential equations typically describe nonstationary, that is, time-dependent problems. This is indicated by the use of the independent variable t in the 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 should assume at a given starting time t 0 . These initial conditions could amount to u(x 1 ,x 2 ,x 3 ,t 0 ) and ˙u(x 1 ,x 2 ,x 3 ,t 0 ) for the wave equation and u(x 1 ,x 2 ,x 3 ,t 0 ) for the diffusion or heat equa- tion. If no constraints are given to confine the solutions to particular spatial coordinates, that is, −∞ <x 1 ,x 2 ,x 3 < +∞, the situation represents a pure initial-boundary problem. In cases where additional spatial conditions are required, such as u(x 1 0 ,x 2 0 ,x 3 0 ,t) for the wave equation, and u(x 1 0 ,x 2 0 ,x 3 0 ,t) or (∂u/∂x 1 )(x 1 0 ,x 2 0 ,x 3 0 ,t), (∂u/∂x 2 )(x 1 0 ,x 2 0 , x 3 0 ,t), and (∂u/∂x 3 )(x 1 0 ,x 2 0 ,x 3 0 ,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-value problems. Appropriate boundary conditions for the Laplace or stationary heat and diffusion equation, respectively, ∆u =0, can be formulated as Dirichlet boundary conditions or as Neumann boundary conditions. Dirichlet boundary conditions mean that solutions for the state variable u are given along the spatial boundary of the system. Neumann boundary conditions mean that solutions for the first derivative ∂u/∂x n are given normal to the spatial boundary of the system. If both the function and its normal derivative on the boundary are known, the border conditions 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 a limited 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 to obtain 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 the transformation of the continuous derivative into its discontinuous counterpart, that is, its finite difference quotient. Using these discretization steps amounts to recasting the continuous prob- lem expressed by differential equations with an infinite number of unknowns, that is, function values, into a discrete algebraic one with a finite number of unknown parameters which can be calculated in an approximate fashion. 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 in this book are discrete not only in time but also in space. Finite difference methods approximate the derivatives that appear in differential equations by a transition to their finite difference counterparts. This applies for the time and the space derivatives. Finite difference methods do not 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 and numerical precision, it is not reasonable to generally favor one out of the many possible finite difference solution techniques for applications in computational materials science. For instance, parabolic large-scale bulk diffusion or heat transport problems can be solved by 320 COMPUTATIONAL MATERIALS ENGINEERING [...]... isotropic, homogeneous, linear, and continuous behavior 1 Each finite element can have a different volume 324 COMPUTATIONAL MATERIALS ENGINEERING More sophisticated methods, which nowadays prevail in advanced computational materials science, increasingly consider more appropriate constitutive materials laws, which take into consideration material heterogeneity, crystal anisotropy, and nonlinear material... connected 322 COMPUTATIONAL MATERIALS ENGINEERING The finite element method approximates the real course of the state variables considered within each element by ansatz or interpolation polynomials This approach of interpolating the variable within each cell amounts to assuming a piecewise polynomial solution over the entire domain under consideration In the case of elastic and large-strain plastic materials. .. element from the physical coordinate system (x1 , x2 ) to the mapped coordinate system 326 COMPUTATIONAL MATERIALS ENGINEERING 9.3.4 Assemblage of the Stiffness Matrix This section describes the assemblage of the element and system stiffness matrix and the consideration of elastic and elastic–plastic constitutive materials response By defining the spatial derivatives Ki,j = ∂Ki /∂xj along xj , j = 1, 2,... 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 mesoscopic and macroscopic levels Of course, a strict subdivision of the numerous methods that exist in computational materials science according to the length scales that they address is to a certain extent arbitrary and depends on the... partial differential equations and whose solution is the discrete solution at the nodes [Cou43, Liv83, ZM83] While most of the early finite element methods used in computational materials science postulated isotropic, homogeneous, linear, and continuous materials properties [ZT89], a number of advanced methods consider material heterogeneity, crystal anisotropy, nonlinear material response, and nonlinear geometrical... where m = 3 n is the number of degrees of freedom of the system, that is, the number of nodes multiplied by the dimension (2D, 3D), F the vector of the externally imposed nodal loads, 328 COMPUTATIONAL MATERIALS ENGINEERING F the internal force vector arsising from the internal stresses among the elements, and F − F the minimization criterion imposed by the requirement for equilibrium.2 Since the total... with the prescribed deformation state The eigenvalues λi represent the incremental rotation-free deformation parallel to the principal axes ni The strain tensor E can be written 3 E= i=1 330 COMPUTATIONAL MATERIALS ENGINEERING εi ni nT i (9.33) where εi are the strain components parallel to the principal axes ni Typically, the strain components are nonlinear functions of the eigenvalues, that is, εi... Beaudoin, and K K Mathur In Proceedings 15th RISØ International Symposium on Materials Science, eds.: S I Andersen, J B Bilde-Sørensen, T Lorentzen, O B Pedersen, and N J Sorensen, p 33 RISØ National Laboratory, Roskilde, 1994 J W Dettman Mathematical Methods in Physics and Engineering Wiley, New York, 1969 P L DeVries A First Course in Computational Physics Wiley, New York, 1994 G Engeln-M¨ llges Formelsammlung... Finite Difference Methods for Partial Differential Equations Wiley, New York, 1960 J Gittus and J Zarka Modeling Small Deformations of Polycrystals Elsevier Applied Science, London, 1986 332 COMPUTATIONAL MATERIALS ENGINEERING [KK96] [Liv83] [Mar76] [MAS93] [NNH93] [RM67] [RSHP91] [SD96a] [SD96b] [SW96] [ZM83] [ZT89] A S Krausz and K Krausz Unified Constitutive Laws of Plastic Deformation Academic Press,... formulation of the interatomic potential It is clear that the accuracy of the underlying 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 and the level at which the microstructure is incorporated in the finite element grid determine the predictive relevance of the simulation . volume. 324 COMPUTATIONAL MATERIALS ENGINEERING More sophisticated methods, which nowadays prevail in advanced computational materials science, increasingly consider more appropriate constitutive materials. for applications in computational materials science. For instance, parabolic large-scale bulk diffusion or heat transport problems can be solved by 320 COMPUTATIONAL MATERIALS ENGINEERING using. 13:1097, 1965. [LK87] J. L ´ epinoux and L. P. Kubin. Scr. Metall., 21:833, 1987. 314 COMPUTATIONAL MATERIALS ENGINEERING [MK81] H. Mecking and U. F. Kocks. Acta Metall., 29:1865, 1981. [Moh96]