Visualization of the damage evolution for Ti–3Al–2Mo–2Zr alloy during a uniaxial tensile process using a microvoids proliferation damage model

2.1 Theoretical basis of GTN damage model

GTN model, as one of the classic microscopic mechanical models, has been widely used to describe the effect of ductile damage on the plastic yielding of materials [12]. The ductile damage process that contains the microvoids evolution can be schematically described as shown in Figure 1. When the material is deformed under a tensile stress, microvoids existing in the material begin to self-proliferate and expand. As the deformation continues, some microvoids begin to merge with each other, forming microcracks. Then, macrocracks come into being with the propagation and coalescence of the microcracks, resulting in dramatic performance degradation.

Figure 1

Schematic illustration of ductile damage process.

In the theory of GTN model, the impact of microvoids on the plastic yielding potential Φ of materials can be expressed as equation (1) [13].

where σ _eq is the macro von Mises equivalent stress; σ _h is the macro hydrostatic stress; σ _m is the equivalent stress of matrix material; q ₁, q ₂ and q ₃ are the calibration parameters of the model, and q ₃ = q ₁ ². Here q ₁ and q ₂ are 1.5 and 1, respectively, for this alloy [14]; f ^∗ is the damage factor which can be expressed as the function of the voids volume fraction (VVF) as equation (2).

where f represents the VVF; f ₀ represents the initial VVF; f _f represents the final VVF, and the material will lose its loading capacity completely once f reaches to f _f; f _c represents the critical VVF, below which the voids will not proliferate and merge. f u ∗ is the accelerating factor for the propagation and coalescence of microvoids. Usually, f u ∗ can be calculated by q ₁ as f u ∗ = q ₁ ⁻¹.

The VVF includes the contribution of newly nucleated microvoids and the growth of existing microvoids, and their relationships can be expressed as equation (3).

where df represents the total increment of VVF; df _nucleation represents the increment of VVF that comes from newly nucleated microvoids; df _growth represents the increment of VVF that comes from the growth of existing microvoids. d ε ¯ m pl is the microplastic strain increment of the matrix material; d ε kk p is the macroplastic strain increment. A is the nucleation coefficient of microvoids and it can be expressed as equation (4).

where f nucleation represents the VVF of the newly nucleated microvoids; ε ¯ p represents the equivalent plastic strain of matrix material; S N represents the standard deviation and ε N represents the equivalent plastic strain for the newly nucleated microvoids. According to refs [13,15,16], the values of ε N and S N can be taken as 0.3 and 0.1, respectively.

Based on the theoretical description of GTN damage model, the control parameters of GTN model include q ₁, q ₂, q ₃, f ₀, f _c, f _f, f _n, ε N and S _N. Since some of the parameters have been obtained from references, the GTN parameters to be solved are f ₀, f _c, f _f, f _n.

2.2 Theoretical basis of RSM

RSM, as a regression analysis method, was introduced by George E. P. Box and K. B. Wilson in 1951 [17]. It is dedicated to characterizing the quantified connection between the response variables and the independent variables. The main operation procedure of RSM is as follows. First, design a sequence of experiments to get the basic data for establishing response surface. Second, choose suitable regression equations to represent the response relationship. This relationship can be expressed as equation (5), where y represents the response variables and x _i (i = 1, 2,…,n) represent the factors.

Finally, fit the regression equation using the obtained experimental data by least square method, working out the coefficients in the regression equation. Usually, for the sake of saving calculation time and avoiding over-fitting, linear or quadratic polynomial functions are widely favored for the regression equations. Quadratic polynomial functions are adopted in the present work. The function of the response surface can be expressed as follows:

where R represents the response variables; b 0 , b i , b i i and b i j are the coefficients of response functions; X i and X j represent the independent variables.

2.3 Method for solving the GTN parameters based on RSM

One of the aims of this work is to figure out the values of GTN parameters. Hence, the GTN parameters can be regarded as the independent variables of the response surface. As introduced in the theory of GTN model, the GTN parameters determine the yielding behavior of alloys directly. Therefore, the indicators related to stress can be considered as the response variables. Since the stress–strain curve for a material gives the direct relationship between stress and deformation, the difference between experimental stress–strain curves and numerical ones can be defined as the response variables of RSM. The detailed expressions of the response values R are as follows:

where ε E P and ε N P represent the experimental peak strain and numerical one, respectively; σ E P and σ N P represent the experimental peak stress and numerical one, respectively; ε E F and ε N F represent the experimental breaking strain and numerical one, respectively; σ E F and σ N F represent the experimental breaking stress and numerical one, respectively. These indicators are schematically shown in Figure 2. In Figure 2, the experimental stress–strain curve can be obtained from the tensile experiments, while the numerical stress–strain curve would be extrapolated from the FE simulation of the tensile processes. Thereupon, for each GTN parameter combination, there will be a specific numerical stress–strain curve and a set of corresponding response value R. The solution process of the GTN parameters can be schematically described as shown in Figure 3. Based on the tensile tests, the experimental curves can be obtained. Then, simulate the tensile process using different GTN parameter combinations; meanwhile, acquire the corresponding response variables based on the difference between the experimental curves and numerical ones. Construct the response surface using the input GTN parameters and the corresponding response variables. Calculate the accurate GTN parameters based on the established response surface.

Figure 2

Schematic description of the related indicators in the response variables of RSM.

Figure 3

The solution procedures of GTN parameters.

4.1 Numerical model of uniaxial tensile process

To calculate the response variables corresponding to various GTN parameters, it is necessary to carry out the numerical simulation of the uniaxial tensile processes. The simulation is carried out on the FE platform ABAQUS. For saving simulation time, the FE model of the uniaxial tensile process was simplified as an axisymmetric model. The simplified tensile model was meshed with the quadrilateral element (CAX4R), and 657 nodes and 548 elements were obtained. The simplified model and the boundary condition of the tensile simulation are shown in Figure 7. In order to ensure that the strain rate in the numerical simulation is consistent with that in the tensile experiments, the tensile speed exerted on the specimens should be calculated by v = l 0 ε ̇ exp ( ε ̇ t ) , where l ₀ represents the gauge length of the specimens, 25 mm here. GTN damage model is applied in the tensile simulation. The mechanical property parameters that are required for the tensile simulation of Ti–3Al–2Mo–2Zr alloy are presented in Table 2 [19,20]. According to the tensile experiments, it has been found that the deformation span is very short for the tensile process of this alloy. Therefore, in order to obtain more details on the damage evolution during the tensile process, the simulation is performed using the explicit analysis arithmetic with an automatic time step. The computational cost for the FE simulation is a personal computer with an eight-core processer. Four threads were applied during the computation of the uniaxial tensile FE model. The computational time lasts about four hours for each uniaxial tensile process.

Figure 7

FE model and boundary condition of the uniaxial tensile simulation.

4.2 Solution of the GTN parameters based on RSM

In order to construct the response surface of GTN parameters versus response variables, a series of simulation schemes need to be designed. Central composite design (CCD) method is applied in this work to design the simulation schemes. Based on the CCD method, a four-factors-three-levels simulation scheme was designed, as shown in Table 3. These factors correspond to the GTN parameters to be solved, i.e., the initial VVF ( f 0 ), critical VVF ( f c ), nucleation VVF ( f N ) and the final VVF ( f f ).

Taking the strain rate of 0.01 s⁻¹ for instance, the detailed solution process of GTN parameters will be demonstrated. First, the uniaxial tensile process was simulated using the designed GTN parameter combination in Table 3 under the strain rate of 0.01 s⁻¹. The response values corresponding to different GTN parameters were calculated according to the previously introduced procedure, and the results were recorded as shown in Table 3.

Based on the factors and corresponding response values in Table 3, the final functions of the response surfaces were achieved by least squares method. The expressions are shown in equation (8).

Due to the fact that the response variables R in equation (8) represent the difference between experimental and simulated stress–strain curves, the minimum R-values correspond to the most possible GTN parameter values. In other words, the closer the input GTN parameter is to the actual values, the smaller the response variables R-values will be. Consequently, the GTN parameters will be solved by searching for a suitable parameter combination that makes the response variables reach the minimum almost simultaneously. Considering the significant difference among R ₁, R ₂, R ₃ and R ₄, the response variables should be replaced by dimensionless variables R 1 ′ , R 2 ′ , R 3 ′ and R 4 ′ , as shown in equation (9).

Subsequently, the GTN parameters were solved by inputting the modified response functions into Matlab software. The solved GTN parameters are as follows: f 0 = 0.005, f N = 0.0412, f c = 0.0686 and f f = 0.3016.

Repeating the above calculation procedures, the GTN damage parameters at different strain rates were figured out, and the results were recorded as shown in Table 4.

According to Table 4, the strain rate has significant influence on the GTN parameters, especially for the final VVF. With the increase in the strain rate, the value of the final VVF decreases. Whereas, the nucleation VVF and critical VVF increase with the increase in the strain rate. As for the initial VVF, there exists little discrepancy under different strain rates.

When the specimens deform under high strain rate, the slip speed of dislocations is rapid. The deformation will not be coordinated, leading to local stress concentration on the matrix. The high interior stress will stimulate the deformation and expansion of the microvoids [21,22]. In addition, as the deformation process continues, the microvoids will be more easy to connect under the concentrated interior stress. When the necking occurs, the VVF reaches a critical value. The critical VVF is higher at high strain rate because the nucleation and expanding rate of voids is quicker at high strain rate. It is remarkable that the final VVF is smaller at higher strain rate, because there is no sufficient time for the nucleation and growth of the microvoids. For clearly displaying the impact mechanism of strain rate on the voids evolution, the scanning electron micrographs of the fracture morphologies for the tensile specimens are shown in Figure 8. There are obvious dimples in Figure 8 and it means that ductile damage is the dominant degradation mechanism for this alloy under the strain rate range of 0.1–10 s⁻¹. There are fine dimples in Figure 8a and b. In Figure 8c and d, the dimples are coarse and sparse. Therefore, a conclusion can be drawn that the higher strain rate contributes to the expansion and coalescence of the microvoids; meanwhile, the number of the microvoids will be less under high strain rate because of the insufficient deformation time.

Figure 8

The scanning electron micrographs of the fracture morphologies for the tensile specimens deformed at different strain rates: (a) 0.01 s⁻¹, (b) 0.1 s⁻¹, (c) 1 s⁻¹, and (d) 10 s⁻¹.

4.3 Characterization of the damage degradation behavior of Ti–3Al–2Mo–2Zr alloy

In order to dynamically characterize the damage degradation process of Ti–3Al–2Mo–2Zr alloy, the uniaxial tensile processes were numerically simulated using the solved GTN parameters. Subsequently, the damage degradation behaviors of Ti–3Al–2Mo–2Zr alloy were discussed taking the strain rate of 0.01 s⁻¹ as an example. Figure 9 shows the distribution and evolution of the VVF in the tensile process. According to the simulation results, it can be seen that the microvoids nucleate in the interior of the specimen at the beginning of the tensile process, and the microvoids begin to proliferate and propagate with the proceeding tensile deformation. When the true strain reaches 0.1 s⁻¹, necking occurs and the VVF increases to a critical level. With the ongoing tensile deformation, the VVF keeps increasing until the maximum VVF arrives at the final VVF (f _f). At this time, the specimen loses its overall loading capacity. The fracture position of the tensile specimen in the simulation is in good agreement with the experimental results.

Figure 9

The distribution of void volume fraction at the true strain of (a) 0 s⁻¹, (b) 0.05 s⁻¹, (c) 0.1 s⁻¹, (d) 0.15 s⁻¹, (e) 0.2 s⁻¹, (f) 0.229 s⁻¹, and (g) 0.23 s⁻¹.

In order to understand the damage process deeply, it is essential to analyze the stress evolution of the tensile process. Figure 10 shows the simulation result of equivalent Mises stress in the tensile process. It is obvious that the distribution of the equivalent stress is uniform at the initial stage of the tensile process. When the true strain reaches 0.1 s⁻¹, the equivalent stress shows a non-uniform distribution. As the deformation continues, the non-uniform stress distribution becomes more and more obvious, and higher stress concentrates in the middle and lower part of the specimen. It has been introduced in the theory of the GTN model that the stress concentration will promote the proliferation of microvoids. On the contrary, the increase in the VVF will make the loading capacity of materials decrease. Consequently, the mutual improvement between stress concentration and microvoids proliferation leads to the equivalent stress decrease in the later stage of the tensile deformation.

Figure 10

The distribution of Mises equivalent stress at the true strain of (a) 0 s⁻¹, (b) 0.05 s⁻¹, (c) 0.1 s⁻¹, (d) 0.15 s⁻¹, (e) 0.2 s⁻¹, (f) 0.229 s⁻¹, and (g) 0.23 s⁻¹.

In order to reveal the relationship between damage degradation and plastic deformation, the simulation results of equivalent plastic strain are presented in Figure 11. It is apparent that the plastic deformation zone is very small, and it mainly occurs at the middle and lower part of the specimen, where VVF is higher. The evolution of equivalent plastic strain indicates that the strain is sensitive to the damage degradation of this alloy. Comparing Figure 11a–g with each other, it can be found that the strain distribution is relatively uniform at the initial stage of the tensile deformation. When the true strain reaches 0.1 s⁻¹, the plastic deformation area intends to contract inside because the VVF is higher at the center of the specimen; meanwhile, necking appears at the area where the equivalent plastic strain is high. Generally, deforming is more easy at the areas with higher VVF because the deformation resistance is lower. When the true strain reaches 0.15 s⁻¹, the maximum of plastic strain keeps basically stable. Further tensile deformation will make the specimen break off.

Figure 11

The distribution of equivalent plastic strain at the true strain of (a) 0 s⁻¹, (b) 0.05 s⁻¹, (c) 0.1 s⁻¹, (d) 0.15 s⁻¹, (e) 0.2 s⁻¹, (f) 0.229 s⁻¹, and (g) 0.23 s⁻¹.

Not only the equivalent stress and plastic strain, but also the stress state is a key impact factor for the damage degradation behaviors of materials. Stress triaxiality is widely applied to represent the stress state of materials. High stress triaxiality values correspond to the tensile stress state, while low stress triaxiality values correspond to the compressive stress state. As for uniaxial tensile stress state, the stress triaxiality value is close to 0.3. Figure 12 exhibits the distribution and evolution of the stress triaxiality in the tensile process of Ti–3Al–2Mo–2Zr alloy. It is obvious that the stress triaxiality is 0.3 in most areas of the tensile specimen. The stress triaxiality at the fracture position is relatively high, which means the failure mechanism of this alloy is ductile damage. It is in agreement with the result that is observed from the EBSD images. More interestingly, the variety of stress triaxiality is complex at the fracture position. The center of the fracture position shows a higher stress triaxiality value, while the vicinities along the tensile axis show a lower stress triaxiality value. It indicates that the damage intents to initiate at the center of the specimen, and then expand along the direction perpendicular to the tensile axis. In addition, there is a visible change in the stress triaxiality distribution at the true strain of 0.1 s⁻¹. This suggests that the microvoids begin to proliferate and expand quickly at the true strain of 0.1 s⁻¹; meanwhile, the VVF reaches a critical value.

Figure 12

The distribution of stress triaxiality at the true strain of (a) 0 s⁻¹, (b) 0.05 s⁻¹, (c) 0.1 s⁻¹, (d) 0.15 s⁻¹, (e) 0.2 s⁻¹, (f) 0.229 s⁻¹, and (g) 0.23 s⁻¹.

In summary, the VVF is closely related to the stress, strain and stress state of the materials. The existence of microvoids will lead to the stress concentration, uneven plastic deformation and the change in local stress state. And the latter will contribute to the increase in VVF in return. This is the damage degradation process of materials during the uniaxial tensile deformation.

Since the evolution of VVF in the tensile process has been visualized by FE simulation, the degradation process of this alloy can be evaluated dynamically. Here the damage degradation degree is defined as the percentage of the current VVF to final VVF. Correspondingly, the damage degradation curves of Ti–3Al–2Mo–2Zr alloy at different strain rates were plotted as shown in Figure 13. According to Figure 13, it is apparent that all the degradation curves show a similar tendency, and the degradation speed is faster at higher strain rate. This point can be explained by the damage mechanism of this alloy. As discussed in the above Sections, the nucleation and proliferation of the microvoids is the dominant damage mechanism for this alloy. Therefore, in the early stage of the deformation, the progress of degradation process is slow because of the low nucleation speed of microvoids. In the middle and later stages of the tensile deformation, the degradation is faster because the nucleation and expansion of microvoids are promoted once the VVF reaches the critical value (f _c). Particularly, the promoting effect is enhanced under higher strain rates because the stress concentration and uneven plastic deformation is more severe at higher strain rate.

Figure 13

The damage degradation curves of Ti–3Al–2Mo–2Zr alloy at different strain rates.