ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
Internal Stress and Elastic Strain Energy in Pearlite and Their Accommodation by Misfit Dislocations
Nobuo Nakada Masaharu Kato
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2016 Volume 56 Issue 10 Pages 1866-1873

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Abstract

Internal stress and elastic strain energy in pearlite caused by misfit between ferrite and lamellar cementite were theoretically analyzed based on micromechanics while taking into account an accommodation mechanism of the misfit strain between ferrite and cementite on pearlitic transformation. Two-dimensional large deformation analysis reveals that the Pitsch-Petch orientation relationship is most appropriate among already reported crystal orientation relationships under the condition that the interface contains a lattice invariant direction. However, the micromechanics analysis using a periodic function proves that the misfit strain generates very large elastic strain energy in pearlite even when the Pitsch-Petch orientation relationship is satisfied, which is almost comparable to the chemical driving force for pearlitic transformation. Assuming that an interval of misfit dislocations dynamically introduced at ferrite/cementite interface upon pearlitic transformation depends on the growth rate of pearlite, the total elastic strain energy reduces more effectively as the growth rate becomes lower. As a result, the elastic strain energy in pearlite changes widely depending on interlamellar spacing.

1. Introduction

Pearlite, defined as eutectoid structure in the Fe–C system, is characterized by fine lamellar structure composed of ferrite and cementite. It is well known that the two-phase fine lamellar structure contributes to high strength of pearlitic steel. Many studies have been performed for a long time in order to understand the various aspects of pearlite and its transformation. For instance, the growth rate and the interlamellar spacing of pearlite have been evaluated for demonstrating the rate-controlling process of the pearlitic transformation in terms of thermodynamics and kinetics.1,2,3,4,5,6,7) In addition to the interlamellar spacing, morphologic and crystallographic features of substructures, i.e. pearlite colony and block (nodule), have been analyzed for understanding how the hierarchical substructures form.8,9,10,11,12,13,14,15,16,17,18,19,20,21) Furthermore, many researchers have investigated the relation between mechanical properties and the unique microstructure with great interest.12,13,14,22) From these researches, it is commonly recognized as a basic guideline on microstructural control of pearlitic steel that the lower the transformation temperature is, the finer the interlamellar spacing becomes, thereby leading to the improvement of strength. However, there have been few studies paying attention to transformation strains, since pearlite is a diffusional transformation product.

Using electron backscattered diffraction (EBSD) and X-ray diffraction (XRD) techniques, one of the present authors and his colleagues have discovered that a significant amount of lattice strain remains in pearlite.23) They have demonstrated that the lattice strain originates from the misfit between ferrite and cementite.24) In addition, the lattice strain increases with decreasing the interlamellar spacing. As a result, it is interesting to find that the strength of pearlitic steel has a good linear relationship with the lattice strain in pearlitic ferrite rather than with the interlamellar spacing.25) These results strongly suggest that the elastic strain and internal stress are very important factors affecting the microstructure and the mechanical properties of pearlitic steel. It seems that the internal stress in pearlite is inhomogeneously distributed among pearlite colonies, and each colony possesses different anisotropic internal stress.24) Therefore, a technique with a high spatial resolution much finer than the colony size (normally less than 5 μm) is required to analyze the internal stress distribution in pearlite. For example, convergent-beam electron diffraction in transmission electron microscopy (TEM) may be most appropriate for this purpose. However, it is always argued that thinning process for TEM samples causes significant stress relaxation.26) In summary, it is very difficult to evaluate the internal stress in pearlite experimentally so far.

In this study, the internal stress and elastic strain energy in pearlite caused by misfit between ferrite and lamellar cementite will be analyzed theoretically based on micromechanics. In addition, the interlamellar spacing dependency of elastic accommodation will be discussed by taking into account the arrays of misfit dislocations which are dynamically introduced on pearlitic transformation.

2. Calculation of Misfit Strain by Two-dimensional Large Deformation Theory

Let us first evaluate the elastic state of pearlite by assuming that coherency is maintained between ferrite and cementite in pearlite. For this purpose, we must know the interface orientation composed of crystal planes of the two phases.

Figure 1 displays unit cells of (a) cementite (orthorhombic crystal,) and (b) ferrite (body-centered cubic crystal) graphically-represented by a software Crystal Maker. To evaluate misfit strains at pearlitic transformation temperature, the lattice constants of cementite (C) and ferrite (F) at 873 K (600°C) (aC=0.45235 nm, bC=0.50895 nm, cC=0.67433 nm and aF=0.28665 nm, respectively) will be used.27) Figure 2 shows iron atom configurations of (a) (001) and (b) (101) in cementite and (c) (112) and (d) (215) in ferrite.18) This figure reveals intuitively that both phases have good lattice correspondence on these planes. Indeed, each set of them corresponds to the following crystal orientation relationships (OR).

Fig. 1.

Unit cells of (a) cementite and (b) ferrite.

Fig. 2.

Fe atomic configurations of (a) (001) and (b) (101) planes of cementite and (c) (112) and (d) (215) planes of ferrite.18)

a–c; Bagaryatsky OR  (001)C//(112)F, [100]C//[110]F, [010]C//[111]F28)

b–c; Isaichev OR        (101)C//(112)F, [101]C//[110]F, [010]C//[111]F29)

a–d; Pitsch-Petch OR (001)C//(215)F, [100]C2–3°[311]F, [010]C2–3°[131]F30,31)

b–d; Unknown OR      (101)C//(215)F, [10-1]C2–3°[311]F, [010]C2–3°[131]F

Except for the set of b–d, they had been actually identified in the previous researches.

As an example and for reasons mentioned below, we will consider the Pitsch-Petch OR and misfit strains at the interface will be evaluated using two-dimensional large deformation analysis.

To change the iron rectangular lattice of cementite (Fig. 2(a)) into that of ferrite (Fig. 2(d)) under the Pitsch-Petch OR, the lattice transformation matrix T is expressed as the product of a simple tensile and compressive deformation matrix A and a simple shear deformation matrix B, as shown in Fig. 3. When mutually perpendicular x1 and x2 axes are fixed along [010]C and [100]C, respectively together with the x3 axis being normal to the ferrite/cementite interface, T is written as   

T=BA=( 1 tanθ 0 1 ) ( 11 a F / (2 b c ) 0 0 11 a F cosθ / (2 a c ) ) =( 11 a F / (2 b c ) 11 a F sinθ / (2 a c ) 0 11 a F cosθ / (2 a c ) ) =( T 11 T 12 0 T 22 ) . (1)
from the plane correspondence in Fig. 2, where θ=5.2° from Fig. 2(d).
Fig. 3.

Lattice deformation from cementite to ferrite in Pitsch-Petch orientation relationship.

Our next task is to find the principal distortions of the transformation T. Since T is non-symmetric, it does not have mutually perpendicular principal axes of distortion. However, a regular matrix T is known to be decomposed into a rotation (orthogonal) matrix R and a distortion (symmetric) matrix D as32)   

T=RD. (2)
The symmetric distortion D can now be diagonalized to find its principal axes and principal strains. From Eq. (2), we have   
T tr T= (RD) tr (RD)= D tr R tr RD= D tr D (3)
where Ttr is the transposed matrix of T. Equation (3) indicates that the principal strains of D, and, thus, those of T, are obtained as square roots of the diagonal components of the matrix (TtrT)dia, the diagonalized matrix of TtrT. Therefore, we write   
S= T tr T=( T 11 0 T 12 T 22 ) ( T 11 T 12 0 T 22 ) =( T 11 2 T 11 T 12 T 11 T 12 T 12 2 + T 22 2 ) (4)
Then, the rotation of the matrix Sdia with respect to the x3 axes results in   
S dia =( cosα sinα -sinα cosα ) ( T 11 2 T 11 T 12 T 11 T 12 T 12 2 + T 22 2 ) ( cosα -sinα sinα cosα ) =( T 11 2 cos 2 α+2 T 11 T 12 sinαcosα+( T 12 2 + T 22 2 ) sin 2 α ( T 12 2 + T 22 2 - T 11 2 )sinαcosα+ T 11 T 12 ( cos 2 α- sin 2 α) ( T 12 2 + T 22 2 - T 11 2 )sinαcosα+ T 11 T 12 ( cos 2 α- sin 2 α) T 11 2 sin 2 α-2 T 11 T 12 sinαcosα+( T 12 2 + T 22 2 ) cos 2 α ). (5)
where α is the angle of rotation. In order for Sdia to be a diagonal matrix, the following equation must be satisfied.   
( T 12 2 + T 22 2 - T 11 2 )sinαcosα+ T 11 T 12 ( cos 2 α- sin 2 α)=0. (6)
Solving Eq. (6) by substituting T11, T22 and T12 obtained in Eq. (1) with lattice parameters of cementite and ferrite and θ=5.2 degree, α is either 71.26 or −18.74 degrees, both being crystallographically equivalent. Furthermore, substituting these values into (5), the detail Sdia is   
S dia =( 1.1345 0 0 0.8422 ) . (7)
The square root of the diagonal components in (7) indicates principal distortions η1C = 1.06512 and η2C = 0.91769. Here, it should be noted that the lattice deformation explained above is defined by setting cementite as a starting reference state (see again in Fig. 3). This is due to the simplification of deformation matrices, since the iron rectangular lattice in cementite can be described by the orthogonal coordinate system. On the other hand, when ferrite is regarded as a reference phase, the principal distortions η1, η2 from ferrite to cementite is obtained as the inverse of those from cementite to ferrite (η1 = 1/η1C, η2 = 1/η2C). Principal misfit strains at the interface are now obtained as ε11* = η1−1 and ε22* = η2−1.

By considering the similar analysis for other ORs, the principal strains were obtained as listed in Table 1. Here, the parameter M is nearly proportional to the elastic strain energy and is defined as   

M= ( ε 11 * ) 2 + ( ε 22 * ) 2 + 2 3 ε 11 * ε 22 * . (8)
From the energy point of view, ORs with smaller M values are more favorable than those with larger M values.33,34) Then, it appears from Table 1 that the Isaichev OR is most favorable followed in order by the Pitsch-Petch and Bagaryatsky ORs. Another factor often discussed for interface crystallography and energetics is that whether the interface contains the invariant-line direction.33,34,35) It is known that the necessary and sufficient condition for the existence of the invariant line is ε11* < 0 < ε22*.32) As can be seen in Table 1, only the Pitsch-Petch OR satisfies this condition among reported ORs.
Table 1. Principal strains of lattice transformation and M value for respective orientation relationships.
Orientation relationshipCombinationprincipal strainM value
ε11*ε22*
Bagaryatskya–c0.025080.115850.01599
Isaichevb–c0.001510.025080.00066
Pitsch-Petcha–d−0.061140.089680.00813
Unknownb–d−0.151260.081880.02133

As a result, it is concluded that the Pitsch-Petch orientation relationship is the most appropriate one to optimize the plane matching at the interface between ferrite and cementite among already reported crystal orientation relationships. In fact, the Pitsch-Petch OR has been observed most frequently in pearlitic steels.15,18,23,24) In the following session, the principal strains for the Pitsch-Petch OR is used as the misfit strains and the eigenstrains in the lamellar cementite.

3. Unaccommodated Internal Stress and Elastic Strain Energy in Pearlite

3.1. Analysis of Internal Stress by Solving of Periodic Function

Pearlite is regarded as a periodic lamellar structure of period L along the interface normal x3, as shown in Fig. 4(a). When the thickness and the volume fraction of cementite are 2a (L > a) and f, respectively, the distribution of eigenstrains as a function of x3 εij*(x3) illustrated in Fig. 4(b) can be expressed as a step function by the following Fourier series.   

ε ij * ( x 3 ) = ε ij * { f+( 2 π ) n=1 sin( n π f ) cos( nπ L x 3 ) } (9)
Here, εij* are given as follows.   
ε ij * =( ε 11 * 0 0 0 ε 22 * 0 0 0 ε 33 * ) (10)
where ε11* and ε22* are the same misfit strains as mentioned above and given in Table 1. ε33* is unknown at this moment but will soon be found immaterial for the present analysis.
Fig. 4.

Schematic illustration of pearlite with a periodic lamellar structure and distributions of (a) eigenstrains and (b) σ11 stress component.

By adopting the method of finding elastic states for periodically distributed eigenstrains36) together with isotropic elasticity, the stresses in cementite (C) and ferrite (F) caused by the above eigenstrain distribution are obtained as   

σ 11 (C)=-(1-f)( 2μ 1-ν ) ( ε 11 * +ν ε 22 * ) σ 22 (C)=-(1-f)( 2μ 1-ν ) (ν ε 11 * + ε 22 * )in   cementite σ 33 (C)=0 (11)
  
σ 11 (F)=f( 2μ 1-ν ) ( ε 11 * +ν ε 22 * ) σ 22 (F)=f( 2μ 1-ν ) (ν ε 11 * + ε 22 * )in   ferrite σ 33 (F)=0 (12)
where μ is the shear modulus and ν is the Poisson ratio.

3.2. Elastic Strain Energy Caused by Misfit Strain in Pearlite

Now that eigenstrains and internal stresses in the two phases are found, elastic strain energy per unit volume of pearlite E0 can be calculated according to the following equation.37)   

E 0 =- 1 2L 0 L σ ij ε ij * d x 3 (13)
Considering that eigenstrains and internal stresses in cementite are given by Table 1 and Eq. (11), respectively, E0 in pearlite is   
E 0 =- 2a 2L { σ 11 (C) ε 11 * + σ 22 (C) ε 22 * } =f(1-f)( μ 1-ν ) { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * }. (14)
Cementite fraction in pearlite is 0.12 in a conventional eutectoid Fe-0.8mass%C alloy. Therefore, substituting f = 0.12, μ = 80 GPa, ν = 0.3, ε11* = −0.0611 and ε22* = 0.0897 into Eq. (14), E0 is calculated to be 1.04×108 J/m3 (737 J/mol). This value is as high as the free energy change from austenite to pearlite at 873 K (600°C) in this alloy (approximately 1150 J/mol) calculated by a thermodynamics calculation software Thermo-calc (Ver. 3.0 SSOL database). This proves that the ideal coherency strain energy of pearlite is almost comparable to the chemical driving force of pearlitic transformation, even when the most stable Pitsch-Petch OR is kept between ferrite and cementite. In addition to periodic function, another calculation method for internal stress analysis using Eshelby’s inclusion problems37) with Mori-Tanaka method38) will be shown in Appendix.

4. Reduction of Elastic Strain Energy in Pearlite by Introduction of Misfit Dislocations

If elastic strain energy generated via phase transformation is comparable to a chemical driving force, total driving force is so low that the phase transformation may not progress spontaneously. Therefore, coherency strain energy should be reduced through a strain accommodation mechanism in pearlite in order to complete pearlitic transformation. In fact, the group of Shiflet performed microstructural observations of pearlitic steels using a high resolution TEM and reported that a lot of misfit dislocations, direction steps and structural ledges are located on ferrite/cementite interfaces to accommodate lattice misfit between ferrite and cementite. And then, they proposed a model in which these structural accommodations are dynamically introduced at the growing interface of pearlite.15,16,17,18) Therefore, the accommodation mechanism of elastic strain energy, especially the introduction of misfit dislocations, will be discussed in connection with interlamellar spacing of pearlite.

The internal stresses and elastic strain energy of plate-shaped second phase like cementite in pearlite are possible to be zero through a complete accommodation.39) However, since the lattice strain is measurable experimentally, it is now clear that the complete accommodation is not achieved in pearlite. Therefore, in the following section, a partial accommodation mechanism that does not accommodate completely but at least partially accommodates elastic strain energy will be considered. Although the partial accommodation is caused by not only the misfit dislocations but also the atomic diffusion, it could be important to consider the accommodation by misfit dislocations in the context of studies by Shiflet et al.

When the interfaces surrounding inclusions are characterized by semi-coherent interface with misfit dislocations, the existence of misfit dislocations leads to additional eigenstrains Δεij* and internal stresses Δσij in inclusions. According to Eqs. (11), (12), (13), (14) or Eqs. (24), (25), (26), (27), (28), elastic strain energy considering misfit dislocations E 0 mis are described by the following equation.39)   

E 0 mis =f(1-f) μ 1-ν { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * +2 ε 11 * Δ ε 11 * +2 ε 22 * Δ ε 22 * +2ν ε 11 * Δ ε 22 * +2νΔ ε 11 * ε 22 * + (Δ ε 11 * ) 2 + (Δ ε 22 * ) 2 +2νΔ ε 11 * Δ ε 22 * } (15)
Since the misfit dislocations are introduced on interfaces to accommodate misfit strains, edge dislocations are arranged perpendicular to each principal axis. Thus, two kinds of edge dislocation arrays are mutually arranged perpendicularly on the interface in principle. In the case of the Pitsch-Petch OR, because of ε11* < 0 < ε22*, extra half planes for edge dislocations parallel to the x2 direction are in cementite while those parallel to the x1 direction are in ferrite, as schematically illustrated in Fig. 5. In the following section, change in elastic strain energy by the introduction of misfit dislocations is discussed under two different conditions (Fig. 6); one is the condition in which misfit dislocations are arranged with a constant interval (Type 1) regardless of the pearlite growth rate, and the other is that in which the interval of misfit dislocations array changes with the growth rate of pearlitic transformation related with temperature (Type 2).
Fig. 5.

Schematic illustration showing the arrangement of misfit dislocations on ferrite/cementite lamellar interface.

Fig. 6.

Schematic illustration explaining the two difference conditions in the interval of misfit dislocations introduced upon pearlitic transformation.

4.1. Constant Interval of Misfit Dislocations Array; Type 1

Assuming that the interval of misfit dislocations array is constant at d, Δε11* and Δε22* are written as b/d and –b/d with Burgers vector b, respectively. Here, we approximate that b is the same for all the misfit dislocations. Substituting these values into Eq. (15), elastic strain energy per unit volume E1mis is modified as   

E 1 mis =f(1-f) μ 1-ν { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * +2(1-ν) b d ( ε 11 * - ε 22 * )+2(1-ν) ( b d ) 2 }. (16)
  
ε 11 * -b/d    or    ε 22 * b/d
In addition to E1mis, elastic strain energy of misfit dislocations themselves E1dis must be considered. In pearlite with interlamellar spacing L, total area of ferrite/cementite interface per unit volume is counted as 2/L. Taking into account that misfit dislocations are arranged along both directions of x1 and x2 (see Fig. 5), the density of misfit dislocations per unit volume is 4/(L×d). Thus, E1dis is given as   
E 1 dis = μ b 2 Ldπ ln( R r 0 ) . (17)
Here, r0 and R indicate the radius of dislocation core and stress field, respectively. Given that r0 = 5b and R = d/2, the total elastic strain energy per unit volume in Type 1 E1 is described as   
E 1 = E 1 mis + E 1 dis =f(1-f) μ 1-ν { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * +2(1-ν) b d ( ε 11 * - ε 22 * )+2(1-ν) ( b d ) 2 } + μ b 2 Ldπ ln( d 10b ) . (18)
Figure 7 shows the change in E1 as a function of interlamellar spacing, with f = 0.12, μ = 80 GPa, ν = 0.3, ε11* = −0.0611 and ε22* = 0.0897. In addition, b is treated as the Burgers vector of ferrite (0.25 nm), while d is set to be 20 nm by reference to the observation results by Shiflet et al.18) As can be seen, E1 is smaller than E0, which demonstrates that the introduction of misfit dislocations accommodates misfit strains. However, most of the elastic strain energy still remains in pearlite. This agrees well with the results in previous researches23,24,25) and suggests that the misfit dislocation interval of 20 nm is insufficient to effectively reduce the coherency strain energy. On the other hands, it is understood that E1 increases as interlamellar spacing becomes finer due to the increase in E1dis. But, the magnitude value of E1dis is much lower than that of E1mis, thus the dependency of interlamellar spacing on E1 is very small.
Fig. 7.

Relation between elastic strain energy and interlamellar spacing. Elastic strain energies E1 is given by Eq. (18).

4.2. Variable Interval of Misfit Dislocations Array; Type 2

If d is changeable depending on the growth rate of pearlite υ, it is reasonable to consider from the kinetics point of view that d is an increasing function of υ. Here, let us assume for simplicity that d is proportional to υ, as written by Eq. (19).   

d= d min υ min υ (19)
dmin and υmin are the minimum interval of misfit dislocations and the minimum growth rate to fully accommodate misfit strains, respectively. From geometry, dmin can be estimated as either –b/ε11* or b/ε22*. Some researchers including Ridly investigated the relation between υ and L,3,4,8,11) and reported that υL2 is constant (k), which emphasizes that pearlitic transformation is controlled by volume diffusion of carbon in austenite near the growing interface. According to this empirical equation, d is expressed as a function of L, as shown in Eq. (20).   
d= k υ min L 2 d min =| kb υ min L 2 ε * |( ε * = ε 11 *    or    ε 22 * ) (20)
Then, substituting Eq. (20) into (18) with ε22* leads to the total elastic strain energy per unit volume in Type 2 E2.   
E 2 = E 2 mis + E 2 dis = f(1-f) μ 1-ν { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * +2 υ min L 2 k (1-ν)[ ε 11 * ε 22 * - ( ε 22 * ) 2 ]+2(1-ν) ( υ min L 2 ε 22 * k ) 2 } + μb υ min L ε 22 * kπ ln( k 10 υ min L 2 ε 22 * ) (21)
Figure 8 displays the change in E2 as a function of interlamellar spacing using the same numerical values as above. Although k and υmin are varied depending on composition of steels, they could be considered as constant values in a steel. Thus, k and υmin were determined to be 1.0×10−19 m3/s and 50 nm/s, respectively, by adopting the experimental data reported by Bolling & Richmann using a low alloy eutectoid steel.3) E2 is lower than E0 within all region of interlamellar spacing, however, the lowering behavior is completely different from E1 in Fig. 7. It is obvious from Eq. (21) that the first term E2mis depends more heavily on L than the second term E2dis since the former contains higher power exponent of L. This means that the accommodation of misfit strains by misfit dislocations is more effective when the interlamellar spacing becomes coarser for slower pearlite growth rate. In this calculation, E2 significantly decreases when interlamellar spacing is larger than 200 nm. As a result, E2 varies more significantly than E1 depending on interlamellar spacing. This means that Type 2 accommodation is more reasonable to explain the fact that ferrite lattice strain in pearlite clearly decreases with increasing interlamellar spacing.25) E2 is concave down function of L and becomes almost constant when interlamellar spacing is finer than 200 nm. This is because we have assumed for simplicity that the interval of misfit dislocations is simply proportional to the pearlite growth rate, as expressed by Eq. (19). In practice, the relation between them should be more complicated.
Fig. 8.

Relation between elastic strain energy and interlamellar spacing. Elastic strain energies E2 is given by Eq. (21).

5. Conclusions

For theoretical analysis of internal stress and elastic strain energy in pearlite caused by misfit between ferrite and lamellar cementite, micromechanics was applied. The obtained results are as follows.

(1) Two-dimensional large deformation analysis reveals that the Pitsch-Petch orientation relationship is the most appropriate one to optimize the plane matching at the interface between ferrite and cementite among already reported crystal orientation relationships in pearlite.

(2) Micromechanics analysis using a periodic function demonstrates that the misfit strain generates very large elastic strain energy in pearlite even when the Pitsch-Petch orientation relationship is satisfied, which is almost comparable to the chemical driving force for pearlitic transformation. Therefore, the coherency strain energy should be reduced through the strain accommodation mechanism, such as the introduction of misfit dislocations.

(3) When misfit dislocations are dynamically introduced at ferrite/cementite lamellar interface upon pearlitic transformation, the elastic strain energy is surely reduced, although most of that still remains. Especially, given that the interval of misfit dislocations depends on the growth rate of pearlite, the internal stress and the elastic strain energy in pearlite change widely depending on interlamellar spacing.

Appendix

Analysis of internal stress by Eshelby’s ellipsoidal inclusion problems37) with the Mori-Tanaka method.38)

When a penny-shaped ellipsoidal single inclusion Ω perpendicular to the x3 axis with uniform eigenstrains εij* of Eq. (10) is within an isotropic infinite body (Fig. 9(a)), internal stresses in the inclusion σijΩ can be expressed by the following equation, as reported by Eshelby.   

σ ij Ω = C ijkl ( S klmn ε mn * - ε kl * ) (22)
Here, Cijkl and Sklmn are the elastic constants and Eshelby’s tensors, respectively. For a penny-shaped inclusion, Sklmn are given as follows.   
S 1111 = S 2222 =0,       S 3333 =1,       S 1122 = S 2211 =0,       S 1133 = S 2233 =0,       S 3311 = S 3322 = ν (1-ν) ,       S 1212 =0,       S 1313 = S 2323 = 1 2 (23)
(22) through (23) gives   
σ 11 Ω =-( 2μ 1-ν ) ( ε 11 * +ν ε 22 * ), σ 22 Ω =-( 2μ 1-ν ) (ν ε 11 * + ε 22 * ), σ 33 Ω =0. (24)
Here, μ and ν are the shear modulus and Poisson’s ratio, respectively. In this study, we assume for simplicity that both the matrix and inclusion have the same isotropic elastic constants. This assumption may be justified since Young’s modulus of cementite27) is almost the same as that of ferrite (approximately 200 GPa).
Fig. 9.

Schematic illustration of penny-shaped inclusion(s) within an isotropic infinite body.

As the next step, many penny-shaped cementite plate are considered in the ferrite matrix (Fig. 9(b)). When the volume fraction of inclusions is f and average internal stresses of ferrite matrix and cementite are <σij>M and <σij>Ω, respectively, the following equation is realized.   

(1-f)< σ ij > M +f< σ ij > Ω =0 (25)
This is because internal stress averaged over the entire volume is always zero.

Suppose a single cementite plate is newly added within the body. Since there are already many cementite plates, changes in <σij>M and <σij>Ω by this cementite addition must be negligibly small. Therefore, <σij>Ω can be expressed as the sum of <σij>M and the generating internal stress in a newly added cementite plate, as written in Eq. (26).   

< σ ij > Ω =< σ ij > M + σ ij Ω . (26)
From Eqs. (25) and (26), Eq. (27) can be obtained.   
< σ ij > Ω =(1-f) σ ij Ω (27)
By using eigenstrains given by Table 1 and average internal stresses in cementite obtained by Eqs. (25) and (27) with f = 0.12, μ = 80 GPa, ν = 0.3, elastic strain energy in a body with infinite number of cementite plate <E0> can be described as   
< E 0 >=- 1 2 Ω < σ ij > Ω ε ij * dx=- 1 2 Ω (1-f) σ ij Ω ε ij * dx =f(1-f)( μ 1-ν ) { ( ε 11 * ) 2 + ( ε 22 * ) 2 +2ν ε 11 * ε 22 * }. (28)
This is the same as the elastic strain energy given by solving the periodic function (Eq. (14)).

References
 
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