WITHDRAWN: Dynamic failure of titanium: Temperature rise and adiabatic shear band formation

Accepted Manuscript

Dynamic failure of titanium: temperature rise and adiabatic shear band formation Yazhou Guo , Qichao Ruan , Shengxin Zhu , Q. Wei , Jianan Lu , Bo Hu , Xihui Wu , Haosen Chen , Daining Fang , Yulong Li PII: DOI: Reference:

S0022-5096(18)30795-6 https://doi.org/10.1016/j.jmps.2019.01.014 MPS 3549

To appear in:

Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

30 September 2018 15 January 2019 17 January 2019

Please cite this article as: Yazhou Guo , Qichao Ruan , Shengxin Zhu , Q. Wei , Jianan Lu , Bo Hu , Xihui Wu , Haosen Chen , Daining Fang , Yulong Li , Dynamic failure of titanium: temperature rise and adiabatic shear band formation, Journal of the Mechanics and Physics of Solids (2019), doi: https://doi.org/10.1016/j.jmps.2019.01.014

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Highlights The adiabatic shear failure process of solids was investigated for the first time by dynamic tests synchronically combined with high-speed photography and infrared temperature measurement.



The key characteristics of ASB, such as temperature, critical strain, propagation speed and cooling rate were systematically studied.



The experimental results shows that the apparent temperature rise occurs after ASB initiation, indicating it could not be the causation but the consequences of ASB.



This discovery clarifies the causality of ASB and may lay the foundation for further physical, mechanistic and mathematic studies

AC

CE

PT

ED

M

AN US

CR IP T



ACCEPTED MANUSCRIPT

Dynamic failure of titanium: temperature rise and adiabatic shear band formation

AN US

CR IP T

Yazhou Guo1,3,4, Qichao Ruan1,3,4, Shengxin Zhu2, Q. Wei1,5, Jianan Lu1,3,4, Bo Hu1,3,4, Xihui Wu1,3,4, Haosen Chen2, Daining Fang2, Yulong Li1,3,4 1 School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China 2 Institute of Advanced Structure Technology, Beijing Institute of Technology, 100081, Beijing, China 3 Shaanxi Key Laboratory of Impact Dynamics and Engineering Application, Northwestern Polytechnical University, Xi’an 710072, China 4 Joint International Research Laboratory of Impact Dynamics and Engineering Application, Northwestern Polytechnical University, Xi’an 710072, China 5 Department of Mechanical Engineering, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, USA

Abstract

AC

CE

PT

ED

M

One of the most important issues related to dynamic shear localization is the correlation among the stress collapse, temperature elevation and adiabatic shear band (ASB) formation. In this work, the adiabatic shear failure process of pure titanium was investigated by dynamic shear-compression tests synchronically combined with high-speed photography and infrared temperature measurement. The time sequence of stress collapse, ASB initiation, temperature rise and crack formation was recorded. The key characteristics of ASB, such as temperature, critical strain, propagation speed and cooling rate were systematically studied. The propagation velocity of ASB is dependent on the impact velocity and the maximum velocity is found in this work to be about 1900m/s, about 0.6Cs (Cs is the shear wave speed). The maximum temperature within ASB is in the range of 350-650℃, while the material close to ASB is also heated. One important observation is that the apparent temperature rise occurs after ASB initiation, which indicates it could not be the causation but the consequences of ASB. Key words: adiabatic shear, ASB, titanium, temperature, high speed photography, shear compression.

1.Introduction Adiabatic shear localization (ASL) is one of the failure mechanisms that are commonly observed within ductile materials under dynamic loading. ASL process is always accompanied with the formation of ASB and followed by the final fracture of the material. As ASL phenomena are generally encountered in industries such as metal-forming, high-speed machining, car crash, debris impact on aircraft, ballistic impact and penetration, etc., it is of great importance to reveal the nature of ASB. (Antolovich and Armstrong, 2014; Bai and Dodd, 1992; Xu et al., 2008)

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

A commendable description of ASB was given in Wright’s book (Wright, 2002): very large shear strain (100~102) occurs in a narrow (100~102μm)，nearly-planar region of the material within extremely short time (100~102μs), accompanied by severe local temperature rise (as high as 103 K). Obviously, understanding the formation of ASB, which is both “transient” and “local”, involves the synergistic interactions of mechanics, thermodynamics and material science. Early investigations on ASB mostly concerned about the prediction of its initiation(Bai, 1982; Batra and Kim, 1990; Clifton et al., 1984; Molinari and Clifton, 1987; Walter, 1992; Wright and Batra, 1985; Wright and Walter, 1987; Zhou et al., 2005) and the microstructure characterization after its formation(Chen and Vecchio, 1994; Chichili et al., 2004; Meyers and Pak, 1986; Meyers et al., 2003; Timothy and Hutchings, 1985; Xue et al., 2002; Yang and Wang, 2006). As for ASB initiation, since the pioneer work of Zener and Hollomon (Zener and Hollomon, 1944), ASB has been widely recognized as a process of thermal-plastic instability, where local disturbance such as temperature fluctuation was believed to be the trigger of ASL (Bai, 1982; Batra and Kim, 1990; Clifton et al., 1984; Molinari and Clifton, 1987; Walter, 1992; Wright, 1992; Wright and Batra, 1985; Wright and Walter, 1987). Up to this point, the mechanics community believed that it is the temperature rise that initially softens the material and leads to the final formation of ASB. Recently, microstructure inhomogeneity and geometric variation were introduced by D. Rittel (Rittel et al., 2008b; Rodríguez-Martínez et al., 2015) and Y. Guo (Guo et al., 2010) into their numerical models, where dynamic recrystallization and grain rotation were claimed to be the softening mechanisms respectively. The evolution and distribution of stress, strain, and temperature etc. within ASB could also be obtained taking advantages of numerical modelling. On the other hand, the materials community usually focuses on the microstructure characteristics related to ASB after its formation, including phase transformation, grain refinement, recrystallization, texture and etc. (Cerreta et al., 2013; Chichili et al., 2004; Meyers et al., 2003; Pérez-Prado et al., 2001; Rittel et al., 2017b; Wei et al., 2002; Xu et al., 2008; Yang and Wang, 2006). To study the microstructure evolution during ASB formation process, interrupted tests were adopted by some researchers (Andrade et al., 1994; Nemat-Nasser et al., 1998; Rittel et al., 2008a; Yuan et al., 2015). Temperature rise within ASB is a key issue for investigating ASL, as it is closely related to both mechanical behavior (such as softening and stress collapse) and microstructure (such as phase transformation and recrystallization). However, this temperature was estimated by most researchers using the relationship of work-heatconversion (Andrade et al., 1994; Hines and Vecchio, 1997; Nemat-Nasser et al., 1998; Wei et al., 2003; Zheng et al., 2017). The in-situ experimental measurements of temperature as well as shear band evolution are rather difficult due to the transient and local nature of ASL. To the best of these authors’ knowledge, the first attempt to measure the temperature of a propagating shear band was made by Costin et al. (Costin et al., 1979), where they used a high-speed single-element infrared (IR) InSb (indium antimonide) detector to measure the average temperature of a spot containing a shear band. Since the size of the spot is 1mm that is far larger than the shear band width, their measured temperature rise of 100K is naturally an

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

underestimation of the real quantity. After that, Duffy and coworkers (Duffy and Chi, 1992; Hartley et al., 1987; Liao and Duffy, 1998; Marchand and Duffy, 1988) performed a series of study on the dynamic shear behavior of thin-walled cylinders, where they developed the IR system and brought in high-speed photography. Ranc et al. (Ranc et al., 2008) also tested thin-walled tubular specimens by torsional Kolsky bar (Split Hopkinson bar) and measured the temperature rise within shear localization area. They used an InSb IR detector array and an intensified CCD camera to detect different temperature ranges simultaneously. According to their experimental results, the maximum temperature within ASB of a Ti-6Al-4V alloy is about 1373K. Zhou et al. (Zhou et al., 1996) and Guduru et al. (Guduru et al., 2001) examined the shear band propagation and mode Ⅱ crack evolution in a pre-notched plate subjected to impact loading. The shear band velocity as well as temperature distribution was measured. The maximum temperature reached 1700K for C-300 Steel, which is about 90% of its melting point, while it was only 720K for Ti-6Al4Valloy (Zhou et al., 1996). Rittel et al. (Rittel et al., 2002) designed a shearcompression-specimen (SCS) for dynamic shear tests and did a series of experiments on the ASL behavior of metals (Rittel et al., 2008a; Rittel et al., 2006; Rittel et al., 2017b). They measured the temperature rise by high-speed InSb IR detector array and found that the maximum temperatures were about 380K and 470K for AM50 and Ti-6Al-4V alloys, respectively (Rittel and Wang, 2008). Based on the above, we found that although the in situ temperature measurement has been attempted with certain degree of success, its correlation with ASB formation and material failure was never experimentally specified. In other words, the causality has not been clarified. Combining high-speed photography and temperature measurement into one mechanical test is very helpful for investigating the dynamic failure mechanism of materials. The roles of impact loading, deformation and temperature in ASL process could be intuitively obtained by comparing the sequence of important observations such as stress drop, strain localization, temperature rise, ASB initiation etc. In this work, we will investigate the shear localization behavior of a commercially pure titanium by a Kolsky bar system combined synchronically with a high-speed camera and IR temperature measurement systems. Benefiting from fast development of high-speed photography technique in recent years, high resolution deformation fields around ASB are able to be presented. The primary objective of this work is thus to clarify the causality associated with ASB processes of elasto-plastic materials in general, and for the commercial purity Ti in particular.

2. Experimental techniques 2.1 Material and specimen The material used in this work is commercial titanium with grade Ⅱ purity. The composition of the material was given in Table 1.

ACCEPTED MANUSCRIPT Table 1 Main ingredients of commercial purity titanium (ωt%) Element Ti Fe C N H Content Bal. 0.08 0.01 0.009 0.008

O 0.1

AC

CE

PT

ED

M

AN US

CR IP T

The specimens are similar to those of Rittel et al. (Rittel et al., 2002; Rittel et al., 2017b). The dimensions of the specimen is presented in Figure 1. The advantages of this specimen include that it can be directly tested by Kolsky bar (or SHPB) system and can be applied in both dynamic and quasi-static tests without change in dimension. Parallel stripes were prefabricated on one side of the specimen for accurate determination of the shear strain. These stripes were carved by laser and spaced 0.25mm apart.

(a)

ACCEPTED MANUSCRIPT (b)

CR IP T

(c) Figure 1. (a) Design and dimensions of the shear-compression-specimen (SCS) used in this work and the arrangement of the IR detector elements, (b) the side view of an SCS before loading, and (c) an SCS after impact loading.

2.2 Mechanical testing

CE

PT

ED

M

AN US

The shear tests were conducted by an SHPB apparatus that is conventionally used for dynamic compression test. The theory and technique of SHPB were introduced in details elsewhere (Chen and Song, 2010). In the present work, the diameter of the bars of the SHPB system is 12.7mm. The procedure of testing SCS specimen by SHPB is the same as conventional SHPB experiments except for data processing. The nominal shear stress is deduced by the shear component of the compression force divided by the area of the gauge section, while the nominal shear strain is derived by the displacement along the groove direction divided by the width of the groove. The equations are listed below. 𝜏=

𝑃 sin 𝜃 cos 𝜃 𝑎𝑡ℎ 𝑣

𝑣

𝛾̇ = 𝑤 cos 𝜃 ∆𝑡

γ = ∫0 𝛾̇ 𝑑𝑡

1 2 3

AC

where P is the average force applied on the specimen. 𝑣 and 𝑣𝑡 are the axial velocities of the two specimen surfaces in contact with the incident and transmit bars respectively; a is the length of the cross-section square of the specimen; 𝑡ℎ is the thickness of the gauge section; 𝜃 is the angle between the slot and the loading direction (𝜃 = 45° in this work); 𝑤 is the width of the slot. According to the one-dimensional stress wave theory, P, 𝑣 and 𝑣𝑡 can be calculated: 𝑃=

𝑃 +𝑃 2

4

𝑣 = 𝐶(𝜀 − 𝜀 ) 5 𝑣𝑡 = 𝐶𝜀𝑡 6 where 𝑃 = 𝐸𝐴[𝜀 + 𝜀 ] and 𝑃𝑡 = 𝐸𝐴𝜀𝑡 are the forces from incident and transmit bars respectively. 𝜀 , 𝜀 and 𝜀𝑡 are the incident, reflected and transmitted strain. E, A

ACCEPTED MANUSCRIPT

CR IP T

and C are the Young’s modulus, cross-sectional area and longitudinal speed of the stress wave, respectively. Note that all the above variables such as 𝜏, γ, 𝛾̇ , 𝑃,𝑣 , 𝑣𝑡 ,𝜀 , 𝜀 and 𝜀𝑡 are functions of time t. It should be noted that the specific design of SCS will introduce nonuniform deformation and complex stress state within the gauge section of the specimen. The true strain and stress state may deviate far from the nominal values. Dorogoy et al.(Dorogoy et al., 2015) studied the deformation characteristics of this specimen geometry by using numerical simulations. Details of the stress state and the process of precise determination of equivalent stress and strain could be found there (Dorogoy et al., 2015). In this work, the mechanical properties are not the uppermost concern, and so nominal values are used in places where qualitative descriptions of mechanical properties are needed. However, when more precise values are required for some critical quantities like shear strain, complementary methods, such as high speed photography, are adopted.

AN US

2.3 High-speed temperature measurement and photography

AC

CE

PT

ED

M

The temperature measurement system includes an optical system with 1:1 magnification and an eight channel IR detector. The detector is made of InSb (indium antimonite) that responds to radiation in the 1μm to 5.5μm range corresponding to temperature from 60℃ to 1200℃, which is adequate for measuring shear band temperatures. The response time for the detector is less than 1μs, fast enough for SHPB tests. Each of the elements in the detector array is square with 0.15mm side. The separation between two adjacent elements is 50μm. Thus, the total length of the array is 1.55mm. The deformation process of the specimen is recorded by a highspeed camera with a maximum framing rate of 5 million fps (frames per second). The flash, camera and IR system are all triggered by the incident pulse of the stress wave. The time sequence for each device can be easily derived by calculating the time period during which the loading pulse travels from the strain gauge to the specimen. The arrangement of the SHPB, temperature measurement and high-speed camera is shown in Figure 2. It should be noted that the high-speed photography needs very high brightness illumination, which interferes the IR temperature measurement system. As such, these two systems should be completely isolated during the entire test. However, the temperature signals are no longer accurate as soon as cracks form, because the light from flashes may elevate the signals sharply.

ACCEPTED MANUSCRIPT

High-speed Camera Flash Strain gauge

Strain gauge

Incident Bar

CR IP T

Transmit Bar

Optical System

IR

AN US

Control Computer

IR Detector

M

Data Acquisition System

PT

3. Results

ED

Figure 2. Arrangement of the SHPB apparatus combined with high-speed photography and temperature measurement system.

CE

3.1 Calibration

AC

Calibration of the IR system is of vital importance to the reliability and accuracy of temperature measurement. A relationship between temperature and voltage needs to be built before the actual test. Ideally speaking, each of the elements in the IR detector should have identical response to the same thermal signal if they are used individually. However, when all the elements are connected to the system simultaneously, their responses could be quite different because of the so called cross-talk effect (Zehnder and Rosakis, 1991). In the present work, each of the eight elements are individually calibrated at the same time. The calibration process includes the following steps: heating the specimen to a prescribed temperature, measuring the temperature by the IR system and a thermal couple simultaneously as the specimen cools down, recording the temperature reading from the thermal couple and the voltage from the IR system, and establishing the relationship by fitting the temperature and voltage data. In the present work, an exponential relation was

ACCEPTED MANUSCRIPT

ED

M

AN US

CR IP T

derived between the temperature and voltage signal. It should be noted that the surface finish of the specimen could change during loading, especially at large deformation, which may result in variation of emissivity. However, according to Hartley et al. (Hartley et al., 1987), this variation is not significant in the temperature range of adiabatic shear localization. Surface oxidation may also affect the emissivity of the material and this could be evaluated by repeating the heating and cooling process (Ranc et al., 2008). In this work, the readings of the IR measurements are identical, which demonstrates that the surface oxidation has little effect on the emissivity within the temperature range concerned.

PT

Figure 3. Calibration curves for the IR detector. The eight channels are separately calibrated.

AC

CE

3.2 Mechanical properties

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 4. Typical shear stress-strain curves of the dynamic shear-compression tests.

AC

CE

PT

ED

M

A total of thirty dynamic tests with IR temperature measurement system operating were conducted in this work, among which eight tests were equipped with highspeed photography simultaneously. Representative shear stress-nominal shear strain curves (derived by equation 1 and 3) are shown in Figure 4. Similar to most ductile metals, the dynamic response of commercial purity titanium can be divided into three sections: elastic, plastic (with or without strain hardening) and failure. Taking test 1 as an example, we will examine the process of shear deformation by means of high-speed photography. Figure 5 presents the shear stress evolution with respect to time. The deformation of the shear-compression specimen could also be divided into three stages, i.e., small deformation (Ⅰ), nonuniform large deformation (Ⅱ) and shear localization (Ⅲ). The characteristics of deformation at different stages are given in Figure 6. In stage Ⅰ, the stress increases dramatically but the deformation of the specimen is very small. Most part of the gauge section remains elastic. Toward the end of this stage, apparent deformation can be observed and the stripes become curved, as shown in Figure 6a. After that, the shear deformation is concentrated in the mid-section of the gauge, which is mainly due to the geometry design of the specimen. It is interesting to notice that the shear deformation within the midsection remains uniform until ASB initiation (indicated by the straight and parallel lines in this section), as shown in Figure 6a~f. The work hardening of the material surpasses the softening mechanisms in this stage and the maximum stress (or load) is achieved at f. It should be pointed out that no visible mismatch of the strips (or severe shear localization) was observed at the maximum stress, which indicates that ASB may initiate after the peak stress.

ACCEPTED MANUSCRIPT

d

e

f

gh i

c

j

b a

Ⅰ

Ⅲ

Ⅱ

CR IP T

ASB

crack

k

AN US

l

ED

M

Figure 5. Stress and temperature evolution of a shear-compression test. The arrows a~l indicate the time of camera shot, as shown in Figure 6. 1~8 represent the elements of IR measurement. T-Cal is the calculated temperature based on the conversion of mechanical work to heat (taking 1.0 as Taylor-Quinney factor). The dashed green line and red line denote the appearances of ASB and crack, respectively.

AC

CE

PT

a 23.5μs

b 33.5μs

d 48.0μs

e 53.5μs

f 58.5μs

M

AN US

c 40.5μs

CR IP T

ACCEPTED MANUSCRIPT

h 67.5μs

PT

ED

g 66.0μs

CE

ASB

j 73.5μs

AC

i 70.5μs

ASB

ASB

ACCEPTED MANUSCRIPT

l 88.5μs

k 78.5μs

crack

CR IP T

crack

AN US

Figure 6. Deformation of an SCS under dynamic loading. Pictures a-l are corresponding to those in Figure 5. The time in each picture indicates the loading time at which the photo is taken.

AC

PT

CE

a 23.5μs

ED

M

The deformation field of the SCS can be calculated based on each high-speed snapshot. Figure 7 presents the evolution of shear strain for Test1, corresponding to the time point given in Figure 5 and Figure 6. Quantitative information could be derived from these deformation patterns. Again, the mid-section of the gauge carried most of the shear deformation, which makes its shear strain much larger than that its vicinity. Apparently, the local shear strain at the mid-section is higher than the nominal shear strain at the same loading stage. Therefore, the values of true local shear strain will be used in discussion in the following parts of this paper.

b 33.5μs

ACCEPTED MANUSCRIPT

c 40.5μs

f 58.5μs

AN US

e 53.5μs

CR IP T

d 48.0μs

h 67.5μs

PT

ED

M

g 66.0μs

AC

CE

Figure 7. Shear strain field of the gauge section of a shear-compression specimen under impact loading. Labels of a~h correspond to those in Figure 5 and 6. The time within each contour indicates the loading time at which the picture is taken.

4. Discussion 4.1 Formation of ASB The initiation of ASB has always been one of the most important issues related to shear failure under dynamic loading. However, the exact value of critical strain at which ASB initiates is very difficult to obtain, because it usually occurs within a few microseconds and the dimension of ASB is too small to be detected with certainty. Furthermore, the random location of ASB initiation in conventional compressive or torsional tests makes the observation even harder. Therefore, the process of ASB

ACCEPTED MANUSCRIPT formation was mostly studied by analytical or numerical methods (Bai and Dodd, 1992; Wright, 2002). For example, the most popular criterion of ASB formation, proposed by Culver, is called Culver criterion (Culver, 1973). This criterion is based on the assumption that ASB initiates at the strain where maximum stress is achieved. By assuming a constitutive relation of 𝜏 = 𝐾𝛾 𝑛 , a simple expression of critical shear strain was derived: 𝛾𝑐

𝐶𝑢𝑙𝑣𝑒

=

𝑛𝜌𝑐𝑉 𝛼

7

where 𝜌 is the density and 𝑐𝑉 is the specific heat. For pure Ti, 𝑐𝑉 is 528 J kg-1 K-1 and ρ 𝜕𝜏

CR IP T

is 4.5 g cm -3 (Yin et al., 2013); n is the strain hardening exponent and 𝛼 = 𝜕𝑇 is the thermal softening rate, respectively. Bai introduced strain rate and temperature effects into the constitutive model, i.e. 𝜏 = 𝐾𝛾 𝑛 𝛾̇ 𝑚 𝑇 𝜈 , and the above criterion could be expressed as (Bai, 1990): 𝛾𝑐

𝐵𝑎

=

(𝑛+𝑚)𝜌𝑐𝑉 𝑇 𝜈𝜏

8

AC

CE

PT

ED

M

AN US

where m and 𝜈 are the strain rate hardening exponent and thermal softening exponent, respectively. Based on similar assumptions, many other forms of 𝛾𝑐 could be obtained given different forms of constitutive relations (Walley, 2007). For commercial purity titanium, a group of parameters given in the literature (Guo et al., 2017; Walley, 2007) could be used to estimate the critical shear strain for ASB initiation: 𝑛 = 0.104, 𝑚 = 0.026, 𝜈 = 0.202, 𝛼 = 0.595 𝑀𝑃𝑎/𝐾. Introducing these parameters into equation 7 and 8, 𝛾𝑐 𝐶𝑢𝑙𝑣𝑒 = 0.44 and 𝛾𝑐 𝐵𝑎 = 1.34. The large difference between the two predictions arouses the question: which one represents the intrinsic fracture characteristics of the material? In this work, we have summarized the nominal shear strain at maximum shear stress, which has been assumed by many scholars to be the critical point of ASB initiation, as shown in Figure 8. The true strain before ASB initiation was obtained by analyzing the photos of the high-speed photography. Due to the non-uniform deformation of the gauge area, the observed shear strain is much larger than the nominal one. The observed critical shear strains, together with the predicted ones, are also given in Figure 8. What is interesting is that the Culver’s prediction is well below the nominal shear strain at maximum shear stress, while Bai’s prediction coincides with the true critical shear strain. Therefore, Bai’s criterion reflects the intrinsic characteristics of ASB initiation of titanium and Culver’s criterion may be useful for engineering prediction. Although it needs quite a number of experiments to acquire the parameters in equation 8 for a specific material, it gives a quantitative evaluation of the ASB susceptibility of the material.

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 8. The measured and predicted critical shear strains for ASB initiation. Predictions are based on Culver’s model and Bai’s model, respectively.

4.2 Propagating speed of ASB

AC

CE

PT

ED

M

The velocity of ASB propagation can be directly estimated from the high-speed photography. The evolution of ASB is presented in Figure 9. Different from single notch specimen impact tests that were adopted by Zhou et al. (Zhou et al., 1996) and Guduru et al. (Guduru et al., 2001), ASB in shear-compression test initiates randomly along the slot. It should be noted that ASB generates at the interior of the slot for most of the tests rather than near the edge. This fact indicates the shearcompression specimen has its advantage in eliminating stress concentration for studying ASB initiation. The evolution of ASB length is shown in Figure 10. The length of the slot is ~9.9mm and the time duration from ASB initiation to running through the slot is less than 10μs. Taking into account the two-way propagation, the average velocity of ASB is no less than 500m/s. The variation of ASB velocity with respect to time is given in Figure 11. Clearly, this velocity is not constant but generally increases with loading process. The maximum velocity reaches 1900m/s which is about 0.6 C s (Cs is the speed of shear wave, about 3100m/s for titanium). This result agrees with those from Zhou et al. (Zhou et al., 1996), except that the ASB stopped propagating in their tests due to the geometry of the specimen. A collection of the ASB velocities from the literature is presented in Figure 12. Note that these limited data are derived from different types of experiments for different kinds of materials. The general trend is that the ASB speed increases with increasing impact speed, regardless of materials or test methods. One deviation is the results from thick-wall-cylinder (TWC) tests (Xue et al., 2002), where the ASB velocities were estimated from initial velocity of the internal boundary which was moved by explosive loading, whereas in all other

ACCEPTED MANUSCRIPT tests such as single-notch impact, double-shear, dynamic torsion and shearcompression, the specimens were loaded by direct impact. Actually, Mercier and Molinari (Mercier and Molinari, 1998) studied various factors that might affect ASB velocity by numerical analysis. They found that in a one-dimensional slab under simple shear, the velocity of ASB was closely related to the form of material constitutive model and the dimension of ASB. For example, for rigid-perfect-plastic (RPP) material, ASB velocity could be expressed as: 𝜍

𝜆 𝑡

𝐶 = 𝜌𝑉𝑓 g (ℎ , ℎ) 9

AN US

CR IP T

where 𝜎𝑓 is the flow stress, 𝜌 is the density and 𝑉 is the applied loading velocity. g is a function of 𝜆/ℎ, the normalized length of the process zone, and 𝑡/ℎ, the normalized ASB thickness. This equation predicts an inverse proportionality between the applied velocity and ASB velocity. While for elastic-viscoplastic material, the applied velocity had little effect on ASB velocity (see Figure 17 in the reference (Mercier and Molinari, 1998)). Apparently, neither of the predictions agrees with Figure 12. The reason may lie in the complexity of the experiments. The stress state, the load form, the geometry of the specimen and the material may all affect the shear band propagation. A unified description and comparison of the ASB velocity needs further investigation. b 67.0μs

AC

CE

c 68.0μs

PT

ED

M

a 66.0μs

d 69.0μs

ACCEPTED MANUSCRIPT

e 70.0μs

CR IP T

f 71.0μs

CE

PT

ED

M

AN US

Figure 9. ASB propagation in a shear-compression test. The time within each picture indicates the loading time at which the picture is taken.

AC

Figure 10. Length of ASB with respect to time. The ASB velocity can be derived from the slope of the curves.

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 11. Speed of ASB propagation with respect to time.

Figure 12. ASB speed with respect to impact velocity from references and this work. The materials and test techniques are given in the legend. Another issue worthy of discussion is the shear strain rate inside the ASB. From the high-speed photography, we can observe clearly the process from ASB initiation to

ACCEPTED MANUSCRIPT crack formation (see Figure 5, 6 and 9). The true shear strain inside the ASB could be estimated by the mismatch of the strips divided by the width of ASB. The average shear strain rate could then be calculated given the deformation time. In this work, the shear strain inside ASB is as large as 35.7 (250μm /7μm) and the shear strain rate is estimated to be 4.46x106/s (35.7/(8μs)). This value is hundreds of times larger than the nominal shear strain rate.

4.3 Width of ASB

ASB

ASB

ED

M

Microviods

AN US

CR IP T

Interrupted dynamic shear-compression tests were conducted to examine the microstructure of ASB at different stage of loading. Different stop-rings were used to limit the deformation to nominal shear strains of 43.3%, 50.0%, 53.3%, 56.7% and 60.0%, respectively. ASB was not observed until the specimens reached a nominal shear strain of 56.7%. The characteristics of ASBs within specimens of 56.7% and 60.0% strains were identical in terms of width and morphology. The microstructure of ASB is shown in Figure 13. The width of the ASB is measured and is about 5.5~8μm.

AC

CE

PT

Crack

ASB

ACCEPTED MANUSCRIPT Figure 13. Morphology of ASB formed within titanium under impact loading. Many researchers have attempted to derive the width of ASB by theoretical analyses. For example, Bai and Dodd (Dodd and Bai, 1989) came to the ASB width as: 𝜆𝑇 1/2

δ = 2( ) 𝜏𝛾̇

10

where 𝜆 is the thermal conductivity and for pure titanium 𝜆 = 14.63 𝑤𝑚 1 𝐾 1 . 𝑇, 𝜏 and 𝛾̇ denote the temperature, shear stress and shear strain rate inside the ASB. Based on our experimental results (𝑇 = 500℃, 𝜏 = 450𝑀𝑃𝑎, 𝛾̇ = 4.46𝑥106 /𝑠),

CR IP T

equation 10 gives an estimation of δ = 3.8μm. This estimation is a little smaller than the observed width of ASB (5.5~8.0μm). Grady also gave an expression to predict the width of ASB (Grady, 1994; Xu et al., 2008):

AN US

δ=( 𝜆

where 𝑘 = 𝛼/𝜎𝑦 is the thermal softening parameter, 𝜒 = 𝜌𝑐

𝑉

16𝜌 3𝑐𝑉 2𝜒3 𝜏 3 𝑘 2𝛾̇

1/4

)

11

is the thermal

PT

ED

M

diffusivity coefficient. This equation predicts the shear band width of titanium to be only 0.8μm. The inconsistencies of the ASB width between the experimental and predicted values may be due to the simplification of the models, which were derived based on simple shear, while in this work the ASB formed under shear-compression stress state. In fact, we also measured the width of ASB in a titanium specimen loaded by uniaxial compression and by pure shear (hat-shape specimen), and the band width in the compressed specimen is about 20μm, while that in the hat-shape specimen is only 5.5 μm in good keeping with Bai’s prediction. Based on the work by Dodd and Bai (Dodd and Bai, 1989) , the time required for ASB formation could also be predicted: 𝛾

𝑡 = 𝛾̇ ∗ = ∗

𝜌𝑐𝑣 𝛽𝜆

𝛿 2 12

AC

CE

Substituting the relevant parameters, 𝑡 = 0.173 𝛽 1 𝛿 2 , where t is in microsecond and 𝛿 in micrometer. Since the width of the ASB in this work is about 5.5~8.0μm and 𝛽 = 0.25~0.5, the time needed for shear band formation is about 10~44μs. This calculated time duration agrees well with the observed one in this work (about 10μs).

4.4 Temperature rise before shear localization Impact loading on solids usually leads to temperature rise and if the loading rate is high enough, heat transfer could be ignored and the process is regarded as adiabatic. In this case, the mechanical work due to impact loading converts to two main parts: heat and internal energy of the solid. The distribution of mechanical work was firstly studied by Taylor and Quinney (Taylor and Quinney, 1932, 1934) and followed by many researchers (Bever et al., 1973). Most of the early efforts examined the strain

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

energy (or cold stored energy) within the materials derived by observing their microstructures after impact loading, because direct measurement of the heat generation was difficult due to the transient nature of the loading process. Very recently, Rittel and coworkers (Rittel et al., 2017a) conducted a series of experiments on a few metals, including CP Ti, to investigate the loading mode effect on the fraction of mechanical work into heat, i.e., Taylor-Quinney factor (TQF), where they measured the temperature rise directly by high speed IR detector. Their results show that the TQF for CP Ti is highly dependent on loading mode and is about 0.79-0.84 for shear compression tests. In the present work, we also measured the temperature rise before and after intense shear localization, as shown in Figure 14. Similar with previous studies (Rittel and Wang, 2008), there is hardly any detectable temperature rise at the initial stage of plastic loading and apparent temperature rise is only observed after the maximum shear stress is achieved. We summarized the temperature rise measured at the maximum shear stress, i.e., before ASB initiation, as shown in Figure 15. All of the temperature rises are in the range of 50-90℃ with acceptable errors. Since this temperature rise is measured before ASB formation, it is relatively uniform at the site of measurement and could be used to derive the TaylorQuinney factor.

Figure 14. Shear stress and temperature evolution at the center of the slot of a SCS test. The nominal shear strain rate is about 10500/s. s1 to s6 denote different IR channels.

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 15. Summary of the measured and calculated (𝛽 = 1.0) temperature rise at the maximum shear stress for the SCS tests.

∆T =

𝛽𝑊𝑃 𝜌𝑐𝑉

13

ED

M

The temperature rise within the deformed materials could be estimated by using the relationship between mechanical energy and adiabatic heat (Taylor and Quinney, 1934).

where 𝛽 is the Taylor-Quiney factor, 𝑊𝑃 the specific plastic work. It should be noted

or

CE

PT

that the shear-compression specimen undergoes both shear and compressive stresses during impact, and the plastic work density 𝑊𝑃 should be expressed as: 𝑊𝑃 = ∫ 𝝈: 𝑑𝜺 14

𝑊𝑃 = ∫(𝜎11 𝑑𝜀11 + 𝜎22 𝑑𝜀22 + 𝜎33 𝑑𝜀33 + 2(𝜎12 𝑑𝜀12 + 𝜎23 𝑑𝜀23 + 𝜎31 𝑑𝜀31 ))

AC

where 𝜎 𝑗 and 𝜀 𝑗 are the components of the stress and strain tensor (ignoring the elastic contributions). For a shear compression test in this work, suppose directions 1 and 2 are perpendicular and parallel to the groove, respectively, then 𝜎22 = 𝜎33 = 𝜎23 = 𝜎31 = 0. Moreover, the widths of the slots on the specimen are carefully measured after the tests and the results indicate that the residual compressive strain 𝜀11 is only about 5%, much smaller than the shear strain 𝜀12 that is on the order of 100%. Therefore, ∫ 𝜎11 𝑑𝜀11 could be neglected without introducing significant error. As a result, only the shear component 𝑊𝑃 = 2 ∫ 𝜎12 𝑑𝜀12 is left. It should be pointed out that the shear stress and strain used for calculating temperature rise are both local values corresponding to the site of temperature measurement, which is quite

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

different from the nominal ones. In the present work, the local shear strains are derived from the images of high-speed photography. A comparison between the specific plastic work calculated by using local and nominal shear strain values indicates that the former ones are about 1.65~1.95 times larger than the latter ones. Therefore, the derived temperature rises are 1.65~1.95 times larger correspondingly. Taking 𝛽 = 1.0, most of the calculated temperature rises are between 140~210℃, much higher than the measured ones, as shown in Figure 15. This large inconsistency indicates the choice of 𝛽 is questionable. The true value of 𝛽 is equal to the ratio of the measured temperature rise to the calculated ones. Figure 16 presents the values of 𝛽 with respect to strain rate. Apparently, the Taylor-Quinney factor is between 0.25 and 0.55, and is dependent on the loading rate. Similar tendency was also found by Zhang et al. (Zhang et al., 2017), where they derived 𝛽 values to be from 0.3 at strain rate 1100/s to 0.96 at 4200/s for 7075 aluminum alloy. The 𝛽 for CP Ti obtained in this work is much smaller than that in the literature (Rittel et al., 2017a), where the authors calculated 𝑊𝑃 by using equivalent stress and strain values derived by numerical modeling. Apparently, 𝛽 is related to both the measured and calculated temperatures, which are affected by many factors such as strain rate, strain, surface condition, loading mode and the way of deriving true shear stress and strain. The differences between these results need further studying.

Figure 16. The calculated Taylor-Quinney factor as a function of shear strain rate for pure titanium.

4.5 Temperature within ASB As described in Section 2, the size of one pixel for the IR temperature detector is

ACCEPTED MANUSCRIPT 150μmx150μm, which is larger than the width of ASB in titanium. Since the magnification of the optical system is 1:1, the ASB covers only part of the pixel, which deviates from the calibration condition. On the other hand, ASB may overlap with more than one pixel, see Figure 17. Based on the thermal equilibrium, the reading of temperature of the ith pixel could be written as (Marchand and Duffy, 1988): 𝐴 𝑇 4 = 𝐴ℎ 𝑇ℎ4 + 𝐴𝑐 𝑇𝑐4 14

ED

M

AN US

CR IP T

where 𝐴 ,𝐴ℎ , and 𝐴𝑐 are areas of pixel i, “hot” and “cold” region of the pixel, respectively. Ti, Thi and Tci are the overall temperature, temperature of the hot region (ASB) and of the cold region, respectively. Based on the measured data by different pixels, the temperature of ASB could be derived. It should be noted that the “cold” region is not completely cold. The temperature near ASB is relatively high compared to that of the undeformed material (or ambient temperature), as shown in Figure 5 where all of the eight pixels have recorded temperature elevation. If the energy from the cold region is ignored (as in reference (Marchand and Duffy, 1988)), the calculated temperature within ASB would be greatly overestimated, as shown in Figure 18. Therefore, we choose to use the third highest temperature of the eight channels as Tci and derived the average ASB temperature T-ASB to be around 500℃ (350-650℃). While, for comparison, the temperature within ASB is between 9501400℃ if Tci is ignored. This large difference would affect significantly our understanding of the mechanism and microstructure within ASB, such as, for example, recrystallization. The temperature rises measured by different researchers are summarized in Table 2. Apparently, the temperature rises depend on the materials and test methods. However, large disagreements are found in different works even for the same material and the same tests.

PT

i+1

Hot region ASB

0.05mm

CE

i

AC

Cold region

i-1 0.15mm

Figure 17. Illustration of the relative position of ASB and the IR detector

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

Figure 18. Temperature within ASB. Ti and Ti+1 are the directly measured temperature of pixel i and i+1. T-ASB is the temperature within ASB calculated based on equation 14 and T-ASB-Ignoring Tci is the calculated temperature ignoring Tci.

ACCEPTED MANUSCRIPT Table 2. Comparison of temperature measurements within ASB Observed spot size (mm)

206-240 235-430

0.25*0.77 0.02*0.062

AISI 1018 Steel

175-240 150-455

0.25*0.77 0.02*0.062

HY100 Steel

875-1140

0.035*0.12

HY100Steel

575

AISI 4340 Steel (425 temper)

460

AISI 4340 Steel (200 temper)

570

Ti-6Al-4V

440-550

PT

Dynamic torsion

Hartley and Duffy (1987)

Dynamic torsion

0.08*0.08

0.017*0.053

Dynamic torsion

Merchand Duffy (1988)

and

Duffy (1992)

Chi

and

Mode Ⅱ dynamic Zhou et al. (1996) fracture Dynamic torsion

Liao and (1998)

Mode Ⅱ dynamic Guduru (2001) fracture

Duffy

et

620

0.10*0.10

950

0.043*0.043

Dynamic torsion

Mg AM50A-F Ti-6Al-4V

110 300

0.045*0.045

Dynamic shear- Rittel and Wang compression (2008)

CP Titanium

350

Ø1.40

CP Titanium

350-650

0.15*0.15

CE

C-300 Steel

M

900-1400 450

0.017*0.053

ED

C-300 Steel Ti-6Al-4V

References

AN US

AISI 1020 Steel

Test method

CR IP T

Materials

Max Temperature （℃）

AC

Ti-6Al-4V

Dynamic (hat-shape specimen)

al.

Ranc and Taravella (2008)

shear Fu et al. (2015)

Dynamic shearThis work compression

ACCEPTED MANUSCRIPT

4.6 Cooling rate Whether the ASB is adiabatic or not is another issue in dispute. The fundamental question is how fast the thermal diffusion could be. Me-Bar and Shechtman proposed an equation to predict temperature evolution within ASB (Me-Bar and Shechtman, 1983):

𝑇=

𝑇0 𝛿 2(𝜋𝜒𝑡)1/2

15

AC

CE

PT

ED

M

AN US

CR IP T

where 𝑇0 is the initial temperature of ASB. They derived that the cooling rate of an ASB in Ti-6Al-4V is in the order of 107℃/s. Since the time needed for ASB formation is in the order of 101μs, temperature drop could be as high as hundreds of degrees, which indicates this process should not be considered adiabatic. Upon a close look at equation 15, one may find that if 𝜒 approaches zero (or in other words, adiabatic case), 𝑇 would be infinity. This prediction is apparently incorrect because 𝑇 cannot be higher than 𝑇0 . In this work, double loading tests were carried out. We measured the temperature evolution after the first loading and calculated the average cooling rate. Figure 19 presents the original signals of the loading pulses and responses of IR detectors. Subjected to the first loading pulse, the titanium sample was gradually heated to a maximum temperature and then cooled down at a much lower rate. Figure 20 shows an example of the stress and temperature evolution during the first and second loading processes. We calculated the cooling rate between the first and second loading and that after the second loading, and results are listed in Table 3. Severe localization has not formed within the first loading and the temperature rise was 5 relatively uniform. The average cooling rate is in the order of 10 ℃/s, while at the initial stage it is about four times higher. After the second loading, where the ASB had formed, the initial cooling rate is close to that of the first loading. Considering that the local temperature within the ASB is higher than the directly measured one, the cooling rate could be in the order of 106℃/s, which is still an order of magnitude lower than that proposed by Me-Bar and Shechtman. This result indicates that it needs a few hundred of microseconds for the ASB to cool down to the ambient temperature. This period of time should be useful for those theoretical works, grain growth and recrystallization, for example.

ACCEPTED MANUSCRIPT

2ndloading

CR IP T

1stloading

PT

ED

M

AN US

Figure 19. Original signal of a shear-compression test. The red and green curves represent the incident and transmitted stress wave, respectively. The other four curves represent temperature measurement channels. Time for the first and second loading are marked by dashed yellow lines. Note the different span of the red and green curves.

(a)

AC

CE

(b) Figure 20. Temperature evolution of a specimen during the first (a) and second (b) loading.

ACCEPTED MANUSCRIPT Table 3. Cooling rate of SCS for double loading tests. Initial cooling rate after 1st loading (105℃/s) 4.0

3.0

1.71

3.0

5.50

4.0

5.33

7.0

Test Number

Max T in 1st loading (℃)

T17

163

113

248

1.25

T25

146

130

260

0.40

T28

181

136

257

1.12

T30

166

130

324

0.90

AN US

T before Max T in 2ndloading 2ndloading (℃) (℃)

CR IP T

Average cooling rate after 1st loading (105℃/s)

Initial cooling rate after 2nd loading (105℃ /s)

4.7 Roles of ASB, temperature and load capacity in the deformation process

AC

CE

PT

ED

M

The roles that ASB initiation, temperature rise and stress collapse play in the deformation process of a material have always been an issue under debate. On the one hand, thermal softening was believed to be the dominant mechanism that triggers ASB; on the other hand, initiation of ASB at the maximum stress was the basic assumption of most analytical criterions such as those by Culver (Culver, 1973), Bai (Bai, 1990) and others (Walley, 2007). It is also confusing to identify the initial causation of the failure process. However, the causality of these events might be interpreted by analyzing the time sequence of their occurrence. In this work, the time for ASB initiation, maximum stress and maximum temperature are recorded by a synchronous system incorporating photography, temperature and dynamic loading measurements. A summary of their time sequences is displayed in Figure 21. Combining Figure 5, 6 and 21, it is clear that ASB initiates after the maximum shear stress. A possible argument on this notion is that the ASB may have already initiated at the maximum shear stress, but on the opposite side of the specimen, which is the reason why it is not observed. However, the time of ASB initiation is always a few microseconds behind the time of maximum stress for all of our tests equipped with high-speed photography. The possibility of ASBs of all the tests initiating on the opposite surface is about (1/2)8 = 0.39%, practically impossible. Moreover, the stress state within the gauge section is approximately plane stress, which makes the ASB tend to initiate simultaneously through the thickness. This experimental evidence indicates that the predicted critical shear strain from

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

most of the criterions underestimate the strain of ASB initiation. It also indicates that ASB may not be the first reason for stress collapse. On the contrary, ASB initiates following stress drop and might be a consequence. As for the reason of stress drop, there are two potential candidates: thermal softening and damage induced by large strain. From our temperature measurement, as shown in Figure 15, the temperature rise at the maximum stress is about 50-90℃, corresponding to a stress drop of 3054MPa (Guo et al., 2017; Sun et al., 2016). This thermal softening is uniform and relatively small compared to the overall flow stress. Moreover, the strain rate begins to increase with the localization process and the strengthening due to strain rate hardening is in the range of 50-60MPa for titanium (Guo et al., 2017; Sun et al., 2016), not to mention strain hardening. These calculations suggest that thermal softening itself may not be enough for the stress drop or ASB initiation. The micro damage induced by the large strain could be another origin of softening. For example, Dodd et al. (Dodd and Atkins, 1983) found that the softening effect of micro voids in the shear localization process was significant. Xu et al. (Xu et al., 1996) also believed that the sharp drop in the load-carrying capacity was associated with the growth and coalescence of the microcracks rather than the occurrence of the shear localization. However, the shear localization would accelerate the growth of the microcracks and in turn speed up the elevation of temperature. We believe that it is the joint effect of thermal softening and micro damage that causes the stress drop, which further triggers the initiation of ASB. About 37μs after the stress collapse, the maximum temperature rise was measured. The accelerated temperature elevation corresponds to the formation and propagation of ASB. As pointed out in section 4.2, it needs less than 10μs for the shear band to go through the entire specimen. Further deformation promotes the development of ASB which leads to further increase in temperature. It is not until the macro crack formation that the temperature reaches its maximum. If we order these events chronologically, it should be: stress collapse, ASB initiation, ASB evolution and temperature elevation, maximum temperature and macro crack formation. The observed evidence that temperature rise is quite behind ASB initiation suggests that it could not be the trigger for ASB formation. Therefore, the analytical and numerical analyses based on thermal perturbation or thermal softening lose their foundation.

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 21. Time sequence of the occurrence of typical events in the dynamic shear failure process.

M

Summary and Conclusion

AC

CE

PT

ED

The thermal mechanical behavior of pure titanium during dynamic shear failure was studied in this work. With the help of synchronized high-speed photography, highspeed temperature measurement and the SHPB system, the initiation and propagation of ASB, temperature elevation and stress evolution were recorded simultaneously. The characteristics of ASB were thoroughly investigated and the roles of stress collapse, thermal softening and ASB formation in the failure process were studied. The main findings of this work are summarized as follows. 1. The local critical shear strain for ASB initiation is as large as 1.3, much larger than the nominal shear strain. ASB always initiates after the stress peak, indicating it should not be the first cause for the stress collapse. 2. The propagating velocity of ASB is not constant but varies with loading process. It is positively dependent on the impact velocity. The maximum velocity of ASB reaches 1900m/s in this work, which is about 60% of shear wave speed for titanium. 3. The fraction of mechanical work converting into heat, i.e., the Taylor-Quinney factor, was derived to be 0.25 to 0.55 and it is also positively dependent on the loading rate. This experiment result indicates the commonly used values of 0.9~1.0 should be re-evaluated. 4. The measured temperature rise at the maximum stress (or before ASB initiation) is in the range of 50-90℃, which by itself is not enough to outplay other strengthening mechanisms (such as strain rate hardening). Micro damage induced by the large shear strain should also be responsible for the stress drop.

ACCEPTED MANUSCRIPT 5. The measured temperature within the ASB is about 350-650℃, whereas the material close to the ASB is also heated. About 37μs after the stress collapse, the temperature reaches its maximum, which is due to the continuous development of the shear band until macro crack formation. The observed fact that temperature rise is quite behind ASB initiation suggests that it could not be the trigger for ASB formation. Therefore, the analytical and numerical analyses of ASB initiation based on temperature perturbation or thermal softening lose their foundation.

Andrade,

U.,

Meyers,

M.A.,

AN US

CR IP T

Acknowledgement Yazhou Guo and Yulong Li acknowledge the financial support by National Natural Science Foundation of China (NSFC Contracts Nos. 11672354, 11472227 and 11527803) and the 111 Project (No. B07050). Shengxin Zhu thanks the financial support from NSFC (No. 11802029) and the State Key Laboratory of Explosion Science and Technology (ZDKT18-03). Valuable discussions with Professor D. Rittel are also acknowledged. References Vecchio,

K.S.,

Chokshi,

A.H.,

1994.

Dynamic

recrystallization in high-strain, high-strain-rate plastic deformation of copper. Acta Metallurgica et Materialia 42, 3183-3195.

Antolovich, S.D., Armstrong, R.W., 2014. Plastic strain localization in metals: origins and consequences. Progress in Materials Science 59, 1-160.

M

Bai, Y., 1990. Adabatic shear banding. Res Mechanica 31, 133-203. Bai, Y.L., 1982. Thermal-plastic instability in simple shear. Journal of the Mechanics and Physics of Solids 30, 195-207.

ED

Bai, Y.L., Dodd, B., 1992. Adiabatic Shear Localization, Occrrence, Theories and Applications. Pergamon Press, Oxford, New York, Seoul, Tokyo. Batra, R.C., Kim, C.H., 1990. Adiabatic shear banding in elastic-viscoplastic

PT

nonpolar and dipolar materials. International Journal of Plasticity 6, 127-141. Bever, M.B., Holt, D.L., Titchener, A.L., 1973. The stored energy of cold work. Progress in Materials Science 17, 5-177.

CE

Cerreta, E.K., Bingert, J.F., Gray Iii, G.T., Trujillo, C.P., Lopez, M.F., Bronkhorst, C.A., Hansen, B.L., 2013. Microstructural examination of quasi-static and dynamic shear in high-purity iron. International Journal of Plasticity 40, 23-

AC

38.

Chen, R., W., Vecchio, K., S., 1994. Microstructural characterization of shear band formation in Al-Li alloys. J. Phys. IV France 04, C8-459-C458-464. Chen, W.W., Song, B., 2010. Split Hopkinson (Kolsky) bar: design, testing and applications. Springer Science & Business Media. Chichili, D.R., Ramesh, K.T., Hemker, K.J., 2004. Adiabatic shear localization in alpha-titanium: experiments, modeling and microstructural evolution. Journal of the Mechanics and Physics of Solids 52, 1889-1909. Clifton, R.J., Duffy, J., Hartley, K.A., 1984. On critical conditions for shear band formation at high-strain rates SCRIPTA METALLURGICA 18, 443-448.

ACCEPTED MANUSCRIPT Costin, L.S., Crisman, E.E., H., H.R., Duffey, J., 1979. On the localization of plastic flow in mild steel tubes under dynamic torsional loading, in: J., H. (Ed.), Second Conference on the Mechanical Properties of Materials at High Rates of Strain, London, pp. 90-100. Culver, R.S., 1973. Thermal instablity strain in dynamic plastic deformation, in: R.W.Rohde, B.M. Butcher, J.R. Holland, Karnes, C.H. (Eds.), Metallurgical Effects at High Strain Rates. Plemun Press, New York, pp. 519-530. Dodd, B., Atkins, A.G., 1983. Flow localization in shear deformation of voidcontaining and void-free solids. Acta Metallurgica 31, 9-15. stresses. Materials Science and Technology 5, 557-559.

CR IP T

Dodd, B., Bai, Y., 1989. Width of adiabatic shear bands formed under combined Dorogoy, A., Rittel, D., Godinger, A., 2015. Modification of the Shear-Compression Specimen for Large Strain Testing. Experimental Mechanics 55, 1627-1639.

Duffy, J., Chi, Y.C., 1992. On the measurement of local strain and temperature during the formation of adiabatic shear bands. Materials Science and Engineering: A 157, 195-210.

AN US

Grady, D.E., 1994. Dissipation in adiabatic shear bands. Mechanics of Materials 17, 289-293.

Guduru, P.R., Rosakis, A.J., Ravichandran, G., 2001. Dynamic shear bands: an investigation using high speed optical and infrared diagnostics. Mechanics of Materials 33, 371-402.

Guo, Y.Z., Li, Y.L., Pan, Z., Zhou, F.H., Wei, Q., 2010. A numerical study of effect

on

adiabatic

M

microstructure

shear

instability:

Application

to

nanostructured/ultrafine grained materials. Mechanics of Materials 42, 1020-1029. Guo, Y.Z., Sun, X.Y., Wei, Q., Li, Y.L., 2017. Compressive responses of ultrafine-

ED

grained titanium within a broad range of strain rates and temperatures. Mechanics of Materials 115, 22-33.

Hartley, K.A., Duffy, J., Hawley, R.H., 1987. MEASUREMENT OF THE TEMPERATURE

PT

PROFILE DURING SHEAR BAND FORMATION IN STEELS DEFORMING AT HIGH-STRAIN RATES. Journal of the Mechanics and Physics of Solids 35, 283-301. Hines, J.A., Vecchio, K.S., 1997. Recrystallization kinetics within adiabatic shear

CE

bands. Acta Materialia 45, 635-649. Liao, S.-c., Duffy, J., 1998. Adiabatic shear bands in a TI-6Al-4V titanium alloy. Journal of the Mechanics and Physics of Solids 46, 2201-2231.

AC

Marchand, A., Duffy, J., 1988. An experimental study of the formation process of adiabatic shear bands in a structural steel. Journal of the Mechanics and Physics of Solids 36, 251-283. Me-Bar, Y., Shechtman, D., 1983. On the adiabatic shear of Ti-6Al-4V ballistic targets. Materials Science and Engineering 58, 181-188. Mercier, S., Molinari, A., 1998. Steady-State shear band propagation under dynamic conditions. Journal of the Mechanics and Physics of Solids 46, 1463-1495. Meyers, M.A., Pak, H.-R., 1986. Observation of an adiabatic shear band in titanium by high-voltage transmission electron microscopy. Acta Metallurgica 34, 2493-2499. Meyers, M.A., Xu, Y.B., Xue, Q., Perez-Prado, M.T., McNelley, T.R., 2003.

ACCEPTED MANUSCRIPT Microstructural evolution in adiabatic shear localization in stainless steel. Acta Materialia 51, 1307-1325. Molinari,

A.,

localization

Clifton,

in

R.J.,

1987.

thermoviscoplastic

Analytical

materials

characterization

JOURNAL

OF

APPLIED

of

shear

MECHANICS-

TRANSACTIONS OF THE ASME 54, 806-812. Nemat-Nasser, S., Isaacs, J.B., Liu, M., 1998. Microstructure of high-strain, highstrain-rate deformed tantalum. Acta Materialia 46, 1307-1325. Pérez-Prado, M.T., Hines, J.A., Vecchio, K.S., 2001. Microstructural evolution in adiabatic shear bands in Ta and Ta-W alloys. Acta Materialia 49, 2905-2917. in

titanium

alloy

during

high

strain

rate

phenomenon. Mechanics of Materials 40, 255-270.

CR IP T

Ranc, N., Taravella, L., Pina, V., Herve, P., 2008. Temperature field measurement loading--Adiabatic

shear

bands

Rittel, D., Landau, P., Venkert, A., 2008a. Dynamic Recrystallization as a Potential Cause for Adiabatic Shear Failure. Physical Review Letters 101, 165501.

Rittel, D., Lee, S., Ravichandran, G., 2002. A shear-compression specimen for large strain testing. Experimental Mechanics 42, 58-64.

AN US

Rittel, D., Wang, Z.G., 2008. Thermo-mechanical aspects of adiabatic shear failure of AM50 and Ti6Al4V alloys. Mechanics of Materials 40, 629-635. Rittel, D., Wang, Z.G., Dorogoy, A., 2008b. Geometrical imperfection and adiabatic shear banding. International Journal of Impact Engineering 35, 1280-1292. Rittel, D., Wang, Z.G., Merzer, M., 2006. Adiabatic Shear Failure and Dynamic Stored Energy of Cold Work. Physical Review Letters 96, 075502.

M

Rittel, D., Zhang, L.H., Osovski, S., 2017a. The dependence of the Taylor–Quinney coefficient on the dynamic loading mode. Journal of the Mechanics and Physics of Solids 107, 96-114.

ED

Rittel, D., Zhang, L.H., Osovski, S., 2017b. Mechanical Characterization of ImpactInduced Dynamically Recrystallized Nanophase. Phys. Rev. Appl. 7, 7. Rodríguez-Martínez, J.A., Vadillo, G., Rittel, D., Zaera, R., Fernández-Sáez, J.,

PT

2015. Dynamic recrystallization and adiabatic shear localization. Mechanics of Materials 81, 41-55.

Sun, X., Guo, Y., Wei, Q., Li, Y., Zhang, S., 2016. A comparative study on the

CE

microstructure and mechanical behavior of titanium: Ultrafine grain vs. coarse grain. Materials Science and Engineering: A 669, 226-245. Taylor, G.I., Quinney, H., 1932. The plastic distortion of metals. Philos. Trans.

AC

R. Soc. Lond. A-Contain. Pap. Math. Phys. Character 230, 323-362. Taylor, G.I., Quinney, H., 1934. The Latent Energy Remaining in a Metal after Cold Working.

Timothy, S.P., Hutchings, I.M., 1985. The structure of adiabatic shear bands in a titanium alloy. Acta Metallurgica 33, 667-676. Walley, S.M., 2007. Shear Localization: A Historical Overview. Metallurgical and Materials Transactions A 38, 2629-2654. Walter, J.W., 1992. Numerical experiments on adiabatic shear band formation in one dimension. International Journal of Plasticity 8, 657-693. Wei, Q., Jia, D., Ramesh, K.T., Ma, E., 2002. Evolution and microstructure of shear

ACCEPTED MANUSCRIPT bands in nanostructured Fe. Applied physics letters 80, 1240-1242. Wei, Q., Jiao, T., Mathaudhu, S.N., Ma, E., Hartwig, K.T., Ramesh, K.T., 2003. Microstructure and mechanical properties of tantalum after equal channel angular extrusion

(ECAE).

Materials

Science

and

Engineering

a-Structural

Materials

hardening

materials.

Properties Microstructure and Processing 358, 266-272. Wright,

T.W.,

1992.

Shear

band

susceptibility:

Work

International Journal of Plasticity 8, 583-602. Wright, T.W., 2002. The physics and mathematics of adiabatic shear bands. Cambridge University Press, Cambridge, New York, Port Melbourne, Madrid, Cape Town. bands. International Journal of Plasticity 1, 205-212.

CR IP T

Wright, T.W., Batra, R.C., 1985. The initiation and growth of adiabatic shear Wright, T.W., Walter, J.W., 1987. On the stress collapse in adiabatic shear bands. Journal of the Mechanics and Physics of Solids 35, 701-720.

Xu, Y.B., Bai, Y.L., Xue, Q., Shen, L.T., 1996. Formation, microstructure and development

of

the

localized

shear

deformation

in

Materialia 44, 1917-1926.

low-carbon

steels.

Acta

AN US

Xu, Y.B., Zhang, J.H., Bai, Y.L., Meyers, M.A., 2008. Shear localization in dynamic deformation: Microstructural evolution. Metall. Mater. Trans. A-Phys. Metall. Mater. Sci. 39A, 811-843.

Xue, Q., Meyers, M.A., Nesterenko, V.F., 2002. Self-organization of shear bands in titanium and Ti-6Al-4V alloy. Acta Materialia 50, 575-596.

Yang, Y., Wang, B.F., 2006. Dynamic recrystallization in adiabatic shear band in

M

α-titanium. Materials Letters 60, 2198-2202.

Yin, W., Xu, F., Ertorer, O., Pan, Z., Zhang, X., Kecskes, L., Lavernia, E., Wei, Q., 2013. Mechanical behavior of microstructure engineered multi-length-scale

ED

titanium over a wide range of strain rates. Acta Materialia 61, 3781-3798. Yuan, F., Bian, X., Jiang, P., Yang, M., Wu, X., 2015. Dynamic shear response and evolution mechanisms of adiabatic shear band in an ultrafine-grained austenite–

PT

ferrite duplex steel. Mechanics of Materials 89, 47-58. Zehnder, A.T., Rosakis, A.J., 1991. On the temperature distribution at the vicinity of dynamically propagating cracks in 4340 steel. Journal of the Mechanics and

CE

Physics of Solids 39, 385-415. Zener, C., Hollomon, J.H., 1944. Effect of Strain Rate Upon Plastic Flow of Steel. Journal of Applied Physics 15, 22-32.

AC

Zhang, T., Guo, Z.-R., Yuan, F.-P., Zhang, H.-S., 2017. Investigation on the plastic work-heat conversion coefficient of 7075-T651 aluminum alloy during an impact process based on infrared temperature measurement technology. Acta Mechanica Sinica.

Zheng, Y.P., Zeng, W.D., Wang, Y.B., Zhou, D.D., Gao, X.X., 2017. High strain rate compression behavior of a heavily stabilized beta titanium alloy: Kink deformation and adiabatic shearing. Journal of Alloys and Compounds 708, 84-92. Zhou,

F.H.,

Molinari,

J.F.,

Ramesh,

K.T.,

2005.

A

cohesive

model

based

fragmentation analysis: effects of strain rate and initial defects distribution. International Journal of Solids and Structures 42, 5181-5207.

ACCEPTED MANUSCRIPT Zhou, M., Rosakis, A.J., Ravichandran, G., 1996. Dynamically propagating shear bands

in

impact-loaded

prenotched

plates--I.

Experimental

investigations

of

temperature signatures and propagation speed. Journal of the Mechanics and Physics

AC

CE

PT

ED

M

AN US

CR IP T

of Solids 44, 981-1006.