The current work is based on the three-dimensional multi-phase-field model for an elastically and structurally inhomogeneous system [24] of Ti-6Al-4V alloy. Within the framework of the multi-phase-field model, 12 order parameters are employed to distinguish β matrix and α phase consisting of 12 alpha variants (V1 to V12) with different spatial orientations due to symmetry reduction during phase transformation. Two sets of order parameters, i.e., {ci(x,t)}i=Al,V and {φp(x,t)}p=1∼12, are adopted for the description of composition and structural variation. The order parameter φp(x,t) takes the value of 0 within the β phase, 1 within pth variant Vp, and varies from 0 to 1 while crossing the interfaces. For example, φ2(x,t)=1 indicates field point x is occupied by V2 at time step t. The total free energy consists of the chemical free energy Echem, the elastic strain energy Eel and the work done by the external field W:

(1)Etot=Echem+Eel+W.

This study explores the equilibrium shape of α plate, and only V1 is allowed to evolve from a preset nucleus, i.e., the order parameter {φp(x,t)}p=1∼12 is now reduced to φ(x,t) as the other parameters {φp(x,t)}p=2∼12 are all set as zero during the whole simulation. Following Cahn et al. [25], the chemical free energy function consists of the bulk free energy and the gradient energy:

(2)Echem=1Vm∫{fm[T,ci(x,t),φ(x,t)]+∑i=Al,Vκi2|ci(x,t)|2+∑p=1N=12κφ2|∇φ(x,t)|2}dx,

where Vm is the molar volume of Ti-6Al-4V. The first term in the integrand is the local chemical free energy density that results from short-range chemical interactions, depending on temperature T, compositions ci(x,t), and structural field φ(x,t). The second and third terms denote the gradient terms in the concentration and structure order parameters, with κi and κφ being the gradient coefficients. These two terms contribute to the interfacial energy due to the nonuniformity in both the composition and structure and are nonzero only at and around the interfaces. For Ti-6Al-4V alloy, the local chemical free energy can be formulated by the summation of the molar Gibbs free energies of α and β phases fmα and fmβ:

(3)fm(T,c,φp)=h(φ)fmα(T,ciα)+(1−h(φ))fmβ(T,ciβ)+ωφ2(1−φ)2.

fmα and fmβ are obtained from the thermodynamics database CALPHAD (CALculation of PHAse Diagrams) [26]. h(φ)=φ3(6φ2−15φ+10) is the interpolation function used in this resesrch, characterizing the smooth transition of phases across the interface. For the general cases where all α variants are allowed to evolve, the h(φ) function should be replaced by h(∑pφp), which summarizes the molar free energy of all existing α variants.

For a system with the assumption that the precipitate and matrix have identical elastic constants Cijkl0, the elastic energy can be calculated by Khachaturyan-Shatalov (K-S) microelasticity through analytical solutions based on the formulation of elastic strain energy density in the Fourier space [15]. However, the elastic modulus and symmetry of the α phase are different from those of the β matrix, which suggests that Cijkl(x) varies across the α/β interface. Following Wang et al. [18], for cases of modulus inhomogeneity, a virtual equivalent system with a homogeneous reference modulus Cijkl0 is introduced, which is usually the value of the bulk phase or the average modulus of the body. The displacement field u(r) (and thus the strain field ϵ(r)) and the stress field of the reference system are identical to the real system of which this study is concerned. The two systems can be bridged by the following relationship:

(4)Cijkl(x)[∂uk∂xl−εklT(x)]=Cijkl0[∂uk∂xl−εkl0(x)],

where εklT(r) is the eigen strain field associated with the stress-free transformation strain from β→α and the distribution of α phase, εkl0(x) is the eigen strain field of the virtual system. Based on Eq. (1), the calculation of stress field or elastic strain energy within the modulus inhomogeneity system is converted to the calculation of εkl0(x) within the virtual system. To initiate the numerical calculation, a guess of εkl0(x) is taken as the input of Eq. (4), the stress field corresponding to the left-hand side of Eq. (4) can be directly obtained. At the same time, the elastic strain energy of the target system can be formulated as (see details of the deduction in [18]):

(5)Eel=∫12Cijkl0εij0(x)εkl0(x)dx+V2Cijkl0ϵ¯ijϵ¯kl−Cijkl0ϵ¯ij∫εkl0(x)−{12∫k≠0σ~kl0(k)njnl⋅[Ω(n)]ikσ~ij0∗(k)dk(2π)3}k→r+12∫{−Cpqkl0ΔSpqmn(x)Cmnij0−Cijkl0}[ϵkl0(x)−ϵklT(x)][ϵij0(x)−ϵijT(x)]dx

where ΔSpqmn(x)=ΔCpqmn−1(x) with Cijkl(x)≡Cijkl0+ΔCijkl(x), k is a vector in the reciprocal space, σ~ij0(k)≡Cijkl0εkl0(k) with σ~ij0∗(k) being the complex conjugate. ϵ¯ij is the homogeneous part of total strain in the virtual system that is determined by the boundary condition. εij0(x) keeps updating and minimizing the elastic strain energy Eel until equilibrium is reached:

(6)∂εij0(x,t)∂t=−LδEelδεij0(x,t).

L is a dynamic parameter characterizing the convergency rate of evolving εkl0(x). Additional work performed by applied external stresses will also contribute to the total free energy through the interaction of stresses with the stress-free transformation strain of V1, εij0:

(7)W=−∫σijapp(r)εij0φ(x,t)dx.

The concurrent evolution of composition and structure follow the general form of Cahn-Hilliard diffusion equation and the Allen-Cahn equation in the multi-phase-field model by Steinbach et al. [27]. When orientations of precipitates are considered, the governing equations in this work is derived as:

(8)1Vm2∂ck(x,t)∂t=∇∑j=Al,VMkj(T,ci,φp)∇δEchemδcj(x,t)+ζc(x,t),

(9)∂φp(x,t)∂t=−Mφ{1N~∑q≠p(δFchemδφp(x,t)−δFchemδφq(x,t))+δ(Eel+W)δφp(x,t)}+ζφ(x,t),

where N~ is the number of phases that co-exist locally, Mkj denotes the chemical mobility, Mφ is the mobility of the structural order parameters characterizing interface kinetics, ζc(r,t) and ζφ(x,t) are the Langevin noise terms for composition and the long-range order parameter. This model’s parameters are identical to those used in the previous work for the same Ti-6Al-4V alloy system [24].

Using α precipitate in the β matrix as an example, the total elastic energy and stresses can be calculated with three sets of elastic moduli:

Case-I: homogeneous modulus of the matrix phase: Cijkl(x)→Cijklβ.

Case-II: homogeneous modulus of the precipitate phase: Cijkl(x)→Cijklα.

Case-III: inhomogeneous modulus that takes the corresponding values of modulus according to the order parameter: Cijkl(x)≡[1−h(ϕ)]Cijklβ+h(ϕ)Cijklα.

During the numerical process of finding the equilibrium elastic state, the elastic moduli concerning the two phases should not be set strictly equal due to the iteration of Eq. (5), where ΔSijkl(x)=ΔCijkl−1(x) is required. In addition, with a smaller ΔCijkl, longer iteration time is needed.

During the β→α phase transformation, the α precipitates usually exhibit a specific OR with the β matrix, referred to as the Burgers OR [28], i.e., (101)β||(0001)α and [1¯1¯1]β||[1¯1¯20]α. The coordinate of the simulation unit in this study is set to x||[ 010 ]β−y||[ 1¯01 ]β−z||[ 101 ]β. Under three different modulus approximations, the α precipitate grows and relaxes in the phase-field model described in Section 2. Minimizing the total free energy determines the equilibrium shape (size and habit plan orientation). The elastic constants used for Case-III are: C11α=115.1GPa, C12α=75.0GPa, C13α=95.0GPa, C33α=157.0GPa, C44α=26.0GPa; C11β=97.7GPa, C12β=82.7GPa, C44β=37.5GPa [29]. To approach the two cases with homogeneous modulus, in Case-I, the Cijklα=1.005Cijklβ is assigned and in Case-II Cijklβ=1.005Cijklα. The assignment of ΔCijkl(x) with 0 will lead to iterative divergence. For all cases the simulation box is 64×64×64nm3, with the initial nucleus of α phase set as a small sphere with radius of 3 nm. Upon aging at 1073 K, such nucleus starts to evolve with the habit plane gradually forming and a precipitate of plate shape comes into being, as shown in Fig. 1, where inhomogeneous modulus is set in the model. Subsequently, Fig. 2 compares the shape of α precipitates at t = 20 (where t is the dimensionless time) for all three cases mentioned above with different modulus assumptions. It indicates that the cross sections (d–f) corresponding to the 3D morphology (a–c), the system with homogeneous modulus of α phase (Case-II) sprouts the sharpest α plate. At the same time, thickened α plate is observed for Case-I. The volume change during aging is shown in Fig. 3a, which confirms that α plate in Case-I with β modulus occupies the largest volume fraction. The black solid lines in Figs. 2d through 2f represent experimentally observed habit plane normal of α plate, i.e., (11¯,13¯,11)β [30]. Compared to the habit planes of the simulated α plates (indicated by dashed white lines), the orientation of the habit plane for Case-II is closest to that of the experimental observation. In contrast, Case-I with homogeneous β modulus shows the largest deviation. It should be mentioned that in most cases it is more likely to assume the system with modulus of the matrix phase, which occupy the largest proportion. However, for α/β phases with different crystal structures, information concerning the crystal symmetry can be lost under homogeneous modulus assumption. For example, five independent elastic constants in α phase with α structure are now reduced to three when homogeneous β modulus is assumed, which can cause larger deviation.

External stresses can also alter the equilibrium shape or size of α precipitate by adding further work to the elastic strain energy. Therefore, stress along the x||[ 010 ]β direction is applied with the magnitude of σ11app=±500MPa to the simulation unit upon the growth of α. The volumes of α at t = 20 are then calculated and compared in Fig. 3b with cases where applied stresses are absent. For Case-I and Case-III, while compressive stress of 500 MPa suppresses the growth of α precipitate, tensile stresses show the opposite effect. However, the α precipitate in Case-II with homogeneous α modulus is insensitive to the applied load. The different choice of the elastic assumptions will not affect the work done by the external field. It will indeed later the elastic strain energy expressed by Eq. (5), where the elastic modulus appears directly and indirectly (involved in the ϵ¯ij through general Hooke’s law). Fig. 3 suggests that for Case-II (with homogeneous α modulus), the work performed by the current external stress with the magnitude of 500 MPa is so trivial, as compared to the elastic strain energy expressed in Eq. (5), that it is unable to promote or suppress the growth of the alpha plate. Significant change in the alpha size would be possible if a much larger stress is applied.

Due to the mismatch of lattice parameters of α precipitate and β matrix, stresses are concentrated at the α/β interface, which depends on the elastic moduli of two phases and the shape of α precipitates. In order to validate the accuracy of the current model, an isotropic medium is considered to contain a spherical cavity (with a modulus of zero) at the center with radius R=(1/10)l0, with l0 being the length scale. Under uniaxial loading of σzz0, the three-dimensional profile of the stress field σzznum calculated using the current model based on Eq. (4) is shown in Fig. 4a. The simulation results indicated that the stresses are concentrated at the edges of the cavity, where the modulus undergoes a sharp variation from zero to that of the isotropic media. The corresponding cross-section of the stress is shown in Fig. 4b. The calculation result is compared to the analytical solution derived by Lee et al. concerning the identical problem [31]. The relative stresses represented by the rates of σzz/σzz0 within the cross section of z=0, along x direction and away from the origin of 0.2R (the blue line in (b)) and 1.1R (the red line in (b)) are respectively present in Figs. 4c and 4d. The relative stress along the blue line increases fast when approaching the interface and suddenly decreases to zero within the cavity region. However, along the red line outside the cavity, the rate varies more smoothly and reaches a minimum of around 0.35 at the position right above the cavity. The calculation results of the developed model agree well with those of the analytical solution indicated by circles.

For titanium alloy with α precipitate embedded in β matrix, the evolution of stress fields of component σ11 are shown in Fig. 5 with t =1, 10 and 20. At the nucleus stage, both tension and compression stresses are observed. This stress concentration field keeps expanding along with the growth of the plane. At t = 20, the stresses are concentrated at the tips of the α plates and present certain symmetry due to the crystallography of α crystals. More specifically, stresses across the plate (along the white line in Figs. 4a–4c) are out-extracted and shown in Fig. 5d, indicating that the magnitude of stresses reaches the maximum when approaching the α/β interface, and almost keeps constant within the α plate.

However, when the homogeneous modulus assumption is applied, the stress field can be distinct from that with the inhomogeneous modulus shown above due to the variation in ΔCijkl(x) (as well as ΔSijkl(x)) and thus the elastic strain energy in Eq. (5). In Fig. 6, the stress fields of different components across the α/β interface associated with cases using α modulus or β modulus are shown and compared to the case where inhomogeneous moduli of α and β phases are strictly assigned, where σ23 and σ13 are almost zero, and therefore, only the other four components are present for comparison. For all these components, stresses using α modulus exhibit small deviations within the α phase region. Especially for σ33, the magnitude is significantly large within the α plate when β modulus is used. Matrix stresses using β modulus exhibit small deviations. All the stresses are approaching zero when moving away from the interface.