2016 Volume 56 Issue 10 Pages 1866-1873
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.
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.
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) (2 15) 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).
Unit cells of (a) cementite and (b) ferrite.
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//(2 15)F, [100]C2–3°[311]F, [010]C2–3°[131]F30,31)
b–d; Unknown OR (101)C//(2 15)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
(1) |
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)
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
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
(8) |
Orientation relationship | Combination | principal strain | M value | |
---|---|---|---|---|
ε11* | ε22* | |||
Bagaryatsky | a–c | 0.02508 | 0.11585 | 0.01599 |
Isaichev | b–c | 0.00151 | 0.02508 | 0.00066 |
Pitsch-Petch | a–d | −0.06114 | 0.08968 | 0.00813 |
Unknown | b–d | −0.15126 | 0.08188 | 0.02133 |
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.
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.
(9) |
(10) |
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
(11) |
(12) |
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)
(13) |
(14) |
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
(15) |
Schematic illustration showing the arrangement of misfit dislocations on ferrite/cementite lamellar interface.
Schematic illustration explaining the two difference conditions in the interval of misfit dislocations introduced upon pearlitic transformation.
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
(16) |
(17) |
(18) |
Relation between elastic strain energy and interlamellar spacing. Elastic strain energies E1 is given by Eq. (18).
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).
(19) |
(20) |
(21) |
Relation between elastic strain energy and interlamellar spacing. Elastic strain energies E2 is given by Eq. (21).
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.
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.
(22) |
(23) |
(24) |
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.
(25) |
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).
(26) |
(27) |
(28) |