Interaction of defects,martensitic transformation and slip in metastable body centred cubic crystals of Ti-10V-2Fe-3Al: A study via crystal plasticity finite element methods (CPFEM)

Abstract

Metastable β titanium alloys are widely applied in many industries. These alloys can have plastic deformation via dislocation slip, twinning, stress-induced martensite (SIM), or a combination of these. These alloys fail in a ductile manner via a process of void nucleation, growth, and coalescence. Inherent defects, such as voids, are commonly attributed to poor mechanical properties. In this study, aspects of plastic anisotropy in damage accumulation are investigated for metastable crystals that deform by combined slip and SIM. The focus of this study is to understand the evolution of damage due to inherent voids in metastable Ti-10V-2Fe-3Al single crystals. This investigation is conducted using crystal plasticity-based 3D ﬁnite element (FE) calculations. A unit-cell FE model involving a spherical void is deformed under constant stress triaxiality and lode parameter. We investigated four triaxiality values at differing lode parameters in three crystal orientations. The void growth was found to be heavily dependent on crystal orientation at low triaxialities. At higher triaxialities, SIM is found to inhibit the void growth via accommodation of the required deformation in the surrounding material. Orientations aligned favourable with SIM undergo significantly less void growth. The accommodation of deformation in the surrounding matrix was found to help preserve the integrity of the void, preventing the localisation of deformation around the void. At lower lode parameter and at higher stress triaxiality this impedes the exponential growth of the void. While, at higher lode parameter with low triaxiality SIM was found to delay the collapse of the void into a crack like morphology. This study not only deepens our understanding of the mechanical behaviour of metastable β titanium alloys, but also unveils the complex interplay between inherent defects, stress-induced martensite, and slip-based plasticity within their crystalline structure, offering fresh perspectives on enhancing material performance.

Keywords

Computational mechanics crystal plasticity damage mechanics materials modelling multiscale modelling

Introduction

Metastable β titanium alloys have become one of the most popular classes of titanium alloys. For example, metastable $β$ titanium alloy Ti-10V-2Fe-3Al, is used in Boeing 777 aircraft’s landing gear structure, and in similar application on Airbus A340-500/600. Metastable $β$ alloys now find wide-ranging application in many industries including chemical process, power generation, marine, biomedical, and many more. They can be processed to have superior mechanical properties and workability (Lütjering and Williams, 2007). They can display numerous deformation mechanisms such as dislocation slip, twinning, stress-induced martensite (SIM), or a combination of these. These alloys and deformation mechanisms can be tailored to produce many sort-after properties, such as improved strength and ductility, or effects like shape-memory and super-elasticity. Furthermore, the wide variety of microstructures and properties achievable, result in both ductile and brittle failure being possible in metastable titanium alloys (Zhao et al., 2019). However, the majority of metastable β alloys fail in a ductile manner via a process of void nucleation, growth, and coalescence. It is well understood that voids can nucleate at material defects (such as inclusions (Garrison and Moody, 1987; Hu et al., 2020)) and areas of non-homogeneity within the material (Das, 2013; Garrison and Moody, 1987). For titanium alloys areas such as, grain boundaries, intracrystalline α precipitation areas, shear bands, lamellar α interface, β flecks, and regions of martensite can also act as nucleation sites (Asim et al., 2019; Das, 2013; Garrison and Moody, 1987; Guan et al., 2015; Li et al., 2016; Shao et al., 2013; Tan et al., 2018; Terlinde et al., 1983; Zhao et al., 2019). Inherent defects are commonly attributed to poor mechanical properties in titanium processed via additive manufacturing (AM) (Fujita et al., 1996; Gong et al., 2015; Leuders et al., 2013; Liu et al., 2016; Wang et al., 2010). A technique which is increasingly popular with metastable titanium alloys (Colombo-Pulgarín et al., 2021). These defects, such as lack-of-fusion or gas entrapment pores, increase the porosity of the components and are often introduced during AM processing (Fujita et al., 1996; Gong et al., 2015; Leuders et al., 2013; Liu et al., 2016; Wang et al., 2010). Such pores can act as regions of stress concentration and failure (Brandl et al., 2012; Leuders et al., 2013; Liu et al., 2016; Renzo et al., 2021).

Significant research has been performed in the past for modelling ductile failure in metals and alloys by void nucleation, growth, and coalescence. It is commonplace to use 3D voided cell models to investigate the complex interaction between loading conditions, deformation mechanisms, crystal anisotropy, grains boundary effects and void growth and coalescence at a mesoscale. A technique first pioneered by Tvergaard, Koplik and Needleman (Koplik and Needleman, 1988; Tvergaard, 1982), it has since been used to investigate ductile failure of many materials. In the past, such approaches were used at macroscale where anisotropy was ignored. However, in our previous works we demonstrated the use of stress triaxiality and lode parameter for anisotropic single crystalline materials (Asim et al., 2017; Guo et al., 2020; Ha and Kim, 2010; Liu et al., 2007; Srivastava and Needleman, 2013). A spherical void growth and coalescence for various crystal orientations, stress triaxiality, initial porosity, lode parameter, activated slip and elastic anisotropy was investigated in FCC single crystals by (Asim et al., 2017; Guo et al., 2020; Ha and Kim, 2010; Liu et al., 2007; Srivastava and Needleman, 2013). Asim et al. (2017) also compared the void growth using local and non-local CPFEM models for spherical and ellipsoidal voids in aluminium alloys. Similar studies for spherical voids in BCC single crystals has been conducted in the recent past (Guo and Li, 2019; Yerra et al., 2010; Yu et al., 2010). Guo and Li (Guo and Li, 2019) presented a CPFEM study of void growth in a unit cell of BCC single crystal containing a void, before applying the results to polycrystal studies. Savage et al. (Savage et al., 2018) extended this to analyse BCC polycrystals under axisymmetric tension and compression to study the void growth. Yang and Dong (Yang and Dong, 2009) and Jeong et al. (Jeong et al., 2018) explicitly modelled a void at the interface of two grains with different orientations, for an FCC and BCC structure, respectively. Asim et al. (Asim et al., 2019) investigated the void growth in single crystal HCP with slip and extended on the grain interface work of (Jeong et al., 2018; Yang and Dong, 2009) by modelling a void at a BCC and HCP interface in dual phase Ti-10V-2Fe-3Al. Selvarajou et al (Selvarajou et al., 2019) investigated the effects of deformation twinning and slip on void growth and coalescence in HCP single crystals. Subsequently, many models have been proposed to account for these effects in crystal plasticity formulations. Recently, more complex interactions such as phase boundary inclination in dual phase materials have been also been incorporated (Asim et al., 2019). However, there has not been any investigation on the effect of SIM on damage evolution due to void growth and hence we present first of its kind CPFEM study on this.

As previously stated, metastable β titanium alloys can exhibit several deformation mechanisms. Modern design of these alloys exploits these deformation mechanisms to produce desirable material attributes. The solid-state martensitic transformations from body centered cubic (BCC) $β$ to orthorhombic $α ″$ martensite in particular (Bignon et al., 2019; Castany et al., 2018; Chen et al., 2021, 2022; Danard et al., 2019; Li et al., 2015). It has also been shown that SIM formation in this class of alloy can produce an improved compromise between strength and ductility (Grosdidier et al., 2000; Neelakantan et al., 2009; Sun et al., 2002). It is suggested that stress-induced martensite delays the material failure by fracture in Ti-7Cr-1Al-1Fe alloy by relaxing areas of locally intense strain fields (Danard et al., 2021). This is supported by the Bhattacharjee et al. (Bhattacharjee et al., 2006) and Paradkar and Kamat (Paradkar and Kamat, 2011) investigation of fracture toughness in Ti-10V-2Fe-3Al and Ti–15Al–8Nb, respectively. Both authors found SIM to increase the fracture toughness. Paradkar and Kamat (2011) observed the transformation in the surrounding $β$ matrix and grains accommodating the deformation and preventing extension/propagation of the crack. However, upon further applied deformation the alloy undergoes plastic deformation at the crack tip and void nucleation at the $α ″ / β$ interfaces. Void nucleation at the $α ″ / β$ interfaces is commonly observed in metastable $β$ titanium alloys (Paradkar et al., 2008; Reck et al., 2019; Xiao et al., 2021; Zhao et al., 2019). Xaio et al. (2021) reports a dual impact of SIM on the damage behaviour of metastable Ti-10V-2Fe-3Al. It was found that in larger grains with larger $α ″$ lathe spacing, SIM facilitates the nucleation of cracks via dislocation pileup at the $α ″ / β$ interface. In contrast, with smaller $α ″$ lathe spacing SIM inhibited the crack nucleation. Understanding of the influence SIM has on damage propagation in these alloys is of great importance. This is precisely the aim of this study. We look to investigate the void growth in metastable $β$ titanium alloys and the influence of SIM in pre-existing voids and defects.

Methodology and void cell model details

Finite element based void cell model

A spherical void encased in a cubic matrix is modelled. Taking advantage of symmetry (Lin et al., 2006; Tekog˜ lu, 2014), we model 1/8 of the full void/cube cell (as shown in the Figure 1). This model was selected to reduce the computational time which was available for the presented work.

Figure 1.

3D cell model with spherical void at the centre, also showing 1/8 symmetry model with mesh.

Figure 1 shows a cross section of the void model. It also shows relevant geometric parameters. $R_{1}$ and $R_{3}$ are the void radius in the 1-axis and 3-axis directions. While $L_{1}$ and $L_{3}$ are the cube parameters. Out of plane dimensions ( $R_{2}$ and $L_{2}$ ) are orthogonal to the 1–3 plane. The following aspect ratios ( $W$ ) can be used to characterise the void shape throughout the simulation:

W_{1} = \frac{R_{1}}{R_{3}}, W_{2} = \frac{R_{1}}{R_{2}}

(1)

where,

R_{2}

is the radius of the void in the 2-axis direction (out of plane in Figure 1). As the void is initially a sphere encased in a cubic matrix,

W_{1} = W_{2}

and

L_{1} = L_{2} = L_{3}

prior to any applied deformation. The initial porosity

f_{0}

of the void cell is equal to 0.001 (0.1%), in keeping with previous void studies on Ti-1023 (Asim et al., 2019a, 2019b).

The model was meshed using ﬁrst-order brick elements with reduced integration (C3D8R) (Ds Simulia, 2019). A total of 1775 elements are used in the model. This was found to give optimal results with respect to convergence and simulation time. The mesh is reﬁned in the region surrounding the void while keeping the element regular and undistorted. Symmetry boundary conditions are applied to the three internal faces of the 1/8 void model. A multi-point constraint (MPC) was applied as an ABAQUS user subroutine to the three external faces. Springs are applied in conjunction to the MPC which impose displacements to the free surfaces keeping the triaxiality ( $T$ ) and lode parameter ( $L$ ) constant throughout the simulation. The methodology used is adapted from (Asim et al., 2017, 2019a, 2019b; Lin et al., 2006; Tekog˜ lu, 2014), the reader is directed to (Tekog˜ lu, 2014) for further details. The following stress triaxialities are investigated, $T = 1 / 3, 1, 2, 3$ , where $T$ is defined at the ratio of hydrostatic stress ( $σ_{h}$ ) to effective von mises stress ( $σ_{v m}$ ):

T = σ_{h} / σ_{v m}

(2)

A triaxiality of $1 / 3$ represents a uniaxial stress state, while $T = 3$ is more representative of the stress state observed at a crack tip (Asim et al., 2019). Furthermore, three lode parameters are also investigated, $L = - 1, 0, 1$ , where L is defined by:

L = \frac{2 σ_{2} - σ_{1} - σ_{3}}{σ_{1} - σ_{3}}

(3)

Here, $σ_{1}$ , $σ_{2}$ , and $σ_{3}$ represent the 1st, 2nd, and 3rd principal stresses, respectively. The significance of these three lode parameters is discussed in section ‘Effect of lode parameter, SIM, and slip on void growth’. Three orientations are considered, representing the outmost orientations on a typical stereographic plot (Kim et al., 2006). Visual representation of the orientations considered is given in Figure 2.

Figure 2.

Visual representation of crystal orientation in relation to loading (Christie et al., 2024).

Material constitutive law

The crystal plasticity theory used during this work is based on the framework of Christie et al. (Christie et al., 2024). A summary of the framework is discussed in the following for brevity. The total deformation gradient is given by $F = V^{e} R^{e} {F^{p} F}^{t r}$ , where $F^{p}$ and $F^{t r}$ are conventional (slip based) plastic and transformation deformation gradients. While $V^{e}$ and $R^{e}$ are the left symmetric elastic stretch tensor and rigid body rotation, respectively. The velocity gradient is expressed as,

L = {\dot{V}}^{e} {V^{e}}^{- 1} + V^{e} L * {V^{e}}^{- 1}

(4)

where,

L *

describes both the plastic flow due to crystallographic slip, martensitic transformation, and the rotation of the lattice.

L^{*} = {\dot{R}}^{e} {R^{e}}^{T} + R^{e} (\sum_{i = 1}^{N_{s}} {\dot{γ}}^{(i)} {\bar{s}}^{(i)} \otimes {\bar{n}}^{(i)} + F^{p} (\sum_{α = 1}^{N_{t}} {\dot{ξ}}^{(α)} {\hat{b}}_{0}^{(α)} \otimes {\hat{m}}_{0}^{(α)}) {F^{p}}^{- 1}) {R^{e}}^{T}

(5)

where, vectors

{\bar{s}}^{(i)}

and

{\bar{n}}^{(i)}

are, unit vectors describing the slip direction and the normal to the slip plane of the slip system

(i)

, respectively;

{\dot{γ}}^{(i)}

is the effective shear rate on that same system.

N_{s}

is the total number of active slip systems.

Furthermore, ${\dot{ξ}}^{(α)}$ is the rate of transformation, ${\hat{b}}_{0}^{(α)}$ and ${\hat{m}}_{0}^{(α)}$ are the martensite shape strain and habit plane normal vector, for each transformation system, determined using the theory of martensitic transformations (Ball et al., 1987; Hane and Shield, 1998). $N_{t}$ is the total number of transformation systems; a total of 96 transformation systems are possible for this BCC to orthorhombic transformation. The following conditions must also be obeyed for transformation,

0 \leq ξ^{α} \leq 1, 0 \leq {\sum_{α = 1}^{N_{t}} ξ^{α} \leq 1 and ξ_{A}^{} = 1 - \sum_{α = 1}^{N_{t}} ξ^{α}}^{}

(6)

Here, $ξ^{α}$ is the volume fraction for transformation systems $α = 1, 2, \dots N_{t}$ and $ξ_{A}^{}$ denotes the volume fraction of the parent phase.

The shear rate on slip system (i) is determined using the well-known power law (Asim et al., 2019a, 2019b):

{\dot{γ_{A}}}^{(i)} = {\dot{γ}}_{0}^{} {(\frac{| τ_{s}^{(i)} |}{{k_{s}^{(i)}}_{}^{}})}^{\frac{1}{m}} sign (τ_{s}^{(i)})

(7)

where,

{\dot{γ}}_{0}^{}

and

m

are the reference shear strain rate and strain rate sensitivity parameter, respectively.

τ_{s}^{(i)}

is the resolved shear stress in slip system (i), defined as

τ_{s}^{(i)} = \tilde{S} : sym (R^{e} [{\bar{s}}^{(i)} \otimes {\bar{n}}^{(i)}] {R^{e}}^{T})

. The effects of transformation on the plastic deformation is considered by weighting the effect of slip by the remaining volume fraction of the parent (untransformed) phase

ξ_{A}^{}

(Tjahjanto et al., 2008),

{\dot{γ}}^{(i)} = \frac{ξ_{A}^{}}{J_{t r}} {\dot{γ}}_{A}^{(i)}

(8)

The hardening law is taken from Kocks and Mecking (1979):

{\dot{κ}}_{s}^{} = Θ_{κ} (κ_{s}, {\dot{γ}}^{i}), Θ_{κ} (κ_{s}, {\dot{γ}}^{i}) = h_{0} (\frac{κ_{s, S} - κ_{s}}{κ_{s, S} - κ_{s, 0}}) \sum_{i = 1}^{N_{s}} |{\dot{γ}}^{i}|

(9)

where,

h_{0}

is the initial hardening rate,

κ_{s, 0}

is the initial critical resolved shear stress (CRSS) and

κ_{s}

is the current CRSS. Lastly,

κ_{s, S}

is the saturation strength given by the relationship of Follansbee and Kocks (Follansbee and Kocks, 1988):

κ_{s, S} = κ_{s, S 0} {[\frac{\sum_{i = 1}^{N_{s}} |{\dot{γ}}^{i}|}{\dot{γ_{S 0}}}]}^{m^{'}}

(10)

with

κ_{s, S 0}

\dot{γ_{S 0}}

, and

m^{'}

all being material parameters.

The evolution of martensite ( ${\dot{ξ}}^{(α)}$ ) is modelled using the following power based, rate-independent, kinetic law:

{\dot{ξ}}^{(α)} = \{\begin{array}{l} {\dot{ξ}}_{0}^{} {(\frac{f^{(α)}}{f_{c r}^{(α)}})}^{1 / m}, 0 \leq \sum_{α = 1}^{N_{t}} ξ^{α} < 1 and \frac{f^{(α)}}{f_{c r}^{(α)}} \geq 0 \\ 0, otherwise \end{array}

(11)

where,

{\dot{ξ}}_{0}^{}

is the reference transformation rate, m is a strain rate sensitivity parameter,

f^{(α)}

is the driving force for transformation,

f_{c r}^{(α)}

is the critical value and

ξ^{(α)}

is the total volume fraction of each system. The following simple formulation for transformation resistance evolution is used throughout in this work (Christie et al., 2024)

{\dot{f}}_{c r}^{(α)} = Ψ (R^{(α)}, {\dot{ξ}}^{(α)}), Ψ (R^{(α)}, {\dot{ξ}}^{(α)}) = R^{(α)} {\dot{ξ}}^{(α)}

(12)

R^{(α)} = \{{(R}_{0} - R_{s}) {sech}^{2} (\frac{{(R}_{0} - R_{s}) ξ_{}^{(α)}}{v_{0}}) + R_{s}\} [1 + R_{h} {(\sum_{α = 1}^{N_{t}} ξ^{(α)})}^{η}]

(13)

where,

v_{0}

is a normalising resistance parameter (MPa) and

R_{0}

is the initial resistance rate (MPa);

R_{s}

is saturation resistance rate (MPa). In the second term of equation (13),

R_{h}

and

η

are all material parameters used to increase the resistance to transformation once it has begun due to the interaction between transformation systems. The initial critical value for driving force is assumed same for all systems. The driving force is defined as (Christie et al., 2024):

f^{(α)} = \tilde{S} : [R^{e} (sym ({\bar{C}}^{e} F^{p} \sum_{α = 1}^{N_{t}} \hat{b_{0}^{(α)}} \otimes \hat{m_{0}^{(α)}} {F^{p}}^{- 1},)) {R^{e}}^{T}] + \frac{1}{2} (C^{A} - (1 + δ_{T}) C^{α}) {\tilde{E}}^{e} : {\tilde{E}}^{e} + \frac{1}{2} c_{k} (μ_{A} - (1 + δ_{T}) μ^{(α)}) ϵ_{s}^{2}

(14)

where,

δ_{t r} = b_{}^{(α)} ∙ m_{}^{(α)}

is the volumetric growth associated with each transformation system.

C_{}^{A}

and

C^{(α)}

are elasticity tensor of parent (

β

) and martensitic phases, respectively.

μ_{A}^{}

and

μ^{(α)}

are the isotropic shear moduli parent and martensitic transformation system

α

, respectively.

c_{k}

is a scaling material parameter for the elastic lattice strain energy.

ϵ_{s}^{}

is a strain-like variable assumed to be related to the elastic lattice strain fields generated on the slip systems by dislocations.

{\tilde{E}}^{e}

is the elastic Green–Lagrange strain, conjugate to the 2nd Piola-Kirchhoff stress

\tilde{S}

Material parameter identification

The material parameters used for this study are acquired via an inverse modelling approach (see (Asim et al., 2017, 2019a, 2019b), and references therein). Manual fitting of the experimental stress–strain data is used to determine the material parameters. The result of the inverse modelling (Figure 3) show good agreement between the experimental and simulated responses during all stages of deformation.

Figure 3.

Model predictions for metastable $β$ titanium alloy Ti-10V-2Fe-3Al after parameter identification; experimental data taken from (Duerig et al., 1982).

All identified material parameters are presented in Table 1. The elastic constants for the BCC crystal are obtained from data for similar materials in the literature (Fisher and Dever, 1970; Ledbetter et al., 2004; Petry et al., 1991; Pinilla Ducreux et al., 2021; Raghunathan et al., 2007; Sun et al., 2007; Zhang et al., 2016, 2021), which were then iteratively adjusted (within the range identified in the literature) to match the experimental results. The elastic constants for the $α ″$ martensite are taken from the first principle calculations of Sun et al. (Sun et al., 2007). The properties are further scaled in the formulation to fit the experimental data. Similar approaches have been used for the shape memory alloy CPFEM (Anand and Gurtin, 2003; Ju et al., 2022; Thamburaja and Anand, 2001, 2002).

Table 1.

Elastic, plastic, and transformation parameters identified.

Elastic parameters
	C11 [GPa]	C12 [GPa]	C13 [GPa]	C22 [GPa]	C23 [GPa]	C33 [GPa]	C44 [GPa]	C55 [GPa]	C66 [GPa]
Ti-10V-2Fe-3Al (BCC)	134.5	115.5	–	–	–	–	20.4	–	–
$α^{″}$ Martensite [63]	129.00	91.12	126.81	148.22	69.32	135.59	28.38	23.13	39.65
Martensite Scaling Parameter, $χ$ :	0.5
Plastic deformation parameters (BCC Slip)
	System	m	${\dot{γ}}_{0}, [s^{- 1}]$	$h_{0}$ [MPa]	$κ_{s, 0}$ [MPa]	$κ_{s, S 0}$ [MPa]	$κ_{0}$ [MPa]	$m'$	${\dot{γ}}_{S 0}, [s^{- 1}]$
Ti-10V-2Fe-3Al:	{110}<111>	0.003	0.001	3500.0	210.0	400.0	212.0	0.005	5e10
	{112}<111>	0.003	0.001	3500.0	230.0	440.0	232.0	0.005	5e10
	{123}<111>	0.003	0.001	3500.0	290.0	520.0	294.0	0.005	5e10
Transformation parameters
	$f_{c r}$	$m$	${\dot{ξ}}_{0}$ [ $s^{- 1}$ ]	$v_{}$	$R_{0}$	$R_{s}$	$R_{h}$	$η$	$ξ_{total}$
Ti-10V-2Fe-3Al:	8.0	0.005	0.0001	2.0	20.0	5.0	100	6.0	0.90

The {110}<111>, {112}<111> and {123}<111> slip families are all used in this model. Each of these families are possible in the $β$ phase titanium (Castany et al., 2012; Gysler et al., 1974; Wu et al., 2022). However, the {110}<111> and {112}<111> families are more dominant (Castany et al., 2012; Gysler et al., 1974; Wu et al., 2022). $c_{k}$ , in equation (14), is kept constant at 0.5.

Results and discussion

Effect of triaxiality on the overall stress-strain behaviour

Figures 4 and 5 show the effective stress-strain curves and the volume averaged total martensite volume fraction (MVF) for the three orientations under two different constant triaxiality T (plotted results of $T = 1 / 3, 2$ are not shown for brevity). Figure 4 compares the stress-strain response and the volume averaged total MVF at $T = 1$ for the three orientations. Figure 4 (left) shows [111] is the ‘hardest’ in both the transformation and slipping. The trigger stress ( $σ_{T r}$ ) for SIM in [111] is 250 MPa compared to 150 MPa and 180 MPa in [011] and [001], respectively. While the slip yield stress ( $σ_{Y s}$ ) is 505 MPa in [111] and 480 MPa and 500 MPa in [011] and [001]. Table 2 shows the $σ_{T r}$ and $σ_{Y s}$ for all the orientations at $T = 1 / 3, 1, 2, 3$ . We can see from Table 2, as the triaxiality is increased the response becomes softer, i.e., both SIM and slip are initiated sooner. Taking the [111] orientation as an example, $σ_{Y s}$ and $σ_{T r}$ reduce from 675 MPa to 340 MPa and from 300 MPa to 190 MPa at $T = 1 / 3, 3$ , respectively. Furthermore, the difference in response between orientations becomes less, especially regarding the trigger and peak stresses. At $T = 1 / 3$ , Table 2 shows a difference of 145 MPa and circa (c.) 195 MPa between orientations [111] and [011] for $σ_{T r}$ and $σ_{Y s}$ , respectively. This is reduced to 62 MPa and 110 MPa at $T = 3$ , showing the orientational differences reduce with increased triaxiality.

Figure 4.

Equivalent stress strain (left), martensite volume fraction vs equivalent strain (right). At T = 1, L = −1 for three orientations.

Figure 5.

Equivalent stress strain (left), martensite volume fraction vs equivalent strain (right). At T = 3, L = −1 for three orientations.

Table 2.

Trigger stress ( $σ_{T r}$ ) for SIM and the slip yield stress ( $σ_{Y s}$ ) for all orientations at $T = 1 / 3, 1, 2, 3$ .

T	[001]		[011]		[111]
T	$σ_{T r}$ [MPa]	$σ_{Y s}$ [MPa]	$σ_{T r}$ [MPa]	$σ_{Y s}$ [MPa]	$σ_{T r}$ [MPa]	$σ_{Y s}$ [MPa]
1/3	200	500	155	∼480^a	300	675
1	180	500	150	480	250	505
2	160	490	135	460	210	380
3	145	485	128	450	190	340

No clear separation between the end of SIM and start of Slipping.

Figures 4 and 5 also show [011]-orientation exhibits the largest strain contribution from transformation. Furthermore, at $T = 3$ the transformation influence is stronger; the stress plateau is more pronounced in Figure 5 (right) as compared to that in Figure 4 (left).

Effect of stress triaxiality on the void growth

The void growth can be efficiently tracked with the normalised void volume fraction (NVVF) and the aspect ratios. The NVVF is the void volume fraction, normalised against the initial void volume fraction ( $f_{0}$ ), this shows the evolution of void volume. While the aspect ratios allow us to tract the void shape throughout the deformation. Figures 6 and 7 present the evolution of aspect ratios ( $W_{1}$ and $W_{2}$ ) for all orientation (left), and the evolution of NVVF against the volume average equivalent strain (right), for a triaxiality of 1 and 3 respectively.

Figure 6.

Aspect ratio ( $W_{1}$ and $W_{2}$ ) (left) and NVVF (right) for all orientations at T = 1, L = −1.

Figure 7.

Aspect ratio ( $W_{1}$ and $W_{2}$ ) (left) and NVVF (right) for all orientations at T = 3, L = −1.

Figure 6 (left) presents the evolution of aspect ratios ( $W_{1}$ and $W_{2}$ ) for three orientations at $T = 1$ . We can see that for [001] $W_{1} = W_{2}$ , and both are initially increasing linearly. From equation (1), this shows uniform void elongation in the primary loading direction (1-axis in Figure 1). For [011] and [111] $W_{1} \neq W_{2}$ , both aspect ratios are also increasing linearly at first. This shows non-uniform elongation in different directions, as $W_{2}$ increasing faster suggests $R_{2}$ is reduced more. The initial elongation of the void is during the elastic and transformation regions of deformation. However, as slip becomes dominant ( $ϵ_{e q} = c . 0.04$ ) the aspect ratio stops increasing, suggesting a volumetric increase. This is confirmed by inspecting the NVVF in Figure 6 (right), which shows significant void growth in the [111] orientation at $T = 1$ . The [001] orientation also has noticeable growth and [011] has comparatively little growth; with a maximum NVVF of c.1.21 at $ϵ_{e q} = 0.10$ as compared to c.3.55 in [111]. This highlights the strong orientational effect of SIM and transformation strain associated with metastable $β$ titanium alloys (Bhattacharjee et al., 2005; Chen et al., 2021; Christie et al., 2024; Duerig et al., 1982; Kim et al., 2006).

Figure 7 (left) presents the evolution of aspect ratios ( $W_{1}$ and $W_{2}$ ) for all orientations at T = 3. For all orientations, the void aspect ratios remain close to 1.0 meaning the void stays close to spherical in nature, consistent with higher triaxial loading. Initially, Figure 7 (left) shows the aspect ratios increasing. Then starts to decrease below 1.0 at an equivalent strain of circa (c.) 0.04 for [001] and [111] and at c.0.06 for [011]. The void is becoming an oblate spheroid shape as both $W_{1}$ and $W_{2}$ fall below 1. Figure 7 (right) shows the NVVF for $T = 3$ . As the triaxiality is increased the orientation effect becomes less and all orientations have significant void growth. This result aligns with previous single crystal void growth studies (Potirniche et al., 2006; Yerra et al., 2010). However, there is still a substantial difference in the response of the orientations. While they all experience exponential void growth, the point at which this start is significantly different between the orientations. Figure 7 (right) shows the exponential void growth starts at an equivalent strain of 0.035, 0.04, and 0.056 in the [111], [011], and [011] orientations respectively. As with the stress-strain response, this difference is associated with the SIM transformation.

The influence of SIM on the void evolution can be seen more clearly in Figures 8 and 9. Figures 8 and 9 present a magnified view of the aspect ratios and NVVF during the initial transformation at T = 3, respectively. Initially the elongation of the void is halted as the aspect ratios begins to saturate. This is shown well in the [001] orientation, Figure 8 (top). The same can be observed in [011] and [111], the initiation of SIM around the void stops the void elongation. This is seen to a drastic extent for $W_{2}$ in the [111] orientation. Figure 9 also shows during this stage there is a rise in the NVVF during this initial transformation. The contour plots presented in Figures 10 to 12 show the evolution of total martensite volume fraction in all orientations at T = 3. This shows initially the transformation is initiated at the void ( $ϵ_{e q} = c . 0.003$ ); however, it quickly spreads to the surrounding matrix. Figure 10 gives the evolution in [001], we can see the transformation is initiated in the 2-axis and 3-axis direction equally. This local transformation promotes $W_{1}$ and $W_{2}$ to plateau, halting of the void elongation. Inspection of Figure 9 also reveals an increase in void volume fraction during the initial transformation at the void surface ( $ϵ_{e q} = c . 0.002 - c . 0.005$ ). A result of the transformation initiating locally around the void (Figures 10 to 11). From Figure 8 and contour Figures 10 to 12, at $ϵ_{e q} = c . 0.01,$ we can then see SIM in the surrounding matrix proceeding this and the void beginning to elongate once more. This transformation of the surrounding material also accommodates the required deformation and greatly inhibits the void growth in all orientations, as shown in Figure 6 (right) and Figure 7 (right). These results are in line with experimental studies on the damage and fracture toughness of metastable $β$ titanium alloys. Danard et al (2021) suggested that SIM delays the material failure by fracture in Ti-7Cr-1Al-1Fe alloy by relaxing areas of locally intense strain fields. This is also supported by the Bhattacharjee et al. (2006) and Paradkar and Kamat (2011) investigation of fracture toughness in Ti-1023 and Ti–15Al–8Nb, respectively. Both authors found SIM to increase the fracture toughness. Higher triaxiality ( $T = 3$ ) is comparable to the stress state at a crack tip. Xaio et al. (2021) observed SIM forming at crack tips of Ti-1023, and indicated that the SIM is triggered by the propagation of the crack. The author also observed SIM creating a blunted crack tip morphology. It is then summarised that as transformation is initiated the local stress is relaxed and the propagation is halted. Paradkar and Kamat (2011) also observed transformation in the surrounding $β$ matrix and grains accommodating deformation and preventing extension/propagation of a crack. Therefore, our CPFEM studies discussed above, further explain and confirm the experimental suggestions.

Figure 8.

Aspect ratio ( $W_{1}$ and $W_{2}$ ) for all orientations at T = 3, magnified at initial transformation. [001] orientation (top), [011] orientation (bottom left), [111] orientation (bottom right).

Figure 9.

NVVF and MVF for all orientations at T = 3, magnified at initial transformation.

Figure 10.

Contour plots of total martensite volume fraction ( $ξ_{T}^{}$ ) evolution in orientation [001] at T = 3, L = −1. Plots show a mirrored image of the 1/8 model to give a physical representation of the void.

Figure 11.

Contour plots of total martensite volume fraction ( $ξ_{T}^{}$ ) evolution in orientation [011] at T = 3, L = −1. Plots show a mirrored image of the 1/8 model to give a physical representation of the void.

Figure 12.

Contour plots of total martensite volume fraction ( $ξ_{T}^{}$ ) evolution in orientation [111] at T = 3, L = −1. Plots show a mirrored image of the 1/8 model to give a physical representation of the void.

The slip has the larger, more obvious, influence on the evolution of the void. At a triaxiality of 3 the slip promotes an oblate spheroid shaped void in all orientations; we can see this as the aspect ratio drops below 1.0 (Figure 7 (left)). Figure 13 shows the evolution of total shear strain from slipping at $T = 3$ . The total shear strain ( $γ_{T}$ ) is defined as the sum of the shear across all slip systems:

{γ_{T}}^{} = \sum_{i = 1}^{N_{s}} γ^{(i)}

(15)

Figure 13.

contour plots for total shear strain ( $γ_{T}$ ) evolution at T = 3, L = −1. a) [001] orientation, b) [011] orientation, c) [111] orientation. Plots show a mirrored image of the 1/8 model to give a physical representation of the void.

where, $γ^{(i)}$ is the accumulated shear from the $i^{t h}$ slip system. Each orientation experiences different amounts of shear strain from slip as shown from the scales on the right-hand side (can also be inferred from Figure 5). The contours show the majority of the of the shear strain concentrated around the 2-axis and 3-axis, this will promote the oblate spheroid shaped void. Unlike the initial response to SIM, slip remains concentrated around the void promoting an ever more oblate shape. Comparing the orientations in Figure 13 shows [111] (Figure 13(c)) exhibits the most slip shear strain followed by [001] (Figure 13(c)). Figure 13(b) shows the response of [011], this orientation shows the least slip shear strain at 0.06 equivalent strain. A twofold effect is in place here; firstly, there is an inherit orientational effect to the relative strength of the crystals under a specific loading. Secondly, the effect of transformation delaying the onset of slip. Following the region of transformation, the void begins to grow exponentially during this final stage of deformation ( $ϵ_{e q} = c . 0.04 - c . 0.1$ ). This is due to the plastic deformation from slip. Figure 13 shows the shear strain is localised about the void, promoting void growth to accommodate the required deformation. In contrast to what is seen with the SIM transformation. The plastic deformation from slip around the void is the driving force for the exponential void growth.

Effect of lode parameter, SIM, and slip on void growth

The lode parameter describes the distribution of deviatoric stresses in a material RVE (Danas and Ponte Castañeda, 2012). Figure 14 illustrates the effects of lode parameter on the applied loading conditions, and the stress state of the material. It shows the normalised stress component against the lode parameter at a Triaxiality of zero ( $T = 0$ ), corresponding to purely deviatoric loading. The direction of the stress components in this purely deviatoric state, tells us how the lode parameter effects the loading state of the material. At $L = - 1$ , we see one positive and two negative stresses, biaxial compression with single axial tension. While at $L = 0$ , the stress state is in-plane shear with one stress component equal to zero. Finally, at $L = 1$ we see biaxial tension with uniaxial compression (two positive and one negative stresses). These three conditions are graphically represented in Figure 14(a) to (c). Figure 14 also shows how the increasing the triaxiality shifts all the stress components upward equally. Increasing the mean stress while preserving the deviatoric stress state in the material.

Figure 14.

Normalised principal stresses as a function of the lode parameter L. Illustrating the effect of lode parameter on the loading conditions applied, a) L = −1, b) L = 0, and c) L = 1. Furthermore, the effects of increasing the stress triaxiality is also highlighted. Adapted from (Danas and Ponte Castañeda, 2012).

Figures 16 and 17 show the stress-strain response for the 3 orientations at a lode parameter of 0 and 1, respectively. These figures present the effect of lode parameter at a stress triaxiality of 1.0. Figure 15 shows the stress-strain response and the evolution of NVVF in the [111] orientation, at a triaxiality of 1 and 3. From Figure 15 we observed the effects of the lode parameter are diminished as the triaxiality is increased. Hence further results will concentrate on $T = 1$ to better analyse the effects of the lode parameter, results at $T = 2, 3$ are omitted for brevity. This can be justified, in respect to transformation metastability, with the increase in volumetric stress. It has been shown by Christie et al. (Christie et al., 2024) that volumetric loading activates all transformation systems equally for this Ti-10-2-3 alloy. This results in all the orientations becoming more similar as triaxiality, and mean stress, is increased.

Figure 15.

Equivalent stress strain (left), NVVF (right). At T = 1 (top) and T = 3 (bottom), for [111] orientation. Plot shows how the effects of lode parameter are diminished as the triaxiality is increased.

Comparing Figures 4, 16, and 17 shows the lode parameter has a more significant effect on the orientational response than the triaxiality. In contrast to the increase in triaxiality, the increased lode parameter separates the response of the three orientations. Firstly, the difference in [001] and the [011] and [111] orientations are greatly increased, during the initial transformation region. However, Figures 16 and 17 both clearly show the [011] orientation becomes more like the [111] during the transformation region, and both the trigger stress and stress plateau become very similar at $L = 1$ (Figure 17).

Figure 16.

Equivalent stress strain (left), martensite volume fraction vs equivalent strain (right). At T = 1, L = 0 for three orientations.

Figure 17.

Equivalent stress strain (left), martensite volume fraction vs equivalent strain (right). At T = 1, L = 1 for three orientations.

The plastic response, due to slip, is also different between orientations and lode parameters. As shown in Figure 14 the lode parameter alters the deviatoric stress state throughout the simulation. Effectively changing the crystals deviatoric loading conditions, as explained in Figure 14. The deviatoric stress has a direct influence on the transformation and slip systems activated/favoured. Hence, due to the anisotropic nature of theses single crystals, it is not surprising that the lode parameter has a strong influence at $T = 1$ .

Figures 6, 18 and 19 present the evolution of aspect ratios ( $W_{1}$ and $W_{2}$ ) for all orientation (left), and the evolution of NVVF against the volume average equivalent strain, for a Triaxiality of 1 and a lode parameter of −1, 0 and 1, respectively.

Figure 18.

Aspect ratio ( $W_{1}$ and $W_{2}$ ) (left) and NVVF (right) for all orientations at T = 1, L = 0.

Figure 19.

Aspect ratio ( $W_{1}$ and $W_{2}$ ) (left) and NVVF (right) for all orientations at T = 1, L = 1.

As discussed earlier, at $L = - 1$ (Figure 6) in all orientations, voids elongate along the primary loading direction (Figure 1) during the first stages of deformation. As the strain is localised around the void during slip, the voids begin to become a more spherical or oblate spheroid shape. This localisation corresponds to an exponential growth in the void volume fraction.

Figure 19 (left) shows the biaxial tension of $L = 1$ keeping $W_{2} ≅ 1$ for [001], i.e., $R_{1} ≅ R_{3}$ . While $W_{1}$ is increasing, showing the 2nd axis is compressing/expanding less. This is constant trend thought out the triaxialities assessed and provides conformation as to the previously described applied loading conditions of the lode parameter. The [011] and [111] show $W_{1}$ increasing and $W_{2}$ remaining close to 1. Danas and Ponte Castañeda (Danas and Ponte Castañeda, 2012) observed a similar trend with the void shape at $L = 1$ and found it to be indicative of void collapse. Inspection of Figure 19 (right) shows how this observation is also experienced in this study. All orientations at this lode parameter experience far less void growth. The [111] orientation experiences a reduction in the void volume. Figure 20 shows a magnified view of NVVF for the [111] orientation at $T = 1$ , $L = 1$ .

Figure 20.

magnified view of NVVF for [111] at T = 1, L = 1. Points a), b), and c) relate to different points in the void growth. a) during transformation, b) at maximum void size, c) point at the end of the void contraction.

Figure 20 also present points of interest along the NVVF evolution. Point a) is the NVVF during SIM transformation, b) is the void at is maximum size, while c) is the point at the end of the void contraction. Graphical representation of the void shape during these points of interests are given in Figure 21. Figure 21 shows how the voids growth remains largely spherical during the transformation, up to the point of maximum void size. We can also see after contraction the void is distinctly shorter in the 2nd axis. Consistent with what is seen in Figure 19 (left). The point c) represents the last useable data point. Figure 21(d) shows the place where the simulation finished, this point was accompanied by distorted finite elements in a penny shaped, collapsed void, structure. We can see this is caused by the collapse of the void on the 2nd axis. This collapse changes the void into a crack like morphology.

Figure 21.

Showing 3D representation of 1/2 of the void and corresponding projected contours of the whole void onto the axes walls. a) to c) shows a graphical representation of the void shape at the critical points highlighted in Figure 20(d)) shows the void at the point where the simulation finished (finite elements are distorted into a penny shaped, collapsed void, structure).

The results at L = 0 given in Figure 18, for the aspect ratios and NVVF, Figure 18 left and right respectively. The void evolution for $L = 0$ is somewhere in between that of $L = - 1$ and $L = 1$ , depending on the orientation.

To investigate the effect of transformation with the lode parameter and the collapse of the void in the [111] case. A comparation between the simulations for [111] at T = 1, L = 1; with and without the transformation mechanism have been conducted. Figure 22 shows this comparison, the top figure is the NVVF plot; the bottom left shows the equivalent stress strain response; and the bottom right shows the aspect ratio ( $W_{1}$ and $W_{2}$ ). The equivalent stress strain response clearly shows the transformation extends the equivalent strain and delays the slip plasticity. The shape of the void, as shown in the aspect ratio plot, remains similar in both cases. The NVVF shows how the transformation from $ϵ_{e q} = c . 0.005 - c . 0.04$ facilitates the continued volumetric void growth during this region. This shows how the transformation prolongs the void integrity and delays the collapse of into a crack which is caused by the localised plastic deformation from slip. The transformation impedes the localisation of plastic deformation around the void; allowing the surrounding matrix to accommodate required deformation. This once again is constant with the experimental observations of transformation relaxing areas of localised strain (Danard et al., 2021; Paradkar and Kamat, 2011; Xiao et al., 2021).

Figure 22.

Comparison between [111] at T = 1, L = 1; with and without the transformation mechanism. Showing the NVVF plot (top), the equivalent stress strain response (bottom left), and the aspect ratio ( $W_{1}$ and $W_{2}$ ) (bottom right).

Conclusions

3D unit-cell CPFEM simulations were performed to study the effect of lattice orientation and stress induced martensite (SIM) on the evolution and growth of a pre-existing spherical void in a single BCC crystal of Ti-10V-2Fe-3Al. The simulations were subjected to variation in both lode parameter and stress triaxiality. An effort was made to understand plastic anisotropy and damage accumulation in metastable β titanium alloy single crystals, while bringing new insights into the effects of stress-induced martensite and slip-based plasticity on void growth. The main ﬁndings of this investigation are summarised below:

Void growth rate increases exponentially with the increase in stress triaxiality. The void growth is heavily dependent on the crystal orientation. At low triaxiality this dependence is far greater. The crystal orientation with higher yield strength tends to have a higher void growth due to higher stress concentration around the void and pressure at the same stress triaxiality.

The void growth is also significantly affected by the lode parameter. The void growth rate was found to decrease for all but the [011] crystal orientations, with the increase in lode parameter. The void growth in [011] was found to significantly increase at a lode parameter of 0. This is attributed to the interaction between initial orientation and changing in loading conditions imposed at this lode parameter.

At higher triaxiality, the influence of crystal orientation and lode parameter becomes less significant for the void growth.

SIM was also found to have a profound effect on the void evolution for all orientations, lode parameter, and stress triaxialities tested. Transformation is highly orientational and the influence vary greatly between orientations and lode parameters. Increasing the stress triaxiality reduces this orientational effect from transformation.

Initially, SIM is found to halt void elongation and produce a more spherical or oblate spheroid shaped void when initiated around the void. This proved consistent with observations of crack tip blunting in metastable $β$ titanium alloys exhibiting SIM, while bringing new insights.

The SIM is then found to accommodate the required deformation in the surrounding material. The accommodation of deformation in the surrounding matrix was found to help preserve the integrity of the void, preventing the localisation of deformation around the void. At lower lode parameter and at higher stress triaxiality this delays the exponential growth of the void. At higher lode parameter with low triaxiality this delays the collapse of the void into a crack like morphology.

On the other hand, slip is initiated at the void and proceeds to concentrate around the void perimeter. The void growth is found to be accommodating the required deformation. This produces the exponential increase in void volume at low lode parameter and/or high triaxiality, and the collapse of the void into a crack at higher lode parameter with lower stress triaxiality.

While this study has considered the effects of crystal orientation, lode parameter, stress triaxiality, and plastic anisotropy in metastable Ti-10V-2Fe-3Al. There remains scope to investigate the effects of initial void porosity and shape. Furthermore, local microstructural effects such as grain boundaries and void location presents an interesting avenue for future research of damage mechanisms and SIM using this CPFEM.

Supplemental Material

sj-pdf-1-ijd-10.1177_10567895241275373 - Supplemental material for Interaction of defects, martensitic transformation and slip in metastable body centred cubic crystals of Ti-10V-2Fe-3Al: A study via crystal plasticity finite element methods (CPFEM)

Supplemental material, sj-pdf-1-ijd-10.1177_10567895241275373 for Interaction of defects, martensitic transformation and slip in metastable body centred cubic crystals of Ti-10V-2Fe-3Al: A study via crystal plasticity finite element methods (CPFEM) by P Christie, MA Siddiq, RM McMeeking and ME Kartal in International Journal of Damage Mechanics

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Authors would like to acknowledge the funding received from university of Aberdeen for this project.

ORCID iD

MA Siddiq

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