Crystal Plasticity Analysis Considering Dislocations’ Behavior in Ferrite/Cementite Lamellar Structure

Abstract

Mechanical properties of ferrite/cementite lamellar structure model in pearlite steel with the Pitsch-Petch orientation relationship are examined by a strain gradient crystal plasticity analysis. Dislocations’ behavior in the lamellar structure is especially considered in the analysis and characteristic lengths of the structure used in its constitutive equations are approximately determined for each slip system. From the analysis results for different lamellar thicknesses, a scale effect on the yield stress of the pearlite model is smaller than that in case of uniform estimation of the characteristic length by the lamellar thickness because of a reduction of the Orowan stress. The yield stress shows better agreement with an experimental result. On the other hands, a scale effect on the strain hardening rate is larger than that of the uniformly estimation. This is attributed to obvious change of active slip systems in the ferrite layer associated with the thickness. It is found that while slip systems with the largest Schmid factor act in case of large thickness, slip systems which have small critical resolved shear stress act in case of small thickness due to the increase of slip anisotropy. And the change of active slip systems arises large shear incompatibility in each layer and makes the stress fields in the layer non-uniform for the model with Pitsch-Petch relationship.

1. Introduction

A lamellar structure is composed of more than two different kinds of materials and shows better mechanical properties than a prediction from each single material.^1,2,3,4,5) For example, its yield stress becomes larger with the reduction of lamellar thickness. Pearlite steel is one of the typical metallic materials which have a fine lamellar structure in a crystal grain and composed from laminated ferrite and cementite which are piled up alternately within sub-micron intervals.^1,4,5,6) Due to a high strength and reasonable ductility, it is widely used as a structural material such as a cable wire for suspension bridges. However, details of the mechanisms which provide superior mechanical properties to the lamellar structure, include pearlite steel, are still not known.

As factors achieve both strength and ductility in the pearlite steels, strengthening of the ferrite layers^1,4,7,8,9) and a certain amount of stable plastic deformation of brittle cementite layers in the pearlite steel^10,11) can be considered. The ferrite layers are strengthened by the mechanisms of a dislocation accumulation,⁴⁾ an increase of resistance to dislocation emission and slip,⁹⁾ formation of subgrains,^1,7) decomposition of the cementite layers⁸⁾ and others. On the other hand, the stable plastic deformation of cementite layers is realized by two mechanisms of a dislocation supply from the ferrite layers¹²⁾ and a relaxation of plastic strain concentration in the cementite layers.¹³⁾ The relaxetion is occurred by an increase of strain hardening rate of the ferrite layers.⁹⁾ Although the scale effect on the yield stress of the pearlite steel has been attracting much attention, scale effects on the ferrite layer are also important in terms of beneficial interactions between ferrite and cementite layers which make a plastic deformation of the cementite stable.

The strengthening of layers by a formation of subgrains or a decomposition is, so to speak, an accessory mechanism for the lamellar structure because the strengthening changes the crystal structure of layer itself. On the other hand, the strengthening brought by dislocations’ behavior which is influenced by interfaces between layers^3,9) is a general phenomenon for the lamellar structure. For example, the lamellar structure with smaller thickness shows higher yield stress because a resistance of dislocation emission from dislocation sources increases with decrease of the space where the sources exit.¹⁴⁾ Additionally, although emitted dislocations in a normal crystal grain of polycrystal stop their movement after cutting with other dislocations, the emitted dislocations in the lamellar structure are caught by the interfaces between layers before the stop by the cutting.^9,14) The thinner lamellar thickness is, the more dislocation is accumulated. Therefore, the strain hardening rate of the lamellar structure has a scale effect.

Because a movement of dislocations is brought by a slip deformation of crystal, the yield stress and strain hardening rate depend on active slip systems and its configuration. In case of a tensile deformation of single crystal, slip deformations occur on the slip systems which have the highest Schmid factor in all the slip systems because the highest shear stress applies on the systems. However, the slip deformations do not necessarily occur on the systems which have the highest Schmid factor. One of these examples is a hexagonal metal. It is because each slip system of the hexagonal metal greatly differs in the critical resolved shear stress¹⁵⁾ and shows a directivity of the slip deformation.

A shape of each phase in lamellar structure is flatter than that in normal polycrystal and a distance between phases calculated on each slip system is quite different in the lamellar structure while that is almost identical in the polycrystal. Because the distance is closely related to dislocations’ behaviour in the lamellar structure, it is considered that the lamellar structure also shows the directivity of slip deformation. And the directivity should change and stand out with a reduction of the lamellar thickness. As observed in molecular dynamics simulations of the lamellar structure, dislocation moves toward the phase boundary and accumulates around it in case of large thickness while dislocation gradually bows and moves toward lamellar alignment direction with a reduction of lamellar thickness.³⁾ This is also considered as an indication of the change of the directivity of slip deformation.

As one of methods to analyse the mechanical properties with a consideration for the dislocations’ behaviour, there is a crystal plasticity analysis.¹⁶⁾ We have studied the mechanical properties of the pearlite steel by using it.⁹⁾ In the analysis, minimum shear stresses to emit a dislocation loop in the ferrite layer and distances to stop movement of the dislocation for each slip system are uniformly evaluated by the thickness of ferrite layer. However, as described above, these stresses and distances are thought to have different values between each slip system with a reduction of the lamellar thickness. To discuss the mechanical properties of lamellar structure, it is essential to take these differences into the analysis.

In this study, we approximately represent the dislocations’ behavior in the lamellar structure on constitutive equations of the crystal plasticity analysis. Then we evaluate the mechanical properties and stress distributions of a pearlite lamellar model and each layer in it. These are reviewed with paying attention to the activities of the slip systems which is influenced by the lamellar thickness. And the advantages of the ferrite/cementite lamellar structure are discussed from the viewpoint of the interactions between ferrite and cementite layers.

2. Constitutive Equations Considering Dislocations’ Behavior in Lamellar Structure

Analyses are made by a crystal plasticity finite element method¹⁶⁾ for body-centered cubic (abbreviated to bcc, hereafter) crystal because thickness of a bcc ferrite layer is about 3.6–9 times larger than that of a cementite layer in a pearlite steel⁴⁾ and change of mechanical property of the ferrite caused by a scale effect mainly influences the mechanical property of the pearlite.⁹⁾ The number of slip systems is 24 which is composed of {110}<111> and {112}<111>. {123}<111> slip systems do not take into account here because these are considered to come from the superposition of {110}<111> and {112}<111>.¹⁷⁾ For their activation, Schmid’s law is assumed and given as   

$${\theta }^{(n)}=P_{ij}^{(n)}{\sigma }_{ij},$$(1)

  

$${\dot{\theta }}^{(n)}=P_{ij}^{(n)}{\dot{\sigma }}_{ij}.$$(2)

Here θ⁽ⁿ⁾, $P_{ij}^{(n)}$, and σ_ij are the critical resolved shear stress (abbreviated to CRSS, hereafter), the Schmid tensor, and the stress tensor, respectively. Superscripts with parenthesis (e.g. (n) in Eq. (1)) denote the slip system number, hereafter. Quantities with (˙) indicate their increments. The Schmid tensor is defined by the slip plane normal vector ${\nu }_i^{(n)}$ and the slip direction vector $b_i^{(n)}$ as   

$$P_{ij}^{(n)}=\frac{1}{2}\left({\nu }_i^{(n)}{b}_j^{(n)}+{\nu }_j^{(n)}{b}_i^{(n)}\right).$$(3)

That is, the right-hand side in Eq. (1) shows the resolved shear stress (abbreviated to RSS, hereafter). The constitutive equation for the slip deformation based on the infinitesimal deformation theory is given by¹⁸⁾   

$${\dot{\sigma }}_{ij}={\left[S_{ijkl}^e+\sum _{n=1}^{24}\sum _{m=1}^{24}{\left\{h^{(nm)}\right\}}^{-1}{P}_{ij}^{(n)}{P}_{kl}^{(m)}\right]}^{-1}{\dot{\epsilon }}_{kl},$$(4)

where the summation is made over the active slip systems and $S_{ijkl}^e$, ε_kl, and h^(nm) denote the elastic compliance, total strain, and strain hardening parameter, respectively.

In our previous studies,^9,14) the CRSS was evaluated not just for a polycrystal but for a lamellar structure by the extended expression of Bailey-Hirsch type model,¹⁴⁾   

$${\theta }^{(n)}={\theta }_0+\sum _{m=1}^{24}{\mathrm{\Omega }}^{(nm)}a\mu \tilde{b}\sqrt{{\rho }_S^{(m)}}+3\beta \frac{\mu \tilde{b}}{d},$$(5)

where ${\theta }_0^{(n)}$, μ, $\tilde{b}$, ${\rho }_S^{(m)}$, and d denote the lattice friction stress, the elastic shear modulus, the magnitude of the Burgers vector, density of the statistically stored (abbreviated as SS, hereafter) dislocations, and characteristic length of a microstructure. a and β are numerical coefficients of the order of 0.1^20,21) and 1,¹⁴⁾ respectively. Ω^(nm) defines the intensity of the interaction of slip systems n and m.^16,20) First, second, and third term of right side in the equation represent a resistance of obstacles with the exception of dislocation, the Taylor’s strain hardening caused by increase of accumulated dislocations, and a minimum shear stress to emit a dislocation loop (the Orowan stress) in the space with length scale d, respectively. For the case of polycrystal, the uniform evaluation of the CRSS for all slip systems by the characteristic length d is a good approximation because the size of space to restrict the emission of a dislocation loop is almost constant with slip system. However, for the case of a flat crystalline texture like the lamellar structure in pearlite steel, the size of space to restrict the dislocation emission is significantly different according to the slip system as shown in Fig. 1. In the lamellar structure, a dislocation which is emitted from dislocation source is prevented its movement perpendicularly to the direction of lamellar alignment at the interface between layers and then moves parallel to the direction of the interface²²⁾ as shown in Fig. 2. Therefore, in the lamellar structure with lamellar thickness d, we approximately define the characteristic length by minimum distance between layers on each slip system ${\hat{d}}^{(n)}$ which is shown by the red line in the figure as follows,   

$${\hat{d}}^{(n)}=k^{(n)}d,$$(6)

  

$$k^{(n)}=\frac{1}{\sqrt{1-{({\nu }_n^{(n)})}^2}},$$(7)

and evaluate the CRSS by   

$${\theta }^{(n)}={\theta }_0+\sum _{m=1}^{24}{\mathrm{\Omega }}^{(nm)}a\mu \tilde{b}\sqrt{{\rho }_S^{(m)}}+3\beta \frac{\mu \tilde{b}}{{\hat{d}}^{(n)}}.$$(8)

${\nu }_n^{(n)}$ in Eq. (7) is a normal component of the slip plane normal vector on a lamellar interface. For the ferrite layer, we use its lamellar thickness for d in Eq. (6). For the cementite layers, on the other hand, we use d→∞ and do not consider the Orown stress for the cementite. This is because dislocations are supplied from ferrite layers¹²⁾ before dislocation sources inside the cementite layers are activated and a curvature radius of the dislocation transferred from the ferrite layer is assumed to be sufficiently large compared to the lamellar thickness.⁹⁾

Fig. 1.

Conceptual diagram. Largeness which restricts expansion of dislocation arc is different for each slip system in lamellar structure. d<d₁<d₂.

Fig. 2.

Schematic illustration of dislocation behavior in ferrite layer and their characteristic lengths.

The increment of the SS dislocation density is calculated by^16,20)   

$${\dot{\rho }}_S^{(n)}=\frac{c{\dot{\gamma }}^{(n)}}{\tilde{b}{L}^{(n)}}.$$(9)

Here, γ⁽ⁿ⁾ denotes the plastic shear strain. c which is defined by the ratio of moving distance of edge and screw dislocations is 2.¹⁶⁾ L⁽ⁿ⁾ is the mean free path (abbreviated to MFP, hereafter) of moving dislocations and denotes distance until moving dislocation on slip plane is stopped by obstacles. In the pearlite microstructure, accumulated dislocations and lamellar boundaries are considered as these obstacles. Therefore, in the previous study,⁹⁾ we define the MFP by the following model,   

$$L^{(n)}=\text{Min }\left[\frac{c^{*}}{\sqrt{\sum _{m=1}^{24}{\varpi }^{(nm)}\left({\rho }_S^{(m)}+\Vert {\rho }_G^{(m)}\Vert \right)}},d\right],$$(10)

where $\Vert {\rho }_G^{(m)}\Vert$ is the density norm of the geometrically necessary (abbreviated to GN, hereafter) dislocations.^19,23) ω^(nm) and c^* are a weight matrix and material constant, respectively. c^* is a material constant of the order of 10-100^21,24) and the higher c^* means lower strain hardening rate. However, because it can be considered that the extra half planes which move with dislocation are prevented by interfaces between layers as shown in Fig. 2, the MFP must also precisely be defined by using minimum distance between layers which is calculated toward slip direction on each slip system d^*(n) which is shown by blue line in Fig. 2 as follows,   

$$L^{(n)}=\text{Min }\left[\frac{c^{*}}{\sqrt{\sum _{m=1}^{24}{\varpi }^{(nm)}\left({\rho }_S^{(m)}+\Vert {\rho }_G^{(m)}\Vert \right)}},d^{*(n)}\right],$$(11)

  

$$d^{*(n)}=l^{(n)}d,$$(12)

  

$$l^{(n)}=\frac{1}{\left|b_n^{(n)}\right|}.$$(13)

$b_n^{(n)}$ in Eq. (13) is a normal component of the slip direction vector on the lamellar interface.

From above equations, the hardening coefficient is given by²¹⁾   

$$h^{(nm)}=\frac{{\mathrm{\Omega }}^{(nm)}a\mu c}{2L^{(m)}\sqrt{{\rho }_S^{(m)}}}.$$(14)

3. Three-layered Pearlite Model and Its Directivity of Slip Deformation

To compare results with the previous study,⁹⁾ we employ the same pearlite model where a ferrite (denoted as α, hereafter) layer is sandwiched by two cementite (denoted as θ, hereafter) layers as shown in Fig. 3(a). The dimension of the model is 5l×l×5l. Thickness d of the α-layer is 5l/7 and d=50 or 500 nm.^1,4) Displacement and surface traction are continuous at the phase interfaces and we do not introduce any special layers between the α and θ-layers. The entire model (denoted as θ/α/θ, hereafter) is divided into eight-node finite elements and the total number of the elements is 50176. Uniform tensile displacement is given to the lateral surface at x₁=5l, while the displacement of the opposite surface is fixed in the x₁-direction as shown in Fig. 3(a). Other lateral surfaces are free boundary.

Fig. 3.

Schematic illustrations of (a) three-layered pearlite model, its slip system in the ferrite layer and boundary condition and (b, c) crystal orientation relationships for numerical models. Pitsch-Petch relationship of (b) cementite and (c) ferrite layers.⁹⁾

Material constants and numerical coefficients are also same as in the previous study.⁹⁾ We assume that α and θ-layers are elastically isotropic and use E=200 GPa and ν=0.3. The model is assumed to be annealed and its initial dislocation density is 10¹¹ m⁻².²⁵⁾ Because we do not consider the initial GN dislocation density, the initial SS dislocation density for each slip system is ${\rho }_S^{(m)}$=4.17×10⁹ m⁻². The magnitude of the Burgers vectors $\tilde{b}$ for the α and θ-layers are 2.48×10⁻¹⁰ m²⁶⁾ and 5.09×10⁻¹⁰ m,²⁷⁾ respectively. The lattice friction stress for the α and θ-layers are 23 MPa and 1304 MPa, respectively. As numerical coefficients, we employ a=0.1 and β=1. The interaction matrix Ω^(nm) is assumed as Ω^(nm)=1 for the diagonal components and Ω^(nm) $\cong$1 for the off-diagonal components.¹⁶⁾ We suppose that there is no interaction between dislocations on same slip system and use ω^(nm)=0 for the diagonal components and ω^(nm)=1 for the off-diagonal components. A material constant c^* is 20 for the α-layer. On the other hand, we use artificial value c^*=2000 for the θ-layer to make its strain hardening rate exceedingly small. This enables us to express the cementite layer as inherently brittle.

Crystallographic orientations for the α and θ-layers are determined by the Pitsch-Petch relationship^28,29,30) where (-2-15)_α parallels (001)_θ on the lamellar boundary and [010]_θ parallels [131]_α; the minor deviation of 2.6° is neglected. Here, subscripts denotes the layer name. And we assume that a drawing direction coincides with the x₁-axis of loading direction and make [-110]_α coincide with the x₁-axis.^5,31) Figures 3(b) and 3(c) show the crystallographic orientations of the θ and α-layers in the model with this relationship. When each single phase material with this relationship is deformed under uniaxial tensile strain, the cementite shows equivalent double slip by (101)[-111] and (-101)[1-11]. And the ferrite shows single slip by (1-12)[-111] which is expressed by a blue dashed line in Fig. 3(a). Here, in the case of the Bagaryatsky relationship^30,32) which is another well-known crystallographic orientation, the ferrite shows equivalent double slip for the tensile deformation.⁹⁾ The reason of choosing the Pitsch-Petch relationship in this study is that observation of changes of slip systems is simpler for the single slip than the double slip.

Let us discuss the influence of Eqs. (5) and (8) on a scale effect of initial CRSS. In these equations, we replace terms of constants at initial state as followings:   

$$\begin{array}{c}b\text{'}={\theta }_0+\sum _{m=1}^{24}{\mathrm{\Omega }}^{(nm)}a\mu \tilde{b}\sqrt{{\rho }_S^{(m)}},\\ a\text{'}=3\beta \mu \tilde{b}.\end{array}$$(15)

Thus we can write Eq. (5) as   

$${\theta }^{(n)}=b\text{'}+\frac{a\text{'}}{d},$$(5)’

and, using Eq. (6), we can write Eq. (8) as   

$${\theta }^{(n)}=b\text{'}+\frac{a\text{'}}{d^{(n)}}=b\text{'}+\frac{a\text{'}}{k^{(n)}d}.$$(8)’

Both equations are linear functions of 1/d. Figure 4(a) shows the changes of the CRSSs of the α-layer with 1/d. Only four slip systems, (1-12)[-111], (0-11)[-111], (-112)[1-11] and (-101)[1-11] which have a possibility to slip in the α-layer are shown. And these configurations are expressed in Fig. 3(a). The initial CRSSs which are evaluated by Eq. (5) are the same for all the slip systems as shown by dotted line. On the other hand, the initial CRSSs which are evaluated by Eq. (8) show different scale effects as expressed by different colored line in Fig. 4(a). The CRSSs are same only for 1/d=0. These are because Eqs. (5)’ and (8)’ have the same intercept b’. And the inclination of Eq. (5)’ is constant for all slip systems while the inclination of Eq. (8) changes depending on 1/k⁽ⁿ⁾ for each slip system. The higher k⁽ⁿ⁾ means the lower initial CRSS. And the value of Eq. (8)’ is smaller than that of Eq. (5) due to k⁽ⁿ⁾≥1. k⁽ⁿ⁾ for (1-12)[-111], (0-11)[-111], (-112)[1-11], and (-101)[1-11] are about 1.35, 1.58, 1.75, and 2.34, respectively. Therefore, the initial CRSS of (-101)[1-11] shows the highest value while that of (1-12)[-111] shows the lowest value in the four slip systems. The difference between these two values increases with the decrease of lamellar thickness d. In the case of the θ-layer, the differences which are discussed above don’t occur due to the assumption of d→∞.

Fig. 4.

Changes in (a) CRSS and (b) ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ of ferrite layer in the pearlite model. Dotted lines show the results whose characteristic lengths are evaluated by d.

Secondly, we discuss the changes in uniaxial stress ${\sigma }_{11}={\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ to yield the slip system n with 1/d. Influences of other stress components on the yield are ignored here. For the CRSS which is calculated by Eq. (5), σ₁₁ becomes   

$${\sigma }_{11}=\frac{{\theta }^{(n)}}{\left|P_{11}^{(n)}\right|}=\frac{1}{\left|P_{11}^{(n)}\right|}\left(b\text{'}+\frac{a\text{'}}{d}\right),$$(16)

and for the CRSS which is calculated by Eq. (8),   

$${\sigma }_{11}=\frac{{\theta }^{(n)}}{\left|P_{11}^{(n)}\right|}=\frac{b\text{'}}{\left|P_{11}^{(n)}\right|}+\frac{a\text{'}}{\left|P_{11}^{(n)}\right|k^{(n)}d}.$$(17)

Figure 4(b) shows these changes in ${\sigma }_{11}={\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ for the slip systems (1-12)[-111], (0-11)[-111], (-112)[1-11], and (-101)[1-11] with 1/d. Dotted and solid lines denote the results from Eqs. (16) and (17), respectively. The slip systems are distinguished by colors. The lowest value of ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ in all slip systems for arbitrary lamellar thickness correspond to the yield stress in uniaxial stress state for the thickness. When the lamellar thickness is sufficiently large, d→∞, ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ is $b\text{'}/\left|P_{11}^{(n)}\right|$ for both Eqs. (16) and (17). Therefore, the only magnitude of the Schmid factor $\left|P_{11}^{(n)}\right|$ for each slip system influences the result. The less $\left|P_{11}^{(n)}\right|$ is, the more ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ is. $\left|P_{11}^{(n)}\right|$ of (1-12)[-111], (0-11)[-111], (-112)[1-11], and (-101)[1-11] are 0.497, 0.446, 0.416, and 0.342, respectively. Therefore, (1-12)[-111] whose $\left|P_{11}^{(n)}\right|$ is closest to 0.5 shows the least value. ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ of (-101)[1-11] shows the highest value in the four slip systems.

When the CRSS is evaluated by Eq. (5), lines of ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ in Fig. 4(b) do not cross each other. As you can see from Eq. (16), this is because b’+a’/d does not change with slip system and only $\left|P_{11}^{(n)}\right|$ changes. Therefore, the slip system which shows the least ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ is always (1-12)[-111], regardless of the lamellar thickness. And (1-12)[-111] becomes the primary slip system and the yield stress is proportional to 1/d. On the other hand, when the CRSS is evaluated by Eq. (8), lines of ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ cross each other. From Eq. (17), this is because the interception is affected only by $\left|P_{11}^{(n)}\right|$ while the inclination is affected by $\left|P_{11}^{(n)}\right|$ and k⁽ⁿ⁾. Therefore, the slip system which shows the least ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ changes with lamellar thickness. (1-12)[-111] whose $\left|P_{11}^{(n)}\right|$ is closest to 0.5 is the minimum ${\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ until about the lamellar thickness of 730 nm. Then the slip system which shows the minimum changes from (1-12)[-111] to (0-11)[-111] and (-112)[1-11] with decrease of the lamellar thickness. For the lamellar thickness of 515 nm or less, (-101)[1-11] is the minimum. Hence, the relationship between the yield stress and lamellar thickness does not bear a simple proportional with 1/d.

4. Results and Discussions

Figures 5(a)–5(c) show stress-strain responses of θ/α/θ model, θ, and α-layer in the model. We evaluate these responses by averaging stress and strain in entire model or each layer.⁹⁾ Circle and square symbols in the figure show the results whose lamellar thickness is 500 nm and 50 nm, respectively. And filled symbol represents the result⁹⁾ (named Isotropic slip type, here, and abbreviated to Is-type, hereafter) whose CRSS and MFP for each slip system is evaluated by Eqs. (5) and (10) , respectively, in which characteristic lengths for all the slip systems are uniformly represented by the lamellar thickness. Open symbol represents the result (named Anisotropic slip type, here, and abbreviated to As-type, hereafter) whose CRSS and MFP for each slip system is evaluated by Eqs. (8) and (11), respectively, with a consideration of anisotropic dislocations’ behavior in the lamellar structure. Figure 5(a) shows that the yield stress and strain hardening rate of the θ/α/θ model increase with reduction of the lamellar thickness and have a scale effect.

Fig. 5.

Averaged stress-strain curves for (a) θ/α/θ and (b) θ and (c) α in θ/α/θ. Circle and square symbols show the results whose lamellar thickness is 500 nm and 50 nm, respectively. Filled and open symbols represent the results for Is-type⁹⁾ and As-type, respectively.

For the Is-type, the 0.2% offset stresses of the θ/α/θ model with lamellar thickness 500 nm and 50 nm are about 486 MPa and 2522 MPa, respectively. On the other hand, the 0.2% offset stresses of the As-type are about 426 MPa for 500 nm and 1870 MPa for 50 nm. Even with the same lamellar thickness, the yield stress of the As-type is lower than that of the Is-type. Especially, difference of the stresses for 50 nm is significant. Experimental results⁴⁾ are 1286 MPa for 56 nm and 1614 MPa for 45 nm. Therefore, the 0.2% offset stress for 50 nm can be assumed as 1467 MPa and the result of the As-type is closer to the experimental result than that of the Is-type.

Let us examine influences of the stress-strain responses of θ and α-layer on the θ/α/θ model. As shown in Fig. 5(b), the yield stresses of the θ-layer with d=500 and 50 nm for the Is-type are about 2685 MPa and 2667 MPa, respectively. For the As-type, these are about 2696 MPa for 500 nm and 2676 MPa for 50 nm. The strain hardening rates of both types slightly increase with the reduction of the lamellar thickness. Therefore, it can be said that differences of the lamellar thickness and evaluation of characteristic length do not influence on stress-strain response of the θ-layer although there is a little difference caused by a gap of plastic deformation between the θ and α-layer.⁹⁾

In the α-layer, on the other hand, the yield stress and strain hardening rate are significantly increased with the reduction of lamellar thickness as shown in Fig. 5(c). That is, the differences of stress-strain responses of the θ/α/θ model are due to the change of the mechanical response in the α-layer. For the Is-type, the yield stresses are about 283 MPa for 500 nm and 2386 MPa for 50 nm. While the yield stresses are about 219 MPa for 500 nm and 1512 MPa for 50 nm for the As-type. As explained in Fig. 4, this is because the CRSS of the As-type is always smaller than that of the Is-type. Additionally active slip systems of the As-type changes with the lamellar thickness.

When the lamellar thickness is 500 nm, the strain hardening rate of the As-type and Is-type are much the same. On the other hand, the strain hardening rate of the As-type is much larger than that of the Is-type for the lamellar thickness 50 nm. If same active slip systems in the layer are deformed by same amount, the strain hardening rate of the As-type is smaller than that of the Is-type because the MFP of the As-type is larger than that of the Is-type. In spite of it, the As-type shows high strain hardening rate. This is because the number of active slip systems is different.

To examine activities of slip systems under multiaxial stress, changes of the SS dislocation densities in the α-layer with tensile deformation are shown in Fig. 6. This is because the accumulation of SS dislocation for each slip system means that the slip deformation of the system occurs. Figures 6(a) and 6(b) are the results for the lamellar thickness 500 nm and 50 nm, respectively. Dotted and solid lines in the figures show the results for the Is-type and the As-type, respectively. Shapes of symbols denote differences of the active slip systems. By comparing Figs. 6(a) and 6(b), it is found that the α-layer of the Is-type always shows single slip by (1-12)[-111] regardless of the lamellar thickness. On the other hand, types of active slip systems and quantity of the activity in the α-layer of the As-type depend on the lamellar thickness.

Fig. 6.

Averaged SS dislocation density-strain curves for α in θ/α/θ when lamella thicknesses are (a) 500 nm and (b) 50 nm. Dotted and solid lines show the results for Is-type⁹⁾ and As-type, respectively.

When the lamellar thickness is 500 nm, a primary slip of the As-type is (1-12)[-111]. Along with that, (-112)[1-11] and (0-11)[-111] slightly act. Although accumulated dislocation density of (1-12)[-111] of the As-type is slightly lower than that of the Is-type, the strain hardening rate of the As-type is equivalent to that of the Is-type as shown in Fig. 5(c) because the As-type shows a multiple slip. When the lamellar thickness is 50 nm, the primary slip of the As-type is (-101)[1-11]. The slip system becomes easier to slip than (1-12)[-111] with the reduction of lamellar thickness as discussed in Fig. 4. Other slip systems, (-112)[1-11] and (0-11)[-111] also slip. Especially, the accumulation of (-112)[1-11] is remarkable. Because the effect of the multiple slip is strong, the strain hardening rate of the As-type is larger than that of the Is-type.

Let us examine an influence of the difference of active slip systems on a stress distribution. Figure 7 shows stress distributions of σ₁₁, σ₃₃ and σ₃₁ on x₁x₂ cross section at the center of the model when tensile strain ε₁₁ is 1.4%. We show the results only for the lamellar thickness 50 nm which brings a noticeable difference of the active slip systems between the As-type and Is-type. Figures 7(a)–7(c) are the results of the Is-type and Figs. 7(d)–7(f) are the results of the As-type. Because other stress components σ₂₂, σ₁₂ and σ₂₃ are exceedingly small, these are not shown here. The distributions of σ₁₁, σ₃₃ and σ₃₁ are nearly homogeneous in the layers except for neighborhood of the model surface.

Fig. 7.

Stress distributions of (a–c) Is-type⁹⁾ and (d–f) As-type of θ/α/θ with lamella thickness of ferrite layer is 50 nm when ε₁₁= 1.4%. Each stress components are (a, d) σ₁₁, (b, e) σ₃₃ and (c, f) σ₃₁. The distributions are depicted on x₁x₂ cross section at the center of models: x₃=5l/2.

As shown in Figs. 7(a) and 7(d), normal stress σ₁₁ in the α-layer is about 2450 MPa (Is-type) or 1790 MPa (As-type) while σ₁₁ in the θ-layers is about 2660 MPa (Is-type) or 2690 MPa (As-type). These large differences of σ₁₁ in the α and θ-layers are due to the differences of the yield stress.⁹⁾ And the yield stress of the α-layer for the Is-type is considerably larger than that for the As-type as discussed above.

From Fig. 6(b), the α-layer for the Is-type shows a single slip by (1-12)[-111] and that for the As-type approximately shows a single slip by (-101)[1-11] when the tensile strain ε₁₁ is 1.4%. A contraction of the α single material along x₃ direction by slip deformation of (1-12)[-111] or (-101)[1-11] is much smaller than that of θ by the Poisson shrinkage. By constrained these difference of deformation at the interface, σ₃₃ shows a compressive stress for the α-layer and a tensile stress for the θ-layer.⁹⁾ The main reason for higher values of the As-type is that the strain incompatibility due to the plastic strain in the α-layer is larger for the As-type.

As shown in Figs. 7(c) and 7(f), σ₃₁ of the As-type also shows a higher value than that of the Is-type due to the large strain incompatibilities between the layers. The slip deformation of (-101)[1-11] of the α accompanies large plastic shear strain ${\epsilon }_{31}^p$ while the plastic shear strain ${\epsilon }_{31}^p$ by the slip deformation of (1-12)[-111] of the α is quite small. From Figs. 7(d)–7(f), these strain incompatibilities also affect all other stress components and disturb the stress fields.

Let us discuss influences of the difference of active slip systems on a ductility of the pearlite steel. Heavy line in Fig. 5(c) shows the stress-strain response of ferrite (named as ferrite5 in [13]) which can stabilize plastic deformation of the θ-layers in the pearlite lamellar structure. The larger the strain hardening rate is, the more stabilized the plastic deformations of θ-layers are.¹³⁾ When the lamellar thickness is 50 nm, the strain hardening rate of the α-layer for the Is-type is the smaller than that of ferrite5. On the other hand, the strain hardening rate of the α-layer for the As-type is more than that of ferrite5. From the view point of this high strain hardening property, the As-type is effective to increase the ductility of the pearlite steel. On the other hand, the non-uniform stress fields of the As-type due to the incompatibilities of the shear strains are adverse for the ductility. For the Pitsch-Petch orientation relationship, therefore, an advantage is coexistent with a demerit. However, the most important point from above results is that there is a possibility to change active slip systems depending on the lamellar thickness for the case of a flat crystalline texture like the pearlite lamellar structure. Therefore, if we find conditions to activate more slip systems in the α-layer, the strain hardening rate will significantly increase, as Eq. (14) suggests. And ductility of pearlite will increase furthermore. As an example to find these conditions, we use Fig. 4(b) which shows the changes in uniaxial stress ${\sigma }_{11}={\theta }^{(n)}/\left|P_{11}^{(n)}\right|$ to yield the slip system n with 1/d. In the section enclosed by the pink dotted line, four lines intersect each other and there is a high possibility of multiple slip by these four slip systems. Equation (17) shows that the intersection of lines changes depending on $\left|P_{11}^{(n)}\right|$, k⁽ⁿ⁾, a’, and b’. That is, it is possible to produce the multiple slip for arbitrary lamellar thickness by changing a crystal orientation or dislocation density.

5. Conclusions

Mechanical properties of ferrite/cementite lamellar structure model in pearlite steel with the Pitsch-Petch orientation relationship were examined by a strain gradient crystal plasticity analysis. In particular, we approximately represented dislocations’ behavior in the lamellar structure on the constitutive equations. Obtained results are as follows.

(1) The scale effect on yield stress of the pearlite model was smaller than that in the case of uniform estimation of the characteristic length of the lamellar thickness because of reduction of the Orowan stress. And the yield stress showed better agreement with experimental results. On the other hand, the scale effect on strain hardening rate was larger than that of the uniform estimation. These mechanical differences of the pearlite were attributed to the ferrite layer.

(2) In the ferrite layer, the slip systems with the largest Schmid factor acted in the case of large thickness. However, the slip systems which have small critical resolved shear stress acted in the case of small thickness because the systems showed strong slip anisotropy with the reduction of lamellar thickness. Additionally, the number of active slip systems increased with the reduction of thickness and the strain hardening rate showed higher than the case of single slip.

(3) Changes of active slip systems with decrease of the lamellar thickness arose large shear incompatibility in each layer and made the stress fields in the layer non-uniform.

Acknowledgement

This research was supported by Japan Science and Technology Agency (JST) under Collaborative Research Based on Industrial Demand “Heterogeneous Structure Control: Towards Innovative Development of Metallic Structural Materials”.

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