The formation mechanism of the hierarchic microstructure in lath martensite
phase and the microstructural evolution in steel
ラスマルテンサイト相の階層構造形成機構の解明とその組織発展過程
Yoshinori MURATA, Graduate School of Engineering, Nagoya University
研究代表者

名古屋大学大学院工学研究科

村田純教

1. Introduction
Martensite phase is classified into several kinds based on the morphology and latticedefects, i.e., thin plate, lens, and lath. Thin plate martensite consists of internal twins
and lens martensite contains both internal twins and dislocations. Lath martensite
contains only dislocations. In the recent heat-resistant steels, which has high creep
strength at elevated temperatures, the carbon content is about 0.1 mass%, where lathmartensite always appears. The lath martensite has the unique hierarchicalmicrostructure with high dislocation density and the mechanical properties are related tothe microstructure evolution.
It is reported that the lath martensite phase forms in steels having relatively highmartensite start (Ms) temperatures and that the habit plane is near {557} different by
about 10 degrees from 111 [1] [2].

Its crystal-orientation relationship is nearly(111) //(011)  , [101] //[1 11]  , where
are close-packed planes and 1 01 and 1 1 1 ' are

K-S, composed of the relationships

111

and011 'close-packed directions in the austenite phase (γ) and the martensite phase (α′),
respectively.

A total of 24 crystallographic variants satisfy this orientation relationshipas shown in Table 1.
The lath martensite phase possesses a hierarchic structure as follows: (i) A prioraustenite grain is composed of packets; (ii) A packet is composed of an ensemble ofgrains, called blocks, which have the same 111 plane as does the habit plane; and(iii) A block is composed of an ensemble of single martensite crystals called laths,
which have nearly the same crystal orientation and high dislocation densities [3] [4].
Crystal orientation in the lath martensite phase was recently analyzed over a
relatively wide area using the electron back-scattering pattern (EBSP) measured with ascanning electron microscope (SEM). Morito et al. found that crystal orientation in
low-carbon steels deviates locally at each point in a block, and that blocks are composedof not just one but rather a combination of two specific crystallographic variants [2] [3] .
For example, grains belonging to variant 4 (V4) are observed only in blocks that arecomposed predominantly of variant 1 (V1); similarly, grains belonging to variant 5 (V5)－ 93 －

are observed only in blocks that are composed predominantly of variant 2 (V2) and soon. The boundary between V1 and V4 in the figure, drawn is the block boundary andthis is called as sub-block boundary and is distinguished from conventional blocks [2][3].
Table 1

All 24 crystallographic variants that satisfy the K-Sorientation relationship

Many theories regarding the deformation geometry of martensitic transformation,such as Bowles–MacKenzie (BM) theory and Wechsler–Lieberman–Read (WLR)
theory, have been proposed [5][6]. They are collectively called the phenomenologicaltheory of martensite crystallography (PTMC). PTMC is based on the experimental
observation that deformation with martensitic transformation is invariant plane
deformation, because the martensite phase maintains continuity with the surrounding
austenite phase. As a result, the martensite phase has the invariant plane as the habit
plane. In PTMC, all deformations induced by martensitic transformation are explainedby a combination of lattice deformation (as crystal structure changes), lattice-invariant
deformation occurring from shear deformations and rigid-body rotation. Well-known
examples applying PTMC to lath martensite include studies by Sandvik and Wayman,－ 94 －

and Kelly [7][8]. However, it is unlikely that transformation actually occurs in thissequence. In other words, PTMC describes the result of transformation well but does
not explain the mechanism of transformation.
Khachaturyan has presented a model describing deformation with martensitic
transformation [9]. His model considers all deformations to be a combination oflattice deformation (as crystal structure changes from fcc to bct) and lattice-invariant
deformations with plastic deformations by slip. This model is reasonable in its
explanation of the transformation mechanism, but has not yet been verified by detailed,quantitative analyses. Also, as far as we know, there is no report of the plastic
deformation that considers several slip systems independently. Furthermore, it remains
unclear why the lath martensite phase should contain sub-blocks as reported by Morito
et al., although the lath martensitic structure is important for its contribution to thestrength of steel. Thus, it is important for both practical and academic reasons toclarify the mechanism by which the structure forms.
Recently, we clarified the formation mechanism of lath martensite by presenting two
types of slip deformation (TTSD) model [10] by extending Khachaturyan’s slip
deformation model. The purpose of this study is to characterize the microstructure oflath martensite phase by means of simulation. In this simulation, the martensitic
transformation in Fe-0.1C mass% lath martensitic steel is simulated by the elasto-plastic
phase-field model based on the TTSD model. The establishment of this elasto-plastic
phase-field model is to prove the validity of the TTSD model on explaining theformation of lath martensite. On the other hand, the microstructure evolution of lathmartensite is presented in 3-D space by the phase-field simulation.
2. Calculation method
Based on the TTSD model [10], an elasto-plastic phase-field model is developed byconsidering both of the Bain deformation and plastic deformation. For the phase-fieldmodel, a field variable i (r )  i  1, 2,3 is introduced to describe the Bain deformation.
i  1, 2,3 is used to distinguish the three cases of coordinate coincidences and r is thecoordinate vector. The eigenstrain caused by the Bain deformation  klB (i )(i  1, 2,3) islisted in matrix form as 2a  / a  1
 klB (1) 
0

0





2a  / a  10,0c  / a  1

－ 95 －00(1) c  / a  1
0 klB (2) 

0


0
2a  / a  1
0

 2a  / a  1
0



B
0
c  / a  1
 kl (3) 

0
0




0
,

2a  / a  1

(2)



,

2a  / a  1

(3)

0

0
0

where a is the lattice parameter of the austenite phase, a  and c  are the lattice
parameters of the martensite phase, respectively. With respect to plastic deformation, theother field variable pi (r )(i  1, 2, 3) is introduced to characterize the values of local
plastic strain produced by dislocations from a specific slip system, where  is thenumber of slip systems. Whenis equal to 1 or 2, it corresponds to the slip system
101  101 or  101 101 ' , respectively. pi is defined as [11]
'

pi

bi
mi d hkl(i 1, 2,3) .

(4)Here, bi is the absolute value of the Burgers vector, mi is the number of latticeplanes between two adjacent slip planes in each slip system, and d hkl is the distance
between  hkl  ' planes.
The eigenstrain tensor  klP , caused by plastic deformation can be written as [11]bi  ni  ni  bi pi (r )(i 1, 2,3) ,


2 bi

P
kl

(5)

where bi is the Burgers vector, ni is the unit vector of the slip plane normal,and  represents the dyadic product.
The martensitic transformation is a minimization process of the total free energy,
which is defined as the sum of the chemical free energy Echem, gradient energy Egrad, andelastic strain energy Eel [12] :Etotal Echem ({i (r )})  Egrad ({ pia (r )})  Eel {i (r )},{ pia (r )} .

(6)

Eq. (6) indicates that the chemical free energy is involved in Bain deformation, thegradient energy is involved in plastic deformation, and the elastic strain energy is－ 96 －

related both with Bain deformation and dislocation slip.
The dynamics of the martensitic transformation is controlled by the Allen-Cahn
equation:M  r, t  Etotal  LM, M  r, t t(7)where M  r, t  ( M  i , pi ) are the field variables and LM is the kinetic parameter of
each field variable.
3. Results and Discussion
Fig. 1 shows the simulation result of time evolution of three types of blocks in apacket on the <111> plane. The blue areas represent the retained austenite phase and theother colored areas (green, yellow, and red) represent three different block as explainedin the formation mechanism of lath martensite. The occurrence of each block is
determined by the value of the field variable i (r ) . In this simulation, it is assumed thatthe martensite phase appears only when i (r )  0.7 . As shown in Fig. 1 that at t* = 2,
all of the three blocks appear around the dislocation loop set in the center of packet. At
t* = 4, different blocks grow bigger around the existing martensite phase. In the processof martensite growth, when a second order block meets a first order block, it will stop
growing. It results in that the boundary of the first order block is a straight line. In thismanner, the martensite phase becomes coarser until the morphology of the fullmartensite appears. At t* = 8, the packet is almost full of lath martensite phase exceptthat only a few austenite phase can be seen. At t* = 20, the three blocks have occupied
the whole packet, indicating the accomplishment of the martensitic transformation.
Fig. 2 shows the simulation results of time evolution of lath variants in a packet on
the <111> plane and the six colored areas represent the six variants which are illustrated
explicitly in (e). As shown in Fig. 2 (e), each block shown in Fig. 1 is composed of apair of variants, i.e., V1~V4, V2~V5, and V3~V6, which is the so-called sub-block
structure in lath martensite. A block composed of the two martensitic variants definedby the K-S relationship is ideal for the adjustment of plastic deformation by the two
types of slip systems. For comparing, the experimental result is shown in (f), which is
the EBSP observation of lath martensite [3]. In Fig. 2(f), the parallel blocks structure
can be observed and each block is a combination of two variants. By comparing Figs.
2(e) and (f), the qualitative similarity of the sub-block microstructure between thesimulation results and experimental observation can be seen. The ratio of the volume
fraction of the three block in our simulation is calculated to be 1:1:1 by the meshmethod, while the ratio of B1:B2:B3 in (f) is calculated to be 1:2:1. As the lath

－ 97 －

martensite is an inhomogeneous system, it needs a lot of mapping information for theaccurate quantitative analysis, which is unavailable up to present and needs further
effort.

Fig. 1 Time evolution of blocks on <111> plane for (a) t* = 2, (b) t* = 4, (c) t* = 8, and(d) t* = 20 simulated by elasto-plastic phase-field model. The three colors, i.e., blue,
red, and yellow represent three blocks and deep blue represents austenite phase.
Figs. 3 and 4 show the simulation results of the time evolution of plastic strainpi (r ) along the slip systems of [101](101)  and [101](101)  when i  1 in apacket, respectively. The blue areas indicate no slip deformation, while the red areas
represent the most dramatic slip deformation. Each point in Figs. 3 and 4 represent a
local plastic strain for a certain i, while all the points scattered in a packet contain theplastic strain coming from the three cases of lattice corresponding, i.e., i  1 , 2 and 3.
The slip systems [101](101)  and 1 01101 ' correspond to the slip systems wheni  1 . These figures reveal that the plastic deformation originated from the center of theaustenite phase and the range of the slip deformation extends with the progression ofmartensitic transformation. By comparing Figs. 3 and 4, it is found that the plastic
deformation along the two slip systems is complementary with each other. Taking area
“A” in Fig. 3 and area “B” in Fig. 4 as an example, the plastic strain in “A” is very large,
while there is almost no plastic deformation in the same area along the other slip system,
as shown in “B”. These phenomena can be observed at all places and times during
－ 98 －

martensitic transformation. So it is concluded that the slip deformation along the two
slip systems cooperated with each other to assist the plastic accommodation. The
phenomenon that the plastic deformation occurs along the two slip systems alternatively
is consistent with the original formation mechanism of lath martensite. It is to be notedthat the plastic strain shown in Figs. 3 and 4 only represent the local values in the lattice
crystal and all of these values should be integrated within the whole crystal to assess thecontribution of dislocation slip on plastic accommodation for martensitic
transformation.

Fig. 2 Time evolution of lath variants on <111> plane for (a) t* = 2, (b) t* = 4, (c) t* = 8,
and (d) t* = 20 simulated by elasto-plastic phase-field model. (e) is the specificexplanation of variants shown in (d). The six colors represent six different variants in apacket. (f) is the EBSP observation of lath martensite quoted from Ref. [3].
On the other hand, by comparing Fig. 2 and Figs. 3, 4, it is found that themorphology of lath martensite evolution corresponds to the plastic strain evolution. In
particular, the slip deformation starts from the beginning of martensitic transformation,
as is seen in Figs.3 and 4.

－ 99 －

(a)

(c)

(b)

A

(d)

Fig. 3 Time evolution of plastic strain pi (r )(i  1, 2, 3) along [101](101)  slip
system on <111> plane for (a) t* = 2, (b) t* = 4, (c) t* = 8, and (d) t* = 20 byphase-field simulation.

B

Fig. 4 Time evolution of plastic strain pi (r )(i  1, 2, 3) along [101](101)  slip
system on <111> plane for (a) t* = 2, (b) t* = 4, (c) t* = 8, and (d) t* = 20 byphase-field simulation.

－ 100 －

With the progression of phase transition, the plastic strain increases resulting in thegrowth and coalescence of the martensitic variants shown in Fig. 2. In other words, theoccurrence and evolution of the plastic deformation determine the formation andmorphology evolution of the lath martensite phase.
By inserting the local plastic strain along each slip system in Eq. (4), the values of
m1 and m2 can be evaluated. As mentioned previously, the subscripts 1 and 2 correspondto the 1011 01 ' and 1 01101 ' slip system i  1 , respectively. Because of thethree cases of lattice corresponding, there should be three pairs of m1 and m2. Since eachpair is equivalent, only the case in which the [001] -axis coincides with the [001] -axis, i.e., i  1 , is discussed as an example. For this case, the variants of V1 and V4
will appear. For the other two cases, the relationship between the values of m along thetwo independent slip systems should be similar to that of i  1 . By inserting the local
values of the slip deformation p11 and p12 in Eq. (4), the values of m1 and m2 areestimated and the relationship between m1 and m2 within simulation areas is plotted inFig. 5(a), where two thousand local values distributed in the simulation areas arecollected. It is found that the dense points concentrate in one corner due to the close
values as shown in Fig. 5(a). For easy observation, the m1 and m2 maxima are limited towithin 100 and the enlarged part is shown in Fig. 5(b). For comparison, the analyticalresult is also shown in Fig. 5(c), which is obtained from eigenvalues satisfying theinvariant plane deformation of the deformation matrix [10]. A “corner” type contour is
exhibited in Fig. 5(b) and the values of m1 and m2 in the corner are approximately 16,
which is close to the corner point m1 = m2 = 19 in Fig. 5(c). Fig. 5(b) also reveals thatwhen m2 reaches its minimum 10, m1 increases from 20 to infinity. The converse
situation yields a similar result. The contour shown in Fig. 5(b) is consistent with theanalytical results shown in Fig. 5(c). This indicates that m1 and m2 are mutuallydependent, which cannot only be seen from the analytical results, but also from thesimulation results. The relationship between m1 and m2 corresponds to the relationshipbetween the plastic deformation along the two slip systems. Therefore, it is concluded
that the simulation result of the relationship between the two types of slip deformationis consistent with the analytical analysis calculated by the TTSD model. On the other
hand, Eq. (4) suggest that if m1 is larger than m2, the plastic strain along 1011 01 ' islarger than that along 1 01101 ' . It causes the plastic deformation will be along1011 01 ' , resulting in the formation of variant V4. On the contrary, the variant V1
appears. It suggests that the values of m1 and m2 determine the appearance of V1 or V4.
This simulation result is consistent with the analytical result calculated by Iwashita et al.
[10].

－ 101 －

Fig. 5 (a) show the simulation results of the relationship between m1 and m2 when i =1, (b) is the enlarged part of (a) when both of m1 and m2 are limited within 100 and (c)is analytical results the relationship between m1 and m2 quoted from Ref. [10].

Fig. 6 The growth process of lath martensite at (a) t* = 2, (b) t* = 4, (c) t* = 8, and(d) t* = 20 in 3-D space simulated by phase-field model.
Fig. 6 shows the growth process of the lath martensite in 3-D space. The cubic
skeleton represents the prior austenite lattice. The martensite phase grows bigger around－ 102 －

the initial lath martensite nucleus and becomes full martensite at t* = 20, which
occupies the whole austenite cubic as shown in Fig. 6(d).
Fig. 7 exhibits the time evolution of lath martensite in 3-D space seen from theoutside of austenite cubic. The blue cubic represents the austenite phase lattice. The six
coloured areas on the surface of the cubic are the six lath variants. At t* = 2 and 4, thelath martensite only exists inside of the austenite cubic as shown in Figs. 6(a) and (b),resulting in that the lath martensite cannot be seen from the outside. Therefore, theaustenite cubic is still blue as shown in Figs. 7(a) and (b). With the progression of themartensitic transition, the lath martensite phase reaches the surface of the austenite
cubic and can be seen from the outside of the austenite cubic as shown in Fig. 7(c).
When the martensitic transformation is completed, the lath martensite phase spreads allover the surface of austenite cubic as shown in Fig. 7(d).

Fig. 7 Time evolution of lath martensite at (a) t* = 2, (b) t* = 4, (c) t* = 8, and (d) t*= 20 in 3-D space observed from the outside of austenite cubic simulated byphase-field model. Each color represents a lath variant.
4. Conclusion
On the basis of the two types of slip deformation (TTSD) model for the formationof lath martensite, an elasto-plastic phase-field model was constructed. Furthermore, themorphology evolution of lath martensite in Fe-0.1C mass% steel was simulated by the－ 103 －

elasto-plastic phase-field model in 3-D space. It was observed that the full martensite
phase can be obtained by releasing the large elastic strain contained in the Baindeformation, via the two types of independent dislocation slips. Moreover, themorphology of lath martensite such as sub-blocks, as seen in commercial steels, can bewell predicted. The relationship of plastic deformation between the two slip systems
simulated by phase-field model is consistent with the analytical calculation using by
TTSD model. This indicates the validity of the TTSD model for explaining theformation of lath martensite phase.
Acknowledgements
This work was carried out in collaboration with Dr. Yuhki Tsukada (present:
Assistant professor of Nagoya Institute of Technology) and was supported by JFE 21stCentury Foundation.
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