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Material embrittlement in high strain-rate loading

Material embrittlement is often encountered in machining, heat treatment, hydrogen and low-temperature conditions among which machining is strain-rate related. More strain-rate evoked embrittlement is expected in material loading processes, such as in high-speed machining and projectile penetration. In order to understand the fundamental mechanisms of the strain-rate evoked material embrittlement, this study is concerned with the material responses to loading at high strain-rates. It then explores the strain-rate evoked material embrittlement and fragmentation during high strain-rate loading processes and evaluates various empirical and physical models from different researchers for the assessment of the material embrittlement. The study proposes strain-rate sensitivity for the characterization of material embrittlement and the concept of the pseudo embrittlement for material responses to very high strain-rates. A discussion section is arranged to explore the underlying mechanisms of the strain-rate evoked material embrittlement and fragmentation based on dislocation kinetics.


Introduction
Material embrittlement can be due to different causes. Tensile stress triaxiality, heat treatment, and low-temperature conditions, for example, are the causes that can lead to material embrittlement. Table 1 lists the common types of material embrittlement among which high-speed machining is often encountered and characterized by high strain-rate loading. Welding embrittlement is related to material segregation and precipitation. The formation of the 'Cottrell atmosphere' during a welding process impedes dislocation movement and causes precipitation embrittlement [1]. In addition, in a heat treatment process of steel, the blue embrittlement may occur at a certain temperature as dislocations are hindered by the carbon and nitrogen atoms, which leads to an increase in strength of a material [2]. Low-temperature embrittlement is due to the increase in the critical stress for the dislocation movement [3,4]. Hydrogen embrittlement is a result of the decrease in surface energy caused by the generation of hydride or the attachment of hydrogen atoms to crack tips if any [5]. Irradiation embrittlement is common in nuclear containers, as the point defects generated by the irradiation interact with the solute atoms to form precipitates, hindering dislocation movements and resulting in material embrittlement [6][7][8].
Material embrittlement can be a result of an enhanced strength and hardness, and a reduced fracture toughness of a material. In fact, material embrittlement occurs in the processes of, for example, machining, projectile penetration, explosion, and tunnel boring, because of the high strain-rates in the processes, as specified in table 2. Many researchers also suggested that a high strain-rate should lead to material embrittlement. Chilton et al [9] investigated the ductile-tobrittle transition of tungsten at low temperatures, and found the ductile-to-brittle transition temperature of the material increased with strain rate.
In a machining process, strain rate can reach up to 10 7 s −1 at which embrittlement should occur in many materials [10]. Material embrittlement at high strain-rates paves a practical way towards machining of difficult-to-machine materials with high efficiency and high quality. The problems (e.g. built-up edges and wheel loading) that are constantly associated with the conventional machining processes may no longer exist in a high-speed machining process as a material can be temporarily embrittled for improved machinability during such a process.
The difficult-to-machine materials (hard and brittle materials, composites, titanium alloys, nickel alloys, etc) generally have outstanding mechanical and physical properties and are widely used in the aerospace and defense, semiconductor, automotive, and cutting tool industries [24][25][26]. However, these materials are difficult-to-machine because they often create machining problems, such as surface and subsurface damage to the workpiece, poor workpiece surface finish and dimensional accuracies, chatter and vibrations in a machining process, and cutting tool wear. Many researchers recognize that high-speed machining is an effective method to solve the machining problem of the difficultto-machine materials and is helpful in improving machining efficiency and surface quality, and suppressing subsurface damage [27][28][29]. Studies also reported that material embrittlement in the grinding processes helped reduce subsurface damage depth [30,31]. Material embrittlement can be induced at a high strain-rate that is realized in a high-speed machining process. It is expected to solve the typical problems involved in the conventional machining processes, such as built-up edges, grinding wheel loading, workpiece burn, burr formation, surface chipping, and surface integrity (e.g. cracking, residual stresses, plastic deformation, and phase transformation). Unfortunately, the strain-rate evoked material embrittlement is not fully understood, and its fundamental mechanisms are to be investigated [32].
This study is focused on the material embrittlement mechanisms of the engineering materials subjected to loading at high strain-rates. Section 2 describes material responses to loading at high strain-rates; sections 3 is related to material embrittlement at high strain-rates; section 4 provides discussion on the strain-rate evoked material embrittlement; and section 5 summarizes the paper and draws conclusions.

Material responses to loading at high strain-rates
To understand the embrittlement mechanisms under the high strain-rate conditions, it is necessary to study material responses to loadings under such conditions. Generally, experimental methods and constitutive models are established for studying the material responses to mechanical loadings, which is presented in this section.

High strain-rate loading processes
Material responses to loading at high strain-rates can be experimentally investigated and analyzed. One of the widely used techniques is the split Hopkinson pressure bar (SHPB) that consists of a striker and the incident and transmission bars between which a specimen is placed in. Figure 1 shows the schematic diagram of SHPB for studying the mechanical properties of a specimen at high strain-rates. SHPB is widely used for both ductile and brittle materials. By collecting the incident, transmitted, and reflected waves during an impact process, the mechanical properties are obtained for the specimen materials loaded at high strain-rates.
Based on the one-dimensional and the stress uniformity assumptions, the strain, strain rate and stress of the specimen can be obtained from equations (1)-(3) [33], respectively, where e R and e T are the pulse strain amplitudes of the reflected and transmitted waves, respectively; A s indicates the crosssectional area of the specimen; L represents the length of the specimen; A is the cross-sectional area of the incident bar; E represents Young's modulus of the specimen; and C 0 is the speed of the elastic wave in the incident bar. In addition, SHPB is often modified to mimic the mechanical loadings at high strain-rates. Frew et al [34] reported a pulse shaping technique on a conventional SHPB apparatus during the high strain-rate compression tests on brittle materials. By shaping the incident pulse, they were able to load the specimen at a constant strain-rate under the dynamically-balanced stress state during the loading process. The accuracy of the compression tests was thus enhanced. In order to study material responses to higher strain-rates, Liu et al [35,36] built a miniaturized SHPB apparatus with a strain rate of up to 10 4 s −1 by strengthening the material and reducing the dimensions of the bar. Although the SHPB technique is widely used on both ductile and brittle materials, it cannot accurately measure the mechanical properties of a material at high strain-rates, e.g. 10 7 s −1 which is often associated with the high-speed machining processes. Meyer et al [37] performed a high-speed shear experiment on a hat-shaped specimen using a modified SHPB device, as shown in figure 2. They studied the microstructural evolution of the AISI 304 L stainless steel using electron backscatter diffractometry and transmission electron microscopy. In the experiment, the local strain-rate of the specimen was increased to above 10 4 s −1 , which is useful for studying the microstructural evolution in the adiabatic shear band at a high strain-rate. However, the complexity of the stress state within the specimen made it difficult to quantitatively analyze the experimental results.
Sutter et al [29] performed a high-speed orthogonal cutting experiment on TC4 using a modified SHPB device, as shown in figure 3. In the experiment, the researchers studied the effects of cutting speed on chip morphology and the mechanisms of chip formation, and found that the chip changed from continuous to discontinuous morphologies with an increase in cutting speed. The experiment allowed them to achieve the orthogonal cutting at a high cutting speed of approximately 75 m s −1 . However, it was difficult to control depth of cut in the experiment.
Liu et al [38] studied the deformation and fracture behavior of the 7050-T7451 aluminum alloy in an orthogonal cutting experiment at a cutting speed of 120 m s −1 , as shown in figure 4. They found that fragmented chips were formed in the mode of brittle fracture, and the material embrittlement    [18][19][20][21] Strain rate: 10 −3 -10 2 s −1 Less impact effect Bridge vibration, structure vibration, machine vibration, etc Tunnel boring [22,23] Strain rate: 10 1 -10 3 s −1 Less impact effect Highway tunneling, high-speed train tunneling, subway tunneling, underwater tunneling, underground shielding, etc leads to the reductions in cutting temperature and cutting force.
Other techniques, including single-point grinding [40], ultrahigh-speed grinding [31], plate impact test [41], SHPB rod test [42], three-axis high-speed impact test [43,44], expansion ring technique [45], pendulum impact technique [46], projectile penetration [47,48], etc, are also commonly used to study the material responses to high strain-rates. The above experimental techniques generally have certain limitations, such as insufficient strain-rate or poor cutting accuracy. However, with the rapid development of highperformance machine tools, it becomes possible to conduct ultrahigh-speed machining on difficult-to-machine materials.

Constitutive material models at high strain-rates
The material responses to dynamic loadings are related to many influencing factors, such as stress/strain fields, strain rate, temperature field, the pre-existing damage, and the surrounding environment. As the number of the factors is overwhelming, it is difficult to experimentally isolate a single factor from the influencing factors. In this regard, constitutive modeling paves an effective way to understanding the material responses to loading at high strain-rates. Generally, the constitutive material models are classified into two types, empirical and physical. Examples are the Johnson and Cook (J-C) empirical model and the physical models based on the dislocation kinetics.
where s is the equivalent stress; e is the equivalent plastic strain; * e e e =    / 0 represents the dimensionless strain-rate, e  0 is the reference strain-rate; * = - represents the non-dimensionalized temperature, T m is the melting temperature of a material, T r is room temperature; A is the yield strength of a material at the reference strain-rate and temperature; B is the strain hardening coefficient; C represents the strain rate hardening coefficient; n is the strain hardening exponent; and m is the thermal softening coefficient.
The J-C constitutive model considers the effects of strain, strain rate, and temperature on material responses to loading. It has fewer parameters and is widely applied to machining simulations. Being an empirical model, it is different under different material and loading conditions. In order to accommodate material responses to dynamic loading, researchers presented the modified models based on the J-C model, as listed in table 3.
The empirically-based J-C constitutive model is of great significance to solving many practical problems and widely used in the finite element methods at high strain-rates. However, since the model decouples the interplay among strain, strain rate, and heat, which inevitably undermines its prediction accuracy.
For brittle materials, the J-H constitutive model [56,57]    where * s i represents the normalized strength of the intact material; * s f is the normalized fracture stress; and D is the damage factor of the material (0D1). The dimensionless expression is represented by where s HEL is the equivalent stress at Hugoniot elastic limit. The strength * s i of the intact material can be shown in equation (7) * * * * where A, C, and N are the material constants to be determined; * e e e =    / 0 is the dimensionless strain-rate; e  is strain rate; e˙0 is the reference strain-rate; * P and T * are the dimensionless pressure and the dimensionless maximum tensile strength, respectively, 9 HEL where P HEL is the pressure at Hugoniot elastic limit; P is the actual pressure; and T is the maximum tensile strength of a material. The dimensionless strength of the partially damaged material * s f is where B and M are material constants; and * s f max is the dimensionless fracture strength of a material. The J-H constitutive model accounts for damage of a brittle material and has widely been used in engineering design of brittle materials, such as concrete and ceramics.

2.2.2.
Physically-based models. Dislocation proliferation and slip movement cause plastic deformation in a material. Theoretically, slip could reach the sound speed in a material if no barriers (e.g. grain boundaries and second phase) were present in the material. Thus, the flow stress of the material is usually expressed as equation (11)  where s pn represents lattice resistance which is related to temperature and strain rate; s dis represents interaction forces among dislocations, s r = Gb dis where G is shear modulus; b is the Burgers vector; r is dislocation density; s HP represents the resistance of grain boundaries to dislocations which can be expressed by the Hall-Petch equation, i.e. s = -/ k d , y HP 1 2 where k y is a constant; d is the average diameter of material grains; s obs represents the resistance of the dispersive barriers to dislocations such as segregation and point defects.
The thermal activation theory is commonly used in the study of physically-based models. According to the theory, flow stress s y can be decomposed into a short-range component s th which is thermally related and a long-range component s ath which is athermal, Based on the thermal activation theory, the thermal barriers vary with temperature, as shown in figure 5, the stress for dislocations to overcome the barriers decreases as temperature increases. In order to graphically describe the barriers, Follansbee et al [59] proposed the following equation to graphically describe the barriers with two Effect of recrystallization on yield strength s e = + + - Correction for strain-rate sensitivity Consideration of hardness effect Temperature effect on strain rate hardening 5000 3000 500 150 Lin [55] Coupling effect of temperature and strain rate * * s e e e l l e For the specific meaning of each parameter, please check the references.
parameters, p and q [60] where DG denotes the activation energy of the dislocations to overcome the barriers; DG 0 is the energy of the dislocations required to overcome barriers at zero temperature; t 0 is the flow stress at zero temperature; and t is the flow stress. A dislocation overcomes the barrier when its energy exceeds the energy level of the barrier. As the amplitude of the atomic vibration increases with temperature, some dislocations successfully overcome the barriers at a certain frequency v which is given in equation (14) = where v 0 is the vibrational frequency of a dislocation; k is Boltzmann's constant; T is temperature; t w =v −1 , where t w is the time a dislocation spends waiting at a barrier for thermal activation. The dislocation moves in-between the barriers at a velocity of v f that can be shown in equation (15) f y ath Assuming that l is the average distance between the successive barriers, slip time t f of the dislocation in-between the barriers is The average speed of the dislocation is given as At a low strain-rate,  t t , w f the dislocation drag can be ignored, whereas at a high strain-rate, » t t , w f the dislocation drag is significant and cannot be ignored. Table 4 lists some of the physically-based models for material deformation at high strain-rates.
The physically-based models are helpful in understanding the effects of strain rate on material properties and dislocation density. However, the physically-based models are complicated, which limits their practical applications. Figure 6 shows the effect of strain rate on the yield strengths of the materials based on different models. The black data points fitted with the solid lines and the white data points fitted with the broken lines are the calculation results based on the physical and empirical models, respectively. The results from the physical models show that the yield strengths of the materials rapidly increase when strain rate is beyond approximately 10 4 s −1 , which is also verified by many researchers in their experimental studies. The results suggest that the dislocation drag be a dominant factor in material deformation at high strain-rates [68][69][70][71]. From this respect, the physically-based models can more accurately describe the strain-rate effects on material properties than does the J-C model. Figure 7 presents the experimental results of shear stress with shear strain rate for single crystalline aluminum. Although the influences of the second phase and grain boundaries are absent, the results show that the dislocation drag is also significant for single crystalline aluminum, indicating that the dislocation drag is related to the dislocation interactions and lattice resistances.

Material responses to dynamic loading
Numerous results show that material strength (yield strength and ultimate tensile strength, etc) increases with strain rate. Zhou et al [31] suggested that the yield strength of a material should approach the ultimate tensile strength with an increase in strain rate. When the yield strength of a material is larger than the ultimate tensile strength, the material undergoes brittle fracture without plastic deformation. Material embrittlement thus occurs. Assuming that the difference between the ultimate tensile strength and the yield strength is s D where s UT is the ultimate tensile strength; s Y is the yield strength. Figure 8 shows the change in s D with strain rate for different materials. The results in figure 8 are summarized from the research conducted by the different researchers [42,[76][77][78][79][80]. From figure 8, a relationship between s D and strain rate is obtained and shown in equation (19) where k 0 is a material constant; k s represents strain-rate sensitivity on material embrittlement. A material undergoes the ductile-to-brittle transition at s D = 0, k s is different for different materials. According to the values of k s shown in figure 8, materials can be divided into three types: I, II, and III, among which the type I materials are of the largest k s and the most susceptible to embrittlement at high strain-rates, whereas the type III materials are the least sensitive to strain rates and the least susceptible to embrittlement at high strain-rates. For example, TC4 and DP1000 are the type I materials with their strain-rate sensitivities of 35.4 and 45.9, respectively. s D of the type I materials rapidly diminish at an increased strain-rate during a loading process. At s D = 0, a material is completely embrittled. Conversely, 7075-T6 and 6061-T6 belong to the type III materials with their respective k s values of 1.5 and 0.4. Material embrittlement is difficult to occur even if at a very high strain-rate. For the materials with the same k s , the larger the k 0 , the higher the strain rate for material embrittlement.
The effect of strain rate on the property changes of a brittle material is also significant. The commonly used technique is SHPB for studying the property changes of brittle materials at high strain-rates. Harsha et al [81] developed a micro-mechanically motivated constitutive model based on crack growth dynamics, and related crack speed to the instantaneous value of stresses at a crack tip. Both the experimental and simulation results are shown in figure 9. It can be seen that similar to the materials depicted in figures 6 and 7, the material strength of a marble also rapidly increased as strain rate was increased to 10 3 s −1 , which is attributed to the understanding that the interplay of the nonlinear and    inertial effects associated with the propagating crack tip may create a material resistance at the crack tip to interfere with crack propagation [82,83]. Li et al [84] studied the variation of the strength and fragment size of a granite with strain rate under impact loading and found that with an enhanced strain-rate, the strength of the granite went up but the fragment size came down. Figure 10 shows the results of an impact test on the aluminosilicate glass and a similar conclusion was obtained [85]. Grady and Kipp et al [86] explained that at a higher strain-rate, more defects might be activated in the material, resulting in material fragmentation with smaller fragments and debris.
For amorphous materials such as metallic glass, strain rate can also significantly affect the material properties [87,88]. Lu et al [89] studied the static and dynamic mechanical properties of Zr-based bulk metallic glass using quasi-static and dynamic SHPB loading devices, respectively. They showed that the material strength increased with strain rate. Yu et al [90] performed tensile tests on La-based metallic glass with different strain-rates and temperatures, and found that an increase in strain rate promoted embrittlement of the metallic glass. Liu et al [35] used a modified SHPB device to perform dynamic compression experiments on a Zrbased bulk metallic glass and found that the superplasticity of the material disappeared with an increase in strain rate. Therefore, an increase in strain rate can also lead to material embrittlement of an amorphous material.
The above studies show that the material strengths increase with strain rate up to a critical value. In addition, the increase in strain rate leads to material fragmentation, which will be discussed in the next section.

Material embrittlement in machining processes
During a machining process, such as high-speed machining, ultrasonically-assisted machining, and tunnel boring, material embrittlement can be induced. As shown in figure 11, in an orthogonal cutting process, cutting chips change from the continuous to discontinuous types at an increased cutting speed. In high-speed machining, strain rate is up to 10 7 s −1 [91][92][93]. Material removal mechanism in high-speed machining is quite different from that in the conventional machining. Wang et al [94] studied the ductile-to-brittle transition in machining and found that as strain rate increased, chip morphology changed from the continuous to discontinuous types. Ma et al [95] carried out a high-speed orthogonal cutting experiment of TC4 on a modified SHPB and found that the brittle fracture occurred on the free surface of the chips. They also found that brittle fracture increased while ductile fracture decreased with an increase in cutting speed.
Ultrasonically-assisted machining is a viable method for machining of many different materials, especially hard and brittle materials. In an ultrasonically-assisted machining process, a cutting tool is superimposed with high-frequency vibration to obtain a much higher instantaneous cutting speed and acceleration than those in a regular cutting process. The higher speed and acceleration lead to a high strain-rate which may evoke material embrittlement in the machining process. Zhao et al [97] studied the ultrasonically-assisted machining of rocks, and found that an increase in the impact frequency could significantly reduce strength and increase fragmentation of the material; Lv et al [98] studied the ultrasonicallyassisted machining of the BK7 glass, and demonstrated that the ultrasonic vibrations caused material fragmentation and a reduced cutting force. In their experimental study on the K9 glass, Wang et al [99] found that the ultrasonically-assisted machining could reduce the depth of subsurface damage by 30%-40% compared to that in the conventional machining.
A material is embrittled and fragmented in an ultrasonically-assisted machining process because of the high strain-rates involved in the process. As shown in figure 12, the influence of strain rate on material embrittlement and subsurface damage can be explained as follows.
(1) The material resistance ahead of a crack tip increases with strain rate, impeding crack propagation.    (2) The directions of cutting and median crack propagation are different in a machining process. At a higher cutting speed, a median crack, if nucleated, has less time to propagate before the cutting tool passes by, resulting in smaller depth of subsurface damage. (3) The internal defects of a material are activated under impact loading, which triggers nucleation, propagation, and interdigitation of microcracks, and more fragmentation and pulverization in the surface layer of the material.
A tunnel boring machine (TBM) is normally applied to cutting hard and brittle rocks encountered in boring tunnels for subways and trains. Material embrittlement, fragmentation, and pulverization also exist in the boring processes of a tunnel. The brittleness of a rock is enhanced in a boring process due to the strain-rate effect, and can in turn significantly influence the machinability of the rock. Kahraman et al [100] studied the methods of measuring material brittleness and the correlations between brittleness and machinability of rocks, and suggested that there should be an exponential correlation between the boring performance of a TBM and the brittleness of a rock; Gong et al [101] studied the influence of rock brittleness on TBM penetration rate, and found that the crushed zone and radial cracks increased with rock brittleness and the failure element induced by the cutter almost linearly increased with rock brittleness. In order to improve machining efficiency and reduce tool wear, Wu et al [102] presented a rock fragmentation method of shock disturbance, and established a crack model for the damaged rock around an induced hole, and observed that the penetration rate increased with crack density, whereas the cutting force decreased with the increase in the disturbance frequency. Increasing rock brittleness through an enhanced strain-rate is obviously beneficial to improving machinability and efficiency in the tunnel boring process.
In a machining process, high strain-rate evokes not only embrittlement but also fragmentation of a material. The fragment size S of brittle materials at high strain-rates was evaluated by Zhang et al [103], as shown in equation (20). It can be shown that the average fragment size decreases with an increase in strain rate where K IC is the fracture toughness of a material; D f denotes the damage scalar corresponding to the initiation of fragmentation; E is Young's modulus; and t f is the total time to reach the fracture stress. In addition, Grady and Kipp et al [104] conducted an expansion ring impact test for ductile materials, and calculated fragment size S based on the fracture energy theory, as shown in equation (21) g where g s is surface energy; and r 0 is material density.

Effects of stress wave on material fragmentation
Stress wave is an important factor affecting material fragmentation under high strain-rate. Material fragmentation is related to the process of stress wave propagation, reflection, and interaction. Zhang et al [105] believed that stress wave response caused by impact loading was the main cause of material removal and crack formation in hard and brittle materials. Based on the propagation state of stress waves in a solid, there are different modes for stress wave propagation, such as elastic, plastic, and viscoelastic waves. The propagation velocities of the elastic and plastic waves are shown in equations (22) and (23), respectively [106] where E represents Young's modulus of the material; s e / d d is the slope of the tangent line corresponding to the point of stress s on the stress-strain curve; r 0 is the density of the material. It can be understood that propagation of stress waves is related to the mechanical properties of a material under impact loading. Material damage can be caused by the stress waves that are generated during impact loading, and propagate to and are reflected from the free surface of a material. Figure 13 shows the pressure distribution at different times when the compressive wave with a wavelength λ is reflected at the free surface of a rod end [107]. In figure 13(a), the compressive wave is close to the free surface; (b) the incident compressive wave and the reflected tensile wave are superposed to form a composite wave, and tensile stress appears in the overlap region (in blue); (c) the composite wave is in tensile, which represents a critical state. Before the state, the rod is still subjected to partial compressive stress. After the state, the rod is completely subjected to tensile stress; (d) the overlap region and the tensile stress are gradually reduced; (e) the incident compressive wave fades and the stress wave reflection ends. As shown in figure 14, Jiang et al [108] carried out an impact compression experiment on a granite by SHPB and studied the fracture characteristics of the granite under an impact load. The reflection of an incident compression wave on the ground surface can cause a tensile wave and eventually induce tensile stress near the ground surface, resulting in crack nucleation and propagation, and thus material fragmentation in the rear portion of the specimen. Figure 15 shows the stress wave propagation process when top of the cylinder receives an impact load. Cracks are formed during the impact loading process, as shown in figure 15(c). The OA crack is caused by the encounter of the reflected tensile waves in the middle of the cylinder; the BC crack is due to the reflected tensile wave from the cylindrical surface; the EF crack is induced by the reflection tensile wave from the bottom surface; the tensile waves are reflected from the bottom and cylindrical surfaces meet at the bottom corner to form crack LM. The crack system in figure 15 can be evidenced by the material fragmentation of a granite subjected to impact loading, as shown in figure 16. The granite has a crack system of radial cracks that extend outward from the impact center and circumferential cracks that are around the impact center. If the impact load is large enough, the cracks are interdigitated and finally form a massive crumb.
In a ductile material, stress waves can also cause material embrittlement. Liu et al [110] carried out a projectile impact test on a titanium alloy target. Figures 17(a) and (c) display a cross-sectional view and stress field of the target during the projectile impact process, respectively. No plastic deformation was observed in zones I and III, which depicts that brittle fracture was the dominant mode of deformation during the impact process. Meanwhile, the simulation results demonstrated large tensile stresses in zones I and III, which verifies the results obtained in the impact test.
Zheng et al [111] also conducted an armor-piercing projectile test on TC4 at an impact velocity of 820 m s −1 , and studied the failure mechanisms of a TC4 material, and found a large number of microcracks left in the TC4 material by the projectile piercing, as shown in figure 18. They reported that the microcracks propagated and coalesced into macroscopic cracks. The TC4 material was thus embrittled in the surrounding area of the piercing hole. It can be observed from figure 18 that the material surrounding the hole was pushed and squeezed by the projectile and was left with a large number of cracks along the fibrous/lamellar layers in the deformed material. The fibrous/lamellar layers might be due to a combination of the tensile stress caused by the head and the compressive stress of the body of the projectile during the penetration process.

Critical conditions for material embrittlement at high strainrates
Rice and Thomson [112] established a criterion for brittle fracture of crystalline materials in terms of the spontaneous emission of dislocations from an atomically sharp cleavage     where G is shear modulus; and b is the Burgers vector. However, the surface energy of a material is difficult to correctly measure, which limits the practical applications of equation (24).
In a study on ductile-to-brittle transition of a material in a high-speed machining process, Zhou et al [31] indicated that the material should have been embrittled when cutting speed v c exceeded plastic wave propagation speed v , p as presented in equation (25), Wang et al [113] also studied material embrittlement in a high-speed orthogonal cutting process and found that material embrittlement occurred at > v v . c p However, problems occur if the propagation speed of a plastic wave is taken as the critical cutting speed for the ductile-to-brittle transition of a material.
(1) Generally, strain rate is proportional to cutting speed. It can alter the properties of a material, and further change the plastic wave propagation speed of the material. Lawn et al [114] were the first to define brittleness using the ratio of hardness H to toughness K f for hard and brittle materials. Zhang et al [30] studied the relationship between brittleness and subsurface damage depth for hard and brittle materials and were the first to report that subsurface damage depth is inversely proportional to material brittleness.
For ductile materials, Campbell et al [115] investigated the influences of strain rate e  , temperature T, and grain size D on tensile fracture strain of the En2A steel, and found that tensile fracture strain decreased with an increase in strain rate, as shown in figure 19. Similarly, in their experimental study on the ductile-to-brittle transition of an Ni metal, Mahalingam et al [116] found that the toughness of the metal decreased with an increase in strain rate. Abushawashi et al [117] also studied the influence of strain rate on the fracture strain of a material during a cutting process. The fracture strain was calculated based on the J-C damage model, as shown in equation (26) where C 1 and C 2 are the material fracture constants to be determined experimentally; h f represents stress state. Toughness of a material is a function of strain, strain rate, and temperature, and can be obtained by a material constitutive model s e e =  ( ) f T , , and fracture strain e , f as shown in equation (27), . 27 According to section 2.3, an increase in strain rate leads to an increase in yield strength but a decrease in toughness of a material to evoke material embrittlement, as shown in  figure 20. Therefore, material brittleness B can also be expressed by a ratio of yield strength s Y to toughness K f , as presented in equation (28 .
0.2% 0 f It is worth noting that material strength does not always rapidly increase with strain rate, which is schematically shown in figure 20. The red broken line represents the trend of material strength change with strain rate. A material may encounter a strength limit if it is loaded at an extremely high strain-rate.

Crack nucleation and propagation
Brittle fracture of a material originates from the nucleation and propagation of cracks. The atomic bond fracture is an important feature in the process of crack nucleation and propagation and is the dominant factor for brittle fracture of a material. The theoretical fracture strength s m of a material can be calculated in equation (30) where E represents Young's modulus of the material; g s indicates the surface energy; and a 0 is the lattice constant.
In order to explain the reason why theoretical fracture strength is much larger than the actual fracture strength of a material, Griffith et al [118] hypothesized that internal defects reduced the fracture strength of a material, and proposed the Griffith fracture criterion for crack propagation, as shown in equation (31), where q is a coefficient related to the stress state. Moreover, metallic glasses do not typically have dislocations and grain boundaries since their atomic structures are long-range disordered. As such, their deformation and fracture behaviors are different from metallic alloys, but significantly depend on strain rate and temperature [122]. High strain-rate and/or low temperature can lead to inhomogeneous deformations of metallic glasses. In this case,  strain is concentrated in the spatial layers (shear bands) as thin as 10-100 nm [123][124][125]. The formation and propagation of shear bands are the main cause of deformation and crack formation in a metallic glass [88,126].

Discussion
Material embrittlement can occur at a high strain-rate. Generally, a high strain-rate may result in dislocation pile-ups, which may block a dislocation source from further emitting dislocations. In a material deformation process, the material may experience atomic bond rupture to nucleate and propagate a crack. Microscopically, a dislocation overcomes the barriers in a loading process at a low strain-rate. Plastic deformation occurs in the material if given a sufficient time. Conversely, if the dislocation does not have enough time to overcome the barriers in the loading process due to a high strain-rate, dislocations pile-up, which triggers crack nucleation. Conclusively, in a high-speed loading process of a ductile material, if strain rate is so high that no plastic deformation should occur before the material is fractured, the material would be subjected to the complete embrittlement. In this case, the ductile material should exhibit a fracture behavior as if it were a typical brittle material. For a brittle material subjected to loading at a high strain-rate, the embrittlement mechanism is also related to dislocations. An increase in strain rate can lead to material embrittlement and fragmentation. In their grinding research of ceramic materials, Zhang et al [127] confirmed that grinding damage to the ceramic materials decreased with an increase in material brittleness.
The nucleation and propagation of a crack are closely related to dislocations. During a crack propagation process, the crack tip emits dislocations which may be impeded by the microstructural barriers, such as grain boundaries, resulting in dislocation pile-ups and microcrack nucleations. On the other hand, as a crack tip is always accompanied with dislocations, dislocation density becomes a governing factor for crack propagation. Generally, the higher the dislocation density, the easier the crack propagation. The reason is that new dislocations may be obstructed by the existing dislocations at the crack tip, forming dislocation pile-up, stress concentration, and finally crack propagation. This has also been confirmed in a study conducted by Jones et al [128] in which the authors claimed that in a TC4 sheet, strain hardening could slightly accelerate fatigue crack growth for the case when strain hardening was induced prior to crack initiation.
Stress wave propagation is another mechanism that directly contributes to material fragmentation at a high strainrate. A stress wave originates and propagates from an impact loading source, and may get reflected on a free surface to become a reflection wave. As shown in figure 13, the interactions between the incident and reflected waves can induce tensile stresses, and further crack nucleation and propagation. For example, if a machining process is accompanied with impact loading, stress wave can be initiated in the machining zone and reflected from the free surface of the workpiece. For a significantly large impact load, the tensile stress caused by the incident and reflected waves can result in crack nucleation and material fragmentation.
Macroscopically, an increase in strain rate during a loading process leads to an increase in strength and hardness but a decrease in toughness of a material. The material is thus embrittled. As to the TC4 material, it is a typical two-phase alloy (a and b phases) in which the a phase has the closepacked hexagonal structure with few slip systems. At an increased strain-rate, TC4 is easy to have dislocation pile-ups, crack nucleation and propagation, and material embrittlement. Conversely, aluminum is a face-centered cubic metal with more slip systems. This material is typically easy to deform during a loading process. Therefore, it is relatively difficult to cause dislocation pile-ups and material embrittlement in aluminum.
If a material is going through an embrittlement process due to fast loading at a high strain-rate, e.g. high-speed machining, the mobile dislocations corresponding to that strain rate do not have enough time to respond to the fast loading but turn into temporarily immobile dislocations under the specified loading conditions. The material influenced by loading exhibits pseudo embrittlement and is not actually embrittled upon unloading. The pseudo embrittlement may help suppress plastic deformation and machining damage in a machined workpiece, which can greatly preserve surface integrity for machining.
Strain-rate sensitivity k s proposed in this study provides a guide for the determination of material embrittlement in terms of strain rate. For example, TC4 has its strain-rate sensitivity of 35.4 which is the 3rd highest among the metallic materials listed in figure 8. Since TC4 is highly strain-rate sensitive, it should be a good candidate material for high-speed machining. High quality and high efficiency machining is thus anticipated.
A material with low strain-rate sensitivity k s may find applications in energy absorption. An example is an armor made of both aluminum alloy and ceramics that can absorb the energy of a projectile [129][130][131]. Because the aluminum alloys possess the lowest strain-rate sensitivity among all the materials shown in figure 8, they are not readily embrittled when subjected to loading at a high strain-rate. The armor has a combination of energy absorption through the outstanding plastic deformability of the aluminum alloy and the antipenetrability of ceramics for an enhanced protectionability from the strike of a high-speed projectile. Temperature rise is another important factor affecting material embrittlement, especially in a machining process. Plastic deformation directly contributes to temperature rise, which is conducive to dislocation movement but inconducive to material embrittlement. In order to study material embrittlement due to the strain-rate effect, it is necessary to minimize the temperature effect. Salomon [132] was the first to propose that in a machining process, machining temperature was expected to have an increase and then a sharp decrease as cutting speed was increased. Unfortunately, the Salomon prediction never becomes a reality although some researchers tried to use different techniques to verify his prediction. The high-speed machining was one of the techniques used by the researchers.
Therefore, more studies are necessary to the investigation of the strain-rate effect on material embrittlement in high-speed machining.
The strain-rate effect is considered as an important influencing factor in the physical and empirical constitutive modeling of a material. Compared with the J-C model, a physical model is more accurate to predict property change of a material subjected to loading at a high strain-rate. However, the physically-based models are generally sophisticated, which limits their practical applications. In this regard, it is necessary to modify the existing constitutive models or to establish a new constitutive model.

Conclusions
This study discusses material embrittlement, and the critical conditions for ductile to brittle transition of ductile materials based on the strain-rate effect, and arrives at the following conclusions: (1) Material embrittlement and fragmentation occur when loading at a high strain-rate.
(2) Strain-rate sensitivity k s is an inherent property of a material that measures how sensitive the material is to strain-rate evoked embrittlement. (3) Material embrittlement and fragmentation do not readily occur on a material that has low strain-rate sensitivity, and vice versa. (4) Strain-rate sensitivity is an important indicator in selecting a material for a specific application, e.g. armor protection and high-speed machining. (5) Material embrittlement and fragmentation can be caused by stress wave propagation and reflection in a loading process.
Strain-rate evoked material embrittlement can have many industrial applications. Machining, TBM, and armor protection are a few examples that are discussed in this study. More applications are anticipated if the underlying mechanisms are made clear. In machining of difficult-to-machine materials, for example, strain-rate evoked embrittlement may be applicable to many difficult-to-machine materials, such as titanium alloys, aerospace alloys, high temperature alloys, semiconductors, and ceramics. Machining quality and efficiency are expected to have a great improvement.
Although strain-rate evoked material embrittlement has been explored, the fundamental issues are yet to be investigated. Dislocation kinetics, crack nucleation and propagation at high strain-rates are still unclear. Further studies are necessary.