The grain orientation effects on crack-tip fields for additively manufactured titanium alloy: A peridynamic study

Theoretical and Applied Fracture Mechanics 107 (2020) 102555

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The grain orientation effects on crack-tip fields for additively manufactured titanium alloy: A peridynamic study Binchao Liu1, Zhongwei Yang1, Rui Bao

T

⁎

Institute of Solid Mechanics, School of Aeronautic Science and Engineeing, Beihang University, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Peridynamics Crack-tip fields Grain orientation Additive manufacturing

In this work, a macroscale description method for grain orientation effects in α -phase additively manufactured titanium alloy is proposed within peridynamic framework, and peridynamic simulations of grain orientation effects on crack-tip fields are performed. The proposed method describes the grain orientation effects by introducing anisotropic peridynamic parameters which are later calibrated according to strain energy density. Then, stationary crack-tip fields in single-crystal model, stationary crack-tip fields in polycrystalline model, and crack-tip fields at branching in polycrystalline model are respectively simulated, whose results are analyzed in detail as well as compared with experimental observations. In the end, some discussions and further considerations are presented, which indicates our efforts to future works. The proposed method may help develop a way to reveal interactive relationships among grain features, crack-tip fields and crack propagation behaviors.

1. Introduction Additive manufacturing (AM) offers a layer-by-layer strategy to economically fabricate customized parts with complex geometries in a rapid design-to-manufacture cycle, and is developing towards major utilization such as biomedical, automotive and aerospace industries [1]; particularly, AM of titanium alloy has attracted wide interest due to unfriendly workability in traditional processing techniques of titanium alloy [2]. However, the mechanical behavior of AM Ti-alloys must be substantially understood at all scales before the above benefits can be explored in critical load bearing applications, for such advanced techniques actually pose great challenges to traditional structural integrity evaluation [3]. Due to its unique forming process of repeated melting and solidifying, anisotropy and heterogeneity are introduced into material microstructures and thus mechanical properties of AM Ti-alloys [4]. Columnar grains and equiaxed grains are two typical material structures in AM Ti-alloys, and material structures displaying all columnar grains, all equiaxed grains, or specific distributions of columnar grains and equiaxed grains can be achieved by controlling processing parameters [5]. For static properties, super-fined microstructures including basketweave morphology and α′ needles lead to high strength [6,7], while continuous α grains, textures and porosities result in low ductility [8];

additionally, the disparities among different microstructures due to heat exchange process will cause conspicuous heterogeneities in mechanical properties, especially in plasticity [9]. For fatigue crack growth (FCG) properties, the diversities of microstructures and defects have an impact on the fracture mode [10], and the alternately arrayed macrostructures lead to periodic fluctuations in fatigue crack growth rates (FCGRs) [11]; Differences in FCGRs along different directions or within different material structures are also reported [12], and some researchers have deduced that this is related to the impacts of α / β phase boundaries [13,14]. Therefore, it is much more complex to deal with damage tolerance evaluation towards AM Ti-alloys, and alternatives to conventional qualification methods must be found [15]. Thereinto, further studies into crack-tip fields such as microstructure effects and new crack-tip parameters should be conducted, both experimentally and numerically. By processing the intensity and range variation of the recorded images before and after deformation, the digital image correlation (DIC) techniques offers a strategy to experimentally measure full-field displacement and deformation [16], and successful use of the DIC technique has been reported for the experimental measurement of the crack-tip fields in AM Ti-alloys. For example, Wu [17] applied DIC to monitoring fatigue crack displacement variation in laser melting deposited (LMD) Ti-6.5Al-3.5Mo-1.5Zr0.3Si titanium alloy, in which differences of crack-tip strain fields,

⁎

Corresponding author at: Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University (BUAA), No. 37 Xueyuan Road, Haidian District, Beijing 100191, China. E-mail address: [email protected] (R. Bao). 1 Binchao LIU and Zhongwei YANG are co-first authors of the article. https://doi.org/10.1016/j.tafmec.2020.102555 Received 26 December 2019; Received in revised form 21 February 2020; Accepted 21 February 2020 Available online 22 February 2020 0167-8442/ © 2020 Elsevier Ltd. All rights reserved.

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particular, the capabilities of PD to quantificationally study 2D and 3D crack-tip fields are validated in the comprehensive studies conducted by Breitenfeld [50,51], which together with many other researches form the theoretical and applied basis for this work. In this work, a macroscale peridynamic description of grain orientation effects on crack-tip fields for additively manufactured titanium alloy is proposed. The proposed method provides a macroscale way to consider grain orientation effects in peridynamic framework, and can help achieve the goal of revealing interactive and integrated relationships among material effects, crack-tip fields and crack propagation behaviors. The paper is organized as following. In Section 2, the original peridynamic theory is briefly introduced and the displacement extrapolation method is adopted to characterize crack-tip fracture mechanics parameters. In Section 3, the description method of crystal orientations is detailed. With the proposed description method, cracktip fields in three different kinds of models with different grain orientation distributions are obtained in Section 4. In the end, some discussions on the proposed method and further considerations are presented in Section 5.

crack-tip opening displacement (CTOD) and FCGRs are observed between heat affected band (HAB) areas and non-HAB areas; Wang [18] has reported that the orderly distributed αp laths have an impact on secondary crack initiation and crack branching at subsurface of LMD Ti–5Al–5Mo–5V–1Cr–1Fe alloy, which obviously influences crack-tip fields on the specimen surface and could be monitored by DIC techniques. Numerical studies involve revealing the relationships between nonlinear crack-tip parameters (NLP) and FCGRs, as well as the influences of crack blunting and crack closure on FCG behaviors. Antunes [19] has developed an elastic-plastic finite-element model to quantify the CTOD and found that the plastic CTOD could be an interesting alternative to stress intensity factor range ΔK in the analysis of fatigue crack propagation, and Camas [20] has developed numerical modelling of three-dimensional fatigue crack closure as well as plastic wake. Theoretically speaking, numerical simulations of crack-tip fields have the potential to shed new lights on FCG predictions in AM metals, but there has been limited studies at present due to the lack of parameters for diverse kinds of microstructures which are necessary in numerical modelling, and this may rely on experimental measurements; therefore, a combination of experimental study and numerical study is of great significant to bridge. Chernyatin [21] has studied crack-tip mechanics with a multi-approach including DIC full-field displacement information, analytical modelling of the crack-tip field and scanning electron microscope (SEM) fractographies. Most numerical studies involving material effects and crack-tip fields, no matter mentioned above or not, adopt finite element method (FEM) and respective improvements of FEM have been made for corresponding problems. For example, crystal plasticity finite element method (CPFEM) is the combination of crystal plasticity and FEM, which becomes a powerful tool to study deformation behaviors of grains. Once a polycrystalline model is established and a proper constitutive relationship is given after parameter calibrations, the deformation of whole model can be obtained and the microscale anisotropic behaviors can be captured by CPFEM [22]. Since CPFEM provides a link between the dislocation-level physics and macro-scale continuum response [23], extensive studies involving modeling mechanical behaviors of advanced alloys are conducted on topics, such as crystallographic slip [24,25] and grain reorientation [26,27], with varying success. Additionally, for crack growth problems the extended finite element method (XFEM) has been developed [28]. XFEM incorporates local discontinuous enrichment functions with additional degrees of freedom into the standard FEM, in order to capture discontinuity across a crack [29]. XFEM has been adopted to study problems such as dynamic crack growth [30,31], complicated crack patterns [32,33], and failure in composites [34,35]. However, a certain mesh refinement is still required in XFEM to obtain results with satisfactory accuracy, and its accuracy is related to the position of the crack tip [36]; besides, reliable fracture criteria for such applications are still missing [37]. Since there exist complex interactive relationships among material effects, crack-tip fields and crack growth behaviors, it is a better way to adopt a method which can take an integrated consideration into the above three factors. Such an awkward situation for FEM-based methods in dealing with crack or crack growth problems is essentially due to their mathematical formulations [38]; FEM is based on the classical continuum theory, which is formulated using spatial partial differential equations and these spatial derivatives lose their meanings at discontinuities where cracks occur. To overcome such mathematical problems, peridynamics (PD) has been proposed in 2000 [39] and it is a promising method in which the spatial derivatives are replaced by integrals so that the peridynamic governing equations are applicable at fractures and external crack growth criteria are no longer necessary. PD has aroused intense scholarly interest since its birth, and been adopted to study dynamic brittle fracture [40,41], failure in fiber-reinforced composites [42,43], damage in functionally graded materials [44,45], crack path in polycrystal materials [46,47] and many other problems [48,49]. In

2. Peridynamic fundamentals In this section we firstly introduce the basic idea of peridynamics, and rigorous derivation should be referred to [52,53]. Then, we introduce the displacement extrapolation method into peridynamics for stress intensity factor calculation: on one hand, a method to characterize fracture mechanics parameters without computation of stress should be suitable with peridynamics; on the other hand, the ability of peridynamics to describe crack-tip fields is also verified in this way. 2.1. Peridynamic theory Assume that an object occupies a certain spatial domain R in the reference configuration at time t , and the peridynamic equation of motion for material point x in the reference configuration can be written as

ρ (x) y¨ (x, t ) =

∫H (T̲ [x, t ] 〈x′ − x〉 − T̲ [x′, t ] 〈x − x′〉 ) dVx′ + b (x, t ) (1)

where ρ is mass density, and y is the deformation map at time t ; x′ is another material point that is bonded to x and ξ = x′ − x is the bond vector; b (x, t ) is the external body force density at point x . H denotes the neighborhood of material point x ,

H = {x′ ∈ R, ∥x′ − x∥ ⩽ δ }

(2)

where the radius δ is called the horizon, a material property parameter in peridynamics. T̲ [x, t ] 〈x′ − x〉 is a function called the force state at x that maps any bond ξ to the corresponding force density vector. The specific formulation of T̲ actually denotes the material constitutive in peridynamics, and the adopted material constitutive in this paper is referred to [54], which is actually an elastoplastic peridynamic constitutive with von Mises yield criterion. The calibrations for peridynamic elastoplastic constitutive are presented in Section 2.2. The deformation state maps any bond ξ to its deformation vector:

Y̲ [x] 〈x′ − x〉 = y (x′, t ) − y (x, t )

(3)

Damage is characterized by irreversible bond breakage in peridynamics; several kinds of bond breakage criteria are available, among which the most intuitionistic one is that a bond ξ breaks when its bond stretch

ε=

∥Y̲ 〈ξ 〉∥ − ∥ξ∥ ∥ξ∥

(4)

exceeds the critical stretch ε* , and the critical stretch in 2D case can be expressed as 2

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ε* =

(

6 μ π

+

Gc 16 (K 9π 2

)

− 2μ) δ

KI = (5)

∫H μ̲ 〈ξ 〉 dVξ ∫H dVξ

uy

(6)

r

where the damage state μ̲ 〈ξ 〉 is defined to express bond breakage at a given material point x :

{

1, ε < ε∗ μ̲ 〈ξ 〉 = 0, ε > ε∗

(8)

where G is the sheer modulus; κ is the indicator of stress state, whose value is 3 − 4ν for plane strain and (3 − ν )/(1 + ν ) for plane stress where ν is the Poisson’s ratio. Away from the crack tip, u y / r can be fitted by a linear function of r [56]:

where Gc is the critical energy release rate. To characterize the total amount of damage at x , the net damage is defined as the ratio of the total number of broken bonds to the initial number of bonds in a family,

φ (x, t ) = 1 −

2G 2π ⎤ lim ⎡u y (r , θ = π ) κ + 1 r→0 ⎢ r ⎥ ⎦ ⎣

= A + B·r

(9)

Combining Eq. (8) with Eq. (9), we finally obtain

KI, i = (7)

2 2π G A κ+1

(10)

By PD simulations we can obtain the displacement fields and thus extract the opening displacement u y, i of material point i whose distance to the crack tip is ri ; the fitting line can thus be obtained by the least square method while its intercept is actually the value of parameter A . It is worth noting that the displacements of points very close to crack tip should be omitted in the fitting, because the results are with huge errors due to the crack-tip singularity; the remaining points ensure that the linear correlation coefficient is above 99% in this work. A center cracked plate subjected to remote tension is modelled as shown in Fig. 2(a). The elastic modulus is set as E = 110 GPa , and the Poisson’s ratio is set as ν = 0.33, which are typical values for Ti-alloys; the applied load is taken as S = 150 MPa , and the PD spacing size is set as dx = 0.5 mm . The half crack length varies, and the stress intensity factor K obtained by this method is compared with the analytical one calculated by:

When the net damage φ (x, t ) reachs critical value φ∗, it is regarded that crack occurs on this material point. Parameter studies in [55] have pointed out that simulation results are most accurate with δ = 3.015dx where dx is the uniform spacing size of PD material points, and the critical net damage value for bulk material points is correspondingly φ∗ ≈ 0.39. 2.2. The capability of peridynamics to study crack-tip fields As mentioned above, a peridynamic elastoplastic material constitutive model with an isotropic hardening rule [54] is adopted in this work, whose detailed derivation is omitted here due to limited length of the paper. Since we found nowhere the open-source code for the above model, a simple validation for tension response of Ti-6Al-4V is shown in Fig. 1, and the results show clearly that the stabilized cyclic response can be well described. The implementation of elastoplastic constitutive within state-based peridynamics is detailed in [54], which can be easily achieved on the basis of open Fortran code in [38]. Despite that peridynamics has been applied to many fracture problems with varying success, there are few studies in discussing crack-tip parameters in detail. Breitenfeld [50,51] obtained the stress intensity factor K and validated the capabilities of PD to quantificationally study 2D and 3D crack-tip fields by the computation of PD stress, but it is somewhat contradictory to the intention of PD for it is a theory based on deformation or displacement fields without such concept of stress. Therefore, we adopt the displacement extrapolation method here to verify the capability of peridynamics to characterize crack-tip fields, i.e. to obtain the stress intensity factor K . For mode I problems, the stress intensity factor can be obtained by the opening displacement perpendicular to crack:

KI =

sec

πa σ πa 2W

(11)

It can be seen from Fig. 2(b) that the results agree well with the analytical solution for a wide range; the displacement extrapolation method in peridynamics is thus verified, which validates the capabilities of peridynamics to estimate crack-tip fields and also provides a method to obtain crack-tip parameters directly from deformation fields without computation of stresses. 3. Description method and parameter calibration According to [38], peridynamic formulations are derived by equaling the strain energy under specific deformation conditions to those of classic continuum mechanics. The strain energy for isotropy materials at some material point in peridynamics is expressed as: N

W(k ) = aθ(2k ) + b ∑ j=1

δ (|y − y(k ) | − |x (j) − x (k ) |)2V(j) |x (j) − x (k ) | (j)

(12)

Zhu [57] has proposed a modified expression of Eq. (12) for modelling of granular fracture in polycrystalline materials, in which they actually modify the parameter b to be direction-dependent. Despite its referential values, there are two maximum principal axes of grain orientations in Zhu’s model, which does not conform to the knowledge of material science; moreover, Zhu’s model considers only 2D crystal lattice effects. In this section, an improved anisotropy modification for 3D crystal lattice effects of LMD Ti–5Al–5Mo–5V–1Cr–1Fe (known as TC18 in China) is presented and calibrated. 3.1. Description method Based on results of the phase map from electron back scatter diffraction (EBSD) patterns shown in Fig. 3, the hexagonal close-packed (HCP) α phase takes the dominating part in the whole materials, including both equiaxed grain areas and columnar grain areas. Thus, the body-centered cubic (BCC) β phase is omitted and only HCP structure is

Fig. 1. The validation of material constitutive model adopted in this paper for tension response of Ti-6Al-4V. 3

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B. Liu, et al.

Fig. 2. (a) The geometric size of numerical center-cracked plate; (b) The peridynamic results by displacement extrapolation method and the analytical solutions of the stress intensity factor. ¯ M

considered in the following modeling. As shown in Fig. 4(a), the anisotropy of average elastic modulus among different crystal plane for HCP structure approximates an elliptical distribution relationship with the plane angle θ :

W(k ) = aθ(2k ) + b¯ ∑ j=1 M1

+ b1 ∑ j=1

E (θ) =

(140 cos θ)2 + (105 sin θ )2

(units: MPa)

M2

(13)

− b2 ∑ j=1

For a certain crystal plane shown in Fig. 4(b), we assume that it is similar to a unidirectional-lamina composite, and according to classic continuum mechanics (CCM) we have

Q11 Q12 0 ⎤ ε11 σ ⎧ ⎫ ⎧ σ11 ⎫ ⎡ 0 ⎥ ε22 22 = ⎢Q12 Q22 ⎨ σ12 ⎬ ⎢ ⎨ε ⎬ ⎥ 0 0 Q ⎩ ⎭ ⎣ 66 ⎦ ⎩ 12 ⎭

δ (|y(j) |x (j) − x (k ) |

− y(k ) | − |x (j) − x (k ) |)2V(j)

δ (|y(j) |x (j) − x (k ) |

− y(k ) | − |x (j) − x (k ) |)2V(j)

δ (|y(j) |x (j) − x (k ) |

− y(k ) | − |x (j) − x (k ) |)2V(j)

(15)

where we assume that the peridynamic parameter b increases to b¯ + b1 in the maximum principal direction, and decreases to b¯ − b2 in the transverse direction; otherwise, b takes the value of b¯ . So far, we have four undetermined PD parameters, i.e. a , b¯ , b1, b2 , which can be calibrated with corresponding engineering material constants in Eq. (14).

(14)

3.2. Calibration method

For peridynamic description we modify Eq. (12) as

Similar to [38], four simple loading conditions are adopted in the calibration as shown in Fig. 5, namely Simple shear: γ12 = ζ Uniaxial stretch in principal direction: ε11 = ζ , ε22 = 0 Uniaxial stretch in transverse direction: ε11 = 0, ε22 = ζ

Fig. 3. The EBSD (a) partterns and (b) phase map for LMD TC18: red for beta phase (5.8%) and blue for alpha phase (94.1%). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 4

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Fig. 4. The schematic for anisotropy: (a) among different crystal plane; (b) within a certain crystal plane.

Biaxial stretch: ε11 = ζ , ε22 = ζ

Q11 Q12 0 ⎤ 0 σ ⎧ ⎫ ⎧ 0 ⎫ ⎧ σ11 ⎫ ⎡ 0 0 ⎥ 0 = 22 = ⎢Q12 Q22 ⎨ σ12 ⎬ ⎢ ⎨ ζ ⎬ ⎨Q ζ ⎬ 0 Q66 ⎥ ⎩ ⎭ ⎣ 0 ⎦ ⎩ ⎭ ⎩ 66 ⎭

3.2.1. Simple shear Due to this loading condition, the stresses in Eq. (14) become

(16)

According to the CCM, the corresponding strain energy density at material point x (k ) can be expressed as

Fig. 5. Four simple loading conditions: (a) simple shear; (b) uniaxial stretch in principal direction; (c) uniaxial stretch in transverse direction; (d) biaxial stretch. 5

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W(CCM = k)

1 Q66 ζ 2 2

M

a+

(17)

2 πhδ 4 ¯ 1 b − b2 δ ∑ |x (j) − x (k ) | V(j) = Q22 4 2 j=1

(30)

While in peridynamics under such deformation condition as shown in Fig. 5(a), the bond deformation can be expressed as

|y(j) − y(k ) | = [1 + (sinϕcosϕ) ζ ]|x (j) − x (k )| = [1 + (sinϕcosϕ) ζ ] ξ

3.2.4. Biaxial stretch Due to this loading condition, the stresses in Eq. (14) become

(18)

For this deformation, the strain energy density in peridynamics can be evaluated as

W(PD k) =

πhδ 4ζ 2 ¯ b 12

Q11 Q12 0 ⎤ ⎧ ζ ⎫ ⎧ (Q11 + Q12 ) ζ ⎫ σ ⎧ σ11 ⎫ ⎡ 0 ⎥ ζ = (Q12 + Q22 ) ζ 22 = ⎢Q12 Q22 ⎨ σ12 ⎬ ⎢ ⎨ ⎬ ⎨ ⎬ 0 Q66 ⎥ ⎩ ⎭ ⎣ 0 0 ⎦ ⎩0⎭ ⎩ ⎭

(19)

According to the CCM, the corresponding strain energy density at material point x (nk ) can be expressed as

Equating Eq. (17) and Eq. (19), we obtain

6Q66 b¯ = . πhδ 4

|y(j) − y(k ) | = [1 + (cos 2ϕ + sin2ϕ) ζ ] ξ

(21)

M1

2 W(PD k ) = 4aζ +

2πhδ 4ζ 2 ¯ b + b1 δζ 2 ∑ |x (j) − x (k ) | V(j) − b2 3 j=1

M2

δζ 2 ∑ |x (j) − x (k ) | V(j)

(22)

(34)

j=1

While in peridynamics under such deformation condition as shown in Fig. 5(b), the bond deformation can be expressed as

|y(j) − y(k ) | = [1 + (cos 2ϕ) ζ ] ξ

Equating Eq. (32) and Eq. (34), we obtain

4aζ 2 +

(23)

2πhδ 4ζ 2 ¯ b 3

=

1 πhδ 4ζ 2 ¯ + b + b1 δζ 2 ∑ |x (j) − x (k ) | V(j) 4 j=1

(24)

1

(25)

1 Q22 ζ 2 2

The above description and calibration are derived only for a single grain, but grain orientations are usually different between two neighbouring grains, and it is of great significance to consider grain boundary effects. However, the macroscale description of grain boundaries has not been fully recognized [58]. To roughly characterize grain boundary PD bonds which connect two respective material points in two different grains in this work, we assume that

(27)

While in peridynamics under such deformation condition as shown in Fig. 5(c), the bond deformation can be expressed as

|y(j) − y(k ) | = [1 + (sin2ϕ) ζ ] ξ

(28)

b(ik, j) =

For this deformation, the strain energy density in peridynamics can be evaluated as 2 πhδ 4ζ 2 ¯ b − b2 δζ 2 ∑ |x (j) − x (k ) | V(j) 4 j=1

2b(ik ) b(ij) b(ik ) + b(ij)

(37)

= b¯, b¯ + b1, b¯ − b2 ; and b(k ) represents corresponding in which parameter for the material point x (k ) , as shown in Fig. 6. bi

M

2 W(PD k ) = aζ +

(36)

3.3. Grain boundary consideration

(26)

According to the CCM, the corresponding strain energy density at material point x (nk ) can be expressed as

W(CCM = k)

+ 2Q12 + Q22

⎧ a = 2 (Q12 − Q66) ⎪ 6Q b¯ = 664 ⎪ πhδ ⎪ ⎪ b1 = MQ111 − Q12 − 2Q66 2δ ∑ (| x (j) − x (k ) |) V(j) ⎨ j=1 ⎪ ⎪ b2 = Q12 − Q22 + 2Q66 M2 ⎪ 2δ ∑ (| x (j) − x (k ) |) V(j) ⎪ j=1 ⎩

3.2.3. Uniaxial stretch in transverse direction Due to this loading condition, the stresses in Eq. (14) become

Q11 Q12 0 ⎤ ζ Q ζ σ ⎧ ⎫ ⎧ 12 ⎫ ⎧ σ11 ⎫ ⎡ 0 ⎥ 0 = Q22 ζ 22 = ⎢Q12 Q22 ⎨ σ12 ⎬ ⎢ ⎨ ⎬ ⎨ ⎬ 0 Q66 ⎥ ⎩ ⎭ ⎣ 0 ⎦ ⎩0⎭ ⎩ 0 ⎭

j=1

)ζ 2 (35)

M

1 πhδ 4 ¯ 1 b + b1 δ ∑ |x (j) − x (k ) | V(j) = Q11 4 2 j=1

1 (Q11 2

Combining Eq. (19), Eq. (24), Eq. (29) and Eq. (34) from the above four simple loading directions, the four PD parameters can be obtained as

where M1 denotes all the interacting material points in the principal direction. Equating Eq. (22) and Eq. (24), we obtain

a+

M2

j=1

=

M

aζ 2

M1

+ b1 δζ 2 ∑ |x (j) − x (k ) | V(j) − b2 δζ 2 ∑ |x (j) − x (k ) | V(j)

For this deformation, the strain energy density in peridynamics can be evaluated as

W(PD k)

(33)

For this deformation, the strain energy density in peridynamics can be evaluated as

According to the CCM, the corresponding strain energy density at material point x (nk ) can be expressed as

1 = Q11 ζ 2 2

(32)

While in peridynamics under such deformation condition as shown in Fig. 5(d), the bond deformation can be expressed as

3.2.2. Uniaxial stretch in principal direction Due to this loading condition, the stresses in Eq. (14) become

W(CCM k)

1 (Q11 + 2Q12 + Q22) ζ 2 2

W(CCM = k)

(20)

Q11 Q12 0 ⎤ ζ Q ζ σ ⎧ ⎫ ⎧ 11 ⎫ ⎧ σ11 ⎫ ⎡ 0 ⎥ 0 = Q12 ζ 22 = ⎢Q12 Q22 ⎨ σ12 ⎬ ⎢ ⎨ ⎬ ⎨ ⎬ 0 Q66 ⎥ ⎩ ⎭ ⎣ 0 ⎦ ⎩0⎭ ⎩ 0 ⎭

(31)

(29)

4. Numerical modelling and results

where M2 denotes all the interacting material points in the transverse direction. Equating Eq. (27) and Eq. (29), we obtain

Based on the description method in Section 3, three plates with 6

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and 4.2. Moreover, crack growth is simulated in the polycrystalline plate shown as Fig. 7(c), with length L = 240 mm , width W = 100 mm and crack length a0 = 25 mm . The PD spacing size is set as dx = 0.5 mm , and the grain distributions are generated based on Voronoi diagram; to roughly characterize brittleness of grain boundaries as compared with grain interior, the critical bond stretch is set as ε∗, trans = 0.055 for transgranular bonds and ε∗, inter = 0.05 for the intergranular ones. Crack branching is observed and the crack-tip fields are studied in Section 4.3. 4.1. Single-crystal model: Stationary crack-tip fields The effect of φ is firstly studied with θ = 0° in which direction the anisotropy is most obvious, as shown in Fig. 8. The crack-tip fields of strain in x-direction, strain in y-direction and shear strain are obtained for φ = 0°, 45° , 90° cases with applied load S = 160 MPa in Fig. 8(a). It shows that for the symmetry cases of φ = 0° and φ = 90° , the crack-tip strain fields are also strictly symmetric; for the asymmetry case of φ = 45° , the εyy strain is relatively smaller in the principal direction (i.e. parallel to φ ) while larger in the transverse direction (i.e. perpendicular to φ ). Besides, compared with φ = 90° case which is somehow similar to the standard pattern in isotropic condition, the φ = 0° cases show relatively limited εxx strain but larger εyy strain at crack tip. It can be concluded that the grain orientation has an impact on the crack-tip strain fields, bringing in anisotropy to be specific. We also extract the von-Mises equivalent strain of material points on the reference line along the crack for φ = 0°, 30°, 45° , 60°, 90° cases shown in Fig. 8(b), and it shows that the order relationships among the von-Mises equivalent strains on the reference line in the above cases are different according

Fig. 6. The schematic for the grain boundary consideration.

grain effects are modeled as shown in Fig. 7; particularly, the area of equiaxed grains is the focus of this paper. The whole plate is firstly set to be of single grain orientation as Fig. 7(a) to study the effects of φ and θ , and then the plate is set as a polycrystalline model to consider orientation differences among all the grains as Fig. 7(b). The geometries for the above two plates are the same, with length L = 7 mm and crack length a0 = 2.1 mm ; the PD spacing size is set as dx = 0.025mm , and the grain distributions are generated based on Voronoi diagram for the second plate. The stationary crack-tip fields for the above two models are studied, and the results are presented respectively in Section 4.1

Fig. 7. (a) The geometry of single-crystal model; (b) The geometry of polycrystalline model; (c) The geometries of single-edge notched tension specimen with polycrystalline microstructures, and the black-dashed rectangle shows the area to observe. 7

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Fig. 8. (a) The crack-tip strain fields for single-crystal model: φ = 0°, 45° , 90° ; (b) the von-Mises equivalent strain of material points on the reference line along the crack: φ = 0°, 30°, 45° , 60°, 90° .

to the location. When the location is near the crack tip, the von-Mises equivalent strain on the reference line is the largest for the φ = 60° case but the smallest for φ = 45° case; as the location becomes farther, it becomes the largest for φ = 90° case and smallest for φ = 30° case; when the location is much farther, it is the largest for φ = 45° , 60°, 90° cases and smallest for φ = 0° case. Besides, according to [59] it can be approximated that the material yields when the von-Mises equivalent strain reached 0.01, and it can be seen from Fig. 8(b) that differences in the von-Mises equivalent strain are more obvious around the crack-tip where plasticity occurs, which also accords with our previous experimental observations reported in [59]. Then the effect of θ is studied by extracting the von-Mises equivalent strain of material points on the reference circle with φ respectively fixed at φ = 0°, 30°, 45° , 60°, 90° , as shown in Fig. 9. The shape of strain distribution is rather alike for it is mainly decided by mechanical factor, while the deviations from isotropic fields are due to the grain orientation effects. In addition, strains are relatively smaller when θ is smaller and vice versa, which is due to the elastic modulus distribution in Fig. 4; when θ is smaller, the elastic modulus is larger according to Eq. (13), and the strain will thus be smaller. An exception occurs when φ = 0°, θ = 60° , in which intense anisotropy is observed with maximum strain towards the polar axis of 150° . This indicates that both θ and φ have an impact on crack-tip fields, and a 2-D modelling method may not be enough to consider the grain orientation effects. It is worth noting that the traditional engineering method to characterize crack tip, i.e. expressions the stress intensity factor K as the combinations of dimensionless geometry correction factor, remote stress and crack length, will give the same estimations of crack tip and cannot reflect the differences caused by material effects in Fig. 9; however, the grain orientation has an impact on the crack-tip fields, and will make a difference to K if it is derived from the deformation around the crack tip. Therefore, it is significant to develop such a method to consider grain orientation effects on crack-tip fields.

S = 100 MPa, 140 MPa, 180 MPa . For model I, the εxx strain is similar to the standard butterfly shape in model I while the εyy strain tends to concentrate towards the left part; however, the εeq strain tends to deviate towards to the right, and the shapes of grains as well as grain boundaries can be identified; the εxy strain displays obvious asymmetry around the crack tip; the crack-tip plastic zone is obviously different from the standard one for isotropic and homogeneous materials. The crack-tip fields of model II seem to be the most similar to those for isotropic and homogeneous materials, but some anisotropy and heterogeneities can still be observed. For model III, a grain boundary dividing the red islands into two parts can be easily observed from the εyy strain fields under S = 180 MPa ; from the standpoints of material science, this is due to the mechanism that the grain orientation differences between two grains lead to different directions in which they respectively tend to deform, but the deformation compatibility condition must be satisfied at the grain boundary. To sum up, the above results show that the differences in orientations among grains lead to anisotropy as well as heterogeneity of crack-tip strain fields, and different grain orientation distributions will also lead to differences in anisotropy and heterogeneity. Besides, it seems that such differences are easier to obtain from the crack-tip plastic zone rather than strain fields, which possibly indicate that the grain orientation distributions may have a greater impact on plasticity. As shown in Fig. 12, we extract the von-Mises equivalent strain of material points on the reference line as well as reference circle for S = 160 MPa . The von-Mises strains on the reference line, as shown in Fig. 12(a), indicate that anisotropy and heterogeneity of crack-tip strain fields are obviously more severe if the material point is closer to the crack tip. Again, we approximate that the material yields when the vonMises strain reaches 0.01, and the conclusion is same to that in Section 4.1: differences in the von-Mises equivalent strains are more obvious around the crack-tip where plasticity occurs. However, the effects of grain boundaries are not clearly reflected in Fig. 12(a), which is possibly due to the inaccuracy of grain boundary characterization Eq. (37). The von-Mises strains on the reference circle, as shown in Fig. 12(b), indicate that grain orientation distributions have an impact on the crack-tip fields. The results are similar between model I and model III, but show a clear difference for grain ① in model II; since the orientation of grain ① is similar to each other between model II and model III, it can be deduced that the grain orientation effects are not only decided by the exact orientation in a certain grain, but also influenced by the grain orientation distributions among the adjacent grains. This is typically

4.2. Polycrystalline model: stationary crack-tip fields To study the grain orientation effects among neighboring grains, three plate models with different orientation distributions are established but the grain boundary distributions are kept unchanged, as shown in Fig. 10. As shown in Fig. 11, the evolutions of stationary crack-tip strain fields are monitored for all three models at three loads 8

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Fig. 9. The von-Mises equivalent strain of material points on the reference circle: (a) schematic for the reference circle; (b–f): φ = 0°, 30°, 45° , 60°, 90° cases.

Mises strain fields are in approximately symmetric butterfly shape when the crack paths are relatively straight, while the grain orientation seems to be of little significance; also, for crack tip (a2) where the local deflection is severe, the crack-tip von Mises strain fields display an obvious asymmetric shape. For crack tip (a4) where a secondary crack occurs due to grain effects, the asymmetry of crack-tip von Mises strain fields becomes more severe, and the maximum equivalent strain concentrates in the direction towards the location of secondary crack; the von Mises strain fields of crack tip (a4) is similar to the von Mises stress fields measured by digital image correlation (DIC) techniques in our previous experiments [18], shown as the second image of crack-tip von Mises stress field in Fig. 13(b). For crack tip (a5), however, the crack-tip von Mises strain fields display an irregular shape due to the crack branching, local crack deflection as well as grain effects. The above results indicate that the crack growth behaviors are closely relevant to crack-tip fields, and the final crack path is affected by both mechanical factors and material factors.

multiscale or nonlocal features, but a further study needs more accurate characterization of grain boundaries. 4.3. Polycrystalline model: crack-tip fields at branching To study the grain orientation effects on the crack path and the corresponding crack-tip fields at different crack-tip locations, the plate model in Fig. 7(c) is adopted and the results are presented in Fig. 13(a). The whole crack paths are relatively straight at first with only slight deflections; then, secondary cracks occur and the crack path will thus display branching patterns, which has been observed in our experiments [18] as shown in Fig. 13(b) despite grain orientation analysis are lacking. The crack branching is caused by the distributions of grain orientations and grain boundaries, because the crack path must be straight without such factors. Such crack branching phenomenon is spontaneously simulated for no extra criteria for crack growth are introduced, which indicates that peridynamics is capable of simulating such complex crack growth behaviors caused by grain orientation effects in AM materials. Furthermore, the crack-tip fields at different crack-tip locations also display some unique characteristics. For crack tip (a1) and (a3), the von

5. Discussions on the proposed method A method to describe grain orientation effects for α -phase AM Ti-

Fig. 10. (a) The original plate and (b) model 1, (c) model 2, (d) model 3 with different grain orietation distributions. 9

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Fig. 11. Evolutions of stationary crack-tip strain fields for (a) model 1, (b) model 2, (c) model 3.

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Fig. 12. The von-Mises equivalent strain of material points on the (a) reference line and (b) reference circle.

compared with CPFEM. Additionally, the theoretical derivation is much simpler and less parameters needs to be calibrated in the proposed method, and further parameter studies are necessary to explore their impacts. As for the modelling, convergence studies of mesh sizes and total numbers of grains have been previously performed in studies such as [57], and are thus omitted in this paper since they are not the focus of this work. However, we indeed found that the modelling geometry does has an impact on the results. To be specific, if merely the black-dashed rectangle areas in Fig. 7(c) are modelled, the obtained deformation fields and the crack paths will be different; even if the whole model is set as isotropic materials without grain orientation effects, such differences still exist. This may be due to the sensitivity to boundary conditions in peridyanmics. Such findings remind that if we hope to conduct peridynamic simulations on crack problems and then compare the results with experimental observations, we must assure that the geometries of peridynamic models should be the same as those of experiment specimens. In fact, many previous studies have not stick to this rule, mainly because of the low computational efficiencies of peridynamic simulations especially for convergence studies and parameter studies. Although some inspiring conclusions can be obtained by this description method for grain orientation effects, there are more works to do. Firstly, with the grain orientation distributions randomly generated like Section 4.2, it is difficult to directly obtain grain orientation distribution effects on crack-tip fields by comparison; more careful studies may be under the conditions that the grain orientation for other grains should be kept unchanged while the grain orientation for only one grain varies. Secondly, the description method for grain boundaries in this paper is rather rough, and their impacts on crack-tip fields are not well reflected; the macroscopic description for grain boundaries surely makes a difference in the simulation results, but a solid or acknowledged description method is still lacking. Thirdly, not only grain orientations but also phases and defects exert obvious effects on crack-tip fields as well as crack growth behaviors, which are not taken into consideration in this work. At last, 3D models should be simulated to better verify the proposed method, and much larger computing scale is inevitable.

alloy in peridynamics is proposed in this paper. To be specific, the method takes an elliptical distribution relationship of the plane angle θ to describe the anisotropy of average elastic modulus among different crystal plane for HCP structure, and assumes that it is similar to a unidirectional-lamina composite within a certain crystal plane for each grain. The peridynamic parameters are then calibrated by equaling the strain energy density under corresponding several simple deformation conditions to those obtained by classic continuum mechanics. For polycrystalline considerations, the grain boundary bonds are roughly characterized with a simple harmonic average rule. Then, three plates are simulated and the results are compared with previous experimental observations. Finally in this section, discussions on the features of the proposed method as well as some further considerations are presented. Compared with FEM models, the proposed method is established in the peridynamic framework, which exploits the advantages of peridynamics to simulate crack propagation problems. FEM-based models need extra criteria for cracks to grow, and different criteria may result in different crack patterns, which brings certain difficulties into the analysis of synergetic interactions among material effects, crack-tip fields and crack growth behaviors. Within peridynamic theory, however, complex crack patterns can be captured by a simple bond breakage criterion; and its capabilities of describing crack-tip fields are also validated by the displacement extrapolation method in Section 2.2. Thus, a method to describe material effects on crack-tip fields in peridynamic framework avoid the difficulties in dealing with simulation techniques of crack growth, which provide convenience for achieving the goal of revealing interactive and integrated relationships among material effects, crack-tip fields and crack propagation behaviors. Besides, the discreteness form and the nonlocality feature of peridynamics also make it easier to perform multiscale simulations. As for parameter calibrations, there exist both similarities and differences between CPFEM and the proposed method. Generally, CPFEM stands on the crystal plasticity theory with its parameters calibrated to tension response of materials; such a method belongs to the scope of mesomechanics and is usually adopted to study dislocation and slip behaviors. Thus, most of the CPFEM simulation models are confined to the level of several hundred micrometers, and temporarily it is not appropriate for CPFEM to study material effects on mechanical behaviors in macroscale. As a contrast, the proposed method stands on macroscopic continuum mechanics with its parameters calibrated to strain energy density in typical loading cases, and the proposed method can be naturally adopted to study crack-tip fields in the specimen level. To summarize, the proposed method can be seen as a macroscopic way to deal with grain orientation effects on mechanical behaviors if

6. Conclusions In this work, a macroscale description method for grain orientation effects in α -phase additively manufactured titanium alloy is proposed within peridynamic framework, and peridynamic simulations of grain 11

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Fig. 13. (a) The process of crack growth, the grain orientations along the crack path, and the crack-tip von Mises strain fields according to different crack lengths; (b) The crack branching observed and the crack-tip von Mises stress measured by DIC techniques [18].

orientation effects on crack-tip fields for equiaxed grains in additively manufactured titanium alloy are performed. Based on the proposed method and corresponding peridynamic simulations in this paper, the main conclusions can be listed as following:

the proposed characterization method. Besides, such impacts are generally more obvious in poycrystalline models than single-crystal models, which indicates that anisotropy and heterogeneity are affected more by the grain orientation differences among adjacent grains rather than the grain orientation itself. 2. For polycrystalline models, the effects of grain orientation distributions lead to more severe asymmetry of plastic crack-tip

1. Anisotropy and heterogeneity introduced by grain orientation into crack-tip strain fields can be well simulated by peridynamics with 12

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parameters than that of elastic ones, and such differences are more obvious with larger differences in orientations among adjacent grains. This indicates that the anisotropy and heterogeneity of cracktip fields are due to the different tendency of material points to yield, which essentially results from mismatch of deformation among grains caused by grain orientation distributions. 3. The grain orientation distributions can result in secondary cracks and crack branching, where crack growth behaviors are closely relevant to crack-tip fields and the final crack path is affected by both mechanical factors and material factors; similar crack-tip fields and crack branching behaviors are also observed in our previous experiments. This indicates that there exist interactive relationships among grain features, crack-tip fields and crack propagation behaviors, and it is better to study these three subjects synergistically rather than separately. 4. The proposed method may help develop a macroscale way to reveal interactive relationships among grain features, crack-tip fields and crack propagation behaviors by peridynamic simulations. To achieve this goal, material factors such as grain boundaries, phases and 3D grain features should be carefully studied in further works, and simulation aspects such as computational efficiencies as well as modelling techniques also need improving.

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CRediT authorship contribution statement Binchao Liu: Conceptualization, Investigation, Writing - original draft, Writing - review & editing. Zhongwei Yang: Methodology, Software, Validation, Formal analysis. Rui Bao: Resources, Supervision, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The NSFC (China) is acknowledged for supporting the project (11672012). References [1] S. Shao, M. Khonsari, S. Guo, et al., Overview: additive manufacturing enabled accelerated design of Ni-based alloys for improved fatigue life, Addit. Manuf. (2019) 100779. [2] M. Liu, S. Liu, W. Chen, et al., Effect of trace lanthanum hexaboride on the phase, grain structure, and texture of electron beam melted Ti-6Al-4V, Addit. Manuf. 30 (2019) 100873. [3] C. Qiu, Q. Liu, Multi-scale microstructural development and mechanical properties of a selectively laser melted beta titanium alloy, Addit. Manuf. 30 (2019) 100893. [4] C.M. Liu, H.M. Wang, X.J. Tian, et al., Microstructure and tensile properties of laser melting deposited Ti–5Al–5Mo–5V–1Cr–1Fe near β titanium alloy, Mater. Sci. Eng. A 586 (2013) 323–329. [5] H. Wang, S. Zhang, T. Wang, et al., Progress on solidification grain morphology and microstructure control of laser additively manufactured large titanium components, J. Xihua Univ. (Natural Science Edition) 37 (2018) 9–14. [6] Z. Zhao, J. Chen, X. Lu, et al., Formation mechanism of the α variant and its influence on the tensile properties of laser solid formed Ti-6Al-4V titanium alloy, Mater. Sci. Eng. A 691 (2019) 16–24. [7] C. Madikizela, L.A. Cornish, L.H. Chown, et al., Microstructure and mechanical properties of selective laser melted Ti-3Al-8V-6Cr-4Zr-4Mo compared to Ti-6Al-4V, Mater. Sci. Eng. A 747 (2019) 225–231. [8] L. Zhou, T. Yuan, et al., Microstructure and mechanical properties of selective laser melted biomaterial Ti-13Nb-13Zr compared to hot-forging, Mater. Sci. Eng. A 725 (2018) 329–340. [9] Y. Zhu, J. Li, X. Tian, et al., Microstructure and mechanical properties of hybrid fabricated Ti–6.5Al–3.5Mo–1.5Zr–0.3Si titanium alloy by laser Addit, Manuf. Mater. Sci. Eng. A 607 (2014) 427–434. [10] R. Konečná, L. Kunz, A. Bača, et al., Long fatigue crack growth in Ti6Al4V produced by direct metal laser sintering, Procedia Eng. 160 (2016) 69–76. [11] S. Lu, R. Bao, K. Wang, et al., Fatigue crack growth behaviour in laser melting

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