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Higher-Order Models of Deformation Processing
CHAPTER
15
Higher-Order Models of Deformation Processing
CHAPTER OUTLINE 15.1 Higher-Order Model of Visco-Plasticity ........................................................................................ 331 15.2 Time- and Space-Dependent Modeling of Texture Evolution........................................................... 337 Summary ............................................................................................................................................ 339
15.1 HIGHER-ORDER MODEL OF VISCO-PLASTICITY In this chapter we introduce texture evolution theory that utilizes the primitive basis approach to model deformation over time and space. We use several of the techniques that were introduced in previous chapters; hence, this chapter will provide a useful mechanism for reviewing earlier work. We do not explicitly discretize the sample or orientation space for the development of the following equations, but one should keep in mind that the aim is to track the evolution of texture in individual cells of both of these spaces during plastic deformation. We first reintroduce a few parameters of interest. The velocity gradient is given by Lij ¼ vi;j
(15.1)
Then, for rigid plasticity (ignoring elasticity), L ¼ DP þ Wp þ W
(15.2)
where Dp is the plastic deformation gradient: Dp ¼
X g$a a
2
ðma 5na þ na 5ma Þ ¼
1 1 p L þ LT ¼ L þ ðLp ÞT 2 2
(15.3)
1 p L ðLp ÞT 2
(15.4)
and Wp is the plastic spin tensor: Wp ¼
X g$a a
2
ðma 5na na 5ma Þ ¼
with slip plane normals na and slip directions ma ; g$a denotes the effective shear rate on this slip system. Microstructure-Sensitive Design for Performance Optimization. http://dx.doi.org/10.1016/B978-0-12-396989-7.00015-0 Copyright 2013 Elsevier Inc. All rights reserved.
331
332
CHAPTER 15 Higher-Order Models of Deformation Processing
Implicit in Eqs. (15.3) and (15.4) is the definition of Lp as the plastic velocity gradient: Lp ¼
X
g$a ma 5na
(15.5)
a
In the absence of elastic deformations, the difference between the applied velocity gradient tensor, L, and that accommodated by the slip processes, Lp, needs to be accommodated by the rigid-body spin, W* (see Section 7.6.1). This rigid-body spin causes a lattice rotation, R*. Therefore, the relationship between the lattice rotation tensor and the plastic velocity gradient tensor can be expressed as $
W ¼ R ðR ÞT ¼ L Lp
(15.6)
We next distinguish between the applied, macroscopic homogeneous velocity gradient, L, and the local velocity gradient, L. Assume that the sample space is divided into cells, us , and that we wish to determine the evolution of the texture within each cell. We need to determine the local velocity to calculate the amount of material entering and leaving the cell, and we need the local lattice spin, W*. The approach is to apply a global L; we then need to calculate the local vi and W*. To calculate the * W we require the local L. We obtain this by using the homogenization relation that links the local L to the global L. Following Adams (1989), let N be the secant modulus; then D sD ij ¼ Nijkl εkl
(15.7)
where D stands for the deviatoric stress and strain. Also, the creep compliance is given by 1 X sD $Pa m1 $0 a a g ðP 5P Þ M ¼ 1 ðsa Þm a
(15.8)
and the secant modulus N ¼ M1 (see also the alternative form in Adams et al. (1989)). For rate-independent plasticity, the slip resistance, Sa, is exactly the same as the critical resolved shear stress (see Section 7.6.1), and m is the rate sensitivity parameter in the slip law: ðkÞ
sðkÞ =s0
¼
gðkÞ =g0
m (15.9)
Furthermore, if n ¼ 1/m, then NðlεÞ ¼ lð1nÞ=n NðεÞ, which reduces the number of times N must be calculated. The Cauchy stress is arrived at from the deviatoric stress by adding the hydrostatic pressure: Tij ¼ Nijkl εkl pdij ¼ Nijkl Lkl pdij
(15.10)
The equilibrium equation requires that Tij;j ¼ Nijkl Lkl ;j p;i ¼ 0
(15.11)
15.1 Higher-Order Model of Visco-Plasticity
333
Now define a reference secant modulus, Z R Nijkl
f h Nijkl L; s0 ; g dg
¼
(15.12)
g˛FZ
where it is assumed that the reference stress s0 may be taken as a constant across the material. A polarized modulus is then given as R (15.13) N~ijkl L; s0 ; g ¼ Nijkl L; s0 ; g Nijkl Then we can write the equilibrium equation as h i R R Lkl;j p;i þ N~ijkl ðL; gÞLkl ; i ¼ Nijkl Lkl;j p;i þ fi ¼ 0 Nijkl
(15.14)
where fi is considered a fictitious body force. This produces the three differential equations. The fourth equation is given by incompressibility: Lii ¼ 0
(15.15)
Now, using the Green’s function approach to solve these equations, with Green’s functions G and H, R Gkm;lj ðx x0 Þ Hm;i ðx x0 Þ þ dim dðx x0 Þ ¼ 0 Nijkl
(15.16)
Gim;i x x0 ¼ 0
(15.17)
leads to solutions for v and p: Z vn ðxÞ ¼ vn ðxÞ þ
Gki ðx x0 Þ fi ðx0 Þdx03
(15.18)
Hi ðx x0 Þfi ðx0 Þdx03
(15.19)
V
Z pðxÞ ¼ pðxÞ þ V
One might rearrange the equation for V, taking the derivative with respect to x and integrating by parts to obtain Z (15.20) Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 ÞN~jlrs ðLðx0 Þ; gðx0 ÞÞLrs ðx0 Þdx03 V
The corresponding Fourier transform of the partial differential equations yields R Nijkl kj k1 G~km ðkÞ iki H~m ðkÞ þ dim ¼ 0
(15.21)
kk G~km ðkÞ ¼ 0
(15.22)
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CHAPTER 15 Higher-Order Models of Deformation Processing
Example 15.1 Find the Green’s function, G, assuming isotropic NR.
Solution If we assume that NR is isotropic then it can be written as (Molinari et al., 1987) N Rijkl ¼
m dik djl þ dil djk 2
(15.23)
Then we can write (15.21) as mr ðdik djl þ dil djk Þkj k1 G~km ðkÞ iki H~m ðkÞ þ dim ¼ 0 2
(15.24)
Substituting Eq. (15.22) into Eq. (15.24), we obtain r m mr ki kk G~km ðkÞ jkj2 G~im ðkÞ þ iki H~m ðkÞ þ dim ¼ 0 2 2
(15.25)
Taking Eq. (15.25) ki and summing over repeated indices, and changing the repeated i’s to ks in the second term, we get r m mr jkj2 kk G~km ðkÞ jkj2 kk G~km ðkÞ þ i$jkj2 H~m ðkÞ þ km ¼ 0 2 2 That is, mr jkj2 kk G~km ðkÞ þ i$jkj2 H~m ðkÞ þ km ¼ 0
(15.26)
i$jkj2 H~m ðkÞ þ km G~km ðkÞ ¼ mr jkj2 kk
(15.27)
Thus,
Substituting Eq. (15.27) into Eq. (15.25) and setting the first term to 0 (Eq. 15.22), we get mr i$jkj2 H~m ðkÞ þ km þ iki H~m ðkÞ þ dim ¼ 0 jkj2 $ 2 mr jkj2 kk i$jkj2 H~m ðkÞ þ km þ iki H~m ðkÞ þ dim ¼ 0 2kk ijkj2 H~m þ iki 2kk
!
km þ dim ¼ 0 2kk
(15.28)
15.1 Higher-Order Model of Visco-Plasticity
Taking Eq. (15.28) ki , we get ijkj2 ki þ ijkj2 H~m 2kk H~m
ijkj2 þ ijkj2 2
!
335
km ki þ km ¼ 0 2kk (15.29)
! km þ ¼ 0 2
km ikm ¼ 2 H~m ¼ 2 k ijkj
(15.30)
Substituting Eq. (15.30) into Eq. (15.25) with Eq. (15.22), we get mr i$km jkj2 G~im ðkÞ þ i $ ki 2 þ dim ¼ 0 2 k mr ki km jkj2 G~im ðkÞ 2 þ dim ¼ 0 2 k 2dim 2ðki km Þ G~im ¼ r 2 r 4 mk mk
(15.31)
(15.32)
1 We now demonstrate how to find the inverse Fourier transform for Z~ ¼ 2. In 3D space the inverse Fourier k transform is given by ZZZ 1 ~ ZðxÞ ¼ eik:x ZðkÞdv (15.33) ð2pÞ3 R3 where x is the position vector in R 3. Changing to spherical coordinates (note that k indicates the length of vector k, and hence this is the radial variable), 9 k1 ¼ k cos q sin f > = (15.34) k2 ¼ k sin q sin f dv ¼ k2 sin fdfdqdk > ; k3 ¼ k cos q It can be shown (Champeney, 1973) that for a spherically symmetric function (letting r denote the radial variable in real space), 1 ð2pÞ3
ZZZ R3
2 ~ ei k:x ZðkÞk sin fdfdqdk ¼
1 ð2pÞ3
ZN
~ sin kr 4pk2 dk ZðkÞ kr
(15.35)
0
Hence, ZðrÞ ¼
4p ð2pÞ3
ZN 0
sin kr 2 1 p 1 k dk ¼ ¼ k3 r 2p2 2r 4pr
(15.36)
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CHAPTER 15 Higher-Order Models of Deformation Processing
To determine the inverse Fourier transform for the remainder of Eq. (15.32) we utilize the often quoted Green’s function for elasticity: dkm ðl þ mÞkk km J~ ¼ ; 2 mjkj mðl þ 2mÞjkj4
J ¼
ðl þ 3mÞdkm ðl þ mÞxk xm þ 8pmðl þ 2mÞjxj 8pmðl þ 2mÞjxj3
(15.37)
1 where jxj ¼ r. Using the previous result for Z~ ¼ 2 it is easy to determine the inverse transform for the k second term in Eq. (15.32): kk km Y~ ¼ ; jkj4
Y ¼
dkm xk xm 8pjxj 8pjxj2
This leads us to 2dim 2 dim xi xm Gð~ xÞ ¼ r 4pmr jxj m 8pjxj 8pjxj3
(15.38)
! (15.39)
Note that this assumes isotropic N. To ensure that this is the case, one may define a new reference tensor that is isotropic, and as close as possible to the original NR. Another alternative is to use the formulation in Adams et al. (1989).
We now employ an iterative scheme to recover L from Eq. (15.20). Write the equation in reduced form as follows: Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 ÞLrs ðx0 Þ
(15.40)
Then, inserting this equation into itself, Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 Þ Lrs ðxÞ þ Grm;sn ðx x0 Þ N~mnpq ðx0 ÞLpq ðx0 Þ
(15.41)
This process may be repeated over and over. If we assume that the polarization terms are small compared to the reference values, then higher-order terms will decrease in value. This equation is analogous to similar equations for elasticity in Chapter 13, such as Eq. (13.11). The theory could be developed in the same way as elasticity to incorporate higher-order correlation functions. We do not fully develop in that direction in this chapter. Instead, we use the results to model the evolution of texture with time. For the following, we only consider the first two terms on the right side of Eq. (15.41). Hence, Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 ÞLrs ðx0 Þ
(15.42)
Note that there is a singularity in Green’s function at x ¼ x0 . One way to deal with this is to take the average value of the function in a small neighborhood of x for the value at x (see Adams et al., 1989). Also note that since we are actually looking for average values of L within each cell of the sample, this scheme is likely to suit our requirements. We must be careful to take account of the nonspherical shape of the sample region if we use the analytical values for Green’s function (see Section 13.1).
15.2 Time- and Space-Dependent Modeling of Texture Evolution
337
The velocity, v, is obtained by integrating L according to Eq. (15.1). To recover the lattice rotation we require Lp. This is obtained by recovering Dp, and therefore g$a , from Eq. (15.3). Then, using Eq. (15.5), we obtain Lp. Finally, we obtain W* from Eq. (15.6). Once W* is available, the actual rotation rate may be obtained (Adams and Field, 1991) (these results are based on linearization of the equations; see Morawiec, 2004): 4$ 1 ¼ W23
sin 42 cos 42 W13 sin F sin F
$
(15.43)
cos 4 þ W sin 4 F ¼ W23 2 2 13 $
4$ 2 ¼ f1 cos F þ W12
We use a different form of these equations in Section 15.2 as it becomes apparent that it is easier to work with Rodriguez angles in the calculations.
15.2 TIME- AND SPACE-DEPENDENT MODELING OF TEXTURE EVOLUTION As originally stated, we are interested in the evolution of the orientation distribution function across a sample; hence, we must employ the microstructure function, which contains information about both space and orientation distribution. Recall the definition of the microstructure function Mðx; gÞ: Mðx; gÞdg ¼ dV=V
(15.44)
This function tells us that the volume fraction density of material lying in the neighborhood of position x has orientation g. Consider a region < of the material surrounded by surface z. The volume of material at time t lying within region <, and having orientation lying within invariant measure dg of g, is ZZZ Mðx; gÞdgdV (15.45) <
Over an increment of time dt, during which deformation is taking place, the orientation g at a given point or at a given particle in < will (in general) change, and some of the particles will travel across the surface z, thereby transporting particles of various orientation into and out of <. The rate of change in the volume of material associated with invariant measure dg of g is given by the notation ZZZ D Mðx; gÞdgdV (15.46) Dt <
The rate of increase of this orientation component is equal to the sum of the rate of increase of this component associated with the particles instantaneously within <, plus the net rate of influx of this component through z into <. Thus, ZZZ ZZZ ZZ D vMðx; gÞ Mðx; gÞdV ¼ Mðx; gÞv$^ ndS (15.47) dV þ Dt vt <
<
z
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CHAPTER 15 Higher-Order Models of Deformation Processing
where n^dS is an infinitesimal patch of the surface z of measure dS and outward normal n^, and v ¼ vðxÞ is the velocity of the current material particle at position x. By applying the divergence theorem to the surface integral in relation (15.47), we obtain
ZZZ ZZZ D vMðx; gÞ Mðx; gÞdV ¼ þ divðMðx; gÞvÞ dV (15.48) Dt vt <
<
It is useful to look at the integrand of relation (15.48) more carefully, by examining it in index form. Note that Mðx; gÞv is a vector of magnitude equal to the velocity of the material point at position x, weighted by the volume fraction density of orientation g at that position. The velocity depends only on position x and not on orientation g. In component form the integrand of (15.48) can be written as vvj vMðx; gÞ vMðx; gÞ vMðx; gÞ þ divðMðx; gÞvÞ ¼ þ Mðx; gÞ þ vj vxj vt vt vxj
(15.49)
Our next task is to evaluate the time derivative of the texture function (i.e., the first term on the right side of relation (15.49)). Assuming that via some model the lattice rotation rate field 6 ¼ 6ðx; gÞ is known, then the orientation flow field at position x is Mðx; gÞ6ðx; gÞ. It is to be understood here that the dependence of 6 on position x is related through the deformation model to the local velocity gradient tensor at that position. Given that crystallite orientations are neither created nor destroyed, a principle of conservation of orientations can be expressed in the form of a continuity relation (Clement, 1982; Morawiec, 2004): vMðx; gÞ ¼ divðMðx; gÞ6ðx; gÞÞ vt
(15.50)
The divergence in relation (15.50) is a divergence with respect to orientation parameters associated with g, and they must be evaluated in accordance with the specialized geometry of the orientation space. The simplest form of the term on the right side of (15.50) is in terms of the Rodriguez n) and angle (q) parameters, for parameters, r1; r 2; r 3, which are conveniently related to the axis (^ a rotation q about n^, by r i ¼ ni tanðq=2Þ
(15.51)
The axis and angle parameters are themselves related to Bunge’s orientation parameters (passive direction cosines) via the relation (15.52) gij ¼ dij cos q þ εijk nk sin q þ ni nj 1 cos q In terms of the Rodriguez vector the divergence term in (15.50) has the form divðMðx; rÞ6ðx; rÞÞ ¼
vðMðx; rÞ6ðx; rÞÞj ðMðx; rÞ6ðx; rÞÞj r j 4 j vr ð1 þ r k r k Þ
v6j ðx; rÞ vMðx; rÞ Mðx; rÞ6j ðx; rÞr j ¼ Mðx; rÞ þ 6j ðx; rÞ 4 j j ð1 þ rk r k Þ vr vr
(15.53)
Summary
339
Thus, in terms of the Rodriguez parameterization of lattice orientation, relation (15.48) becomes
ZZZ ZZZ D vMðx; rÞ þ divðMðx; rÞvÞ dV (15.54) Mðx; rÞdV ¼ Dt vt <
<
And, incorporating the results of relations (15.49) through (15.53), this becomes ZZZ ZZZ vvj D vMðx; rÞ Mðx; rÞ Mðx; rÞdV ¼ þ vj vxj Dt vxj <
<
v6j ðx; rÞ vMðx; rÞ 6j ðx; rÞ vr j vr j
M x; r 6j x; r rj dV þ4 ð1 þ rk r k Þ Mðx; rÞ
(15.55)
At this point in the derivation, readers should note that vvi =vxj ¼ Lij is the well-known velocity gradient tensor, and in incompressible media, Ljj ¼ 0. It follows that relation (15.55) simplifies to ZZZ ZZZ D vMðx; rÞ v6j ðx; rÞ vj Mðx; rÞdV ¼ Mðx; rÞ Dt vxj vr j < < (15.56) j ðx; rÞr j
vMðx; rÞ Mðx; rÞ6 dV 6j ðx; rÞ þ4 vrj ð1 þ rk r k Þ The lattice rotation rate vector, 6ðx; rÞ, is related to the local velocity gradient tensor in a particular way. In terms of the time rate of change of the (passive) parameters of Bunge, the relationship is dgij ¼ dij εijk 6k dt
(15.57)
In terms of the deformation model, assume that the local velocity gradient tensor is LðxÞ. We have shown how to recover the lattice spin, W*. The components of the lattice rotation rate vector are known to be related to W* by the expression 6i ¼ εijk Wjk =2
(15.58)
Therefore, having obtained v and 6ðx; rÞ we may now use Eq. (15.56) to determine the evolution of the microstructure function in small cells represented by the region < in the integrals.
SUMMARY In this chapter readers were exposed to a theoretical framework for capturing the higher-order details of microstructure evolution using higher-order composite theories. It is hoped that readers can appreciate the MSDPO framework’s power in addressing this highly challenging problem. It should be
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CHAPTER 15 Higher-Order Models of Deformation Processing
recognized that process design even using the first-order composite theories is itself highly challenging. Therefore, it is not surprising that process design using higher-order composite theories is substantially more difficult. This chapter showed the essential foundations for addressing higher-order process design. However, it should be clear to readers that additional development is needed before we can obtain practical solutions to the process design problem using higher-order theories.
15
Higher-Order Models of Deformation Processing
CHAPTER OUTLINE 15.1 Higher-Order Model of Visco-Plasticity ........................................................................................ 331 15.2 Time- and Space-Dependent Modeling of Texture Evolution........................................................... 337 Summary ............................................................................................................................................ 339
15.1 HIGHER-ORDER MODEL OF VISCO-PLASTICITY In this chapter we introduce texture evolution theory that utilizes the primitive basis approach to model deformation over time and space. We use several of the techniques that were introduced in previous chapters; hence, this chapter will provide a useful mechanism for reviewing earlier work. We do not explicitly discretize the sample or orientation space for the development of the following equations, but one should keep in mind that the aim is to track the evolution of texture in individual cells of both of these spaces during plastic deformation. We first reintroduce a few parameters of interest. The velocity gradient is given by Lij ¼ vi;j
(15.1)
Then, for rigid plasticity (ignoring elasticity), L ¼ DP þ Wp þ W
(15.2)
where Dp is the plastic deformation gradient: Dp ¼
X g$a a
2
ðma 5na þ na 5ma Þ ¼
1 1 p L þ LT ¼ L þ ðLp ÞT 2 2
(15.3)
1 p L ðLp ÞT 2
(15.4)
and Wp is the plastic spin tensor: Wp ¼
X g$a a
2
ðma 5na na 5ma Þ ¼
with slip plane normals na and slip directions ma ; g$a denotes the effective shear rate on this slip system. Microstructure-Sensitive Design for Performance Optimization. http://dx.doi.org/10.1016/B978-0-12-396989-7.00015-0 Copyright 2013 Elsevier Inc. All rights reserved.
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CHAPTER 15 Higher-Order Models of Deformation Processing
Implicit in Eqs. (15.3) and (15.4) is the definition of Lp as the plastic velocity gradient: Lp ¼
X
g$a ma 5na
(15.5)
a
In the absence of elastic deformations, the difference between the applied velocity gradient tensor, L, and that accommodated by the slip processes, Lp, needs to be accommodated by the rigid-body spin, W* (see Section 7.6.1). This rigid-body spin causes a lattice rotation, R*. Therefore, the relationship between the lattice rotation tensor and the plastic velocity gradient tensor can be expressed as $
W ¼ R ðR ÞT ¼ L Lp
(15.6)
We next distinguish between the applied, macroscopic homogeneous velocity gradient, L, and the local velocity gradient, L. Assume that the sample space is divided into cells, us , and that we wish to determine the evolution of the texture within each cell. We need to determine the local velocity to calculate the amount of material entering and leaving the cell, and we need the local lattice spin, W*. The approach is to apply a global L; we then need to calculate the local vi and W*. To calculate the * W we require the local L. We obtain this by using the homogenization relation that links the local L to the global L. Following Adams (1989), let N be the secant modulus; then D sD ij ¼ Nijkl εkl
(15.7)
where D stands for the deviatoric stress and strain. Also, the creep compliance is given by 1 X sD $Pa m1 $0 a a g ðP 5P Þ M ¼ 1 ðsa Þm a
(15.8)
and the secant modulus N ¼ M1 (see also the alternative form in Adams et al. (1989)). For rate-independent plasticity, the slip resistance, Sa, is exactly the same as the critical resolved shear stress (see Section 7.6.1), and m is the rate sensitivity parameter in the slip law: ðkÞ
sðkÞ =s0
¼
gðkÞ =g0
m (15.9)
Furthermore, if n ¼ 1/m, then NðlεÞ ¼ lð1nÞ=n NðεÞ, which reduces the number of times N must be calculated. The Cauchy stress is arrived at from the deviatoric stress by adding the hydrostatic pressure: Tij ¼ Nijkl εkl pdij ¼ Nijkl Lkl pdij
(15.10)
The equilibrium equation requires that Tij;j ¼ Nijkl Lkl ;j p;i ¼ 0
(15.11)
15.1 Higher-Order Model of Visco-Plasticity
333
Now define a reference secant modulus, Z R Nijkl
f h Nijkl L; s0 ; g dg
¼
(15.12)
g˛FZ
where it is assumed that the reference stress s0 may be taken as a constant across the material. A polarized modulus is then given as R (15.13) N~ijkl L; s0 ; g ¼ Nijkl L; s0 ; g Nijkl Then we can write the equilibrium equation as h i R R Lkl;j p;i þ N~ijkl ðL; gÞLkl ; i ¼ Nijkl Lkl;j p;i þ fi ¼ 0 Nijkl
(15.14)
where fi is considered a fictitious body force. This produces the three differential equations. The fourth equation is given by incompressibility: Lii ¼ 0
(15.15)
Now, using the Green’s function approach to solve these equations, with Green’s functions G and H, R Gkm;lj ðx x0 Þ Hm;i ðx x0 Þ þ dim dðx x0 Þ ¼ 0 Nijkl
(15.16)
Gim;i x x0 ¼ 0
(15.17)
leads to solutions for v and p: Z vn ðxÞ ¼ vn ðxÞ þ
Gki ðx x0 Þ fi ðx0 Þdx03
(15.18)
Hi ðx x0 Þfi ðx0 Þdx03
(15.19)
V
Z pðxÞ ¼ pðxÞ þ V
One might rearrange the equation for V, taking the derivative with respect to x and integrating by parts to obtain Z (15.20) Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 ÞN~jlrs ðLðx0 Þ; gðx0 ÞÞLrs ðx0 Þdx03 V
The corresponding Fourier transform of the partial differential equations yields R Nijkl kj k1 G~km ðkÞ iki H~m ðkÞ þ dim ¼ 0
(15.21)
kk G~km ðkÞ ¼ 0
(15.22)
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CHAPTER 15 Higher-Order Models of Deformation Processing
Example 15.1 Find the Green’s function, G, assuming isotropic NR.
Solution If we assume that NR is isotropic then it can be written as (Molinari et al., 1987) N Rijkl ¼
m dik djl þ dil djk 2
(15.23)
Then we can write (15.21) as mr ðdik djl þ dil djk Þkj k1 G~km ðkÞ iki H~m ðkÞ þ dim ¼ 0 2
(15.24)
Substituting Eq. (15.22) into Eq. (15.24), we obtain r m mr ki kk G~km ðkÞ jkj2 G~im ðkÞ þ iki H~m ðkÞ þ dim ¼ 0 2 2
(15.25)
Taking Eq. (15.25) ki and summing over repeated indices, and changing the repeated i’s to ks in the second term, we get r m mr jkj2 kk G~km ðkÞ jkj2 kk G~km ðkÞ þ i$jkj2 H~m ðkÞ þ km ¼ 0 2 2 That is, mr jkj2 kk G~km ðkÞ þ i$jkj2 H~m ðkÞ þ km ¼ 0
(15.26)
i$jkj2 H~m ðkÞ þ km G~km ðkÞ ¼ mr jkj2 kk
(15.27)
Thus,
Substituting Eq. (15.27) into Eq. (15.25) and setting the first term to 0 (Eq. 15.22), we get mr i$jkj2 H~m ðkÞ þ km þ iki H~m ðkÞ þ dim ¼ 0 jkj2 $ 2 mr jkj2 kk i$jkj2 H~m ðkÞ þ km þ iki H~m ðkÞ þ dim ¼ 0 2kk ijkj2 H~m þ iki 2kk
!
km þ dim ¼ 0 2kk
(15.28)
15.1 Higher-Order Model of Visco-Plasticity
Taking Eq. (15.28) ki , we get ijkj2 ki þ ijkj2 H~m 2kk H~m
ijkj2 þ ijkj2 2
!
335
km ki þ km ¼ 0 2kk (15.29)
! km þ ¼ 0 2
km ikm ¼ 2 H~m ¼ 2 k ijkj
(15.30)
Substituting Eq. (15.30) into Eq. (15.25) with Eq. (15.22), we get mr i$km jkj2 G~im ðkÞ þ i $ ki 2 þ dim ¼ 0 2 k mr ki km jkj2 G~im ðkÞ 2 þ dim ¼ 0 2 k 2dim 2ðki km Þ G~im ¼ r 2 r 4 mk mk
(15.31)
(15.32)
1 We now demonstrate how to find the inverse Fourier transform for Z~ ¼ 2. In 3D space the inverse Fourier k transform is given by ZZZ 1 ~ ZðxÞ ¼ eik:x ZðkÞdv (15.33) ð2pÞ3 R3 where x is the position vector in R 3. Changing to spherical coordinates (note that k indicates the length of vector k, and hence this is the radial variable), 9 k1 ¼ k cos q sin f > = (15.34) k2 ¼ k sin q sin f dv ¼ k2 sin fdfdqdk > ; k3 ¼ k cos q It can be shown (Champeney, 1973) that for a spherically symmetric function (letting r denote the radial variable in real space), 1 ð2pÞ3
ZZZ R3
2 ~ ei k:x ZðkÞk sin fdfdqdk ¼
1 ð2pÞ3
ZN
~ sin kr 4pk2 dk ZðkÞ kr
(15.35)
0
Hence, ZðrÞ ¼
4p ð2pÞ3
ZN 0
sin kr 2 1 p 1 k dk ¼ ¼ k3 r 2p2 2r 4pr
(15.36)
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CHAPTER 15 Higher-Order Models of Deformation Processing
To determine the inverse Fourier transform for the remainder of Eq. (15.32) we utilize the often quoted Green’s function for elasticity: dkm ðl þ mÞkk km J~ ¼ ; 2 mjkj mðl þ 2mÞjkj4
J ¼
ðl þ 3mÞdkm ðl þ mÞxk xm þ 8pmðl þ 2mÞjxj 8pmðl þ 2mÞjxj3
(15.37)
1 where jxj ¼ r. Using the previous result for Z~ ¼ 2 it is easy to determine the inverse transform for the k second term in Eq. (15.32): kk km Y~ ¼ ; jkj4
Y ¼
dkm xk xm 8pjxj 8pjxj2
This leads us to 2dim 2 dim xi xm Gð~ xÞ ¼ r 4pmr jxj m 8pjxj 8pjxj3
(15.38)
! (15.39)
Note that this assumes isotropic N. To ensure that this is the case, one may define a new reference tensor that is isotropic, and as close as possible to the original NR. Another alternative is to use the formulation in Adams et al. (1989).
We now employ an iterative scheme to recover L from Eq. (15.20). Write the equation in reduced form as follows: Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 ÞLrs ðx0 Þ
(15.40)
Then, inserting this equation into itself, Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 Þ Lrs ðxÞ þ Grm;sn ðx x0 Þ N~mnpq ðx0 ÞLpq ðx0 Þ
(15.41)
This process may be repeated over and over. If we assume that the polarization terms are small compared to the reference values, then higher-order terms will decrease in value. This equation is analogous to similar equations for elasticity in Chapter 13, such as Eq. (13.11). The theory could be developed in the same way as elasticity to incorporate higher-order correlation functions. We do not fully develop in that direction in this chapter. Instead, we use the results to model the evolution of texture with time. For the following, we only consider the first two terms on the right side of Eq. (15.41). Hence, Lik ðxÞ ¼ Lik ðxÞ þ Gij;kl ðx x0 Þ N~jlrs ðx0 ÞLrs ðx0 Þ
(15.42)
Note that there is a singularity in Green’s function at x ¼ x0 . One way to deal with this is to take the average value of the function in a small neighborhood of x for the value at x (see Adams et al., 1989). Also note that since we are actually looking for average values of L within each cell of the sample, this scheme is likely to suit our requirements. We must be careful to take account of the nonspherical shape of the sample region if we use the analytical values for Green’s function (see Section 13.1).
15.2 Time- and Space-Dependent Modeling of Texture Evolution
337
The velocity, v, is obtained by integrating L according to Eq. (15.1). To recover the lattice rotation we require Lp. This is obtained by recovering Dp, and therefore g$a , from Eq. (15.3). Then, using Eq. (15.5), we obtain Lp. Finally, we obtain W* from Eq. (15.6). Once W* is available, the actual rotation rate may be obtained (Adams and Field, 1991) (these results are based on linearization of the equations; see Morawiec, 2004): 4$ 1 ¼ W23
sin 42 cos 42 W13 sin F sin F
$
(15.43)
cos 4 þ W sin 4 F ¼ W23 2 2 13 $
4$ 2 ¼ f1 cos F þ W12
We use a different form of these equations in Section 15.2 as it becomes apparent that it is easier to work with Rodriguez angles in the calculations.
15.2 TIME- AND SPACE-DEPENDENT MODELING OF TEXTURE EVOLUTION As originally stated, we are interested in the evolution of the orientation distribution function across a sample; hence, we must employ the microstructure function, which contains information about both space and orientation distribution. Recall the definition of the microstructure function Mðx; gÞ: Mðx; gÞdg ¼ dV=V
(15.44)
This function tells us that the volume fraction density of material lying in the neighborhood of position x has orientation g. Consider a region < of the material surrounded by surface z. The volume of material at time t lying within region <, and having orientation lying within invariant measure dg of g, is ZZZ Mðx; gÞdgdV (15.45) <
Over an increment of time dt, during which deformation is taking place, the orientation g at a given point or at a given particle in < will (in general) change, and some of the particles will travel across the surface z, thereby transporting particles of various orientation into and out of <. The rate of change in the volume of material associated with invariant measure dg of g is given by the notation ZZZ D Mðx; gÞdgdV (15.46) Dt <
The rate of increase of this orientation component is equal to the sum of the rate of increase of this component associated with the particles instantaneously within <, plus the net rate of influx of this component through z into <. Thus, ZZZ ZZZ ZZ D vMðx; gÞ Mðx; gÞdV ¼ Mðx; gÞv$^ ndS (15.47) dV þ Dt vt <
<
z
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CHAPTER 15 Higher-Order Models of Deformation Processing
where n^dS is an infinitesimal patch of the surface z of measure dS and outward normal n^, and v ¼ vðxÞ is the velocity of the current material particle at position x. By applying the divergence theorem to the surface integral in relation (15.47), we obtain
ZZZ ZZZ D vMðx; gÞ Mðx; gÞdV ¼ þ divðMðx; gÞvÞ dV (15.48) Dt vt <
<
It is useful to look at the integrand of relation (15.48) more carefully, by examining it in index form. Note that Mðx; gÞv is a vector of magnitude equal to the velocity of the material point at position x, weighted by the volume fraction density of orientation g at that position. The velocity depends only on position x and not on orientation g. In component form the integrand of (15.48) can be written as vvj vMðx; gÞ vMðx; gÞ vMðx; gÞ þ divðMðx; gÞvÞ ¼ þ Mðx; gÞ þ vj vxj vt vt vxj
(15.49)
Our next task is to evaluate the time derivative of the texture function (i.e., the first term on the right side of relation (15.49)). Assuming that via some model the lattice rotation rate field 6 ¼ 6ðx; gÞ is known, then the orientation flow field at position x is Mðx; gÞ6ðx; gÞ. It is to be understood here that the dependence of 6 on position x is related through the deformation model to the local velocity gradient tensor at that position. Given that crystallite orientations are neither created nor destroyed, a principle of conservation of orientations can be expressed in the form of a continuity relation (Clement, 1982; Morawiec, 2004): vMðx; gÞ ¼ divðMðx; gÞ6ðx; gÞÞ vt
(15.50)
The divergence in relation (15.50) is a divergence with respect to orientation parameters associated with g, and they must be evaluated in accordance with the specialized geometry of the orientation space. The simplest form of the term on the right side of (15.50) is in terms of the Rodriguez n) and angle (q) parameters, for parameters, r1; r 2; r 3, which are conveniently related to the axis (^ a rotation q about n^, by r i ¼ ni tanðq=2Þ
(15.51)
The axis and angle parameters are themselves related to Bunge’s orientation parameters (passive direction cosines) via the relation (15.52) gij ¼ dij cos q þ εijk nk sin q þ ni nj 1 cos q In terms of the Rodriguez vector the divergence term in (15.50) has the form divðMðx; rÞ6ðx; rÞÞ ¼
vðMðx; rÞ6ðx; rÞÞj ðMðx; rÞ6ðx; rÞÞj r j 4 j vr ð1 þ r k r k Þ
v6j ðx; rÞ vMðx; rÞ Mðx; rÞ6j ðx; rÞr j ¼ Mðx; rÞ þ 6j ðx; rÞ 4 j j ð1 þ rk r k Þ vr vr
(15.53)
Summary
339
Thus, in terms of the Rodriguez parameterization of lattice orientation, relation (15.48) becomes
ZZZ ZZZ D vMðx; rÞ þ divðMðx; rÞvÞ dV (15.54) Mðx; rÞdV ¼ Dt vt <
<
And, incorporating the results of relations (15.49) through (15.53), this becomes ZZZ ZZZ vvj D vMðx; rÞ Mðx; rÞ Mðx; rÞdV ¼ þ vj vxj Dt vxj <
<
v6j ðx; rÞ vMðx; rÞ 6j ðx; rÞ vr j vr j
M x; r 6j x; r rj dV þ4 ð1 þ rk r k Þ Mðx; rÞ
(15.55)
At this point in the derivation, readers should note that vvi =vxj ¼ Lij is the well-known velocity gradient tensor, and in incompressible media, Ljj ¼ 0. It follows that relation (15.55) simplifies to ZZZ ZZZ D vMðx; rÞ v6j ðx; rÞ vj Mðx; rÞdV ¼ Mðx; rÞ Dt vxj vr j < < (15.56) j ðx; rÞr j
vMðx; rÞ Mðx; rÞ6 dV 6j ðx; rÞ þ4 vrj ð1 þ rk r k Þ The lattice rotation rate vector, 6ðx; rÞ, is related to the local velocity gradient tensor in a particular way. In terms of the time rate of change of the (passive) parameters of Bunge, the relationship is dgij ¼ dij εijk 6k dt
(15.57)
In terms of the deformation model, assume that the local velocity gradient tensor is LðxÞ. We have shown how to recover the lattice spin, W*. The components of the lattice rotation rate vector are known to be related to W* by the expression 6i ¼ εijk Wjk =2
(15.58)
Therefore, having obtained v and 6ðx; rÞ we may now use Eq. (15.56) to determine the evolution of the microstructure function in small cells represented by the region < in the integrals.
SUMMARY In this chapter readers were exposed to a theoretical framework for capturing the higher-order details of microstructure evolution using higher-order composite theories. It is hoped that readers can appreciate the MSDPO framework’s power in addressing this highly challenging problem. It should be
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CHAPTER 15 Higher-Order Models of Deformation Processing
recognized that process design even using the first-order composite theories is itself highly challenging. Therefore, it is not surprising that process design using higher-order composite theories is substantially more difficult. This chapter showed the essential foundations for addressing higher-order process design. However, it should be clear to readers that additional development is needed before we can obtain practical solutions to the process design problem using higher-order theories.