Simulation of the plastic response of Ti–6Al–4V thin sheet under different loading conditions using the viscoplastic self-consistent model

Abstract

In this article, the capacity of the Visco-Plastic Self-Consistent model (VPSC) as a constitutive model for the simulation of cold-rolled Ti–6Al–4V under a diverse set of loading conditions is investigated. The model uses information about the material crystallographic texture and grain morphology obtained with EBSD together with a grain constitutive model that is sensitive to strain rate and temperature. The ability of the model to capture the macroscopic response and underlying microstructural phenomena is critically assessed considering various sets of hardening parameters, obtained applying different fitting procedures to tensile experiments in different directions on the sheet plane. In order to validate the fitted parameters, the obtained model is applied to the simulation of experiments in a wide range of testing conditions, including uniaxial, plane strain and simple shear loading modes at strain rates ranging from ${10}^{-4}{\mathrm{s}}^{-1}$ to ${10}^3{\mathrm{s}}^{-1}$ and temperatures between $-10°\mathrm{C}$ and $70°\mathrm{C}$. High strain rate experiments were performed using a Split Hopkinson Bar setup, with specimens designed to achieve the desired loading mode, while isothermal experiments were performed with the help of a fluid cell to keep a constant temperature. A good agreement between experimental and simulation results is obtained.

Introduction

Ti–6Al–4V is the most widely used titanium alloy. Due to its excellent combination of strength, ductility and low weight, and its extraordinary resistance to extreme temperatures and corrosive environments, it is the chosen material for applications that go from the fabrication of components for the aerospace, energy and defense industries to the manufacture of sports equipment, electronics, and even, due to its biocompatibility with the human body, for medical devices and implants.

In order to use the material in these applications, accurate models to describe its plastic behavior are required. This task presents some non-trivial challenges in the particular case of Ti–6Al–4V. This multiphase alloy, composed of a predominant hexagonal Ti-$\alpha$ phase but also with a smaller fraction of cubic Ti-$\beta$, is highly sensitive to temperature and strain rate, has a pronounced strength differential effect, and exhibits a highly anisotropic behavior, especially if its crystallographic texture is strong. A strong texture is commonly developed with plastic straining, and therefore generally results from forming processes.

The effect of strain rate and temperature on the hardening behavior of Ti–6Al–4V has been extensively studied. It is well known that thermal softening and strain rate hardening are significantly more pronounced in Ti–6Al–4V than in other engineering materials such as steel [1], [2], [3], [4]. The problem is further complicated by the significant strength differential effect, i.e. the tension–compression asymmetry, of Ti–6Al–4V, related with the polarity in the formation of twins under compression loads [5], [6] and very sensitive to small variations in composition and processing [1].

Several hardening models that account for strain rate and temperature sensitivity have been employed to model the plastic behavior of Ti–6Al–4V. Some of these models follow a purely phenomenological approach, such as the Johnson-Cook (JC) [1], [2], [4], [7], Fields-Backofen (FB) [7], [8], and Khan-Huang-Liang (KHL) [7], [9] models, while other ones such as the Mechanical Threshold Stress (MTS) and Zerilli-Armstrong (ZA) models are based on the deformation mechanisms of the material at the dislocation level [7], [8], [10], [11].

One of the most popular phenomenological models in literature is the one proposed by Johnson and Cook [12], as a result of its good combination of accuracy, simplicity and range of application. Indeed, the model has been successfully employed in simulations from temperatures close to the absolute zero up to more than $1200\mathrm{K}$, and strain rates varying from the quasi-static regime up to ${10}^5{\mathrm{s}}^{-1}$ [2]. Other researchers establish narrower limits of temperature and strain rate at which the model is valid ($T<500°\mathrm{C}$ and $\dot{\epsilon }<7500{\mathrm{s}}^{-1}$), due to the interdependence between both sensitivities at more extreme conditions [9].

As an alternative to empirical hardening models, physically-based modeling approaches have been developed. These models consider to a greater or lesser extent microstructural phenomena, often including texture, lying at the origin of the observed mechanical response. Among them, crystal plasticity (CP) models [13] offer a solution to the problem of predicting the anisotropic mechanical behavior of polycrystal materials taking into account the inherent anisotropy of crystals and crystalline aggregates. CP models can be divided in two main groups: those that use a mean-field approach, such that each grain is subjected to a homogeneous mechanical state; and full-field models, where a single grain is further divided in a number of material points with different mechanical states. Some examples of mean-field models include the classical Taylor [14] and Sachs [15] theories, as well as models based on more complex grain interaction schemes such as the ALAMEL [16] and VPSC (Visco Plastic Self Consistent) [17], [18] models. Current full-field models are mainly characterized by the solver used, which can be based either on Finite-Element (FE) formulations, in what is called CPFEM (Crystal Plasticity Finite Element Method) [13], [19] or use Fast-Fourier-Transforms (FFT), as it is done in the more recent CPFFT (Crystal Plasticity Fast Fourier Transform) models [20], [21].

The use of CP models requires additional data about the materials microstructure and information about the deformation mechanisms at the crystal level. In the case of Ti-$\alpha$, the predominant phase of Ti–6Al–4V, deformation is the result of slip along the $〈a〉$ direction on the prismatic, basal or pyramidal planes, and along the $〈c+a〉$ direction on the first and second order pyramidal planes. Under certain conditions, such as compression loads, very high strain rates, or elevated temperatures, twinning deformation modes can also be activated and even become dominant [22], [23]. It has also been argued that twinning may be present at much lower strains and strain rates than generally expected [23], [24] but, due to the complete reorientation of grains, it goes undetected in microscopic studies. However, this hypothesis cannot be easily tested experimentally. Relatively speaking, it is well known that the shear stress required for slip along $〈a〉$ is one-quarter to one-half of the one required for slip along $〈c+a〉$ [25], [26], [27]. Between the different slip systems in the $〈a〉$ direction, it has been observed that slip on the prismatic and basal planes is more abundant than pyramidal slip [26], [28], [29].

Several authors have attempted to simulate the plastic behavior of Ti–6Al–4V using different crystal plasticity models. While some studies have focused on specific details of plastic deformation using CPFEM methods [25], [30], [31], [32], [33], [34], [35], in other works to use a homogenization CP model as VPSC is preferred, in order to approach a more statistical description of material behavior and avoid the hard requirements of CPFEM for extensive input data and computational power. For example, Lebensohn and Canova used VPSC to study the theoretical influence of microstructural properties in the tensile behavior of Ti $\alpha /\beta$ alloys [36], and Yapici and coworkers [37] employed VPSC to study the behavior of Ti–6Al–4V under severe plastic deformation. The work of Coghe et al., who applied the model to the study of twinning in Ti–6Al–4V under static and dynamic tension and compression loading modes [38], [23], is also remarkable. Their model considers deformation by slip and twinning in a single $\alpha$ phase, as well as strain rate sensitivity. However, since the parameters were fitted using exclusively compression experiments, the model cannot accurately predict tensile behavior.

In this work, it is discussed how the VPSC crystal plasticity model can be applied to the simulation of the plastic behavior of Ti–6Al–4V under a diverse set of loading conditions. The VPSC model provides a unique combination of accuracy, thanks to its grain interaction scheme based on the Eshelby inclusion theory [39], and relative simplicity (and, therefore, computational efficiency), when compared to full-field methods. A range of strain rates between, approximately, ${10}^{-5}$ and ${10}^3{\mathrm{s}}^{-1}$ are investigated, at temperatures going from $-10°\mathrm{C}$ to $70°\mathrm{C}$. Both temperature and strain rate sensitivity are introduced in the VPSC formulation such that they are equivalent to the effect described by the Johnson-Cook model, which, as discussed in the previous paragraphs, can effectively describe the influence of both factors in the material behavior for this range of variation. In particular, strain rate sensitivity is considered taking advantage of the similarities between the strain rate dependent term of the Johnson-Cook model and the strain rate sensitivity approach used in VPSC. Strain hardening parameters of the different slip systems are fitted with tensile experiments in different loading directions using an optimization algorithm specifically developed for this task. Later, the model is validated under a wider range of loading conditions, including plane strain and shear loading modes at quasi-static and dynamic rates. To this purpose, the VPSC90 optimized version of VPSC is used. VPSC90 is also implemented as a user material model in the commercial finite-element program Abaqus™, thus opening the possibility to the simulation of problems with arbitrary geometries and boundary conditions in a finite-element framework [18], [40].

After this introduction, the experimental methods used for the fitting and validation of the model are presented in Section 2, and the VPSC model is briefly described in Section 3, as well as the fitting method employed. The obtained results are shown in Section 4, and further discussed in Section 5. Finally, the conclusions of the study are presented in Section 6.