Towards understanding the brittle–ductile transition in the extreme manufacturing

The BRM is an obvious characteristic of materials, such as stone, advanced engineering ceramic and semiconductor, which are used in manufacturing processes at the conventional deformation scale. However, the DRM has been found in some manufacturing processes, such as single point diamond turning, ultra-precision grinding and micro-milling [40–42], in which the material removal scales are extremely small.

The critical deformation size is defined as the boundary between BRM and DRM in the manufacturing processes of these materials [43]. For example, in single point diamond turning, a diamond tool with a larger radius and extremely sharp cutting edge combined with extremely small cutting depth and feed is the key to acquiring a thin chip [44]. As for ultra-precision grinding, the ultra-fine grain and nanoscale maximum undeformed cutting thickness related to the grain density and grinding parameters guarantee that more material is removal by the ductile mode [45].

The surface finish is one of the important factors to judge the material removal mode. The machined surfaces with DRM are usually damage-free with material accumulation due to plastic flow; and those with BRM show obvious brittle fracture. The results of the high-speed grinding of SiC depicted that the material removal gradually transitions from brittle to ductile modes with maximum undeformed chip thickness decreases, as shown in figure 1 [46].

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The chip morphology is another important factor to judge the material removal mode [47]. The layer-type chips and particle-type chips usually reveal the DRM and BRM, respectively. The chip morphology of a l monocrystalline Si by single point diamond turning is shown in figure 2. It shows the chip morphology transited from particle-type to layer-type as the maximum undeformed chip thickness declined from 690 nm to 20 nm, which indicates the transition of material removal mode [48].

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Thermally assisted machining is another approach to acquire the DRM of those materials which usually present brittle removal characteristics in conventional machining processes [49]. Compared with small-scale machining processes, the primary aim of thermally assisted machining is to achieve DRM at the conventional machining scale [50]. The key to thermally assisted machining is the provision of an external heat source, including plasma [51], laser beam [52], gas torch [53] and induction heating [54]. The focused research is laser assisted machining because it provides high, local temperature increases on the workpiece in front of deformation zone; only the volume of material to be removed is effectively heated [55]. In this way, the effect of an external heat source on the workpiece can be significantly reduced [56].

Numerous studies have been carried out regarding the use of laser-assisted machining to increase the machinability of materials presenting brittle characteristics at the conventional machining scale [56]. The transition from BRM to DRM has been found due to the involvement of laser beam [57]. The comparative study of conventional and laser-assisted grinding of zirconia (ZrO₂) ceramic with a cBN grinding wheel has been implemented. BDT occurred in the subsurface morphology of the machined surface, showed in figure 3, due to the elevated temperature. Obvious cracks can be found on the subsurface due to the BRM in the conventional grinding of ZrO₂ ceramic. However, the cracks on the subsurface were suppressed in the laser-assisted grinding because of the BDT. The piled-up material, found in figure 3(b), also presented non-negligible plasticity of ZrO₂ ceramic during the laser-assisted grinding [58].

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With the development of high-speed cutting machine tools and advanced cutting tool technologies, high-speed machining technologies play an increasingly important role. The advantages of high-speed machining include low cutting forces and temperature, low wear of cutting tools, and high productivity [59, 60]. It is worth pointing out that heat generation in high-speed machining is confined to a thinner area, which produces a high local temperature and temperature gradient. There is less time for heat to enter the workpiece, which results in a lower finished workpiece temperature [61]. However, the chip, cutting tool and tool/chip interface temperature may be more complex [62, 63]. The cutting speed in high-speed machining is significantly improved compared with that of conventional machining. Extremely high cutting speed has been realized by increasing either the diameter of the cutting tools or workpiece and increasing the rotation speed of spindle [64]. In addition, modified light gas guns have been used to study the high-speed machining processes [38, 65]. The chip formation mechanism is one of the study focuses of high-speed machining [66, 67], and Liu et al have conducted a systematic study in this area [68].

The chip morphology is also the main basis for identifying the material removal mode [69]. Cutting experiments of 7050-T7451 aluminum (Al) alloy with different cutting speeds have been carried out. The chip morphology transforms as cutting speed increases, as shown in figure 4. When the cutting speed was 2500 m min⁻¹, the chip root depicted a large area of dimples spreading over the separation zone, and ductile fracture occurred at the separation point. When the cutting speed was 7000 m min⁻¹, the chip was removed as fragments. Cleavage steps and crystalline fractures were observed on the fracture surface, which are typical characteristics of brittle fracture [70].

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The evolution of the finished surface with the increase of cutting speed also proves that BDT occurs during high-speed machining. Figure 5 shows the finished surface of orthogonal cutting experiments of Inconel 718. When the cutting speed was 800 m min⁻¹, the smooth machined surface was obtained through DRM. However, when the cutting speed was 7000 m min⁻¹, BRM induced brittle tearing and degraded the surface quality [71].

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As for the deformation scale dependence of brittle-to-ductile transition for these materials presenting the failure of brittle fracture in conventional deformation scale, small-scale testing techniques make the study possible. These testing techniques include indentation and scratching at the micro- and nano-scales and micro-compression and micro-bending [72–74]. Advanced technologies such as (in situ) scanning electron microscope (SEM) and transmission electron microscope (TEM) make the characterization of plastic deformation at micro- or nano-scale more intuitive and credible [75, 76]. It is worth pointing out that nano-scratching can introduce a scratch with varied deformation scale, which may directly present the critical deformation scale of BDT. It could also be determined by micro-compression and micro-bending using a series of specimens with different sizes.

Micro-scratching has been carried out to study the evolution of the deformation mode of sapphire using Vickers indenter, as shown in figure 6 [77]. Deformation can be divided into three stages depending on the scratching depth. In the brittle fracture dominated stage, irregular debris and spalling occurred because of the intersection of lateral and radial cracks, as shown in figure 6(a). In the coexistence stage of plastic deformation and brittle fracture, initial radial cracks and torn and segmented chips appeared, as shown in figure 6(b). In the pure plastic deformation stage, the scratching grooves and long shavings were smooth, and localized plastic flow resulted in curled shavings ejected from the scratches. The results showed that the ductility of sapphire increases as the deformation scale decreases; and pure plastic deformation occurs at sufficiently small deformation scales.

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BDT due to the decline of the deformation scale has been observed in other materials, such as GaAs, SiC and Si [78, 79]. The mechanisms of deformation scale induced brittle-to-ductile transition have been comprehensively studied. The decline or absence of the natural defect population because of the decreased deformation scale increases the strength of these materials, presenting brittle characteristics in conventional deformation scale, which enables the plastic deformation mechanisms can be activated [80–82]. In general, the material strength and plasticity follows the Weibull distribution, i.e. a larger deformation scale deforms the material in a more brittle fashion due to the higher probability of having a critically oriented flaw in a larger volume [83]. It has been pointed out that cracks could be suppressed, and the plastic deformation could be observed as long as the deformation scale is sufficiently small [84].

Liu et al proposed that brittle materials could undergo considerable plastic deformation when the deformation scale is small enough, which causes the stress intensity factor K_I to be less than the fracture toughness K_C due to the three following effects [18, 85]. Firstly, the small deformation scale will lead to the obvious negative rake angle effect in the single point diamond turning, abrasive machining [86, 87], which places extremely high compressive stress on workpiece materials in deformation zone, significantly reducing K_I. Secondly, at the mesoscale of deformation (0.1–10 μm), yield strength enhancement due to dislocation hardening related to the dislocation elastic interaction increases K_C. Thirdly, at the mesoscale of deformation, strain gradient accommodated by geometrically necessary dislocations strengthens the yield stress, which also increases K_C.

The common plastic deformation mechanisms include the slip, twinning and phase transformation [88]. The most representative example of slip-related plastic deformation is the micro-compression of diamond. It is widely accepted that diamond does not deform plastically at room temperature and often fails in catastrophic brittle fracture. However, a recent study reported the unprecedented dislocation plasticity in the micro-compression of diamond with sub-micrometer pillars [89]. As shown in figure 7, the systematic study showed that the sliding of dislocations primarily determines the plasticity. The defect-free diamond pillar suppresses the cracks and raises the maximum stress level [90], which activates the dislocation.

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The decline of deformation scale may activate other plastic deformation mechanisms. Compared with macroscopic specimens, the micro-compression of polycrystalline ZrO₂ depicted the absence of defects significantly increases compressive strength up to 8 GPa with a failure strain of 7%, which is four times larger than that of macroscopic tests. The higher stresses activate plastic deformation. The TEM and STEM observations suggest that the phase transformation was the main plastic deformation mechanism [91]. In addition, the decline of the deformation scale induces twinning and a combination of the three kinds of plastic deformation mechanisms [92].

The critical deformation size has been proposed, which is expressed as following [79, 83, 93]:

$$\begin{equation}{d_{{\text{crit}}}} = \lambda {\left( {\frac{{{K_{\text{C}}}}}{{{\sigma _{\text{y}}}}}} \right)^2}\end{equation} \tag{ 1 }$$

where the d_crit is the critical deformation scale of brittle-to-ductile transition; K_C and σ_y is the fracture toughness and yield stress respectively; λ is the flaw geometry constant. It has been pointed out that when the deformation scale is less than d_crit, the smaller deformation zone cannot provide sufficient elastic energy for the rapid crack propagation, and material failure occurs by plastic deformation [83].

Bifano et al [94] proposed an empirical equation for the depth of BDT. The critical penetration depth h_c is given by:

$$\begin{equation}{h_{\text{c}}} = \psi \left( {\frac{E}{H}} \right){\left( {\frac{{{K_{\text{C}}}}}{{{\sigma _{\text{y}}}}}} \right)^2}\end{equation} \tag{ 2 }$$

where E is the Young's modulus of materials, H is the hardness, K_C is the fracture toughness, σ_y is the fracture toughness and yield stress respectively and ψ is BDT factor of materials.

The brittle-to-ductile transition induced by the decline of deformation size has been comprehensively studied in experiments, and the representative experimental results are listed in table 1.

In addition to the reduced deformation scale, elevated temperature could also enable plasticity in these materials that present brittle fracture at room temperature. Elevated temperature induced brittle-to-ductile transition has been extensively studied. The micro-compression of monocrystalline Si with a pillar diameter of 2 μm has been carried out at different temperatures and a constant loading rate of 0.5 mNs⁻¹; the results are shown in figure 8 [103]. The micro pillar directly failed in brittle fracture at room temperature. However, a BDT was found in which the micro pillar deformed plastically without catastrophic failure, although some cracking was still observed in figure 8(b). Figure 8(c) depicts a micro pillar that had experienced substantial plastic deformation, which verifies that the cracks were suppressed in the monocrystalline Si micro pillar. These results are consistent with an earlier study, which reported that the BDT temperature was about 500 °C [104]. Similar results have been found in the micro-compression of single crystal MgAl₂O₄ [105] and small punch tests of polycrystalline Al₂O₃ [106]. These results demonstrate that elevated temperature can induce the BDT.

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One of the research focuses of elevated temperature induced plasticity is to expand the ductile regime. The scratching test of RB-SiC ceramics was implemented at varied temperatures. The wedge scratching mode was used with a Vickers indenter to study the ductile-to-brittle transition under different scratching temperatures. Figures 9(c)–(f) shows that the scratch length undergoing plastic deformation increased with the rise of temperature, which indicates a positive correlation between the critical deformation scale of BDT and deformation temperature [107].

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The mechanism of elevated temperature induced brittle-to-ductile transition is related to the thermal activation. The dislocation motion is more difficult because of the high energy of covalent and ionic bonds in some materials at room temperature [108]. Therefore, ionic-bonded and covalent-bonded materials, such as semiconductors and ceramics, always present catastrophic brittle fracture. Plastic deformation involves dislocations that move from one equilibrium position to the next, during which the dislocation must overcome the Peierls–Nabarro lattice resistance [109]. As shown in figure 10, each peak represents the resistance applied on the dislocation during the dislocation motion process. The area under the curve represents the energy consumed during the dislocation motion. The atoms in solids always vibrate randomly near their equilibrium positions. Atoms vibrate more rapidly at higher temperatures, which increases the thermal energy ΔG and decreases the resistance energy consumed during the dislocation motion process. Therefore, dislocations move easily at high temperatures [110, 111]. Increased temperatures facilitate dislocation motion better than brittle fracture, which contributes to the brittle-to-ductile transition [112].

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The competition between dislocation motion and brittle fracture is quantitatively described in the following equations [112, 114]:

$$\begin{equation}{\tau _{\text{y}}} = A{\dot \varepsilon ^{1/n}}{\text{exp}}\left[ { - \frac{{\Delta {H_\tau }}}{{kT}}} \right]\end{equation} \tag{ 3 }$$

$$\begin{equation}{\tau _{\text{f}}} = S{\sigma _{\text{f}}}\end{equation} \tag{ 4 }$$

Where τ_y is the yield stress, A and n are material constants, ΔH_τ is the energy parameter related to the activation of dislocation motion, k is the Boltzmann constant, T is the deformation temperature, σ_f is the fracture stress, τ_f is the corresponding shear stress of σ_f, S is Schmid factor. τ_y and τ_f are the critical shear stresses for dislocation motion and fracture, respectively. There is a negative correlation between τ_y and T. τ_y for a crystal is weakly temperature dependent and can be regarded as a constant [112]. In general, τ_y is larger than τ_f for these materials presenting brittle fracture at room temperature. However, BDT can be acquired by rising the deformation temperature. The brittle-to-ductile transition temperature T_BDT could be obtained by combining equations (3) and (4), which is expressed as [112, 115, 116]:

$$\begin{equation}{T_{{\text{BDT}}}} = \frac{{\Delta {H_\tau }}}{{k{\text{ln}}\left[ {\frac{{S{\sigma _{\text{f}}}}}{{A{{\dot \varepsilon }^{1/n}}}}} \right]}}.\end{equation} \tag{ 5 }$$

The BDT induced by the elevated deformation temperature has been comprehensively studied, and the representative experimental results are listed in table 2.

Table 2. The results of BDT induced by the elevated deformation temperature.

Materials	Test method	Brittle-to-ductile transition temperature	Note	Reference
Single-crystal silicon	Pillar indentation splitting test	About 250 °C	The diamond cube corner indenter was used and a facet edge of the cube corner was kept aligned with the (0 1 0) plane	[117]
GaAs	Micro-pillar compression	300 °C	The diameter of micro-pillar is 2 µm.	[103]
GaAs	Indentation experiment	200 °C	Indenter type: Vickers	[112]
			Crystal plane: (0 0 1)
GaAs	Four-point bend test	300 °C–380 °C	Sample size is 35 × 3 × 1 mm, and the 35 × 3 mm2 top and bottom faces of the sample were cut parallel to the (0 0 1) plane.	[112]
ZrO2and Al2O3	Small punch test	1100 °C (For ZrO2) 1250 °C (For Al2O3)	Disk-shaped specimen with the diameter of about 10 mm and thickness of 0.5 mm.	[106]
4H–SiC	Compression experiments	1100 °C($\dot \varepsilon$ = 3.6 × 10−5 s−1) 1030 °C($\dot \varepsilon$ = 2.6 × 10−5 s−1)	Parallelepiped-shaped samples with the basal (0 0 0 1) plane at 45° to the compression axis.	[118]
MgAl2O4	Micropillar compression	About 200 °C	The diameter of micropillar is 2.5 µm.	[105]

White layer is a common phenomenon in manufacturing processes for these materials presenting obvious plasticity in conventional manufacturing, especially for high-speed machining, in which the grain refinement has been widely observed [119–121]. The grain distribution below the machined surface is shown in figure 11. It depicted that the white layer can be divided into two layers including the recrystallization (RX) layer and transition layer. There were 200 nm sized nanocrystal grains in the RX layer with large angle grain boundaries. A transition layer with low angle grain boundaries and sub grains formed below the RX layer. There were larger grains with small angle grain boundaries in the deformed buck layer [122]. It has been reported that brittleness and hardness in the white layer were significantly higher compared with the buck material due to the severe grain refinement [121, 123], which contributes to the ductile-to-brittle transition in the machining processes.

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The grain size has a significant effect on the mechanical properties of materials, which can be expressed by the Hall–Petch relationship [124, 125]. The following formula quantitatively establishes the relationship between the yield strength and grain size, where the yield stress increases as the average grain size increases [126]:

$$\begin{equation}{\sigma _{\textrm{y}}} = {\sigma _0} + {\raise0.7ex\hbox{${{k_{\textrm{y}}}}$} \!{\left/\right.} \!\lower0.7ex\hbox{${\sqrt d }$}}.\end{equation} \tag{ 6 }$$

where σ_y is the yield stress, σ₀ is a material's constant for the starting stress for dislocation movement, k_y is the strengthening coefficient (a constant specific to each material), and d is the average grain diameter.

The relationship between the yield stress or hardness and the average grain size of copper is shown in figure 12, which was summarized by Tschopp et al [127]. It has been reported that yield strength could be improved by more than five times when the materials transit from micro-sized to nano-sized grains [128] (figure 12).

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However, the elongation of materials decreases with the decrease of grain size. The relationship between elongation and average grain size summarized by Liu et al is shown in figure 13 [129]. They concluded that brittleness increased with the decrease of grain size. Therefore, the ductile-to-brittle transition may occur in materials with extremely small grain size.

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The brittle fracture of nano-grained Cu and Ni with ultrahigh strength has been found in the tensile test, which was identified by fracture morphology [130], as shown in figure 14. The strengthening mechanism for fine-grained materials is that the dislocation slip is substantially suppressed by grain boundaries of the extremely small grains [131]. However, grain boundary sliding or diffusional creep is not active enough to accommodate plastic straining at ambient temperature, which leads to brittle fracture [132].

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It has been summarized that the plastic deformation mechanisms differ according to grain size and transit smoothly with the decline of average grain size [133]. The dominated plastic deformation mechanism is dislocation motion when the grain size is larger than 10 nm, in which the Hall–Petch relationship is widely confirmed [134, 135]. However, strength and brittleness decrease when the grain size is less than about 10 nm, which is called the inverse Hall–Petch effect [136]. This effect is the result of other plastic deformation mechanisms, such as grain boundary sliding and grain rotation [137]. It is worth noting that the size of refined grain during the high speed machining usually larger than 10 nm [138–143]. Therefore, the decline of strength and brittleness with the decrease of grain size is rarely involved in extreme manufacturing.

Machined materials are often subjected to dynamic loadings. The mechanical behaviors of materials subject to high strain loading differ from those subject to static or quasi-static loading [144, 145]. The different mechanical test methods are used to acquire similar strain rates consistent with specific manufacturing processes, as shown in figure 15 [71, 146, 147].

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It has been reported that both the tensile strength and yield strength of these materials presenting obvious increase of plasticity as the strain rate increases, and plastic failure strain declined with the growth of strain rate [148], as shown in figure 16. The connecting curve EDCBA represents the critical condition where the material fails, which was named the 'fracture locus.' figure 16 shows that the brittleness increased as the strain rate increased. The ductile-to-brittle transition occurs as long as the strain rate is high enough.

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The stress wave theory is often used to analyze the mechanical behaviors of materials that have high strain rates. According to the theory, the brittleness induced by high strain rate should be defined as pseudo brittleness. The materials subject to high strain rate deform locally, near the impact positions, and the rest of the part presents no obvious plastic deformation, which is a mechanism of pseudo brittleness [149]. The materials fail in a ductile manner when pseudo brittle fracture occurs. The velocity of the stress wave can be expressed by the following equations [150, 151]:

$$\begin{equation}{c_0} = \sqrt {\frac{1}{{{\rho _0}}}\frac{{\partial \sigma }}{{\partial \varepsilon }}} { }(\sigma \leqslant {\sigma _{\text{y}}},\,\frac{{\partial \sigma }}{{\partial \varepsilon }} = E)\end{equation} \tag{ 7 }$$

$$\begin{equation}{c_{\text{P}}} = \sqrt {\frac{1}{{{\rho _0}}}\frac{{\partial \sigma }}{{\partial \varepsilon }}} { }(\sigma > {\sigma _{\text{y}}},\,\frac{{\partial \sigma }}{{\partial \varepsilon }} < E)\end{equation} \tag{ 8 }$$

where c₀ and c_p are the elastic and plastic stress velocities, respectively. ρ₀ is the density of materials. σ and are the stress and strain, respectively. σ_y is the yield stress, and E is Young's modulus. The above equations suggest that the plastic stress velocity is always less than the elastic stress velocity, which cooperates with unloading waves to make materials deform locally during the impact process.

The unloading waves mainly include the reflected elastic wave generated at the end of specimens and the elastic wave induced by the unloading impact, as shown in figures 17(a) and (b), respectively [113]. As for the effect of the reflected elastic wave generated by the end of specimens on the deformation distribution, it is depicted by a bar with length of l impacting with a rigid object. When the time t is from zero to l/c₀, both the elastic and plastic compression waves propagate from the left end to the right end of the rod. The elastic compression wave propagates faster than plastic compression wave. When the elastic compression wave reaches the right end at time l/c₀, the reflection generates the elastic tensile wave. Then, the elastic tensile wave, functioning as an unloading wave, propagates from the left end to the right end and unloads the elastic and plastic compression waves in turn, which propagate from the left to right. When the elastic tensile wave reaches the left end at time 2l/c₀, the elastic and plastic compression waves are completed unloaded. Finally, the local residual plastic deformation zone with length x₁ forms, which depicts the local deformation of the impacted material. The effect of the elastic wave, induced by the unloading of impact on the deformation distribution is shown in figure 17(b). It is depicted by a bar with semi-infinite length impacting with a rigid object. The elastic tensile wave, functioning as the unloading wave, is generated at time t_u, and the unloading process of earlier elastic and plastic compression waves is similar to those shown in figure 17(a).

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The difference between the elastic wave velocity and the plastic wave velocity and the unloading waves cause the materials subject to impact to deform locally. The most obvious example of this is the Taylor impact test, in which the projectiles are accelerated to impact a rigid wall [152, 153]. Five distinct deformation and fracture modes in the projectile during Taylor impact tests for steels have been reported, as shown in figure 18 [154–157]. The projectiles always deform or fracture locally close to the impact end.

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When a bar with velocity V is impacted, the strain at the impact end is _V., and their relationship can be expressed as the following formula [71, 149]:

$$\begin{equation}V = \mathop {{\text{ }}\int\limits_0^{{\varepsilon _{\text{V}}}} {\sqrt {\frac{1}{{{\rho _0}}}\frac{{\partial \sigma }}{{\partial \varepsilon }}} } }\limits_{}^{} .\end{equation} \tag{ 9 }$$

The plastic wave velocity V_cr is expressed as following:

$$\begin{equation}{V_{{\text{cr}}}} = \mathop {\int\limits_0^{{\varepsilon _{\text{m}}}} {\sqrt {\frac{1}{{{\rho _0}}}\frac{{\partial \sigma }}{{\partial \varepsilon }}} } }\limits_{}^{} \end{equation} \tag{ 10 }$$

where _m is the strain when ∂σ/∂ is zero, which means that the plastic wave velocity is zero. If the velocity of impact V exceeds plastic wave velocity V_cr, the strain at the impact end will be larger than _m, which is the stain corresponding to tensile strength. The materials will fail in the form of pseudo brittle fracture.

Dislocation kinematics provides another perspective to explain the brittle-to-ductile transition when materials subject to impact. Plastic deformation is the combined result of dislocation motion, mechanical twinning and phase transformation [158]. Figure 19 expresses the plastic deformation at different strain rates when only dislocation movement is considered. It is worth pointing out that the variation of stress was due to the change of strain rate. In general, there is a positive correlation between stress and strain rate for a specific strain [159]. The dominant factors affecting the mean dislocation velocity is thermal activation, pure drag and relativistic effect in turn as the stress increases, and the mean dislocation velocity is always less than the shear wave velocity due to the relativistic effect [160]. The velocity of the dislocation relative to the loading tends toward zero with increasing loading velocity. Therefore, the dislocation seems to be fixed when the material has a high enough strain rate, which leads to the ductile-to-brittle transition.

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Severe grain refinement also contributes to the ductile-to-brittle transition of materials presenting obvious plasticity. The grain refinement is much more severe during impact deformation than quasi-static deformation. The results of metal particle impact tests suggest that the impact process can lead to the formation of nano-sized grains, in which the dynamic RX caused by adiabatic shear instability plays a major role [161, 162]. The grain refinement is shown in figure 20.

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