The mechanical performance and a rate-dependent constitutive model for Al 3Ti compound

Yang Cao, Dan-dan Zhang, Jian-xiu Liu, Kun Liu, Jin-guang Du, Wen-bin He, Jun Ma

Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

Keywords:Titanium aluminide JH-2 model Numerical simulation Quasi-static and dynamic compression tests Ballistic test

ABSTRACT The Al3Ti compound has potential application in the high temperature structure materials due to its low density, high strength and stiffness. The mechanical behaviors of the material under different loading rates were studied using compression tests.The results indicate that Al3Ti is a typical brittle material and its compressive strength is dependent on the strain rate. Therefore, a series of rate-dependent constitutive equations are needed to describe its mechanical behaviors accurately. However, it is still short of professional research on the material model for Al3Ti.In this study,the material model was developed on the basis of JH-2 constitutive equations using the experimental data. The model was then applied in simulating the impact process of Ti/Al3Ti metal-intermetallic laminate composites so as to validate the established model. Good agreement between simulation and experiment results shows the constitutive model predict the material responses under high rate and large deformation accurately. This work provides more support for the theoretical and numerical research on the intermetallic.

1. Introduction

Ti-Al intermetallics have low density, high melting point, high specific strength, high stiffness and excellent oxidation and corrosion resistance. Therefore, good performance makes Ti-Al intermetallics have potential applications in many fields, such as industry, aviation spaceflight and military affairs [1,2]. In Ti-Al system there are three major intermetallic compounds: TiAl, Ti3Al and Al3Ti. TiAl has low density and high creep resistance. But its plastic deformability was poor at room temperature. The ability of plastic deformation for Ti3Al is better than TiAl, while it has some disadvantages,such as the lower strength and the weaker oxidation resistance at elevated temperatures.Compared with TiAl and Ti3Al,Al3Ti possesses the lowest density and the highest oxidation because of an impervious film of Al2O3forms on the surface due to the exposure to air[3,4].In addition,the Young’s modulus of Al3Ti is highest in other titanium aluminides [5]. These characteristics made Al3Ti has potential application in the high temperature structure materials. A density intermetallic Al3Ti was successfully synthesized using reactive sintering technique in vacuum by Wei[6]. Then the phase and microstructure were characterized by Energy Dispersive Spectrometer (EDS), X-ray diffraction (XRD) and scanning electron microscope (SEM), and the mechanical properties, including compressive strength and failure strain, were determined by quasi-static compressive tests,which demonstrated that the material exhibit brittle features. The mechanical response of Al3Ti under dynamic loading were also studied.The compressive strength can reach 1300 MPa or more, and it is sensitive to strain rate [7]. However, the lack of ductility in room temperature have greatly limited the application of Al3Ti[8,9].A great effort has been made to improve its ductility by adding ductile reinforcement with particles [10], fibers [11] or layers [12]. A new class of structural armor material,metal-intermetallic laminate(MIL)composites was developed by Harach and Vecchio in 2001 [13,14], have attracted researchers’ attention recently due to the combined desired features of metal and intermetallic [15-17]. This laminate composite has been considered promising for lightweight structural armor plating due to its unique properties, such as high stiffness, high modulus, and low density [18-21]. The ballistic performance for this composite has been evaluated at the U.S. Army Research Laboratory. In the research carried out by David at University of California, San Diego, the Al3Ti (with a trace of Ti) was also processed using hot-press sintering technology [22]. It is well known that numerical simulation is a brilliant solution to investigate the mechanical behaviors of a laminated system,especially in impact dynamics fields such as projectile protective performance [23,24].To obtain accurate simulation results, the mechanical properties of each constituent of the laminate composite must be known. It is easy to find the parameters of constitutive models for titanium.The JC model, for instance, is widely used to simulate the mechanical responses of titanium under high rate and large deformation.However,this is not the case for Al3Ti[25,26].The elastic properties and damage evolution of the Ti/Al3Ti laminate composite was modeled by Li [27]. In his work, Ti was simulated using the Johnson-Cook (JC) plasticity equation and the intermetallic was modeled using the JH-2 equation since the brittle nature of the Al3Ti. The parameters involved in the JH-2 model for Al3Ti were replaced by the ones for aluminum-nitride[28]due to the absence of these constants from Al3Ti. The failure mechanisms of Ti/Al3Ti composite under high-speed impact were analyzed by finite element simulation in the works of Fan[29]. The constitutive data of Al3Ti matrix were also taken from aluminum nitride due to lack of the data for the material.

This paper investigates the mechanical responses of Al3Ti under different compression strain rates and then obtains its constitutive parameters by using JH-2 constitutive equations and fitting test data. The reliability of the results is confirmed through a comparison between finite element analysis and a ballistic test for the Ti/Al3Ti laminate composite.

2. Experiment and constitutive model

2.1. Specimen preparation

The monolithic intermetallic Al3Ti was synthesized through reactive sintering in vacuum.The cleaned foils of Ti and Al,with the thickness of 0.5 mm and 0.9 mm respectively, are stacked in alternating layers and placed in the vacuum furnace.A diffusion reaction between Ti and Al foils has taken place by controlling the pressure and temperature. The furnace was heated to the maximum temperature ～685°C and held for 7 h at this temperature.The pressure was 2-3 MPa when the temperature was below 500°C, and reduced to 1-1.5 MPa in the temperature range of 500-685°C.The pressure was free during the heat preservation [6]. Compression specimens with dimensions of approximately 6.00 mm×6.00 mm×8.00 mm were cut from the fabricated material using a spark machine. Each surface of two foils were polished with silicon carbide papers and cleaned in alcohol bath using ultrasonic cleaning machine to remove the surface contaminations and oxide layers before sintering.

2.2. Experimental setup

The microstructure of Al3Ti was observed by LETCA DM IRM metallographic microscope. The distribution of chemical elements were scanned and analyzed by XFlash5010 Energy Dispersive Spectrometer(EDS)produced by Bruker.The qualitative analysis of Al3Ti components was conducted by XRD experiment (X’Pert Powder, PANalytical, Dutch). The scanning speed was 5°/min and the scanning range was 10°～90°. The data of experiment was processed by X’Pert HighScore Plus software.

The Young’s modulus E and Poisson ratio of Al3Ti were measured with an ultrasonic pulse-receiver (Olympus, 5072 PR,Olympus Co. Ltd., Beijing, China) according to ASTM E494-10 standard.

The quasi-static compression test was performed using an Instron 5500R electronic universal testing machine. The maximum output load of the testing machine is 100 KN, and the punch velocity can be adjusted between 0.05-500 mm/min.

To investigate the mechanical behavior under dynamic loading,a split Hopkinson pressure bar(SHPB)test system was employed to carry out the dynamic compression test.Wave shaping technology was used to tailor the incident pulse to a ramp loading so that the specimen deformed at a relatively uniform strain rate. A piece of copper with a thickness of 0.4 mm and a diameter of 5 mm was placed on the impact end of the striker in each dynamic compression test.

A ballistic test was employed to validate the correctness and practicability of the established material model.The ballistic test is conducted by shooting the projectile from the ballistic gun. The projectile was an armor-piercing projectile of diameter 6.2 mm and length 27.6 mm, which was made of a steel alloy. The projectile then passed through a speed detector, which consisted of two sheets of aluminum foils and a speed measuring circuit. The distance between two aluminum foils was available, and the corresponding time that the projectile passed through the two foils was obtained by the speed measuring circuit to calculate the initial speed of the projectile. The MIL composite target used in the test herein was 73 mm wide, 85 mm long, and 7 mm thick, which consists of 9 layers of Ti and 8 layers of Al3Ti. The material was clamped in the target box. The residual velocity of the projectile was obtained by a high-speed camera.

2.3. Constitutive model for Al3Ti

The Johnson-Holmquist-Ceramics (JH-2) equation proposed by Johnson and Holmquist has been proven to accurately simulate the mechanical responses of brittle materials under impact loading[28].In the compressive test under different strain rates,there are no significant plastic deformation in the failure materials. In addition,the strength of the material is sensitive to the strain rate.Thus the JH-2 equation is suitable for establishing the constitutive relation of Al3Ti material. In this equation, the damage parameter D,defined as the ratio of equivalent plastic strain increment to fracture strain, is used to describe the damage degree of the material.The failure strain is expressed as

where P*and T*are current and maximum hydrostatic pressures,respectively, both of which are normalized by the pressure at the Hugoniot elastic limit (HEL). D1and D2are the constants that depend on the damage. Each different damage parameter corresponds to a different strength model, as shown in Eq. (2).

The normalized strength with different damage, including the normalized intact equivalent stress, the normalized current stress σ*, and the normalized fracture stress, has the general form

where σ is the actual equivalent stress, and σHELis the equivalent stress at the HEL.Other constants in Eq.(2),A,N,C,B,and m are just the ones that need to be worked out in this study. A set of state equations in the JH-2 model are employed to describe the relationships between the volume and pressure with different damage. The hydrostatic pressure for intact material is simply expressed as

where K1,K2,and K3are material constants(K1is the bulk modulus).For the brittle materials, the failure mainly results from tensile damage,and both values of K2and K3are zero at the tensile stress.μ is volumetric strain and is relevant with the densities at initial ρ0and current ρ moment:

After damage begins to accumulate (D>0), bulking occurs due to the increase of pressure and/or volumetric strain [30]. An additional item for pressure increment is added to the state equation

From the energy standpoint,the energy loss ΔU is converted to potential internal energy by incrementally increasing Δp according to the following equation

The first term on the left side represents the potential energy for μ>0 and the second term represents the corresponding potential energy for μ<0. The constant β（0 ≤β ≤1） is friction of elastic energy loss converted to potential internal energy.The value of β was assumed to be one in this study, which meant the complete conversion of the energy.

3. Experimental results and model validation

3.1. Experimental results

The surface morphology of the prepared Al3Ti is shown in Fig.1,which has a uniform appearance without layered structure. However, it is observed that there are some pores in the Al3Ti. The appearance of the pore defect can be explained by Kirkendall effect.The diffusion rates of Ti and Al are different during the preparation process,and the solid solubility of Ti in Al is smaller than that of Al in Ti, which results pore defects. There also many discontinuous black lines in the product,this is related to the reaction mechanism.The reaction temperature is higher than the melting point of Al,and the reacting Al3Ti can move away from the Ti layer due to the existence of molten Al, which maintains a continuous reaction interface.When the reaction interface migrating away from one Ti layer meets its reciprocal moving downward from the next layer,the migration ends and the oxides at each front come together to form the centerline.

Fig.1. The surface morphology of Al3Ti intermetallic observed by SEM.

Based on Ti-Al binary phase diagram, several intermetallic compound, involving AlTi3, TiAl and Al3Ti, can be formed in the temperature range of 500°C-685°C. But only Al3Ti formed in practice due to its lowest free energy. It shows from the results of XRD and EDS. The XRD scanning pattern on the cross section of prepared material is shown in Fig. 2. The Ti had no characteristic diffraction peaks because it consumed completely in the preparing process.The products are mainly Al3Ti and a little Al3Ti0.8V0.2from the results of XRD.The reason is that the foils used to prepare Al3Ti is not pure Ti but titanium alloy (TC4), and the V element can be also observed from the result of EDS (Fig. 3). From the EDS result,the atomic number of Al and Ti are 74.31 At.% and 24.78 At.%,respectively,which is very closed to the theoretical atomic number ratio of 75 At.% and 25 At.% for Al3Ti. It can be concluded that the product of the reaction is Al3Ti.

The density of the Al3Ti was measured with drainage and ρ =3.4 g/cm3, then the Young’s modulus and Poisson’s ratio can be obtained by the ultrasonic pulse-receiver, which are 196 GPa and 0.18, respectively.

Fig. 2. The XRD result of the prepared Al3Ti.

Fig. 3. The EDS result of the prepared Al3Ti.

Fig.4. The failure morphologies of Al3Ti after(a)quasi-static compression test and (b)SHPB dynamic compression test,from which it was found that the crush degrees were more observable in dynamic loading.

The failure morphologies of Al3Ti after quasi-static compression test and SHPB dynamic compression test are given in Fig.4. In the quasi-static test, the material’s surface was broken more seriously than the central location,and the principal crack was at an angle of about 45°to the compressive direction.For the specimen subjected to dynamic compression,the specimen failed catastrophically,with little noticeable plastic deformation, behaving like a typical ceramic-like material. The fracture surface of the tested specimen was observed by SEM to investigate the failure mechanisms of the synthesized Al3Ti intermetallic. As shown in Fig. 5, the microstructure has a rock candy pattern, and the reflective ability is worse than the cleavage fracture. The fracture mode of the Al3Ti under the compressive load is intergranular fracture. In addition,there is a step-like morphology on the fracture surface, and with the increase of the strain rate, this phenomenon is more obvious.The JH-2 equation was therefore employed to simulate the damage evolution of Al3Ti in this study.

The data results in the quasi-static loading condition of Al3Ti are displayed in Table 1,and the typical stress-strain curves are given in Fig.6(a).The strain rate employed here is 10-3s-1.The difference in compressive strength is determined by the brittle nature of the Al3Ti. The average compressive strength of the quasi-static compression test is 778.28 MPa with the standard deviation of 66.1 MPa, and the average failure strain is 0.0075 under the strain rate of 10-3s-1.In the dynamic compression test,the strain pulses transmitted in the incident and transmission bars were captured by the strain gauges.The data was then used to calculate the stress and strain at various moments according to one-dimensional stress wave theory[31].The typical test results of the Al3Ti obtained from SHPB compression tests are displayed in Fig. 6(b), and the data result are listed in Table 2. The average dynamic compressive strength is 1158.12 MPa with the standard deviation of 7.5 MPa.The strength at dynamic condition is higher than the one obtained from the quasi-static test,and the corresponding failure strain is 0.0157.

For the projectile test of Ti/Al3Ti laminate composite,the initial impact velocity was determined by the speed detector as being 803 m/s.The projectile penetrated the composite,and the residual velocity of the projectile was captured using a high speed camera,which was 706 m/s. The damage morphology of Ti/Al3Ti MIL composite after the ballistic test is shown in Fig. 7. The crater diameter in the opposite side is bigger than the front side and the composite produces distortion between layers for stress action.Too large stress leads to a tearing in the layers. There are three main failure modes in the damage materials,including brittle fracture of Al3Ti,plastic deformation and tearing of Ti,and delamination at the interface.

3.2. Parameters characterization

Fig. 5. The SEM results of Al3Ti at the fracture surfaces after compressive test: (a) the strain rate is 10-3 s-1, (b) the strain rate is 103 s-1.

Table 1 The quasi-static compression tests results.

Fig. 6. The typical stress-strain curves obtained from quasi-static and dynamic compressive test: (a) the strain rate is 10-3 s-1, (b) the strain rate is 103 s-1.

Table 2 The dynamic compression tests results.

Fig. 7. The damage morphology on the (a) front side and (b) opposite side of the MIL composite after the ballistic test.

The data results for Al3Ti intermetallic under different strain rates are used to calculate the constitutive parameters. There are many parameters,as listed in Tab.3,need to characterized for JH-2 equations. The shear modulus G and bulk modulus K1could be obtained through the following relationships when the density,Young’s modulus and Poisson ratio are known.

The intact strength σ*iwas firstly determined by the quasi-static and dynamic result data shown in Tables 1 and 2.The linear fitting approach was used to process the experimental data of compressive strength and pressure under the two different strain rates.The fitting results are

The variation of compressive strength ratio (σ1/σ2) with pressure was obtained by dividing the two fitted linear equation.It was found that when the pressure is higher than zero,the strength ratio approaches a constant of 1.149 with the increase in pressure. The two fitted linear equation given in Eq.(10)was substituted into the intact strength model(D=0),and the reference strain rate（˙ε0）was equal to s-1.The value of C was then obtained to be 0.0146 from the following equation.

The maximum tensile stress that the material could be sustained has been measured by Zelepugin S.A from the Russian Academy of Sciences[32,33],which is equal to 0.4 GPa.The stress contained the deviatoric stress and tensor component of hydrostatic pressure,in which the maximum hydrostatic tensor component (T) was assumed to be 0.2 GPa here. All of the mechanical parameters included in the JH-2 strength equation need to be normalized by their corresponding values at HEL. The elastic limit at HEL was estimated by the previous study [34]:

where Ycis the uniaxial compressive strength under the dynamic loading, which could be measured using the SHPB compression test. The hydrostatic pressure has the following expression under HEL:

and the deviatoric stress under HEL(SHEL)is calculated based on the relationship with PHELand σHEL

Once the mechanical parameters at HEL were obtained, the experimental result data was normalized and also listed in Tables 1 and 2 as σ*and P*.At this moment,the constants of A and n in the intact strength model were able to be calculated by two sets of data for σ*and P* under different strain rates.

For the fracture model(D=1),the impact test is required for the debris to determinate the residual strength of the material. Parameters of B and m were taken directly from the AD95 ceramic due to the absence of a relative experiment, and the values would be adjusted according to the ballistic test result.So far the constants in the JH-2 strength model for the Al3Ti intermetallic are all listed in Tab. 3.

There is a damage equation beside strength equation for JH-2 constitutive model, and two material parameters (D1and D2) in the damage equation need to be determined.The failure strain and corresponding pressure results obtained from the quasi-static and SHPB dynamic compression tests were substituted into Eq.(1),and the values of D1and D2were worked out as D1=0.19, D2=3.1.

3.3. Model validation

The parameters of the JH-2 constitutive model for Al3Ti were then applied to simulate the matrix Al3Ti of the Ti/Al3Ti MIL composite in the penetration procedure. FEA software LS-DYNA (Livermore Software Technology Corp.) was used to simulate the penetration process of the MIL composite.The element type is an 8-noded solid element, and the simulation was conducted with the same geometry and boundary conditions as those in the ballistic test. Given that the further the material is from the impact crater,the less the damage that occurs,the mesh of different densities was used to save computing time.The total number of the element was 263935. The CONTACT_ERODING_SURFACE_TO SURFACE contact model was defined between the projectile and target material.The CONTACT_SINGLE_SURFACE contact model was defined for the target material to consider its own contact when deformation was generated.The static and dynamic coefficients of frictional were set to 0.15 and 0.1 respectively.The default hour glassing technology in LS-DYNA was applied in the simulation.

In the simulation,the JH-2 model and JC model were employed to simulate the mechanical response of the intermetallic matrix Al3Ti and the reinforcement of Ti, respectively. The parameters of the JC model for Ti originated from Ref.[35],and the parameters of the JH-2 model for Al3Ti came from the experiments described above. The interfaces between Ti and Al3Ti layers were not considered in the simulation. That is because the interfaces are metallurgical bond,and the thickness of the interface is only 1 μm from the result of electron back-scattered diffraction. In order to simplify the simulation process, the different layers of the MIL composite were connected by common nodes.The final numerical model is shown in Fig. 8.

In LS-DYNA, the damage parameter D was used to describe the damage degree of the material. The distribution of the damage parameter for Ti/Al3Ti MIL composite during the penetrating process is shown in Fig.9(b).The damage around the crater was serious at t=5 μs,while the last layer of the Al3Ti sustained slight damage parallel to the layers. With the penetration depth increased(t=8-13 μs),the stress wave reached the back face of the material and reflected, made the Al3Ti at the bottom tend to have severe damage. When the projectile penetrated at the last layer of the material (t=16 μs), the velocity of the projectile dramatically decreased. The plastic deformation of the material at the bottom was more obvious than the front side. When the projectile penetrate through the material, a large delamination was initiated between the Ti and Al3Ti layers.

From the result of damage process, it has been found that the diameter of the projectile hole had change continuously along thickness direction, and the crater at the bottom had the largest diameter,which shows a similar failure characteristic with the test result,as shown in Fig.10.The residual velocity of the projectile was captured using a high speed camera, which was 706 m/s. The corresponding velocity calculated by the finite element simulation was 712 m/s, which was very close to the test result. The similarity between the FEM results and test data demonstrates that the FEA model is effective and reliable, and also proves the correctness of the material model of Al3Ti.

4. Conclusion

The constitutive parameters of the JH-2 model for the titanium tri-aluminide intermetallic alloy were determined under basic physical and mechanical property tests.The model is established to simulate the mechanical response and damage evolution of the matrix of a potential protecting material Ti/Al3Ti. In order to illustrate the validity of the constitutive model parameters,the ballistic test for Ti/Al3Ti was conducted and compared to corresponding finite element simulation. The following conclusions were drawn.

(1) The material’s quasi-static compressive strength is 778.28 MPa, as measured by an Instron universal testing machine. In addition, the dynamic compressive strength of Al3Ti under the strain rate of 1000 s-1is 1158.12 MPa, as determined by a modified SHPB system.

(2) The compressive tests under different strain rates for Al3Ti showed that the strength and failure strain increase with increasing strain rate. In addition, the Al3Ti is a typical ratedependent material.

(3) The ballistic test showed that there are radial and circumferential cracks in the materials during the impact process,and there is delamination between different layers.

Fig. 8. The numerical model of the MIL composite impacted by the armor-piercing projectile.

Fig. 9. The distribution result of damage parameters in the numerical simulation.

Fig.10. The damage morphology on the(a)front side and(b)opposite side of the MIL composite obtained from numerical simulation.

(4) The JH-2 parameters for Al3Ti have been successfully determined by compressive tests under different strain rates,and the model has been proven to accurately simulate the mechanical responses of Al3Ti under impact loading given the similarity between the experimental and simulation data.

Fund

This work was supported by the National Natural Science Foundation of China (grand number 11602230), the Program for Innovative Research Team in Science and Technology in the University of Henan Province (grand number 18IRTSTHN015), Key Scientific Projects of University in Henan Province(20B430021).

Acknowledgements

The authors gratefully acknowledge the financial support from National Natural Science Foundation of China (No.11602230), the Program for Innovative Research Team in Science and Technology in the University of Henan Province (No.18IRTSTHN015), Key Scientific Projects of University in Henan Province (20B430021).