Prediction of ductile cracking in the titanium alloy forging process

The control of surface cracking in the forming of titanium alloy forgings is a significant problem in the forging industry. For titanium alloys, the formation of surface cracks is related to temperature, strain rate, and stress state. This study selected the widely used medium to high strength titanium alloy Ti-6Al-4V in the field of forging as the research material, and designed six different shapes of specimens for high-temperature tensile and compression tests. The mechanisms underlying crack formation were analyzed at the microscopic level, and the critical fracture displacement of these tests was extracted. Moreover, their critical fracture strains were obtained through simulations, and a High-temperature damage model was established based on the DF2016 model. The research results showed that cracks through void at grain boundaries propagate and aggregate to form, leading to a fracture mechanism characterized by ductile fracture through micro-pore aggregation. Simulation results demonstrate that the established model accurately predicts the crack of forgings.


Introduction
Titanium alloys have been widely utilized for many years due to their high corrosion resistance, high specific strength, and low density (about 60% of the density of steel) [1][2][3].Forging is an advanced shaping process that can produce near-net-shape product and enhance the mechanical properties of alloys, and it is widely employed in aerospace and other related industries [4,5].However, the window for titanium alloy forging process is narrow, considering the cooling effect of the mold, the damage evolution on the surface of titanium alloy forgings under high temperature conditions becomes more complex, making it difficult to achieve accurate predictions [6,7].After the occurrence of surface cracks, not only will substantial time be expended on the process of polishing, but it will also result in a diminishment of product quality and production efficiency.In severe cases, it may even culminate in the disposal of the product.Therefore, it is imperative to elucidate the fracture behavior of titanium alloy forgings under thermal deformation conditions and establish a crack initiation model for predicting the surface cracks.
To predict the ductile fracture of titanium alloy during the forming process, numerous scholars have endeavored various approaches [8,9].In the past, it was customary to predict cracks under the condition of equal proportion loading and isothermal conditions through forming limit diagrams and thermal processing diagrams [10,11].However, for forged products, the strain path is typically nonlinear, and the cooling effect of the mold can result in significant temperature gradients on the surface of the forged part, rendering traditional methods inadequate for accurate prediction of cracks [12].Therefore, the utilization of a ductile fracture criterion to study and predict the cracking behavior of titanium alloys has become a trend.The uncoupled fracture criterion, due to its simplicity and the requirement of fewer calculated parameters, is more favored in industrial applications [13,14].
The ductile fracture criterion for the hot deformation of titanium alloys is generally established by considering the influence of temperature and strain rate on the basis of the cold deformation fracture criterion [15,16].Johnson and Cook pioneered the modeling of high-temperature ductile fracture criteria by introducing two separate terms incorporating temperature and strain rate, establishing the Johnson-Cook criterion [17].
Valoppi et al extended the Johnson-Cook criterion by incorporating the influence of stress triaxiality and deviatoric stress to achieve ductile fracture prediction in titanium alloy sheets [18].Zhu et al integrated the influence of temperature and strain rate on fracture strain through the Z parameter, and incorporated it into the O&K criterion ( ̅ / ) to establish the thermal deformation fracture criterion for Ti40 alloy [19].Feng et al introduced internal state variables such as recrystallization volume fraction, grain size, and stress triaxiality into the GTN damage model to establish a coupled thermal deformation damage criterion for titanium alloys [6].Lou invented the DF2016 fracture criterion, which incorporates the stress triaxiality and the Lode parameter, and has been widely embraced and utilized by scholars [20].Shang et al established a thermal deformation toughness fracture criterion for 316LN stainless steel by incorporating the Z parameter and recrystallization volume fraction into the DF2016 model [21].Scholars have developed various fracture criteria based on different conditions to forecast the ductile fracture during the metal deformation process.Their models have exhibited considerable prediction precision and range under laboratory conditions.However, these models fall short in effectively addressing the toughness cracking within industrially complex deformation conditions.This research endeavors to address the prediction of surface cracks in titanium alloy forging, by establishing a fracture model that is applicable to intricate stress states, wide temperature variations, and strain rates.Moreover, the proposed model possesses a simplistic and user-friendly format.
The present study focuses on the wrought titanium alloy Ti-6Al-4V, conducting high-temperature tensile tests on specimens of various shapes.It establishes a fracture criterion for thermal deformation toughness by numerically simulating the relationship between fracture strain and stress state, temperature, and strain rate.The application of this model in engineering practice enables the prediction of ductile cracking in the forming process of titanium alloy forgings.

Experimental section 2.1. Experimental material
The experimental material is a wrought Ti-6Al-4V titanium alloy, with a chemical composition as shown in table 1.The phase transition point of the alloy is 1008 °C.Its initial microstructure, as showed in figure 1, exhibits a typical wrought equiaxed structure, comprising equiaxed primary α phase, fine secondary α phase, and a small amount of β phase.Because of the material being in a forged state, the material's anisotropy can be negligibly small [22].

Laboratory experiments
To analyze the fracture mechanism and establish cracking criteria for titanium alloys, tensile and compression tests were conducted according to the standards GB/T 4338-2006 and HB 7571-1997.To investigate the effects of temperature and strain rate on alloy fracture, high-temperature tensile tests were performed on T1 specimens  using a Gleeble-3500 thermal simulation system.Additionally, in order to mitigate the negative effects caused by temperature gradients resulting from Gleeble resistance heating, three thermocouples were welded at distances of 3, 5, and 10 mm from the center of the specimen to measure the axial temperature distribution.To evaluate the influence of stress state on alloy fracture at high temperatures, tensile tests were conducted on T2-T6 specimens using a confocal laser scanning microscopy (CLSM) equipped with a tension module.Different shapes of samples represent different stress states in the test.CLSM enables in situ observation of the microstructural evolution of the samples during the tensile process.Furthermore, to assess the cracking behavior of the material under compression deformation, thermal compression tests were performed on C1 specimens using the Gleeble-3500 combined with a camera.All specimens were derived from cylindrical forgings and prepared through mechanical machining.The detailed experimental information is provided in table 2, and the dimensions of each specimen are shown in figure 2. All specimens were heated at a rate of 10 °C/s to the experimental temperature, followed by a 5-minute holding period before the beginning of the tests.After the completion of the experiments, the specimens were allowed to cool naturally to room temperature.The T1 specimens were cut along the axial direction to observe the microstructure and internal cracking using an optical microscope.The microstructure and fracture morphology were further examined using a scanning electron microscope (SEM).

Hot tensile deformation behavior
The true stress-strain curves under different deformation temperatures and strain rates are depicted in figure 3. It can be observed that the deformation parameters have a significant impact on the true stress-strain curve, the peak stress decreases markedly with increasing temperature and decreasing strain rate.At the same time, the yield strength of the material also shows the same rule, as shown in table 3.This is due to the influence of temperature and strain rate on work hardening, dynamic recovery, and dynamic recrystallization.The temperature transformation significantly impacts the phenomena of dynamic recovery and dynamic recrystallization.As the temperature escalates, the thermal  activation process of the material intensifies, leading to an increase in vacancy concentration, a decrease in dislocation density, and a softening effect on dynamic recrystallization, thereby reducing the flow stress.Furthermore, strain rate induces a stronger work hardening effect, generating a substantial number of dislocations that interlace with one another, resulting in a rapid amplification of stress levels [23,24].However, in the temperature range of 400 °C-600 °C, the thermal deformation behavior of alloys is not sensitive to the strain rate.Many literature indicates that the strain rate sensitivity of titanium alloys is influenced by the size of the primary α phase and the content of the β phase [25,26].
Increasing the β phase content can enhance the stability of deformation, while smaller grain size leads to more uniform deformation.Within the temperature range of 400 °C-600 °C, when the β phase content is equivalent and at a low level, poor deformation stability results in insensitivity to the corresponding strain rate.The tensile process of all specimens can be divided into four stages [16]: the stage of elastic deformation, the stage of uniform plastic deformation, the stage of diffuse necking, and the stage of localized necking.During the early stage of uniform plastic deformation, work hardening plays a dominant role, and the load increases with the displacement.After reaching the peak load, the material undergoes softening due to the effects of dynamic recovery and dynamic recrystallization.The load gradually decreases, and higher temperatures enhance the softening effect.In the stage of diffuse necking, the material undergoes necking, resulting in unstable deformation and fluctuation in load.As it enters the stage of localized necking, microvoids appear within the material, and the load rapidly decreases due to the combined effects of localized necking and  internal damage, eventually leading to fracture [27].Therefore, based on the critical damage point, the thermal tensile process can also be divided into two stages: uncracked deformation and cracked deformation, as show in figure 3(d).

Analysis of the fracture mechanism
The initiation of microcracks includes three stages: nucleation of micropores, growth, and aggregation, and further stretching leads to crack propagation resulting in ultimate fracture [28].Figure 4 illustrates the evolution of microcracks during the stretching process of the T5 specimen captured under a confocal microscope.The nucleation of micropores is concentrated in the yielding stage, and as the load increases to its maximum value, the nucleation of micropores is successively observed under the microscope, with the nucleation positions mostly occurring at grain boundaries, with trifurcate grain boundaries being more prone to the formation of micropores.This is due to the lower strength between grain boundaries compared to within the grains.With further stretching, the load continues to decrease, and the micropores gradually expand and aggregate with adjacent micropores to form microcracks. Figure 5 illustrates the fracture morphology of the T1 specimen under a strain rate of 800 °C and 0.01S −1 .From the figure, it can be observed that there are many dimples at the fracture surface, indicative of typical ductile fracture.The appearance of voids is the result of the nucleation, growth, and aggregation of micropores during the deformation process, and from the figure, it is also possible to observe the tearing traces produced by the aggregation of two adjacent voids.These signs indicate that the fracture mechanism at high temperatures for the alloy is a ductile fracture type characterized by the aggregation of micropores.
4. Finite element simulation and establishment of fracture criteria 4.1.Finite element simulation A significant portion of this study is dedicated to the extraction of critical fracture strains under various conditions.Currently, the primary method for measuring fracture strain is hybrid experimental-numerical method [20,29].Given the demanding accuracy requirements of numerical simulations, it is crucial to establish an appropriate material model to describe the plastic deformation behavior of the alloy.The stress-strain curve during tension is obtained through Gleeble tensile tests.During the uniaxial tension process, when necking occurs, there is an uneven deformation within the gauge length, and the flow stress of the material cannot be directly obtained through experimentation.To determine the stress-strain relationship after necking occurs, the finite element aided testing method is employed [30].The other material parameters are presented in table 4.
To obtain the critical fracture strain, a finite element model was established based on the experimental conditions.The specific simulation parameters are shown in table 5.The specimens subjected to tension and compression on the Gleeble-3500 are heated via resistance, which is generated by the Joule effect of the Gleeble thermal simulator.This results in temperature gradients along the axis of the tensile specimens, leading to non-uniform necking deformation, further influencing the accuracy of calculating true stress-strain data through load-displacement and the assessment of specimen damage and cracking [31,32].Therefore, by combining data obtained from thermocouples welded to the specimen surface with finite element simulations, the adverse effects of temperature gradients can be minimized.Figure 6(a) illustrates the axial temperature distribution of the specimen at 800 °C, along with the mesh division and the location of the welded thermocouples.The high-temperature co-focus stretching of the specimens is carried out through halogen lamp irradiation.This heating method ensures relatively stable sample temperature [33], allowing for the assumption of a stable temperature distribution in the deformation zone during simulation.At the same time, in order to ensure the reliability of the simulation calculations, the deformation zone of the notched specimens undergoes grid refinement, as shown in figure 6(b).All specimens are defined as elastic-plastic models.

Extraction of critical fracture strain
The widely accepted method of hybrid experimental-numerical is employed in this study.The method requires obtaining the critical fracture displacement under various deformation conditions [34]. Figure 7 illustrates the comparison of load-displacement curves between experiment and simulation.For the notched tensile specimen, its critical fracture displacement corresponds to the fracture point in the experiment [35].However, for the smooth round bar tensile specimen, damage and cracking occur inside the specimen, resulting in an overestimation of the fracture strain obtained by defining the final fracture point as the critical fracture displacement.Therefore, the use of metallographic microscope to observe the internal damage of the specimen is necessary to redefine the fracture displacement of the smooth round bar tensile specimen.Figure 8 depicts longitudinal sections near the fracture site at various temperatures and strain rates, revealing the existence of numerous micropores, indicating severe internal damage and cracking prior to fracture.At temperatures below 800 °C, the number of voids gradually increases with increasing temperature, This is due to increased atomic motion and increased average interatomic separation at higher temperatures reducing the micro-void-nucleation threshold [36].However, after surpassing 800 °C, the voids sharply decrease due to the recrystallization temperature of the material being close to 800 °C, under the influence of dynamic recrystallization, the nucleation of voids is suppressed.Within the range of 400 °C to 800 °C, the number of voids decreases with increasing strain rate, as the shorter stretching time at high strain rates inhibits the growth and aggregation of voids.However, above the Table 4.The thermal physical parameters of the material.

Elastic modulus (Gpa)
Poisson's ratio Thermal conductivity Coefficient of thermal expansion (10 Specific heat capacity (J Kg -1 •K) 4430 109 0.34 6.8 9.0 611 recrystallization temperature, dynamic recrystallization at high strain rates is inhibited, resulting in a higher number of voids.The critical fracture diameter at each temperature and strain rate is quantitatively evaluated based on the distribution of micropores and the specimen's profile at the point of crack initiation, as indicated in table 6.
Based on the experimental results, critical fracture displacements under different temperatures, strain rates, and stress states were obtained.By leveraging finite element analysis, the evolution of critical fracture strains and stress states corresponding to different deformation conditions can be acquired.
Figure 9 presents the extraction of critical fracture strains under 400 °C and 0.01 S −1 strain rates.The critical fracture diameter of each specimen was determined through metallographic examination, and then the corresponding strain value at this diameter was determined in the simulation.The fracture point was selected as the centroid of the specimen's interior.As a result, the critical fracture strains obtained are shown in table 7.
The essential parameters for describing the stress state are the stress triaxiality (η) and the Lode parameter (L), as given by equations (1) and (2).
Among them, s m denotes the mean stress, ̅ s represents the equivalent stress, and s , 1 s , 2 s 3 refer to the principal stresses, with s 1 > s 2 > s .
3 Stress triaxiality represents the influence of principal stress on ductile fracture, while Lode parameter represents the influence of deviating stress on ductile fracture.
Figure 10 presents the simulated results of the notched tensile specimens, showing a good correlation between the shapes of the specimens and the experimental results, thus confirming the accuracy of the simulation.
During the experimental process, the relationship between triaxiality of stress and effective strain, as well as the relationship between Lode parameter and effective strain, is utilized to describe the loading paths of stress state.Different shapes of specimens yield distinct loading paths.Furthermore, the loading paths are independent of temperature and strain rate, indicating that when samples share the same geometric shape, they exhibit similar evolution of stress state [37].Figure 11 illustrates the loading paths of each specimen at 800 °C and a strain rate of 0.01 S −1 .
As the radius of the specimen's notch decreases, the triaxiality of stress increases, with values ranging between 0.33 and 0.55.Conversely, the Lode parameter demonstrates an opposite trend, increasing as the notch radius decreases, with values between −0.82 and 0. The two stress state parameters for T6 are approximately 0, indicating a shear stress state.
Each specimen represents the average values of the triaxiality of stress and Lode parameter, which are calculated using the equations (3) and (4), depicting the stress state parameters from the onset of deformation to the critical fracture point.From the graph, the critical fracture strains for each specimen can be obtained, as shown in table 8, specimen C1 under compression did not exhibit any cracking.

Construction and validation of fracture criteria 4.3.1. Construction of fracture criteria
In the plastic deformation of metallic materials, the toughness fracture is influenced by the stress state, different stress states lead to different fracture patterns [38].Under high stress triaxiality conditions, toughness fracture occurs due to the necking of internal voids in the material, while under low stress triaxiality conditions, it occurs due to shear localization [39].In recent years, many new uncoupled fracture criteria have been developed, which describe the fracture failure behavior of materials under different stress states [13,40].Among them, in the research by Lou et al a fracture criterion DF2016, which is related to stress triaxiality and the Lode coefficient, is proposed [20].Its expression is shown as equations ( 5) and (6).( ) In this context, η represents the stress triaxiality, L represents the Lode coefficient.This criterion has been proven to be applicable for predicting toughness fracture within the range of compression to biaxial tension.Furthermore, it has fewer parameters and clear physical meanings, making it easy to modify, expand, and incorporate into finite element software for engineering applications.Therefore, this paper establishes a toughness fracture criterion considering temperature, strain rate, and stress state to predict toughness cracking of Ti-6Al-4V under forging conditions.By using the corresponding data in table 8, the relevant parameters of the model are obtained through surface fitting.Among them, C = 0, C1 = 5.03, C2 = 1.51,C3 = 2.29, C4 = 1.08.The fitted surface plot is shown in figure 12.
In order to incorporate temperature and strain rate, DF2016 was redefined as equation (7).The ( )  e f T, ln is obtained through surface fitting based on the data in table 7, and the expression is as shown in equation (7), the  fitting surface graph as shown in figure 13.The e sf represents the fracture strain of T1 under 800 °C and a strain rate of 0.01 S −1 .
As described above, the DF2016 model suitable for hot deformation has been established.In order to accommodate non-proportional loading conditions, the parameter D is represented by an integral form in this criterion.The final expression is as follows: ò e e

Validation of fracture criteria
In this article, an industrial case study is used to validate the effectiveness of the established fracture model.It involves the hot die forging process of large Ti-6Al-4V alloy forgings.In actual production, a multi-step hot forging process is adopted to manufacture the product.The coefficient of friction is established as 0.2, the heat transfer coefficient is set at 0.5 N sec −1 mm −1 C −1 , and the downward velocity is 5 mm s -1 .After two pre-forging steps, cracks appeared on the lower surface of the forging, as shown in the localized schematic diagram in figure 14(c).
In this study, the process is simulated using Deform software.The surface location is identified as a high-risk area for cracking, hence the refinement of the surface mesh of the forging through the use of local grid windows.Following the completion of the forming process, the actual location of cracking exhibits a significant equivalent strain value.During the forming process, this particular location experiences a noticeable decrease in temperature and a high rate of strain under the influence of contact with the mold, resulting in tensile stress.From a simulation perspective, the risk of cracking at this location is high, while the damage values at other locations remain within a safe range.However, if further deformation occurs, there is also a risk of cracking at the elevated convex areas.Upon incorporating the modified simulation into the calculations, the results, as depicted in figure 14(b), demonstrate a satisfactory level of consistency with the actual observations.

Conclusion
(1) Different geometric shapes of specimens were designed to represent different stress states and loading paths during the hot forming process.Microscopic observation and in situ monitoring with a camera were employed to obtain the critical fracture strains during tensile and compressive tests.In addition, uniaxial tensile tests on smooth round bars were conducted to investigate the effect of wider ranges of temperature and strain rates on toughness cracking.The critical failure point of the necking stage was determined through metallographic observation, and the corresponding critical fracture strain was calculated.
(2) The fracture mechanism of Ti-6Al-4V was analyzed, revealing that the fracture mechanism at high temperatures is characterized by ductile fracture with pore aggregation.(3) Based on the experimental results, the critical fracture strains and stress states under various deformation conditions were calculated through numerical simulations.A model reflecting the influence of temperature, strain rate, and stress state was established by fitting the data and using the DF2016 fracture criterion.
(4) By incorporating this model into the secondary development of DEFORM, simulations of the forming process of a titanium alloy forging were conducted.The results demonstrated that the fracture criterion accurately predicted surface cracks during the titanium alloy forging process.

Figure 6 .
Figure 6.(a) The temperature distribution of the T1 after Gleeble heating.(b) finite element mesh dividing of T3 specimens.

Figure 7 .
Figure 7.Comparison of load-stroke curves between experimental and simulated results: (a) Deeply grooved and notched specimens (b) Smooth round bar specimens.

Figure 10 .
Figure 10.The simulated results of the notched specimen.

Figure 11 .
Figure 11.Loading paths of the notched specimens.
According to the aforementioned definition, D = 1 represents the critical condition that triggers cracking.When the fracture parameter D < 1, the tested metal is in a state of stable deformation.If the parameter D > 1, there is a risk of cracking.

Figure 12 .
Figure 12.The fracture locus constructed for Ti-6Al-4V at 800 °C and a strain rate of 0.01 S −1 .

Figure 13 .
Figure 13.The fitting surface graph of critical fracture strain under variations in temperature and strain rate.

Table 7 .
The critical fracture strains under various temperature and strain rate conditions.

Table 8 .
Critical fracture strains under various stress conditions.