108 Simulation of an Austenite-Twinned-Martensite ... - NIST

1. Introduction

Simulation of microstructures is important for deter-mining the behavior of complex materials [1,2,3,4,5,6].Applying macroscopic loads to some materials canreveal their mechanical properties. These properties areassociated with the microscopic structures created dueto the material deformation. Microscopic structures ofinterest are often found near internal surfaces which, inturn are found in systems of liquid-liquid, solid-liquidand solid-solid boundaries [7].

An Austenite-twinned-Martensite interface is anexample of an internal surface. This interface conjoinstwo phases: Austenite and Martensite. Figure 1 showsan example of the twinned-Martensitic microstructures.These microscopic structures are the result of thermalor mechanical loading of the material. When a materialis deformed, the structural organization of the atoms isrearranged. At the microscopic level, this rearrange-

ment can create the crystallographic patterns shown inFigure 1, for example. These patterns represent 3-dimensional microscopic structures. There are manydifferent ways to rearrange the atoms that make up amaterial and each variant corresponds to a particulararrangement.We focus on the two specific variants ofMartensite distinguished by the alternating shadedareas in Figure 1.

One important mechanical property associated withan Austenite-twinned-Martensite interface is the ShapeMemory Effect (SME) characteristic of Shape MemoryAlloys (SMAs). SMAs are materials that, whendeformed for an indefinite period of time, return to theiroriginal shape. The deformed state is a metastable state[8]. In metastability, the total stored energy is at a localminimum, thus requiring energy to induce a transfor-mation. In SMAs, heat will trigger the SME which inturn, returns the SMA to its original shape. AnAustenite-twinned-Martensite interface is observed in

Volume 108, Number 6, November-December 2003Journal of Research of the National Institute of Standards and Technology

413

[J. Res. Natl. Inst. Stand. Technol. 108, 413-427 (2003)]

Simulation of an Austenite-Twinned-MartensiteInterface

Volume 108 Number 6 November-December 2003

A.J. Kearsley and L. A. MelaraJr.

National Institute of Standardsand Technology,Gaithersburg, MD 20899-0001

[email protected]@nist.gov

Developing numerical methods for pre-dicting microstructure in materials is alarge and important research area. Twoexamples of material microstructures areAustenite and Martensite. Austenite is amicroscopic phase with simple crystallo-graphic structure while Martensite is onewith a more complex structure. Oneimportant task in materials science is thedevelopment of numerical procedureswhich accurately predict microstructuresin Martensite. In this paper we present amethod for simulating material microstruc-ture close to an Austenite-Martensite inter-face. The method combines a quasi-Newton optimization algorithm and a non-conforming finite element scheme that

successfully minimizes an approximationto the total stored energy near the interfaceof interest. Preliminary results suggest thatthe minimizers of this energy functionallocated by the developed numerical algo-rithm appear to display the desired charac-teristics.

Key words: Austenite-Martensite inter-face; finite element method; quasi-Newtonmethod.

Accepted: January 6, 2004

Available online: http://www.nist.gov/jres

materials when they are in a metastable state.Therefore, to simulate Martensite, we minimize thetotal stored energy near an internal surface. In thispaper, we present a numerical technique used to obtaina solution which represents the spatial structure of thetwinned Martensite. The technique employs a Q 1 finiteelement discretization of a function representing totalenergy and a limited memory quasi-Newton method tominimize the resulting approximate total energy func-tion. The gradient of the function is calculated bycomputing the partial derivatives using finite differ-ences. The result is an apparently robust method forapproximating what are usually difficult minimizers tolocate.

1.1 Total Stored Energy

A myriad of research methods can be found whichfocus on the simulation of twinned Martensite, andrelated microstructures, e.g. Refs. [9,10,11,12,13,14]).Previous numerical work includes minimization of thetotal stored energy using the conjugate gradient meth-ods [15], the method of steepest descent [15], and adescent method [16]. Further references using andstudying the use of finite element methods to simulateMartensitic microstructures can be found in[17,18,19,20,21,22,23,24]. We are interested in simu-lating the microstructures in the metastable state, thusour work will focus on the static problem.

The functional representing the total stored energy istaken from work by Kohn-Müller in [10,11]. TheMartensite region is represented by the domainΩ = (0, L) × (0, K). Let x = (x, y) ∈ Ω ⊂and u = u(x, y). The double-well energy function isgiven by

(1)

where u equals 0 at x = 0. The x = 0 boundary corre-sponds to the internal surface. The function u = 0 atx = 0, due to the constraint of elastic compatibility[10,11]. The first integral is the elastic energy and thelast integral is the surface energy. We seek a function

which minimizes Eq. (1) where

1.2 Elastic Energy

The 2-D scalar model of elastic energy, in Eq. (1) is

(2)

This term is minimized by a function u such that

(3)


414

Fig. 1. Twinned-Martensite is shown to the lower left of the diagonal line. The field of viewis in the order of µm.

2 2, :u →

( )22 2( ) ( ) ( ) 1 d | | dx y yyJ u u u x u xεΩ Ω

= ∂ + ∂ − + ∂∫ ∫

1,4 ( )u Ω∈W

1,4 4 4 4( ) | ( ), ( ), ( )

and 0 at 0.x yv v L v L v L

v xΩ = ∈ Ω ∂ ∈ Ω ∂ ∈ Ω

= =

W

( )22 2( ) ( ) 1 d .x yu u xΩ ∂ + ∂ − ∫

0.

1u

∇ = ±

Each of the gradients in Eq. (3) corresponds to a stress-free state in one of the two distinct variants ofMartensite [10,11]. Examples of functions satisfyingEq. (3) are plotted in Fig. 2.

In Fig. 2 functions are piecewise linear in the y-direc-tion and constant along the x-direction. Requiring thatu equal zero at x = 0 creates more oscillations. This issimilar to the behavior of Martensite near an internalsurface. The function with Amplitude 1 is closest tozero, on average, than those with Amplitudes 2 and 3.

The number of oscillations is directly related to thenumber of discontinuities in ∂y u. We note that the func-tion with Amplitude 1 also has the largest number ofdiscontinuities, occurring at the peaks of u, where ∂y uchanges from +1 to –1 (or vice-versa). Therefore, adesirable minimizer u has the characteristics of thefunction with Amplitude 1. The consequence of u satis-fying Eq. (2) and u = 0 at x = 0 is the occurrence ofmore discontinuities in ∂y u. At the minimizer u, howev-er, the surface energy penalizes these discontinuities.

1.3 Surface Energy

We employ the surface energy presented by Kohnand Müller in Ref. [10,11],

(4)

where ε is a constant.The definition proposed by Kohn and Müller

replaces the integral of Eq. (4) in the y-direction withthe total variation of ∂yu over (0, K) [10,11]. Thischange designates the role of the surface energy atthe minimizer u as a “counter” becausecounts the number of discontinuities in ∂yu, where∂yu ∈ ±1. Thus, at the minimizer u of Eq. (1), thesurface energy Eq. (4) exhibits opposite behavior to Eq.(2) since we are minimizing Eq. (1). To illustrate thiscounting role of Eq. (4), we briefly review functions ofbounded variation.

1.4 Functions of Bounded Variation

The definition of functions of bounded variation istaken from Refs. [25,26,27]. Let

be any partition of (0, K) and let


415

Fig. 2. Oscillatory behavior of minimizer u.

| | d ,yyu xεΩ∂∫

102 | | dK

yyu y∫ ∂

0 1 , ,..., ly y y=P

1| | max | | .j jjy y+= −P

The partitioning P is a collection of points yj, j = 0, 1,…, l, such that y0 = 0 and yl = K. For each partitioningP, let

(5)

for a fixed ∈ (0, L). The total variation of ∂yu over(0, K) is

(6)

where the supremum is taken over all countablepartitions P of (0, K). If 0 ≤ SP ≤ +∞, then 0 ≤ε ∫Ω | ∂yyu | dx ≤ +∞. If ε ∫Ω | ∂yyu | dx < +∞, then ∂yu isa function of bounded variation.

Next, we describe the substitution ofby Eq. (6). We assume that ∂y u ∈ C1(Ω). Using Eq. (5)and the Mean Value Theorem, we have for someζ j ∈ (yj, yj+1), j = 1, 2, ..., l that

(7)

(8)

and we obtain by Theorem 2.9 in Ref. [25]:

The assumption that u ∈ C1(Ω) is stronger than theassumtion that ; however, the equivalencein Eq. (9) is possible since the changes in ∂y u do notoccur in subintervals of measure zero, [9,10,11,28].Figure 3 shows that counts the numberof sign changes in ∂y u. Let (0, K) = (0, 1).

In Fig. 3a, the piecewise linear function u(x, y) isplotted over (0, 1). In Fig. 3b, the function ∂y u(x, y) isplotted over (0, 1). Clearly, the number of discontinu-ities in ∂y u is 5 from Fig. 3b. For a discontinuity at thepartition point yj, we adopt the following convention:

(10)

Now consider (yj–1, yj+1) = (yj–1, yj) (yj, yj+1) in Fig. 3b.Evaluating at the endpoint of the two subintervals gives

(11)

(12)

A jump occurred in (yj–1, yj) because ∂y u is discontinu-ous at a point in this subinterval. Based on our conven-tional definition of ∂y u(x, yj) at a point yj in Eq. (10), wesay a “jump” occurs at yj if ∂y u is discontinuous at thispoint. This “jump” refers to the change in values of ∂y uat yj from +1 to –1 or vice-versa.

This jump gives the nonzero value in Eq. (11). In thesubinterval (yj, yj+1), no jump occurs in ∂y u, thereforethe difference in Eq. (12) is zero. We compute Eq. (9):

where the magnitude of the jump is 2 and the numberof discontinuities is 5. Table 1 shows the number of dis-continuities for the functions presented in Fig. 2.


416

1

10

( ) | ( , ) ( , ) |,l

y j y jj

x u x y u x y−

+=

= ∂ − ∂∑SP

x

sup ( ),xSPP

( )x

0 | | dKyyu y∫ ∂

1

10

1

10

( ) | ( , ) ( , ) |

| ( , ) | ( ),

l

y j y jj

l

yy j j jj

x u x y u x y

u x y yς

−

+=

−

+=

= ∂ − ∂

= ∂ −

∑

∑

SP

1

1| | 0 | | 0 0

0

sup ( ) lim ( ) lim | ( , ) | ( )

| | d . (9)

l

yy j j jj

K

yy

x x u x y y

u y

ς−

+→ → =

= = ∂ −

= ∂

∑

∫

S SP PP PP

1,4 ( )u∈ ΩW

102 | | dK

yyu y∫ ∂

,

( , ) ( , )( , ) lim .

j j

jy j y y y y

j

u x y u x yu x y

y y→ >

−∂ ≡

−

∪

1

1

| ( , ) ( , ) | |1 ( 1) | 2 and

| ( , ) ( , ) | |1 1| 0.y j y j

y j y j

u x y u x yu x y u x y

−

+

∂ − ∂ = − − =

∂ − ∂ = − =

Fig. 3. Surface energy term as a counter. Pictorial example illustrat-ing the role of surface energy as a counter term. Thenotation u(·, y) means that the graph shows the y-dependence only.

10 | | dyyu y∫ ∂

1

0| ( , ) | d 2 5 10,yyu x y y∂ = × =∫

Table 1. Number of “jumps” in ∂y u

Function Discontinuities

Amplitude 1 24 12Amplitude 2 12 6Amplitude 3 6 3

10 | | dyyu y∫ ∂

Therefore, we have:

and the surface energy succeeds in serving as a “count-er.” The next section describes the numerical approxi-mation to Eq. (1).

2. Description of Method

The total stored energy Eq. (1) consists of elasticplus surface energy. First, we describe the implementa-tion of the elastic energy followed by a discussion ofthe surface energy implementation. We computed Eq.(2) using affine finite elements, [29]. We chose thefinite element space:

(13)

because functions in this polynomial space lie in, which satisfy the criteria in Eq. (3) for a min-

imizer u of Eq. (1). The elements are rectangles withthe function values at the vertices as degrees of free-dom. The parent element is Q = (0, 1) × (0, 1) and forsome , the reference element is

where h1 is the mesh size along the x-axis and h2 is themesh size along the y-axis. The degrees of freedom ofthe elements are the function values at the vertices, andfor Q we label them u(a1), u(a2), u(a3), and u(a4). Figure4 shows the parent element with the vertices also denot-ed by concentric circles.

2.1 Affine Finite Elements

Let F : Q→Qk be given by

(14)

We simplify, by letting F = F(x, y). We use the affinemap Eq. (14) to evaluate the function u, to compute ∇uand to integrate. The function u is spanned by the fourbasis functions defined in each element Qk. Using Eq.(14), we can determine all basis functions by the fourbasis functions in Q. Thus, the approximation to u is:

(15)

where x = x(x, y), y = y(x, y). The nth basis function inQ is

and it satisfies the property that φn(am) = δmn, for n, m =1, 2, 3, 4, where δmn is the Kronecker delta function.

To compute the gradient of u and to integrate u, weneed: ∇F, det∇F and ∇F–1. The gradient of F is

(16)

and, we have

(17)

Using Eqs. (15), (16), and (17) we have

To integrate the elastic energy over Qk we use Eqs.(15)-(17) and obtain


417

1

0| ( , ) | d 2 (number of discontinuities of ),yy yu x y y u∂ = × ∂∫

1 span1, , , ,x y xy=Q

1,4 ( )ΩW

k ∈

( ) ( )1 1 2 2, ( 1) , ( 1) ,k pp

Q ih i h jh j h Q= + × + ⊂ Ω =∪

Fig. 4. Parent Element Q with four degrees of freedom labeled u(a1),u(a2), u(a3), and u(a4).

1 1

2 2

ˆ ˆ0.

ˆ ˆ0h ihx x

Fh jhy y

= +

1 1 2 2 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , ),u x y x y x y x y x yφ φ φ φ= + + +u u u u

( ) ( ) ( ) ( )0 1 2 3ˆ ˆ ˆ ˆ ˆˆ( , ) ,n n n n

n x y x y xyφ α α α α= + + +

1

2

00h

Fh

∇ =

1 ˆ ˆ( , ) ( , ).n nx y F x yφ φ−∇ = ∇ ∇

41

1

ˆ ˆ( , ) ( , ).n nn

u x y F x yφ−

=

∇ = ∇ ∇∑u

242

ˆˆ11

24

ˆ 1 2ˆ11

1 ˆ ˆ ˆ ˆ( ) d d ( , ) det d d ,

1 ˆ ˆ ˆ ˆ( , ) d d . (18)

kx n x nQ Q

n

n x nQn

u x y x y F x yh

x y h h x yh

φ

φ

=

=

∂ = ∂ ∇

= ∂

∑∫ ∫

∑∫

u

u

The procedure is the same for

2.2 Computation of Surface Energy

We evaluate the surface energy term in Eq. (4) usingEq. (6), approximating ∂yu using backward finite differ-ences,

and

Here,

(19)

where N(h2) is the number of nodes along the y-direc-tion. We can integrate along the x-direction to completethe approximation of Eq. (4).

3. Numerical Implementation

The computation of the integral in Eq. (18) is doneexactly in the parent element Q. The symbolic packageMaple1 [30] was employed to integrate this term exact-ly. The algebraic expression consisting of nodes uj andmesh sizes h1 and h2, is evaluated over each element.

The gradient of Eq. (1) is approximated using cell-centered finite differences. Let u ∈ (Ω). TheGateaux derivative is:

(20)

where ϕ ∈ (Ω) [31]. The linear operation G is thedirectional derivative of J in the direction ϕ. From theEuler-Lagrange equations, we see that the gradient of Jis difficult to compute. Considering the Euler-Lagrangeequation we have,

where the gradient is div (∂J/∂∇u). Let J(u) be the dis-cretized representation of the total energy Eq. (1) foru ∈∈ , where N(h) is the total number of nodes in Ω.

With this numerical approximation to ∇J(u), we usethe limited memory BFGS method [32] to numericallyminimize J(u). This inexpensive quasi-Newton methodseeks to build an approximation H(k) to the inverse ofthe Hessian matrix of second derivatives of J at thepoint u(k). These inexpensive approximations are con-structed from a small number of vectors updated atevery iteration.

The k + 1 step of the limited memory BFGS methodis given by:

where the parameter α(k) is a line search parameter andis computed at each iteration using a line-search proce-dure.

The Hessian matrix B(k) satisfies the secant equation:

where

We employ a limited memory BFGS method becausethe Hessian matrix B(k) is expensive to compute sincewe are solving a large scale optimization problem[32,33]. This is done by storing a certain number ofvector pairs s(k), y(k). At each new iterate, the oldestvector pair is deleted and replaced by the new vectorpair, thus preserving curvature information only fromthe most recent iterations. The formula for updating theinverse of the Hessian matrix H(k) = (B(k))–1 is:


418

( )22( ) 1 d d .k

yQu x y∂ −∫

1

2

( , ) ( , )( , ) j j

y j

u x y u x yu x y

h−−

∂ ≈

11

2

( , ) ( , )( , ) .j j

y j

u x y u x yu x y

h+

+

−∂ ≈

2( ) 1

100

| | d | ( , ) ( , ) |N hK

yy y j y jj

u y u x y u x y−

+=

∂ = ∂ − ∂∑∫

[ ]0

1( ) lim ( ) ( )t

G u J u t J ut

ϕ→

= + −

1 Certain commercial equipment, instruments, or materials are iden-tified in this paper to foster understanding. Such identification doesnot imply recommendation or endorsement by the National Instituteof Standards and Technology, nor does it imply that the materials orequipment identified are necessarily the best available for the pur-pose.

0d ( )( ) ( ) | d ,d

div d ,

tJ uG u J u t x

t uJ xu

ϕ ϕ

ϕ

= Ω

Ω

∂= + = ∇∂∇

∂= −∂∇

∫

∫

( )N h

( 1) ( ) ( ) ( ) ( )( ).k k k k ku u H J uα+ = − ∇

( ) ( ) ( )s ,k k kB y=

( ) ( ) ( 1)

( ) ( ) ( 1)

s ( ), andy ( ) ( ).

k k k

k k k

u uJ u J u

−

−

= −= ∇ −∇

1,4W

1,4W

( 1) ( ) ( 1) ( ) 0 ( ) ( 1)

( ) ( 1) ( 1) ( ) ( ) ( 1) ( 1)

( 1) ( 2) ( 1) ( 1) ( 1) ( 2) ( 1)

( 1) ( 1) ( 1

( ... ) ( ) ( ... )( ... ) s (s ) ( ... )

( ... ) s (s ) ( ... )...

s (s

k k m k T k k m k

k m k m k T k m k m T k m k

k m k m k T k m k m T k m k

k k k

H V V H V VV V V V

V V V Vρρ

ρ

+ − − − −

− − + − − − − + −

− + − + − − + − + − + −

− − −

=+

+++ ) ) .T

where (H(k))0 is some initial Hessian approximation,

and

4. Results

In this section, results computed on the various grids,30 × 30, 60 × 60, and 120 × 120, are presented. In allminimization, the initial iterate u0 is a small (≈10–3) per-turbation of pure Austenite, which we take to be u = 0.Density plots of the minimizer u are shown in Figs. 5-10 and demonstrate the existence of the desiredmicrostructures in the twinned-Martensite phase. Thesefigures also demonstrate the role of the surface energyas a penalization term of Eq. (1). We seem to be able tocontrol the number of microstructures by varying thevalues of ε, with a larger value of ε resulting in a small-er number of microstructures in Martensite and vice-versa. Figures 11, 12, and 13 show plots of the profilesof minimizer u at x = 4. Figures 11, 12, and 13 exhibita larger number of discontinuities in ∂y u for ε = 2 ×10–6 than for ε = 2 × 10–2. In Figs. 5-9 one sees theeffect of grid-size comparing results on 60 × 60 gridwith 120 × 120 grid for ε = 2 × 10–6 through ε = 2 ×10–2. The black and white stripes correspond directly tothe saw-toothed behavior of u at x = 4. For small ε val-ues, the number of discontinuities is close to the num-ber of partitions along the y-direction. Tables 2-4 showthe energy values for the minimizers shown in Figs. 5-10. The column with E(u) corresponds to the elasticenergy values of the minimizer while the column withS(u) corresponds to the surface energy values of theminimizer. The energy values presented in Tables 3 and4 appear consistent with these conclusions. The largerthe ε, the larger the value of the surface energy at theminimizer. In Table 3 the value of the surface energyincreases by a larger order than that of the elastic ener-gy.

One final and curious observation is the appearanceof a “diagonal band” structure in Figs. 5-10. It wouldseem that this band, which is apparent for solutions onvarious grid sizes, is related to the competing roles ofthe surface and elastic energies is and may be associat-ed to the “equipartitioning of energy” principle pro-posed in Kohn and Müller in Ref. [10,11]. Futureresearch plans include a deeper investigation of thisdiagonal-band structure.


419

( ) ( ) ( ) ( )( ) ,k k k k TV I y sρ= −

( ) ( )

1 .( )k T ky s

ρ =

Table 2. Energy values for various ε on 120 × 120 grid

ε E(u) εS(u) Total energy

2 × 10–6 6.3014 0.5691e-2 6.30712 × 10–5 6.1122 0.5698e-1 6.16922 × 10–4 7.0859 0.5474 7.63342 × 10–3 8.2222 4.5200 12.74232 × 10–2 23.3218 1.2559 24.5770



2 × 10–6 4.3684 0.3120e-2 4.37162 × 10–5 6.3005 0.2808e-1 6.32862 × 10–4 6.3010 0.2798 6.58092 × 10–3 5.7738 2.7356 8.50952 × 10–2 3.7945 14.6825 18.477



2 × 10–6 4.6093 0.0015 4.61082 × 10–5 4.9832 0.0145 4.99772 × 10–4 6.1073 0.1326 6.14992 × 10–3 5.5335 1.3246 6.85812 × 10–2 8.0650 9.6770 17.7420


420

Fig. 5. Comparison of density plots of u for values of ε = 2 × 10–6 on 60 × 60 grid (bottom) vs 120× 120 grid (top).

Fig. 6. Comparison of density plots of u for values of ε = 2 × 10–5 on 60 × 60 grid (top) vs 120 × 120grid (bottom).


421




422



423

Fig. 10. Density plot of solution u for ε = 2 × 10–6, ..., 2 × 10–2 on 30 × 30 grid.


424

Fig. 11. Comparison of u(4, y) for various values of ε = 2 × 10–6, ..., 2 × 10–4 on 120 × 120 (left column) grid vs 60 × 60 grid (right column).


425

Fig. 12. Comparison of u(4, y) for various ε = 2 × 10–3 and ε = 2 × 10–2 on 120 × 120 grid (left column) vs 60 × 60 grid (right column).


426

Fig. 13. Profiles of u(4, y) for ε = 2 × 10–6, ..., 2 × 10–2 on a 30 × 30 grid.

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About the authors: Anthony J. Kearsley is a memberof the Mathematical and Computational SciencesDivision at NIST. His research involves developingnumerical algorithms. In 1998, he was awarded aPresidential Early Career Award for his work onNumerical Methods for Optimization. Luis A. Melara isa National Research Council Postdoctoral Associate inthe Mathematical and Computational SciencesDivision at NIST. The National Institute of Standardsand Technology is an agency of the TechnologyAdministration, U.S. Department of Commerce.


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