Ultrasonics 96 (2019) 1–9 Contents lists available at ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras Constitutive model of 6063 aluminum alloy under the ultrasonic vibration upsetting based on Johnson-Cook model T ⁎ Zhendong Xie, Yanjin Guan , Jun Lin, Jiqiang Zhai, Lihua Zhu Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China ARTICLE INFO ABSTRACT Keywords: Establishing an accurate constitutive relation in ultrasonic vibration assisted metal forming, can provides a Ultrasonic vibration reliable theoretical basis for analyzing the mechanism of the ultrasonic vibration on materials. A constitutive Johnson-Cook model model of 6063 aluminum alloy under the ultrasonic vibration upsetting at room temperature was constructed Strain rate based on Johnson-Cook Model and experimental results. The influence of amplitude and frequency on the yield Acoustic softening strength, hardening coefficient and exponent were analyzed quantitatively. Results showed that the yield Hardening strength was reduced due to the softening effect induced by the ultrasonic vibration. The maximum decreasing amount was 68.8% when imposing the maximum ultrasonic energy in this study. The hardening coefficient and exponent increased by 10.9% and 16.6% in maximum, respectively. However, the ultrasonic vibration has little impact on the strain rate hardening. The modified Johnson-Cook constitutive model under the ultrasonic vi- bration upsetting was established and was in good agreement with the experimental results. 1. Introduction ultrasonic intensity. Based on elastic-viscoplastic constitutive relation, He et al. [11] studied the variation of the uniaxial tensile stress and Considerable efforts are being devoted to the investigation of the strain under the vibration condition. A one-dimensional mechanical ultrasonic vibration assisted metal plastic forming and its acting me- model was established to describe the volume effect of low and medium chanism on materials in recent decades [1–3], and these developed frequency vibrations. They found that the vibrational tensile stress forming processes, including ultrasonic drawing, ultrasonic extrusion depends on the strain rate and strain history. According to the super- and ultrasonic upsetting [4–6], has been widely used in practice. position principle and the strain variation assumption proposed by Earlier studies [7–9] have shown that the influence of ultrasonic Kirchner, He et al. [12] also expressed a mathematical description of vibration on material properties mainly include two aspects: (1) the the volume effect, and they denoted that the physical nature of the volume effect which can reduce the flow stress, and (2) the surface volume effect was the decrease of the average stress. The influence of effect which affects the friction status between the die and the work- ultrasonic vibration on uniaxial tensile process depended on the fre- piece. Focusing on these two effects, many scholars have conducted quency, amplitude and deformation velocity. Zheng et al. [13] studied various studies. However, the influence and acting mechanism of ul- the special behavior of metals under super-frequency vibration, and trasonic vibration on metals and theirs plastic behaviors are still un- established the elastic constitutive equation of metals and emphasized clear and the quantitative description model has not been established. that the effect of microstructures should be taken into account. Ac- Therefore, many efforts have been devoted to construct theoretical cording to the vibration assisted tensile experiments with applying low models that can quantitatively describe the effect of ultrasonic vibra- frequency vibration, Cai et al. [14] constructed a one-dimensional tion on the plastic deformation behaviors of metals based on certain visco-elastoplastic model based on the Kirchner‘ assumption, and stu- assumptions and simplification. died the mechanism of volume effect. Atanasiu [10] systematically and quantitatively estimated the re- The above mentioned theoretical models only can reveal the influ- lation between the ultrasonic field and material properties, described ence of ultrasonic vibration on material behavior qualitatively and can’t the metal deformation behaviors in the process of ultrasonic assisted describe the stress-strain relationship under the ultrasonic vibration forming based on rigid-viscoplastic model and concluded that the yield condition quantificationally and accurately. Especially, the influence of strength of the material reduced exponentially with the increase of the the frequency and amplitude can’t be included into these theoretical ⁎ Corresponding author. E-mail address: [email protected] (Y. Guan). https://doi.org/10.1016/j.ultras.2019.03.017 Received 19 June 2018; Received in revised form 28 November 2018; Accepted 20 March 2019 Available online 21 March 2019 0041-624X/ © 2019 Elsevier B.V. All rights reserved. Z. Xie, et al. Ultrasonics 96 (2019) 1–9 models. Thus, based on Arrhennius thermal activation model and dis- strain hardening behavior of materials. C3 is the coefficient of strain- location evolution theory [15], Yao [16] proposed an acoustic softening rate hardening, reflecting the sensitivity of material to strain rate. m is model and an acoustic residual hardening model respectively. Upsetting the thermal softening exponent, meaning the softening effect of mate- tests assisted by high frequency vibration were conducted to construct rials subjected to thermal loading. Because all of the experiments in our the constitutive model of which the results showed a good agreement work were carried out at room temperature, T = Tr. Meanwhile, the with the experimental results. Wang [17] divided the softening effect of factors of strain, strain-rate and temperature that in the JC model are ultrasonic vibration into two aspects: acoustic softening and stress su- independent of each other. Thus, the effect of the temperature can be perposition, and then calculated the proportion of the stress reduction ignored and then the JC model can be simplified as induced by acoustic softening effect. Prabhakar et al. [18] also pro- posed a theoretical model based on the dislocation dynamics, which can n ⎡ ε̇⎤ σ =+()1(lnCCε12+ C 3 ) express the effect of strain, strain rate and amplitude on the stress. ⎣⎢ εṙ⎦⎥ (4) However, there are too many parameters that are hard to obtain In order to determine the parameters in (4), the upsetting tests through the experiments in these models. Therefore, it is necessary to under the reference strain-rate were performed firstly. When εε̇= , Eq.̇ construct a relatively convenient and accurate theoretical model to r (4) can be simplified as reflect the stress-strain relationship of the material deformed under the n ultrasonic vibration condition. σ =+()CCε12 (5) In general, an ideal model should be able to describe the materials’ plastic behaviors, such as strain-rate dependence, temperature depen- The value of C1 is determined as the yield stress σs at the strain of dence, strain and strain-rate history dependence, work-hardening or 0.2% at the reference strain-rate. In this study, the reference strain rate -3 −1 strain-hardening behaviors. However, it is extremely difficult and even εṙis 3.33 × 10 s . impossible to completely describe all these phenomena. Therefore, By rearranging Eq. (5) and taking natural logarithm on both sides of many scholars introduced necessary assumptions and tried to describe Eq. (5), it could be transformed into the following form: the deformation behavior in a wide range of strain rate and temperature ln(σσ−=s ) nε ln + ln C2 (6) as accurately as possible. At present, many phenomenological con- stitutive models are proposed which can well represent the materials’ Then, using the true stress-strain data under reference strain-rate plastic deformation behaviors in a certain range, such as Johnson-Cook condition, the relationship between lnε and ln(()σσ− s ) is determined. (JC) model [19], Zerilli-Armstrong model [20], Bodner-Partom model The values of n and C2 can be obtained as the slope and intercept of the [21], and Khan-Huang model [22]. Especially, the JC constitutive fitting line. model has been applied widely because of its simplicity and uni- After that, several upsetting tests under a list of strain rates should – versality [23 25]. be performed to obtain the unknown C3. Therefore, in this paper, the conventional upsetting (C-upsetting) It is found from Eq. (4) that the relationship between σ and ln(εε/̇ṙ) and ultrasonic vibration assisted upsetting (UV-upsetting) of 6063 Al can be decided by flow stresses under different strain rates. The value of ff n alloy under di erent strain rates were conducted, and the parameters in C3 can be calculated from the slope CC31·(+ Cε 2 ) of the fitting line of the JC model were estimated based on the experimental results. The σ-σ0.2. influence of the frequency and amplitude on material’s parameters was analyzed. Consequently, the modified JC model which couples with the 3. Ultrasonic vibration assisted upsetting ultrasonic vibration parameters was established, which can reflect the influence of ultrasonic vibration on the stress-strain relation of 6063 Al 3.1. Establishment of ultrasonic vibration assisted upsetting system alloy. This model provides a reliable theoretical basis for the analysis of the effect of ultrasonic vibration on the material deformation behaviors. The experimental system consists of a universal