Crystal Plasticity Simulation of the Effect of γ Lamellae on the Plastic Behavior of the Core–Shell-like Structured TiAl Alloy

Abstract

The preparation of the core–shell-like structured before hot working can significantly enhance the hot workability of the alloy. In order to research the properties of the alloy, the finite element method combined with the crystal plasticity constitutive theory was used to establish the finite element model of the core–shell-like structured TiAl alloy with (α₂ + γ) lamellae colonies as the core and α₂ matrix as the shell. The research focuses on the influence of the length and number of γ lamellae on the stress–strain distribution and the contribution of slip systems in each phase to the plasticity of the alloy. The results show that when the γ lamella length increases from 12 μm to 16 μm, the overall stress decreases by 12.0%; when the number increases from 6 to 10, the stress decreases by 7.7%. The stress reduction is primarily influenced by the α₂ phase. Increasing the volume fraction of γ lamellae facilitates stress distribution within the α₂ phase and enhances the plasticity of the material. In the γ phase O4, S1 and S7 slip systems contribute the most to the plastic deformation of the γ phase. In the α₂ phase, the B1 slip system is the main contributor to the plasticity of the α₂ phase. And the B1 slip system contributes more significantly to the plastic deformation of the entire model.

1. Introduction

As a new type of high-temperature lightweight structural material, TiAl-based alloy has broad application prospects in aerospace, marine development, automobile transportation and other fields [1,2,3]. Due to its low density, high melting point, excellent high temperature strength and high modulus, outstanding oxidation resistance and good creep properties [4], it has become a candidate material for replacing nickel-based superalloys in the 650–900 °C temperature range [5]. However, the poor room-temperature ductility and limited machinability of TiAl alloys limit their wide application.

At present, ingot metallurgy is the main method for preparing TiAl alloy [6]. However, the castings produced by this process generally have problems such as uneven microstructure and coarse as-cast grains, which affect the mechanical properties of materials [7,8,9]. Yu et al. [10] obtained these (α₂ + γ) lamellae colonies as the core, α₂ matrix as the shell of the core–shell-like structure by controlling the solid-state phase transformation of TiAl alloys, and the γ phase could not disappear completely within the lamellae colonies during the formation process. And through the study, it is confirmed that by obtaining the core–shell-like structure before hot working, the residual lamellae colonies no longer exist in the deformation organization, and the core–shell-like structure can significantly improve the hot working performance of the material.

The crystal plasticity finite element method (CPFEM) is an effective and versatile tool to describe the deformation mechanism and mechanical response of crystal materials [11,12,13,14]. TiAl alloy has been widely studied in the numerical simulation of crystal plasticity. Based on the crystal plasticity theory, Kad [15], Schlogl [16], Marketz [17], and Zambaldi [18] et al. mainly proposed the classical numerical model of two-phase lamellae TiAl alloys, which was used to study the orientation relationship, creep behavior and internal stress change in the two-phase microstructure of lamellae TiAl alloys. Based on CPFEM, Chen et al. [19] systematically explored the microscopic mechanism of plastic anisotropy of lamellae TiAl alloys. Ilyas et al. [20] revealed the intrinsic mechanism of temperature-enhanced strain rate sensitivity in the local deformation behavior of TiAl alloys through numerical analysis. Researchers such as Zhao [21], Selvarajou [22] and Wang [23] mainly explored the complex interaction mechanism between lamellae orientation, structural feature size, temperature and stress state of lamellae TiAl alloy under compression, tension and bending conditions. Wu et al. [24] analyzed the brittle–ductile transition behavior of γ-TiAl alloy at different temperatures by using the crystal plasticity finite element method. Kabir et al. [25] used the crystal plasticity finite element method to analyze the effects of layered orientation and applied load angle on plastic deformation and yield surface evolution in single-phase and two-phase composite systems. Peng et al. [26] mainly studied the plastic anisotropy of TiAl alloys under various stress states, and developed a prediction model that can accurately characterize the anisotropic yield behavior of TiAl alloys under various stress states. Crystal plasticity has been able to provide a micromechanical model for slip-dominated plastic deformation and is used as a constitutive theory.

Previous studies on core–shell-like structured TiAl alloys have primarily focused on the formation mechanism of the core–shell-like structure and its hot workability, while research on the mechanical behavior of such alloys at room temperature remains insufficient. Under room-temperature conditions, factors such as lamellar orientation and interface characteristics significantly influence the performance of core–shell-like structured TiAl alloys. This paper focuses on the effects of lamellae length and quantity on the mechanical response of core–shell-like structured TiAl alloys under uniaxial compressive loading. In this study, the finite element method combined with the crystal plasticity constitutive theory was used to establish the finite element model of the core–shell-like structured TiAl alloy with (α₂ + γ) lamellae colonies as the core and α₂ matrix as the shell. The model can effectively describe the mechanical behavior of the core–shell-like structured TiAl alloy. The research focuses on the influence of the length and number of γ lamellae on the activation of slip systems and stress–strain distribution characteristics of the core–shell-like structured TiAl alloy.

2. Crystal Plastic Constitutive Model

2.1. Deformation Mechanism and Morphological Classification

Figure 1 shows the cell structures of the γ-TiAl(a) and α₂-Ti₃Al(b) phases. Figure 1a is the cell structure of the γ-TiAl phase, which is the L1₀-type face-centered tetragonal structure [27]. Slip occurs on the {111} crystal plane, and the Burgers vector of the four slip systems is b = 1/2⟨110], called ordinary slip. Other slip systems are super slip, with Burgers vector b = ⟨101]. Figure 1b is the cell structure of the α₂-Ti₃Al phase which is the α₂-Ti₃Al phase has a D0₁₉-type hexagonal close-packed structure. The atoms are densely packed on (0001) and the Ti and Al atoms alternate along the [11$\stackrel{\mathrm{-}}{2}$0] direction. Three families of slip systems exist. The basal and prismatic slip systems have slip surfaces (0001) and {10$\stackrel{\mathrm{-}}{1}$0}, respectively, and the Burgers vectors for both slip systems are b = 1/3 ⟨11$\stackrel{\mathrm{-}}{2}$0⟩. The pyramidal slip has slip surfaces {$11\stackrel{\mathrm{-}}{2}1$}.

Table 1. Slip systems in the γ-TiAl and the α₂-Ti3Al phases.

Phase	Slip Systems	Type of Slip	Detail
γ-TiAl	$\left\{111\right\}<110>$	Ordinary slip	O1$\left(111\right)\left[1\stackrel{\mathrm{-}}{1}0\right]$ O2$\left(\stackrel{\mathrm{-}}{1}11\right)\left[\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}0\right]$ O3$\left(1\stackrel{\mathrm{-}}{1}1\right)\left[110\right]$ O4$\left(\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}1\right)\left[1\stackrel{\mathrm{-}}{1}0\right]$
	$\left\{111\right\}<101>$ $\left\{111\right\}<011>$	Super slip	S1$\left(111\right)\left[01\stackrel{\mathrm{-}}{1}\right]$ S2$\left(111\right)\left[10\stackrel{\mathrm{-}}{1}\right]$ S3$\left(\stackrel{\mathrm{-}}{1}11\right)\left[01\stackrel{\mathrm{-}}{1}\right]$ S4$\left(\stackrel{\mathrm{-}}{1}11\right)\left[\stackrel{\mathrm{-}}{1}0\stackrel{\mathrm{-}}{1}\right]$ S5$\left(1\overline{1}1\right)\left[10\overline{1}\right]$ S6$\left(1\stackrel{\mathrm{-}}{1}1\right)\left[0\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}\right]$ S7$\left(\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}1\right)\left[\stackrel{\mathrm{-}}{1}0\stackrel{\mathrm{-}}{1}\right]$ S8$\left(\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}1\right)\left[0\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}\right]$
α2-Ti3Al	$\left(0001\right)\langle 11\stackrel{\mathrm{-}}{2}0\rangle$	Basal slip	B1$\left(0001\right)\left[11\overline{2}0\right]$ B2$\left(0001\right)\left[1\overline{2}10\right]$ B3$\left(0001\right)\left[\overline{2}110\right]$
	$\left\{10\stackrel{\mathrm{-}}{1}0\right\}\langle 11\stackrel{\mathrm{-}}{2}0\rangle$	Prismatic slip	Pr1$\left(01\overline{1}0\right)\left[\overline{2}110\right]$ Pr2$\left(\overline{1}100\right)\left[11\overline{2}0\right]$ Pr3$\left(\overline{1}010\right)\left[1\overline{2}10\right]$
	$\left\{11\overline{2}1\right\}\langle \overline{1}\overline{1}\overline{2}6\rangle$	Pyramidal slip	Py1$\left(11\overline{2}1\right)\left[\overline{1}\overline{1}26\right]$ Py2$\left(1\overline{2}11\right)\left[\overline{1}2\overline{1}6\right]$ Py3$\left(\overline{2}111\right)\left[2\overline{1}\overline{1}6\right]$ Py4$\left(\overline{1}\overline{1}21\right)\left[11\overline{2}6\right]$ Py5$\left(\overline{1}2\overline{1}1\right)\left[1\overline{2}16\right]$ Py6$\left(2\overline{1}\overline{1}1\right)\left[\overline{2}116\right]$

shows the slip systems in the γ-TiAl and α₂-Ti₃Al phases. According to the deformation mechanism of the phase (such as ordinary slip and super slip in the γ-TiAl phase) and crystallographic equivalent slip surface (such as basal, prismatic, and pyramidal slip in the α₂-Ti₃Al phase), the deformation system of the phase is classified. In crystal plasticity models, the {111}<110> slip system is often used to equivalently describe the ordinary slip activity in γ-TiAl, because the combined motion effects of 1/2<110> partial dislocations are equivalent to those of <110> perfect dislocations.

2.2. Crystal Plastic Theory

Based on Huang‘s research results [28], we established a crystal plasticity framework to study the mechanical behavior of biphasic polycrystalline materials under uniaxial compression. The total deformation gradient tensor F of the crystal is divided into two parts according to the tensor multiplication law:

$$F=F^e{F}^p$$

(1)

where $F^p$ is the deformation gradient corresponding to plastic deformation, and $F^e$ is the deformation gradient caused by lattice distortion and rigid rotation. The velocity gradient tensor L can be derived from the time derivative of Equation (1):

$$L=\dot{F}{F}^{-1}=F^e{F}^{e-1}+F^e{F}^p{F}^{p-1}{F}^{e-1}=L^e{L}^p$$

(2)

where $L^e$ and $L^p$ are the lattice distortion and rigid rotation part and plastic slip part of L, respectively. The plastic gradient $L^p$ is expressed as:

$$L^p={\displaystyle \sum _{\alpha =1}^n{\dot{\gamma }}^{\alpha }{s}^{\alpha }\otimes n^{\alpha }}$$

(3)

where ${\dot{\gamma }}^{\alpha }$ is the shear strain rate of the α slip system, $s^{\alpha }$ is the slip direction of the α slip system, and $n^{\alpha }$ is the direction normal to the slip surface of the α slip system.

A viscoplastic power law is used to describe plastic slip [29,30,31], so that the plastic shear strain rate ${\dot{\gamma }}^{\alpha }$ of the α slip system is described by the following equation:

$${\dot{\gamma }}^{\alpha }={\dot{\gamma }}_0^{\alpha }{\left|{\displaystyle \frac{{\tau }^{\alpha }}{g^{\alpha }}}\right|}^{{\displaystyle \frac{1}{m}}}sign\left({\tau }^{\alpha }\right)$$

(4)

where ${\dot{\mathsf{\gamma }}}_0^{\mathsf{\alpha }}$ is the initial plastic shear strain rate, ${\tau }^{\alpha }$ is the decomposed shear stress, $g^{\alpha }$ is the critical shear stress of the α slip system, and m is the strain rate sensitivity factor. When m tends to 0, the model is rate independent. During plastic deformation, the change of $g^{\alpha }$ varies with the change in shear rate ${\dot{\gamma }}^{\beta }$ of all slip systems β and satisfies the following relationship:

$$g^{\alpha }={\displaystyle \sum _{\alpha }{h}_{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|}$$

(5)

where $h_{\alpha \beta }$ is the hardening matrix, calculated by the equation $h_{\alpha \beta }=q_{\alpha \beta }{h}_{\beta }$. Here $q_{\alpha \beta }$ is the matrix form of the potential hardening modulus. $h_{\beta }$ is the self-hardening modulus. The hardening model is as follows:

$$h_{\beta }=h_0{\mathrm{sech}}^2\left|{\displaystyle \frac{h_0\gamma }{{\tau }_s^{\alpha }-{\tau }_0^{\alpha }}}\right|$$

(6)

where $h_0$ is the initial hardening modulus, ${\tau }_0^{\alpha }$ is the initial shear stress in the α slip system, ${\tau }_s^{\alpha }$ is the shear stress in the first stage due to plastic deformation, and γ is the cumulative shear strain in all slip systems. The accumulated shear strain γ for all slip systems is defined as follows:

$$\gamma ={\displaystyle \sum _{\alpha }{\displaystyle \underset{0}{\overset{t}{\int }}\left|{\dot{\gamma }}^{\alpha }\right|}dt}$$

(7)

Based on the above crystal plasticity theory, the crystal plasticity constitutive subroutine of core–shell-like structured TiAl alloy was written in Fortran language for simulation analysis.

3. Finite Element Modeling

3.1. Establishment of Finite Element Model

Figure 2 illustrates the evolution from the lamellar structure to the core–shell-like structure. The core–shell-like structured TiAl alloy shown in the figure contains defects such as lamellar branching. In this study, these features are simplified, and an idealized periodic model is adopted. The model simplifies the lamellar structure and indeed neglects defects commonly present in real microstructures—such as lamellar bending, branching, and termination points—which may lead to overly optimistic predictions of failure behavior and phase transformation kinetics. However, the advantage of an idealized periodic model lies in its ability to isolate the influence of complex microstructural features, thereby focusing on the physical essence of deformation mechanisms and providing a clear and reliable basis for mechanistic interpretation and theoretical prediction.

Based on the typical characteristics of the core–shell-like structured TiAl alloy in the above research [9,32], a finite element model of the core–shell-like structured TiAl alloy with (α₂ + γ) lamellae colonies as the core and α₂ matrix as the shell was established. The length and width of the model are 80 μm and 20 μm. Figure 3 shows the finite element model of the core–shell structure TiAl alloy used in the simulation, and also presents the distribution of each phase and the key microstructure length scale. The yellow part is γ-TiAl and the blue part is α₂-Ti₃Al. After lamellae decomposition, the γ lamellae present a step-like structure, where L₁ is the length of the γ lamellae, λ_γ1 and λ_γ2 are the thicknesses of the narrow and wide ends of the γ lamellae step-like structure.

Table 2. Lamellae length (L1), width (λγ1, λγ2) of γ and the number of γ lamellae used in the simulation.

γ lamellae length, L1 (μm)	12	14	16
γ lamellae width, λγ1 (μm)	0.3	0.35	0.4
γ lamellae width, λγ2 (μm)	0.6	0.7	0.8
The number of γ lamellae, n	6	8	10

shows the parameter ranges of L₁, λ_γ1, λ_γ2 and the number of γ lamellae.

3.2. Simulation Parameter Setting

The model adopts the C3D8 solid element, and the mesh is refined to 0.1 in the key area. In order to simplify the simulation and reduce the influence of other non-important factors, the fixed boundary conditions are set on the corresponding surface of the model loading surface, and then the reverse applied compression displacement −0.8 μm is set on the loading surface along the Z axis of the coordinate axis, and the displacement and its steering along the X and Z axes are set to 0.

The elastic constants of the γ and α₂ phases at room temperature were obtained from first-principles calculations by Yoo et al. [33] and Fu et al. [34], respectively, as shown in

Table 3. Elastic constant of the γ-TiAl and the α₂-Ti₃Al phases (MPa), adapt from Refs. [33,34].

Phase	C1111	C1122	C2222	C1133	C2233	C3333	C1212	C1313	C2323
γ-TiAl	190,000	105,000	190,000	90,000	90,000	185,000	50,000	120,000	120,000
α2-Ti3Al	221,000	71,000	221,000	85,000	85,000	238,000	75,000	69,000	69,000

. The constitutive model parameters of TiAl alloy in different slip systems are shown in

Table 4. Constitutive model parameters of the TiAl alloy in different slip systems [17,34,35,36].

Phase	Slip Systems	τ0/MPa	τs/MPa	$\dot{\mathsf{\gamma }}$	m	h0/MPa	q0	q1
γ-TiAl	Ordinary slip	46	96	0.001	0.1	500	1	1
	Super slip	110	200		0.2	500
α2-Ti3Al	Basal slip	600	800	0.001	0.1	1000	1	1
	Prismatic slip	250	350			800
	Pyramidal slip	1600	2000			1000

. These parameters are primarily based on the research findings of Marketz [17], Yamaguchi [35], Inui [36], and Fu et al. [34]. The meaning of each parameter symbol in Table 4 is as follows: τ₀ is the initial analytical shear stress; τ_s is the first-stage shear stress caused by large plastic flow; $\dot{\mathsf{\gamma }}$ is the plastic shear strain rate; m is the strain rate sensitivity coefficient; h₀ is the initial hardening modulus; q₀ is the initial ratio of potential hardening coefficient to self-hardening coefficient; q₁ is the ratio of potential hardening coefficient to self-hardening coefficient during plastic deformation.

4. Results and Discussion

4.1. The Effect of the γ Lamellae Length on the Core–Shell-like Structured TiAl Alloy

4.1.1. Stress–Strain Analysis

During the formation of the core–shell-like structure, the length and thickness of γ lamellae decrease simultaneously [10], so the length and thickness of the γ lamellae are the same variable in this study. In this paper, the dimensions of γ lamellae are shown in Table 2, and the number of γ lamellae is 8.

Figure 4 shows the von Mises stress–strain curves of the whole model and its γ and α₂ phases with different γ lamellae lengths under uniaxial compression. From Figure 4, at a strain of 0.035 with γ lamellae lengths of 12 μm, 14 μm, and 16 μm, the von Mises stress of the whole model is 2524 MPa, 2341 MPa and 2221 MPa respectively. The von Mises stress of the α₂ phase is 2859 MPa, 2790 MPa and 2730 MPa respectively. And the von Mises stress of the γ phase is 804 MPa, 810 MPa and 818 MPa respectively. The von Mises stress in the γ phase exhibits minimal variation with increasing γ lamellar length. In contrast, the von Mises stress in the α₂ phase decreases as the γ lamellar length increases. Under the condition that the volume of the whole model remains unchanged, the volume fraction of the γ phase increases with the increase in the γ lamellar length, while the volume fraction of the α₂ phase decreases accordingly. This indicates that in the core–shell-like structured TiAl alloy, with the increase in γ lamellar length, the decrease in von Mises stress is mainly affected by the change in the volume fraction of the α₂ phase, but is hardly affected by the change in the volume fraction of the γ phase. This finding is consistent with the observations of Yu et al. [10].

When the strain is below 0.005, the stresses in both phases increase linearly, with the von Mises stress of the α₂ phase consistently higher than that of the γ phase. It indicates that from the early stage of deformation, due to the higher elastic modulus of the α₂ phase, the load distribution is uneven, with the α₂ phase bearing the main load. When the strain exceeds 0.005, the γ phase undergoes plastic yielding, and its von Mises stress remains largely stable, while the von Mises stress of the α₂ phase continues to rise and remains significantly higher than that of the γ phase. These results directly demonstrate that the increase in work hardening rate observed in the model’s stress–strain curve is primarily attributable to the additional load transferred from the yielding γ phase to the α₂ phase. Consequently, the von Mises stress of the model is predominantly governed by α₂.

In Figure 4, the maximum von Mises stress of the γ phase is 800 MPa, which is consistent with the trend of the stress–strain curve and the research of Selvarajou et al. [22] The von Mises stress of the core–shell-like structured TiAl alloy is greater than that of the lamellae TiAl alloy. The critical shear stress of α₂-Ti₃Al is 3–5 times that of γ-TiAl. At the initial deformation, the γ lamellae begin to slip but are quickly hindered by the α₂ phase. In order to make the deformation continue, the dislocation motion in the α₂ phase must be activated, so the stress quickly approaches the high critical shear stress of the α₂ phase.

Figure 5 presents the von Mises stress nephograms of core–shell-like structured TiAl alloys under uniaxial compression, corresponding to different γ lamellae lengths. As shown in Figure 5, the von Mises stress distribution in the central region of the model remains relatively uniform. Compared with the obvious stress concentration in the lamellae TiAl [21], the stress distribution in the core–shell-like structured TiAl alloy is more uniform without obvious stress concentration. It is noted that as the length of the γ lamellae increases, the low-stress region (red dashed area) in the α₂ phase expands, further indicating that the stress in the core–shell structured TiAl alloy is mainly influenced by the α₂ phase. In addition, stress magnitudes at the model edges increase significantly. This observation is mainly attributed to geometric and load discontinuities and mesh discretization.

4.1.2. Analysis of γ Phase and α₂ Phase Slip System

The relative activity is calculated to quantify the contribution of different slip systems to plasticity in the core–shell-like structured TiAl alloys. The formula is as follows [22]:

$${\mathrm{Relative} \mathrm{activity}}^{\kappa }={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{\gamma }_i^{\kappa }{V}_i}}{{\displaystyle \sum _{i=1}^N{\Gamma }_i{V}_i}}}$$

(8)

Here, κ denotes the type of slip system, γ_i^κ denotes the net shear strain of the i-th unit body in the current slip system model, Г_i denotes the total cumulative shear strain of the i-th unit body in the current slip system, V_i is set to the volume of the i-th unit body, and N denotes the total number of units in the current slip system.

Figure 6 is the calculated relative activity of slip systems in the core–shell-like structured TiAl alloy. From Figure 6a, the three slip systems with the highest relative activity are the O4, S1, and S7 slip systems. The relative activity of these three slip systems is between 14% and 15%. The relative activity in the O3, S4, and S8 slip systems is the lowest, ranging between 3% and 5%. The relative activity in the other slip system is between 6% and 11%. The above results show that the slip systems of O4, S1 and S7 exhibit higher relative activity under different γ lamellae length conditions, and the contribution to the plastic deformation of γ phase is the most significant. Based on Figure 6b, in the α₂ phase, only the basal slip system exhibits a non-zero relative activity, while the relative activity of both the prismatic and pyramidal slip systems is zero, indicating that only the basal slip system is activated in this phase. Among these, the B1 slip system shows the highest relative activity, whereas the B2 and B3 slip systems have very low relative activity. Therefore, the B1 slip system is the primary contributor to the plasticity of the α₂ phase.

Figure 7 presents the cumulative shear strain curves of the slip system in the γ phase with different γ lamellae lengths (a, b, c), and (d) is the cumulative shear strain of the slip system in the γ phase with different γ lamellae lengths at a strain of 0.035. As shown in Figure 7, the three slip systems with the highest cumulative shear strain are O4, S1, and S7, respectively. The calculation results in Figure 6 also indicate that the relative activities of these three slip systems are relatively high. Therefore, during the plastic deformation of the γ phase, the O4, S1, and S7 slip systems are the main contributors.

When the γ lamella length is 12 μm at a strain of 0.017, the cumulative shear strains of O4 and S7 are equal. This also holds when the length is 14 μm at 0.024 strain, and when the length is 16 μm at 0.031 strain. Prior to the point where the cumulative shear strains of the two slip systems intersect, the O4 slip system assumes a more substantial role in plastic deformation. After the cumulative shear strain curves of the two slip systems intersect, the S7 slip system plays a more critical role in plastic deformation. The activities of S7 (transversal slip) and O4 (mixed slip) are comparable at room temperature, which is highly consistent with the findings reported by Selvarajou et al. [22], who observed similar relative activities between transversal slip and mixed slip under ambient conditions. From Figure 7d, the cumulative shear strain of the S7 slip system decreases with increasing γ lamellar length. This occurs primarily because a longer lamella leads to a larger γ/α₂ interfacial area, which provides stronger resistance to the movement of the S7 slip system. Consistent with our findings, Zhao [21] et al. demonstrated that the γ/α₂ interface can inhibit the motion of slip systems in the γ phase.

Figure 8 presents the cumulative shear strain of the slip system in the α₂ phase with different γ lamellae lengths at a strain of 0.035. Based on the calculated relative activity of slip systems in the α₂ phase, only basal slip is activated. As shown in Figure 8, the cumulative shear strain of the B1 slip system is highest, while the B2 and B3 slip systems exhibit extremely low cumulative shear strains, approaching zero. This indicates that the B1 slip system is the primary contributor to plastic deformation in the α₂ phase. From Figure 7d and Figure 8, it can be observed that the cumulative shear strain of the B1 slip system is the highest among all slip systems. Considering that the volume fraction of the α₂ phase in the overall model is higher than that of the γ phase, the B1 slip system contributes more significantly to the plastic deformation of the entire model.

4.2. The Effect of γ Lamellae Number on the Core–Shell-like Structured TiAl Alloy

4.2.1. Stress–Strain Analysis

In this paper, the number of γ lamellae is 6, 8 and 10, and the size of the γ lamellae is shown in Table 2, with the lamellae length being 14 μm.

Figure 9 is the von Mises stress–strain curves of the whole model and its γ and α₂ phases with different γ lamellae numbers under uniaxial compression. From Figure 9, when the strain is 0.035 and the γ lamellae number is 6, 8 and 10, the von Mises stress of the whole model is 2491 MPa, 2341 MPa and 2298 MPa, respectively; the von Mises stress of the α₂ phase is 2898 MPa, 2790 MPa and 2695 MPa, respectively; and the von Mises stress of the γ phase is 801 MPa, 810 MPa and 822 MPa, respectively. Under the constraint of a constant overall model volume, an increase in the number of γ lamellae leads to a corresponding rise in the volume fraction of the γ phase, accompanied by a concurrent reduction in the volume fraction of the α₂ phase. The results show that the von Mises stress of the γ phase shows little change with the number of γ lamellae, whereas that of the α₂ phase decreases significantly. This finding indicates that in the core–shell-like structured TiAl alloy, the decline in von Mises stress with the increasing number of γ lamellae is affected mainly by the variation in the α₂ phase volume fraction, with the variation in the γ phase volume fraction contributing minimally to this trend.

When the strain is less than 0.005, the von Mises stress of the α₂ phase is always higher than that of the γ phase, which indicates that the α₂ phase bears a larger load from the beginning of deformation. When the strain is greater than 0.005, the von Mises stress is basically unchanged. The von Mises stress of the α₂ phase continues to increase, which is significantly larger than that of the γ phase. The results show that the work hardening rate of the stress–strain curve of the model increases, mainly due to the significant redistribution of the load from the soft phase to the hard phase. Therefore, the stress of the model is mainly affected by the α₂ phase.

Figure 10 is the von Mises stress nephograms of the core–shell-like structured TiAl alloy with different γ lamellae numbers under uniaxial compression. The stress distribution in the central region of the model is relatively uniform. As the number of γ lamellae increases, the stress-reduced area (marked by the red dashed circle) in the α₂ phase increases. This result is consistent with that presented in Figure 9, indicating that the von Mises stress of the model is dominated by the α₂ phase.

4.2.2. Analysis of γ Phase and α₂ Phase Slip System

Figure 11 shows the calculated relative activity of slip systems in the core–shell-like structured TiAl alloy. Figure 11a shows the calculated relative activity of each slip system in the γ phase. When the γ lamellae number is 6, the relative activity of O4, S1and S7 is the highest, between 11% and 14%. When the γ lamellae number is 8, the relative activity of O4, S1, and S7 is the highest, between 13% and 14%. When the γ lamellae number is 10, the relative activity of O4, S1 and S7 is the highest, between 10% and 13%. The relative activity of lower slip systems is between 4% and 8%. Therefore, O4, S1, and S7 have the greatest influence on the plasticity of the γ phase. Figure 11b is the calculated relative activity of basal slip system in the α₂ phase. In the α₂ phase, the prismatic slip system and the pyramidal slip system are not activated, so only the relative activity of the basal slip system is calculated. The relative activity of the B1 slip system is the highest, and the relative activity of the B2 and B3 slip systems is extremely low, so the B1 slip system is the main contributor to the α₂ phase plasticity.

Figure 12 presents the cumulative shear strain curves of the slip system in the γ phase with different γ lamellae numbers (a, b, c), and (d) is the cumulative shear strain of the slip system in the γ phase with different γ lamellae numbers at a strain of 0.035. As shown in Figure 12, the three slip systems with the highest cumulative shear strain are O4, S1, and S7, respectively. The calculation results in Figure 11 also indicate that the relative activities of these three slip systems are relatively high. Therefore, during the plastic deformation of the γ phase, the O4, S1, and S7 slip systems are the main contributors. Equivalent cumulative shear strains of the O4 and S7 slip systems are observed at strain values of 0.017, 0.024, and 0.036, corresponding to 6, 8, and 10 γ lamellae, respectively. Prior to the point where the cumulative shear strains of the two slip systems intersect, the O4 slip system assumes a more substantial role in plastic deformation. After the cumulative shear strain curves of the two slip systems intersect, the S7 slip system plays a more critical role in plastic deformation.

Figure 13 is the cumulative shear strain of the slip system in the α₂ phase with different γ lamellae numbers at a strain of 0.035. It is evident that the B1 slip system possesses the maximum cumulative shear strain, while the B2 and B3 slip systems show cumulative shear strain approaching zero. This finding confirms that plastic deformation within the α₂ phase is dominated by the activation of the B1 slip system. Moreover, Figure 12d and Figure 13 demonstrate that the B1 slip system has the highest cumulative shear strain among all slip systems. In view of the fact that the α₂ phase accounts for a larger volume fraction than the γ phase in the constructed model, the B1 slip system plays a more significant role in mediating the plastic deformation behavior of the entire model. Given that the volume fraction of the α₂ phase in the model is higher than that of the γ phase, the B1 slip system makes a more pronounced contribution to the plastic deformation of the overall model.

Based on the available data, a preliminary inference regarding the synergistic effect of γ lamellar length and number has been drawn: shorter lamellar length and fewer lamellae correlate with inferior material properties, whereas longer lamellae and greater quantity lead to superior performance. However, the specific influence of increasing lamellar quantity under short-length conditions, as well as the performance variation induced by the same increment in quantity under long-length conditions, remains unclear. These issues require elucidation through systematic comparative studies, which will constitute the focus of our subsequent research.

5. Conclusions

This study is based on crystal plasticity theory and the finite element method, establishing a finite element model of a core–shell-like structured TiAl alloy with (α₂ + γ) lamellar colonies as the core and the α₂ matrix as the shell. The model assumes an idealized periodic distribution of lamellar colonies to simplify the analysis. Although actual core–shell structures deviate from this idealization in several aspects, which may lead to inaccuracies in predicting mechanical properties and deformation behavior, the model still provides valuable theoretical insights into the deformation mechanisms of such alloys. By simulating the effects of different γ lamella lengths and numbers on the mechanical behavior of the alloy under uniaxial compression, this study reveals the deformation mechanisms of the core–shell-like structured TiAl alloy. The research focuses on the influence of the length and number of γ lamellae on von Mises stress–strain curves and nephograms, and relative activity and the activation of slip systems of the core–shell-like structured TiAl alloy. Future work will combine crystal plasticity simulations or in situ experiments based on stress distribution to more accurately identify the dominant slip systems and their underlying mechanisms. The main conclusions are summarized as follows:

(1)

When the γ lamella length increases from 12 μm to 16 μm, the overall von Mises stress decreases from 2524 MPa to 2221 MPa, a reduction of approximately 12.0%. When the number of γ lamellae increases from 6 to 10, the overall von Mises stress decreases from 2491 MPa to 2298 MPa, a reduction of approximately 7.7%. The decrease in the overall von Mises stress is primarily attributed to the reduction in stress within the α₂ phase, while the change in the volume fraction of the γ phase has a negligible effect on this reduction. By decreasing the volume fraction of the α₂ phase, the plastic deformation capacity of the material is effectively enhanced.

(2)

The stress distribution of the core–shell-like TiAl alloy is more uniform. With the increasing of the length or number of γ lamellae, the low stress region in the α₂ phase shows an expanding trend. Increasing the volume fraction of γ lamellae significantly reduces the stress level of the α₂ phase and promotes stress dispersion within the α₂ phase, thereby enhancing the plasticity of the material.

(3)

In the case of changes in the length or number of γ lamellae, the relative activity of the slip systems in the γ phase was analyzed. The results showed that the O4, S1, and S7 slip systems have the highest relative activity. At the same time, the cumulative shear strain of these three slip systems is also significantly higher than that of other slip systems, indicating that O4, S1 and S7 slip systems contribute the most to the plastic deformation of γ phase.

(4)

In the α₂ phase, the B1 slip system exhibits the highest relative activity and cumulative shear strain, while the relative activity and cumulative shear strain of the B2 and B3 slip systems are close to zero. Therefore, the B1 slip system is the main contributor to the plasticity of the α₂ phase. Considering that the volume fraction of the α₂ phase in the overall model is higher than that of the γ phase, the B1 slip system contributes more significantly to the plastic deformation of the entire model.