Direct and Inverse Characterization of the Asymmetric Hardening Behavior of Bulk Ti64 Alloy ^†

Abstract

This work focuses on the evaluation of the calibration strategy of the CPB06 asymmetry parameter k and its influence on the predictive behavior of the model at large deformations for Ti64. The direct identification strategy is based on fitting the model with experimental strain hardening data up to the onset of plastic instability. The inverse strategy is performed by reducing the prediction errors of the load–displacement curves of both the cylindrical bar tensile test and the elliptical cylinder compression test. Both strategies use an orthotropic tensor of the CPB06 criterion previously identified from experiments performed in all three dimensions. The results presented quantify the maximum error achieved by each method in terms of elongation and load predictions for specimens with different stress states.

1. Introduction

To increase the prediction of mechanical deformation of Ti64 in specimens with several combined stress states and high triaxialities, the asymmetric anisotropic pressure-sensitive plasticity-damage model has been proposed [1,2] to be applicable not only for bulk forming operations but also for accurate prediction of sheet metal forming and forming limit diagrams [3].

The complexity of hcp metals and alloys has been investigated by several authors in order the characterize the mechanical behavior for bulk and sheet metal forming processes, additive manufacturing, and design of parts under operational conditions of combined stress states [4,5,6,7,8].

In this work, the focus is on the prediction accuracy of the large deformation behavior of an hcp Ti64, with a tension–compression asymmetric behavior including the plastic instability. The finite element (FE) material model selected to simulate the evolving quasi-static, anisotropic, and tension–compression asymmetric mechanical response until the onset of fracture of bulk Ti64 alloy is based on five distorted CPB06 plasticity surfaces [9]. The model considers that the different yield surfaces follow the Voce hardening law in the reference direction of the alloy, while the other directions are modelled by distortional hardening. Two identification methods are performed and compared: direct and inverse. The direct identification is performed by curve fitting of the model with the actual stress vs. logarithmic strain data from tensile tests under uniform stress states, without considering data generated during necking or under triaxial stress states. The second identification strategy consists of minimizing the load prediction error obtained in the test simulations with respect to the experimental data generated until the complete failure of the specimen. By considering more information in the second strategy, it is expected to obtain higher accuracy in the predictions; however, because of the complexity of the identification due to the asymmetry of the mechanical response in terms of strain hardening between tension and compression, it is not evident to obtain adequate model parameters.

In the following sections, the methods and main results obtained with the two calibration strategies of the CPB06 asymmetry parameter k and strain hardening are reported, including the assessment of the two identification methods and their influence on the predictive behavior of the CPB06-Ti64 plasticity model at large strain levels.

2. Materials and Methods

2.1. Material Model

The orthotropic plastic behavior of Ti64 is described by 5 previously identified distorted CPB06 yielding surfaces [9] with a constant homogeneity degree of 2. The asymmetric tension–compression hardening behavior is adjusted by the k-parameter of the CPB06 criterion and the constants of the Voce law representing the reference tensile hardening. The CPB06 equivalent stress for Ti64 is given by:

$${\overline{\sigma }}^{{\mathrm{CPB}}_{06}}=\tilde{m}{\left\{{\left(\left|{\Sigma }_1\right|-k{\Sigma }_1\right)}^2+{\left(\left|{\Sigma }_2\right|-k{\Sigma }_2\right)}^2+{\left(\left|{\Sigma }_3\right|-k{\Sigma }_3\right)}^2\right\}}^{\frac{1}{2}}$$

(1)

${\Sigma }_1,$ ${\Sigma }_2,$ ${\Sigma }_3$ are the principal values of the tensor $\Sigma =C:S$, where C is the is orthotropy tensor and S the deviator of the Cauchy tensor ($S=L:\sigma$). The tensors C, L, and σ are defined as follows:

$$C=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{22}& C_{23}& 0& 0& 0\\ C_{13}& C_{23}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right];L=\left[\begin{array}{cccccc}2/3& -1/3& -1/3& 0& 0& 0\\ -1/3& 2/3& -1/3& 0& 0& 0\\ -1/3& -1/3& 2/3& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right].$$

(2)

The material constant m̃ describes the equivalent stress as the tensile hardening in the reference direction. The Voce strain hardening law for the reference direction (Equation (3)) is adopted here in the form given by Harbaoui et al. [8], where σ_y corresponds to the highest level of stress that the material withstands (stagnation), α provides information on the state of hardening, being zero for a perfectly plastic material; and β represents the rate at which the stagnation stress is reached:

$$\overline{\sigma }={\sigma }_y\left(1-\alpha \exp \left(\beta {\overline{\epsilon }}_p\right)\right)$$

(3)

2.2. Asymmetry Paramater Identification

The anisotropic plasticity parameters applied to the CPB06 model given in

Table 1. Model parameters for anisotropy (C_ij) as a function of the specific plastic strain energy.

Strain Energy per Unit Volume (W/m3)	Anisotropic Constants of CPB06
	C11	C12	C13	C22	C23	C33	C44 = C55 = C66
1.857	1	−2.37	−2.36	−1.84	1.20	−2.44	−3.61
9.377	1	−2.50	−2.93	−2.28	1.28	−2.45	4.02
48.66	1	−2.43	−2.92	1.65	−2.24	1.00	−4.00
100.2	1	−2.57	−2.88	1.39	−2.38	0.88	−3.93
206.6	1	−2.97	−2.93	0.53	−2.96	0.44	−3.88

have been previously identified with mechanical experiments performed in the three dimensions of the alloy [9]. In order to determine the two sets of constitutive hardening parameters with direct and inverse strategies, tension and compression test data are post-processed according to the procedure described in Tuninetti et al., 2020 [10]. The performed tests, experimental arrangement of machines, and measurement devices, as well as the dimensions of the samples are given in Figure 1. The loading conditions include a controlled non-constant machine head speed computed with the method described in [11] for a constant deformation rate of 0.001 s⁻¹. The determined model constants affecting the tension–compression asymmetry predictions of Ti64 are provided in

Table 2. Identified model parameters for asymmetric hardening with direct and inverse identification methods.

	Asymmetry Parameter k for Each Yield Surface					Strain Hardening Coefficients
	1.857 *	9.377 *	48.66 *	100.2 *	206.6 *	σy (MPa)	α	β
Direct	−0.136	−0.136	−0.165	−0.164	−0.18	1081	0.14801	−15.5
Inverse	−0.136	−0.136	−0.125	−0.114	−0.11	1208	0.24007	−5.8

* Specific plastic strain energy (W/m³).

. The tension–compression asymmetry k-value of the CPB06 criterion and the constants of the Voce law representing the reference tensile hardening were considered in this work, and identified with the direct and inverse strategy. The first identification strategy is mainly based on the experimental actual stress–strain data until the necking point computed with the Considère criterion (CPB06 direct), and the second is performed by reducing the FE prediction errors of load–displacement curves of both tensile test of a cylindrical bar and compression test of an elliptic cylinder sample. The latter strategy allows to obtained a model called here CPB06 (inverse). Digital image correlation has been applied to all the experiments to evaluate the two identification strategies and the model capabilities about the shape prediction (Figure 1). Large prediction errors of area reduction and load in a simple uniaxial tensile cylindrical bar confirm that the second strategy is essential to accurately predict the plastic instability. The validity of this claim has been proved here in this work only for bulk material and should be further verified for sheet metals, such as simulations of plain strain tests, prediction of forming limit diagram, and sheet metal forming processes.

3. Results and Discussion

To evaluate the numerical predictions of load in Ti64, FE results for the tensile of cylindrical bar and compression of elliptic cylinder simulations, both axially loaded in the reference direction are computed. The prediction capabilities of the two models are also assessed in a plain-strain sample.

3.1. Tensile Loading Predictions

The loading prediction results provided by the inversely identified CPB06 model are more accurate than the directly identified CPB06 (Figure 2). From initial plasticity till the onset of necking, load is slightly lower in accuracy for CPB06-inverse than the CPB06-direct. However, CPB06-inverse allows to obtain overall load–displacement curves much closer to the experimental data, with a slightly increasing error near the sample fracture. These results allow us to confirm that to accuracy increase of tensile load prediction during plastic deformation of the Ti64 until fracture, asymmetrical hardening should be identified for values of plastic work higher than the values reached at the onset of necking or stress–strain curves until 0.1 as is the case for the investigated Ti64. For the case of simple plain strain (Figure 3), both strategies yield to accurate results, with slightly better prediction for CPB06-inverse. This is explained by the lower values of strain reached in the plain strain sample at fracture (0.11) which is close to the plastic instability value in the uniaxial tensile test.

3.2. Compressive Loading Prediction

For compression testing, considerable higher load displacement predictions were also reached with the CPB06-inverse (Figure 4). The strain hardening parameters for the reference direction provide closer values in tensile after necking, and, here for compression, the better predictions are in a complete plastic range, due to the modifications of the asymmetry parameter k. Note that inverse identification includes the effect of barreling, friction, and strain gradients on the compression sample, which are not included in the direct strategy.

4. Conclusions

The accuracy of load and shape predictions between neck instability and fracture of the cylindrical tensile specimen was significantly improved by identifying a new set of hardening parameters and k values by inverse modeling. For compression testing, considerable higher load displacement predictions were also reached with the CPB06-inverse.

Finally, to increase the prediction of mechanical deformation of Ti64 in specimens with several combined stress states and high stress triaxialities, further research required an asymmetric anisotropic pressure-sensitive plasticity-damage model.