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Excellent plasticity of a new Ti-based metallic glass matrix composite upon dynamic loading
Materials Science & Engineering A 677 (2016) 376–383
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Excellent plasticity of a new Ti-based metallic glass matrix composite upon dynamic loading R.F. Wu a,b, Z.M. Jiao c, Y.S. Wang a, Z. Wang a, Z.H. Wang c, S.G. Ma c, J.W. Qiao a,b,n a Laboratory of Applied Physics and Mechanics of Advanced Materials, College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China b State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China c Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2016 Received in revised form 13 September 2016 Accepted 15 September 2016 Available online 16 September 2016
Quasi-static and dynamic compressive properties of in-situ Ti60Zr14V12Cu4Be10 bulk metallic glass matrix composites containing ductile dendrites were investigated. Upon quasi-static compressive loading, the composite exhibits a high fracture strength of 2,600 MPa, combined with a considerable plasticity of 40% at room temperature. However, upon dynamic loading, an excellent plasticity of 16% can be obtained due to the abundant dislocations and severe lattice distortions within dendrites and multiplication of shear bands within the glass matrix analyzed by transmission-electron microscopy. A constitutive relationship is obtained by Johnson-Cook plasticity model, which is employed to model the dynamic flow stress behavior. In addition, under dynamic compression, the adiabatic temperature rise increases with increasing strain rates, resulting in that the softening effect within the glass matrix is obviously enhanced during deformation. & 2016 Elsevier B.V. All rights reserved.
Keywords: Bulk metallic glass Dynamic loading J-C model Adiabatic temperature rise
1. . Introduction Since firstly fabricated by rapid cooling from the melts by Duwez et al. [1], the bulk metallic glasses (BMGs) were developed quickly and attracted significant technological and scientific studies all over the world on account of a series of superior mechanical properties at ambient temperature, such as ultrahigh strength, high hardness, large elastic limits, and excellent wear and corrosion resistance [2–6]. BMGs are esteemed as potential engineering structural materials [7]. However, owing to highly localized shear deformation upon loading, BMGs exhibit limited plasticity at room temperature and have subsequent catastrophic failure without apparent macroscopic plasticity by shear softening in one band [5,8]. In order to circumvent the poor plasticity, in-situ formed ductile secondary phases can be introduced into the glass matrix to form the metallic glasses matrix composites (MGMCs) [9,10]. To date, compared with monolithic BMGs, the existence of dendrites within the glass matrix contributes to obvious macroscopic compressive plasticity and tensile ductility, together with the improved toughness upon quasi-static loading for in-situ n Corresponding author at: College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China. E-mail address: [email protected] (J.W. Qiao).
http://dx.doi.org/10.1016/j.msea.2016.09.057 0921-5093/& 2016 Elsevier B.V. All rights reserved.
MGMCs. The dendrites hinder the prompt propagation of shear bands and lead to multiplication of shear bands [11]. Nevertheless, actual structural engineering materials are always performed superb ductility even upon dynamic compression. Therefore, many researches focus on the dynamic loading of MGMCs to meet the engineering applications in recent years, which can be efficaciously applied to strategic fields, such as defense, aerospace, and precision optical machinery [12]. Wang et al. [13] have demonstrated that the dynamic compressive yielding strength and the maximum strain value in Zr-based MGMCs are 2,700 MPa and 4%, respectively. Jeon et al. [14] have investigated the dynamic compressive behavior of Ti-based MGMCs modified from Ti-6Al-4V alloy and the dendrites inside have great impact on the plasticity of alloys. According to previous study by Qiao et al. [15], the multiple shear bands have been discovered for the in-situ Zr-based MGMCs under quasi-static compression, leading to large plasticity, whereas expeditious failure under the dynamic case is due to insufficient time to form profuse shear bands. Previously, researchers have found that brittle fracture or little plasticity occurred for in-situ MGMCs upon dynamic loading [16–18]. However, during quasi-static and dynamic loading, only limited information of the influence of temperature rise on the mechanisms of the deformation from the microstructure is available. In the current study, a new Ti-based MGMC was developed and investigated at quasi-static and dynamic loadings so as to further understand the deformation mechanisms.
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2. . Experimental 2.1. . Sample preparation and characterization The ingot with a nominal atomic percent composition of Ti60Zr14V12Cu4Be10 (Ti60) was prepared by arc melting the mixture of Ti, Zr, V, Cu, and Be with purity higher than 99.99 wt% in a Tigetted high purity argon atmosphere. In order to ensure the chemical homogeneity, the pre-alloyed ingots were re-melted at least four times. The rod like samples with 3 mm in diameter and about 85 mm in length were obtained by suctioning the melt into the copper mould in an argon atmosphere. The structure of phases was checked by X-ray diffraction (XRD) with a monochromatic CuKα radiation. The microstructures of the as-cast samples, and the lateral and fracture surfaces of the deformed samples after quasistatic and dynamic compression tests were investigated using scanning-electron microscopy (SEM). The cross sections of the ascast samples were polished, and etched by a solution of 40 mL HF, 20 mL HNO3, 40 mL HCL, and 200 mL H2O for SEM observation. The structural characteristics of the samples before and after compression were investigated using transmission-electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM) (JEOL-2010). The TEM specimens were obtained by ion thinning with liquid nitrogen cooling. 2.2. . Mechanical tests The uniaxial compressive tests under quasi-static loading were preformed on a 3-mm cylinders at ambient temperature at a strain rate of 5 10 4 s 1. The dynamic compressions were carried out by a split Hopkinson pressure bar (SHPB) [12] on samples with an aspect ratio of 1:1 at room temperature. It is made of a compressed gas-gun, a striker bar, a incident bar, specimen, a transmission bar, strain gauges, and an oscilloscope. The stress-strain curves of the tested specimen were obtained by inputting the recorded strain pulses into computing software. According to the one-dimensional stress wave theory [19], the stress and strain as well as strain rate could be obtained finally. Furthermore, in order to ensure the stability of results, the quasi-static compression tests were repeated at least three times and five times for dynamic compression tests. The detailed process was described elsewhere [20,21].
3. . Results 3.1. . Microstructures of MGMCs The microstructure and the dual-phase structure of the Ti60 are tested by the SEM and the XRD pattern, respectively. Fig. 1 (a) shows the back scattered SEM image of the cross-section of the as-cast composites. It can be seen that the dendrites are homogeneously embedded in the continuous glass matrix. By estimating the contrast in SEM images, the volume fraction of the dendrites is approximately 67%. This means that the secondary phase takes most part of the composites. It has been demonstrated that the size of the dendritic arms in the composites are 3–5 mm. The XRD exhibited in Fig. 1(b) reveals that the body-centered-cubic (bcc) crystalline peaks are overlapped on the broad diffuse scattering amorphous maxima, which means that the matrix has amorphous structure, and the dendrites are β-Ti solid solution. It is certified by the microstructure characterization mentioned above. To further examine the dual-phase structure, TEM and HRTEM analyses are employed on the as-cast composites. It is obviously observed that the bright-field image of the dendrites and the matrix with a low magnification is shown in Fig. 2(a), which is
Fig. 1. (a) SEM image of the microstructure for Ti60 composite. (b) XRD pattern of the composite.
similar to the SEM in Fig. 1(a). The edge of the dendrites is very smooth, indicating the absence of deformation [22]. Fig. 2 (b) displays the HRTEM image of the matrix, and no crystallization can be detected. Only diffuse halos, typical characterization of an amorphous structure, can be found from the selected area electron diffraction (SAED) pattern in the inset of Fig. 2(b). On the contrary, the crystal lattice is presented in Fig. 2(c), and the SAED pattern is − identified as the ⎡⎣ 1 11⎤⎦ zone axis of bcc β-Ti, as presented in the inset of Fig. 2(c). Fig. 2(d) shows the inverse fast-Fourier transform (IFFT) pattern of the area of the dendrites. Lacking of the stress, no lattice distortion and dislocations can be found. Similar results can be observed in other Ti-based MGMCs [22]. 3.2. . Mechanical properties The typical engineering stress-strain curves under quasi-static and dynamic compressions of the present MGMCs are shown in Fig. 3. Fig. 3(a) displays the engineering stress-strain curve of the present composite upon quasi-static compressive loading at room temperature at a strain rate of 5 10 4 s 1. It can be clearly seen that for the present composites, the yielding strength and the plasticity are about 1120 MPa and 43%, respectively. Compared with the typical monolithic BMGs with little plasticity [23,24], the plasticity of present MGMCs has been greatly improved. After obvious linear work-hardening, the MGMCs finally
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Fig. 2. (a) BF TEM image of the composite at low magnification. HRTEM images of (b) the matrix and (c) the dendrites. (Insets) Corresponding SAED patterns. (d) IFFT pattern from the area of the dendrite.
fracture at about 2,600 MPa. Apparently, the compressive mechanical properties are superior to those of most conventional Tialloys. For example, the plastic strain is 7.2 times of that of 6% in the Ti-6Al-4V [25] and 4 times of that of 10% in the Ti-13V-5Al [26]. Fig. 3(b) shows the dynamic results under different strain rates. It should be noted that the present MGMCs possess distinguishingly high yielding stress of about 1,290 MPa, as well as remarkable plasticity of about 16% even at a strain rate of 2250 s 1, which is superior to the previous Ti-based [12,27] and Zr-based MGMCs [15,28]. The significant plasticity of the composites upon high strain rates implies the potential application of present composites under dynamic loading at ambient temperature. Moreover, a slight softening behavior takes place, demonstrating that a temperature rise happens due to adiabatic heating during dynamic compression [29,30]. In addition, the plasticity decreases to about 10%, when the strain rate increases to 4050 s 1. During the dynamic compression, as the increase of the strain rates, the ductile to brittle transition occurs, which is different from that under quasistatic loading [17]. These obvious differences reveal that the strain rate plays a crucial role on the mutual effects between the glass matrix and dendrites. It is noted that the slope in the elastic stage upon dynamic compression is different from that upon quasi-
static loading on account of the desynchrony between the displacement-time and load-time curves. 3.3. . Fracture morphologies of MGMCs Fig. 4(a) displays the SEM images of the deformation under quasi-static compression. Corresponding to the large plasticity at the strain state of 5 10 4 s 1, the length decreases to 1.5 mm, and obvious bulge can be seen. It can be seen that the fracture angle is less than 45°, which is in accord with the theory proposed by Zhang et al. [31]. The fracture angle, θCF , can be calculated by the following equation [32]:
(
θCF = arc tan
1 + (μC )2 − μC
)
(1)
where μC is the slope of the critical compression fracture line. Profuse shear bands within the glass matrix, indicated by the bright arrows with different orientation, and profuse slip bands within the dendrites, denoted by the dark arrows, are distributed on the lateral surface of deformed composites, in agreement with the considerable macroscopic plasticity, as shown in Fig. 4(b). Meanwhile, several microcracks can be observed by the virtual bright arrows along the shear bands due to the severe plastic
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HRTEM image taken near the interface between the dendrites and − glass matrix. The spotty diffraction can be defined as the ⎡⎣ 1 11⎤⎦ zone axis for the dendrites, which implies that no phase transformation occurs in the dendrites. The deformation structure of the dendrites is clear, as shown in the IFFT pattern in Fig. 5(f). The multiple dislocations and severe lattice distortions are present, which corresponds to large plasticity. Moreover, no debonding at the interface, demonstrated as before [33], occurs. But the density of the dislocation is far lower than the samples under quasi-static loading as well as the distortions, which accords with the small plasticity compared to the quasi-static compression Similar results have been reported in a Zr-based MGMC [10] and a Ti-based nanostructure-dendrite composite upon compression [34].
4. . Discussion
Fig. 3. The engineering stress-strain curves of the present composite upon (a) quasi-static compressive loading and (b) dynamic compressive loading.
deformation. Fig. 4(c) exhibits the fracture surface with a large temperature rising in critical shear bands. Fracture morphologies of dynamically deformed samples are shown in Fig. 4(d)–(f). Macroscopically, it can be clearly seen that the fractured specimen is almost the same as the sample under quasi-static loading, as shown in Fig. 4(d). Fig. 4(e) illustrates the lateral surface of the dynamically fractured specimen. Like the quasi-static fracture surface, multiple shear bands can be observed from the fracture feature, as exhibited in Fig. 4(e), which unveils that the sample displays excellent plasticity before fracture. Typical vein patterns can be found, and the abundant resolidified liquid droplets, associated with the adiabatic heating [31], are shown in Fig. 4(f). Fig. 5(a) displays the bright-field TEM image of the dendrites upon quasi-static loading. Obvious plastic deformation takes place within dendrites. The shear steps can be found, which indicates the interaction between the deformation structures and the shear bands [22]. The interface between the glass matrix and dendrites is shown by HETEM in Fig. 5(b). It is confirmed that the interface is atomically sharp. Diffraction patterns are exhibited in the insets of Fig. 5(b) for both the dendrites and glass matrix. The dendrites reveal a bcc pattern, which further confirms that no phase transformation takes place for the dendrites upon quasi-static deformation, whereas the matrix exhibits a typical hallow diffraction, indicating an amorphous structure without nanocrystallization [22]. Fig. 5(c) shows the IFFT pattern within the dendrites, as marked by the red square in Fig. 5(b). It is found that dislocations, denoted by T, and lattice distortions, represented by ellipses are present. The images after dynamic compression are shown in Fig. 5(d)–(f). Fig. 5(d) illustrates the whole dendrites undergo severe deformation, and the shear bands pass into the dendrites and are hindered by the dislocation walls. In order to examine the deformation structure of the dendrites, Fig. 5(e) shows a typical
According to the previous studies [5], the deformation mechanisms for the monolithic BMGs are very different from the conventional crystalline alloys. Because of the rapid propagation of localized shear bands, most BMGs exhibit catastrophic failure and softening, while the conventional crystalline alloys usually possess excellent plasticity due to the multiplication of dislocations. Here, the large plasticity in the present MGMCs is tightly related to the interaction between the shear bands in the matrix and the dislocations in the dendrites upon loading. From the above experimental results and observations, it can be found that the strain rate plays an important role in the density increasement of shear bands and dislocations. Upon quasi-static compression, with the increase of the stress, the dislocations come up within the dendrites in the elastic stage and continually increase and rapidly accumulate. Once the stress up to the critical stress, the dislocations begin to motivate along the fixed slip lines or planes near the interface and pile up, which agrees with the image shown in Fig. 5(c). This feature can be elucidated by such a simple expression as [35] v = Aτ m ( v is the velocity of dislocation movement, A and m are materials constants, and τ is the initial shear stress for the dislocation movement). However, the extension of some dislocations can be hindered by the other crystal imperfections, resulting in obvious work-hardening behavior in the dendrites. In contrast, the primary shear bands nucleate near the interface due to a higher concentration stress, which leads to the generation of shear bands much easier in these areas. Due to the high shearing energy, the primary shear bands propagate in the glass matrix. Once the released energy overcomes the critical plastic energy dissipated in individual shear bands, the shear banding will develop as a runway shear crack. On one hand, the soft secondary phase with a lower shear modulus can effectively absorb the energy and hamper the rapid propagation. On the other hand, the coarsen dendrites or the dislocation clusters hinder the propagation of shear bands, resulting in the formation of multiple shear bands, the furcation of the shear bands, as well as the deflection of orientations. However, upon dynamic loading, there is no enough time to generate the multiple shear bands or dislocations, which induces the decrease of resistance to fracture. The shear bands will cut though the dendrites uninhibitedly, which is mainly responsible for brittle failure. According to the free-volume theory, for the inhomogeneous plastic flows of BMGs, Spacepen [36] has proposed the softening mechanism:
⎛ ⎛ ⎛ τΩ ⎞ αgv* ⎞ ΔG m ⎞ ⎟⎟exp⎜⎜ − ⎟⎟sinh⎜⎜ ⎟⎟ γ⋅pi = 2γvΔf exp⎜⎜ − vf ⎠ kT ⎠ ⎝ ⎝ ⎝ 2kT ⎠
(2)
where γ⋅pi is the plastic strain rate, γv is the frequency of Debye, Δf is the volume fraction, αg is a geometrical factor between 0.5 and 1,
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Fig. 4. The lateral surface of the deformed sample upon quasi-static and dynamic loadings shown in (a) and (d), respectively. The magnified deformation region near the crack upon quasi-static and dynamic loadings shown in (b) and (e), respectively. The fracture surface of the present composite upon quasi-static and dynamic loadings shown in (c) and (f).
v* is the effective hard-sphere size of atoms, vf is the average free volume of an atom, ΔGm is the thermal activation energy, k is the Boltzman's constant, T is the absolute temperature, and Ω is the molar atomic volume. As is reported by Steif et al. [37], the annihilative rate and the productive rate of the free volumes are in a dynamical equilibrium under the lower shear stress. Nevertheless, with the increase of the shear strain rates, the productive rate of the free volumes is exceeding the annihilative rate, indicating that more average free volumes are created. As a consequence, the propagation of shear bands will be promoted. Upon dynamic compression, the secondary phase can not impede the rapid propagation of shear bands. In the end, the cracks would be generated, leading to a catastrophic fracture with lower plasticity, compared with that upon quasi-static loading. For dendrites with the volume fraction more than 50%, the yielding strength of the composite follows the load-bearing model given as Eq. (1) [38]:
σcomposite = σdendrite(1 + 0.5fglass )
(3)
where fglass is the volume fraction of the glassy phases, and σdendrite and σglass are yielding strength of the crystalline and glassy phases, respectively. The relationship between the stress and strain for the dendrites could be expressed according to Taylor dislocation model [39]:
⎛ σy ⎞2n Lε p σdendrite = σref ⎜ d + εdp⎟ + d D ⎠ ⎝ Ed σdy ,
(4)
εdp
where σref , and are the reference stress, the yielding stresses and the plastic strain of the dendrites, respectively. n is the hardening coefficient of the dendrites, L is the intrinsic material length of ductile phase, and D is the average diameter of the dendrites. From the Eq. (1) and Eq. (2), it can be obtained that under the same plastic strain, the dendrite size and the stress of the composites are in a negative relationship, which is in agreement with the previous study [16]. Fig. 6(a) plots the dependence of the dimensionless yielding stress on the strain rate, and the equation can be obtained as follows [40]:
⎛ ε⋅ ⎞m σd = k⎜⎜ ⋅d ⎟⎟ σq ⎝ εq ⎠
(5)
where σd and σq are the yielding stress under dynamic and quasistatic loadings, respectively, k is the strain rate sensitivity coefficient, ε⋅d and ε⋅q are the strain rates upon dynamic and quasi-static compressions, and m is the strain rate sensitivity exponent. An approximately nonlinear relationship is established, which can be simply expressed as follows:
⎛ ε⋅ ⎞0.08 σd = 0.32⎜⎜ ⋅d ⎟⎟ σq ⎝ εq ⎠
(6)
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Fig. 5. The TEM images of the Ti60 composites after quasi-static compression as shown in (a)–(c). (a) BF TEM image of the composite at low magnification. (b) HRTEM image taken near the interface. (Inset) Corresponding SEAD patterns. (c) IFFT image in of the dendrites marked by the red square in (b). (d)–(f) show the images after dynamic compression. (d) BF TEM image of the composite at low magnification. (e) HRTEM image near the interface of the dendrites and glass matrix. (f) IFFT pattern from the area of the dendrites. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the current study, the temperature effect is ignored, due to that the dynamic compression occurs at room temperature. Based on the Johnson-Cook (J-C) plastic model [41], which is widely employed in metals and alloys, the flow stress of ductile materials can be effectively quantified as:
σ = (A + Bεn)(1 + C ln ε⋅*)
(7)
where σ is the von Mises flow stress, A, B, and C are constants, n is the strain hardening exponent, and ε⋅* is the normalized strain rate, i.e., ε⋅* = ε⋅ /ε⋅ 0 ( ε⋅0 is the reference strain rate and equals to 10 3 s 1). According to the experimental results at strain rates of 2250 and 4050 s 1, the parameters of the J-C constitutive model are respectively obtained by fitting experimental flow stress. Further, the corresponding average values of the parameters are calculated and adopted. In order to validate the fitted J-C constitutive model, the intermediate strain rate of 2550 s 1 is used in fitted J-C constitutive model to obtain the predicted plastic flow. Actually, the experimental curve at the strain rate of 2550 s 1 coincides well with the predicted one by the J-C model, which faithfully validates the fitted J-C constitutive model. Using this model and fitting the flow curves in Fig. 3(b) based on the J-C model, the constitutive relationship for the present in-situ MGMCs is obtained as σ = (1120 + 654.5ε0.0065)(1 − 0.0161 ln ε⋅*). For the sake of the consistency between the fitting curves and experimental data, as shown in Fig. 6(b), it further confirms the validity of the constitutive relationship. It can be distinctly seen that the experimental curves are basically consistent with those based on the J-C equations, which signifies, for the current Ti60 MGMCs, the J-C model legitimately forecasts the plastic flows under dynamic compression.
During quasi-static and dynamic compression, the plastic deformation energy may be transferred to heat and led to an adiabatic temperature rise, which can be estimated using the heat diffusion equation [42]:
⎛ ⎞ 1 ⎛ −x 2 ⎞ H ⎟⎟ exp⎜ ⎟ ΔT = ⎜⎜ ⎝ 4α t ⎠ ⎝ 2ρCp πα ⎠ t
(8)
where ρ is the material density, Cp is the specific heat capacity, α is the material thermal diffusivity, and H = βσyδ ( β is a constant of value about 0.3 [43,44], σy is the uniaxial yielding stress, and δ is the shear displacement [44]). For the present Ti-based MGMCs, it is assumed that σy≈1300 MPa at a strain rate of 2,550 s 1 and 1120 MPa at a strain rate of 5 10 4 s 1, ρ ¼ 5,445 kg m 3 [45], Cp ¼520 J kg 1 K 1 [46], α ≈ 10−6 m2 s 1 [47], and δ ≈ 495 nm [44]. The profile of the temperature rise along the x is schematically illustrated in Fig. 7. It should be noted that at a short time after shear band growth, the temperature increase is relatively high. At the same time, the increasement of the temperature under dynamic loading is higher than that under quasi-static loading, which is consistent with the SEM in Fig. 4(c) and Fig. 4(f). Furthermore, the shorter the time, the wider the temperature between quasi-static and dynamic compression, which is shown in Fig. 7(a) and (b), respectively. In addition, at a given strain rate, from the above Eqs. (7) and (8), it can be seen that the adiabatic temperature rise and the strain rate are in a positive relationship. Actually, upon dynamic compression, the dendrite-dominated mechanism associated with dislocation movement within the ductile secondary phase is
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Fig. 6. (a) The dependence of the dimensionless yielding stress on the strain rate. (b) The experimental flow stresses and their corresponding J-C equation for the Ti60 composite.
mainly responsible for work-hardening behavior during plastic flow, and matrix-dominated deformation related to structural and thermal softening leads to strain-softening effect for the MGMC [35]. For the present composite upon dynamic compression, the flow behaviors during deformation mainly depend on the competition between two different deformation mechanisms. Indeed, the work-hardening behavior prevails upon quasi-static compression, indicating that the dislocation movement within the crystalline dendrites assumes more dominance on the plastic flow. Compared to work-hardening behavior upon quasi-static compression, the flow stress is almost the same upon dynamic compression, demonstrating that two different deformation mechanisms basically keep the balance during deformation.
5. . Conclusion In summary, a plastic Ti-based MGMC with the composition of Ti60Zr14V12Cu4Be10, containing homogeneously distributed β-Ti dendrites embedded in the glass matrix, was fabricated by coppermould suction casting. Distinguishing plastic behaviors are observed upon both quasi-static and dynamic loadings. Upon the quasi-static loading, the considerably large plasticity up to 40% was observed due to the abundant dislocations and the multiple shear bands. However, upon dynamic loading, excellent plasticity of 16% was obtained, which was superior to the previous results. The equation the dependence of the dimensionless yielding stress on the strain rate σd/σq = 0.32(ε⋅d /ε⋅q )0.08. The Johnson-Cook (J-C) plasticity model is employed to model the dynamic flow behavior,
Fig. 7. The profile of the adiabatic temperature rise along x upon quasi-static and dynamic compressions are shown in (a) and (b), respectively.
and the constitutive relationship is obtained as σ = (1120 + 654.5ε0.0065)(1 − 0.0161 ln ε⋅*). According to theory of the adiabat, the adiabatic temperature rise increases with increasing strain rates, resulting in that the softening effect within the glass matrix is obviously enhanced during deformation.
Acknowledgements The authors would like to acknowledge the financial support of National Natural Science Foundation of China (Nos. 51371122 and 51501123), the Program for the Innovative Talents of Higher Learning Institutions of Shanxi (2013), the Youth Natural Science Foundation of Shanxi Province, China (Nos. 2015021005 and 2015021006), and the project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology), and the project number is KFJJ1519M. The authors declare that they have no conflict of interest.
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Excellent plasticity of a new Ti-based metallic glass matrix composite upon dynamic loading R.F. Wu a,b, Z.M. Jiao c, Y.S. Wang a, Z. Wang a, Z.H. Wang c, S.G. Ma c, J.W. Qiao a,b,n a Laboratory of Applied Physics and Mechanics of Advanced Materials, College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China b State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China c Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2016 Received in revised form 13 September 2016 Accepted 15 September 2016 Available online 16 September 2016
Quasi-static and dynamic compressive properties of in-situ Ti60Zr14V12Cu4Be10 bulk metallic glass matrix composites containing ductile dendrites were investigated. Upon quasi-static compressive loading, the composite exhibits a high fracture strength of 2,600 MPa, combined with a considerable plasticity of 40% at room temperature. However, upon dynamic loading, an excellent plasticity of 16% can be obtained due to the abundant dislocations and severe lattice distortions within dendrites and multiplication of shear bands within the glass matrix analyzed by transmission-electron microscopy. A constitutive relationship is obtained by Johnson-Cook plasticity model, which is employed to model the dynamic flow stress behavior. In addition, under dynamic compression, the adiabatic temperature rise increases with increasing strain rates, resulting in that the softening effect within the glass matrix is obviously enhanced during deformation. & 2016 Elsevier B.V. All rights reserved.
Keywords: Bulk metallic glass Dynamic loading J-C model Adiabatic temperature rise
1. . Introduction Since firstly fabricated by rapid cooling from the melts by Duwez et al. [1], the bulk metallic glasses (BMGs) were developed quickly and attracted significant technological and scientific studies all over the world on account of a series of superior mechanical properties at ambient temperature, such as ultrahigh strength, high hardness, large elastic limits, and excellent wear and corrosion resistance [2–6]. BMGs are esteemed as potential engineering structural materials [7]. However, owing to highly localized shear deformation upon loading, BMGs exhibit limited plasticity at room temperature and have subsequent catastrophic failure without apparent macroscopic plasticity by shear softening in one band [5,8]. In order to circumvent the poor plasticity, in-situ formed ductile secondary phases can be introduced into the glass matrix to form the metallic glasses matrix composites (MGMCs) [9,10]. To date, compared with monolithic BMGs, the existence of dendrites within the glass matrix contributes to obvious macroscopic compressive plasticity and tensile ductility, together with the improved toughness upon quasi-static loading for in-situ n Corresponding author at: College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China. E-mail address: [email protected] (J.W. Qiao).
http://dx.doi.org/10.1016/j.msea.2016.09.057 0921-5093/& 2016 Elsevier B.V. All rights reserved.
MGMCs. The dendrites hinder the prompt propagation of shear bands and lead to multiplication of shear bands [11]. Nevertheless, actual structural engineering materials are always performed superb ductility even upon dynamic compression. Therefore, many researches focus on the dynamic loading of MGMCs to meet the engineering applications in recent years, which can be efficaciously applied to strategic fields, such as defense, aerospace, and precision optical machinery [12]. Wang et al. [13] have demonstrated that the dynamic compressive yielding strength and the maximum strain value in Zr-based MGMCs are 2,700 MPa and 4%, respectively. Jeon et al. [14] have investigated the dynamic compressive behavior of Ti-based MGMCs modified from Ti-6Al-4V alloy and the dendrites inside have great impact on the plasticity of alloys. According to previous study by Qiao et al. [15], the multiple shear bands have been discovered for the in-situ Zr-based MGMCs under quasi-static compression, leading to large plasticity, whereas expeditious failure under the dynamic case is due to insufficient time to form profuse shear bands. Previously, researchers have found that brittle fracture or little plasticity occurred for in-situ MGMCs upon dynamic loading [16–18]. However, during quasi-static and dynamic loading, only limited information of the influence of temperature rise on the mechanisms of the deformation from the microstructure is available. In the current study, a new Ti-based MGMC was developed and investigated at quasi-static and dynamic loadings so as to further understand the deformation mechanisms.
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2. . Experimental 2.1. . Sample preparation and characterization The ingot with a nominal atomic percent composition of Ti60Zr14V12Cu4Be10 (Ti60) was prepared by arc melting the mixture of Ti, Zr, V, Cu, and Be with purity higher than 99.99 wt% in a Tigetted high purity argon atmosphere. In order to ensure the chemical homogeneity, the pre-alloyed ingots were re-melted at least four times. The rod like samples with 3 mm in diameter and about 85 mm in length were obtained by suctioning the melt into the copper mould in an argon atmosphere. The structure of phases was checked by X-ray diffraction (XRD) with a monochromatic CuKα radiation. The microstructures of the as-cast samples, and the lateral and fracture surfaces of the deformed samples after quasistatic and dynamic compression tests were investigated using scanning-electron microscopy (SEM). The cross sections of the ascast samples were polished, and etched by a solution of 40 mL HF, 20 mL HNO3, 40 mL HCL, and 200 mL H2O for SEM observation. The structural characteristics of the samples before and after compression were investigated using transmission-electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM) (JEOL-2010). The TEM specimens were obtained by ion thinning with liquid nitrogen cooling. 2.2. . Mechanical tests The uniaxial compressive tests under quasi-static loading were preformed on a 3-mm cylinders at ambient temperature at a strain rate of 5 10 4 s 1. The dynamic compressions were carried out by a split Hopkinson pressure bar (SHPB) [12] on samples with an aspect ratio of 1:1 at room temperature. It is made of a compressed gas-gun, a striker bar, a incident bar, specimen, a transmission bar, strain gauges, and an oscilloscope. The stress-strain curves of the tested specimen were obtained by inputting the recorded strain pulses into computing software. According to the one-dimensional stress wave theory [19], the stress and strain as well as strain rate could be obtained finally. Furthermore, in order to ensure the stability of results, the quasi-static compression tests were repeated at least three times and five times for dynamic compression tests. The detailed process was described elsewhere [20,21].
3. . Results 3.1. . Microstructures of MGMCs The microstructure and the dual-phase structure of the Ti60 are tested by the SEM and the XRD pattern, respectively. Fig. 1 (a) shows the back scattered SEM image of the cross-section of the as-cast composites. It can be seen that the dendrites are homogeneously embedded in the continuous glass matrix. By estimating the contrast in SEM images, the volume fraction of the dendrites is approximately 67%. This means that the secondary phase takes most part of the composites. It has been demonstrated that the size of the dendritic arms in the composites are 3–5 mm. The XRD exhibited in Fig. 1(b) reveals that the body-centered-cubic (bcc) crystalline peaks are overlapped on the broad diffuse scattering amorphous maxima, which means that the matrix has amorphous structure, and the dendrites are β-Ti solid solution. It is certified by the microstructure characterization mentioned above. To further examine the dual-phase structure, TEM and HRTEM analyses are employed on the as-cast composites. It is obviously observed that the bright-field image of the dendrites and the matrix with a low magnification is shown in Fig. 2(a), which is
Fig. 1. (a) SEM image of the microstructure for Ti60 composite. (b) XRD pattern of the composite.
similar to the SEM in Fig. 1(a). The edge of the dendrites is very smooth, indicating the absence of deformation [22]. Fig. 2 (b) displays the HRTEM image of the matrix, and no crystallization can be detected. Only diffuse halos, typical characterization of an amorphous structure, can be found from the selected area electron diffraction (SAED) pattern in the inset of Fig. 2(b). On the contrary, the crystal lattice is presented in Fig. 2(c), and the SAED pattern is − identified as the ⎡⎣ 1 11⎤⎦ zone axis of bcc β-Ti, as presented in the inset of Fig. 2(c). Fig. 2(d) shows the inverse fast-Fourier transform (IFFT) pattern of the area of the dendrites. Lacking of the stress, no lattice distortion and dislocations can be found. Similar results can be observed in other Ti-based MGMCs [22]. 3.2. . Mechanical properties The typical engineering stress-strain curves under quasi-static and dynamic compressions of the present MGMCs are shown in Fig. 3. Fig. 3(a) displays the engineering stress-strain curve of the present composite upon quasi-static compressive loading at room temperature at a strain rate of 5 10 4 s 1. It can be clearly seen that for the present composites, the yielding strength and the plasticity are about 1120 MPa and 43%, respectively. Compared with the typical monolithic BMGs with little plasticity [23,24], the plasticity of present MGMCs has been greatly improved. After obvious linear work-hardening, the MGMCs finally
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Fig. 2. (a) BF TEM image of the composite at low magnification. HRTEM images of (b) the matrix and (c) the dendrites. (Insets) Corresponding SAED patterns. (d) IFFT pattern from the area of the dendrite.
fracture at about 2,600 MPa. Apparently, the compressive mechanical properties are superior to those of most conventional Tialloys. For example, the plastic strain is 7.2 times of that of 6% in the Ti-6Al-4V [25] and 4 times of that of 10% in the Ti-13V-5Al [26]. Fig. 3(b) shows the dynamic results under different strain rates. It should be noted that the present MGMCs possess distinguishingly high yielding stress of about 1,290 MPa, as well as remarkable plasticity of about 16% even at a strain rate of 2250 s 1, which is superior to the previous Ti-based [12,27] and Zr-based MGMCs [15,28]. The significant plasticity of the composites upon high strain rates implies the potential application of present composites under dynamic loading at ambient temperature. Moreover, a slight softening behavior takes place, demonstrating that a temperature rise happens due to adiabatic heating during dynamic compression [29,30]. In addition, the plasticity decreases to about 10%, when the strain rate increases to 4050 s 1. During the dynamic compression, as the increase of the strain rates, the ductile to brittle transition occurs, which is different from that under quasistatic loading [17]. These obvious differences reveal that the strain rate plays a crucial role on the mutual effects between the glass matrix and dendrites. It is noted that the slope in the elastic stage upon dynamic compression is different from that upon quasi-
static loading on account of the desynchrony between the displacement-time and load-time curves. 3.3. . Fracture morphologies of MGMCs Fig. 4(a) displays the SEM images of the deformation under quasi-static compression. Corresponding to the large plasticity at the strain state of 5 10 4 s 1, the length decreases to 1.5 mm, and obvious bulge can be seen. It can be seen that the fracture angle is less than 45°, which is in accord with the theory proposed by Zhang et al. [31]. The fracture angle, θCF , can be calculated by the following equation [32]:
(
θCF = arc tan
1 + (μC )2 − μC
)
(1)
where μC is the slope of the critical compression fracture line. Profuse shear bands within the glass matrix, indicated by the bright arrows with different orientation, and profuse slip bands within the dendrites, denoted by the dark arrows, are distributed on the lateral surface of deformed composites, in agreement with the considerable macroscopic plasticity, as shown in Fig. 4(b). Meanwhile, several microcracks can be observed by the virtual bright arrows along the shear bands due to the severe plastic
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HRTEM image taken near the interface between the dendrites and − glass matrix. The spotty diffraction can be defined as the ⎡⎣ 1 11⎤⎦ zone axis for the dendrites, which implies that no phase transformation occurs in the dendrites. The deformation structure of the dendrites is clear, as shown in the IFFT pattern in Fig. 5(f). The multiple dislocations and severe lattice distortions are present, which corresponds to large plasticity. Moreover, no debonding at the interface, demonstrated as before [33], occurs. But the density of the dislocation is far lower than the samples under quasi-static loading as well as the distortions, which accords with the small plasticity compared to the quasi-static compression Similar results have been reported in a Zr-based MGMC [10] and a Ti-based nanostructure-dendrite composite upon compression [34].
4. . Discussion
Fig. 3. The engineering stress-strain curves of the present composite upon (a) quasi-static compressive loading and (b) dynamic compressive loading.
deformation. Fig. 4(c) exhibits the fracture surface with a large temperature rising in critical shear bands. Fracture morphologies of dynamically deformed samples are shown in Fig. 4(d)–(f). Macroscopically, it can be clearly seen that the fractured specimen is almost the same as the sample under quasi-static loading, as shown in Fig. 4(d). Fig. 4(e) illustrates the lateral surface of the dynamically fractured specimen. Like the quasi-static fracture surface, multiple shear bands can be observed from the fracture feature, as exhibited in Fig. 4(e), which unveils that the sample displays excellent plasticity before fracture. Typical vein patterns can be found, and the abundant resolidified liquid droplets, associated with the adiabatic heating [31], are shown in Fig. 4(f). Fig. 5(a) displays the bright-field TEM image of the dendrites upon quasi-static loading. Obvious plastic deformation takes place within dendrites. The shear steps can be found, which indicates the interaction between the deformation structures and the shear bands [22]. The interface between the glass matrix and dendrites is shown by HETEM in Fig. 5(b). It is confirmed that the interface is atomically sharp. Diffraction patterns are exhibited in the insets of Fig. 5(b) for both the dendrites and glass matrix. The dendrites reveal a bcc pattern, which further confirms that no phase transformation takes place for the dendrites upon quasi-static deformation, whereas the matrix exhibits a typical hallow diffraction, indicating an amorphous structure without nanocrystallization [22]. Fig. 5(c) shows the IFFT pattern within the dendrites, as marked by the red square in Fig. 5(b). It is found that dislocations, denoted by T, and lattice distortions, represented by ellipses are present. The images after dynamic compression are shown in Fig. 5(d)–(f). Fig. 5(d) illustrates the whole dendrites undergo severe deformation, and the shear bands pass into the dendrites and are hindered by the dislocation walls. In order to examine the deformation structure of the dendrites, Fig. 5(e) shows a typical
According to the previous studies [5], the deformation mechanisms for the monolithic BMGs are very different from the conventional crystalline alloys. Because of the rapid propagation of localized shear bands, most BMGs exhibit catastrophic failure and softening, while the conventional crystalline alloys usually possess excellent plasticity due to the multiplication of dislocations. Here, the large plasticity in the present MGMCs is tightly related to the interaction between the shear bands in the matrix and the dislocations in the dendrites upon loading. From the above experimental results and observations, it can be found that the strain rate plays an important role in the density increasement of shear bands and dislocations. Upon quasi-static compression, with the increase of the stress, the dislocations come up within the dendrites in the elastic stage and continually increase and rapidly accumulate. Once the stress up to the critical stress, the dislocations begin to motivate along the fixed slip lines or planes near the interface and pile up, which agrees with the image shown in Fig. 5(c). This feature can be elucidated by such a simple expression as [35] v = Aτ m ( v is the velocity of dislocation movement, A and m are materials constants, and τ is the initial shear stress for the dislocation movement). However, the extension of some dislocations can be hindered by the other crystal imperfections, resulting in obvious work-hardening behavior in the dendrites. In contrast, the primary shear bands nucleate near the interface due to a higher concentration stress, which leads to the generation of shear bands much easier in these areas. Due to the high shearing energy, the primary shear bands propagate in the glass matrix. Once the released energy overcomes the critical plastic energy dissipated in individual shear bands, the shear banding will develop as a runway shear crack. On one hand, the soft secondary phase with a lower shear modulus can effectively absorb the energy and hamper the rapid propagation. On the other hand, the coarsen dendrites or the dislocation clusters hinder the propagation of shear bands, resulting in the formation of multiple shear bands, the furcation of the shear bands, as well as the deflection of orientations. However, upon dynamic loading, there is no enough time to generate the multiple shear bands or dislocations, which induces the decrease of resistance to fracture. The shear bands will cut though the dendrites uninhibitedly, which is mainly responsible for brittle failure. According to the free-volume theory, for the inhomogeneous plastic flows of BMGs, Spacepen [36] has proposed the softening mechanism:
⎛ ⎛ ⎛ τΩ ⎞ αgv* ⎞ ΔG m ⎞ ⎟⎟exp⎜⎜ − ⎟⎟sinh⎜⎜ ⎟⎟ γ⋅pi = 2γvΔf exp⎜⎜ − vf ⎠ kT ⎠ ⎝ ⎝ ⎝ 2kT ⎠
(2)
where γ⋅pi is the plastic strain rate, γv is the frequency of Debye, Δf is the volume fraction, αg is a geometrical factor between 0.5 and 1,
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Fig. 4. The lateral surface of the deformed sample upon quasi-static and dynamic loadings shown in (a) and (d), respectively. The magnified deformation region near the crack upon quasi-static and dynamic loadings shown in (b) and (e), respectively. The fracture surface of the present composite upon quasi-static and dynamic loadings shown in (c) and (f).
v* is the effective hard-sphere size of atoms, vf is the average free volume of an atom, ΔGm is the thermal activation energy, k is the Boltzman's constant, T is the absolute temperature, and Ω is the molar atomic volume. As is reported by Steif et al. [37], the annihilative rate and the productive rate of the free volumes are in a dynamical equilibrium under the lower shear stress. Nevertheless, with the increase of the shear strain rates, the productive rate of the free volumes is exceeding the annihilative rate, indicating that more average free volumes are created. As a consequence, the propagation of shear bands will be promoted. Upon dynamic compression, the secondary phase can not impede the rapid propagation of shear bands. In the end, the cracks would be generated, leading to a catastrophic fracture with lower plasticity, compared with that upon quasi-static loading. For dendrites with the volume fraction more than 50%, the yielding strength of the composite follows the load-bearing model given as Eq. (1) [38]:
σcomposite = σdendrite(1 + 0.5fglass )
(3)
where fglass is the volume fraction of the glassy phases, and σdendrite and σglass are yielding strength of the crystalline and glassy phases, respectively. The relationship between the stress and strain for the dendrites could be expressed according to Taylor dislocation model [39]:
⎛ σy ⎞2n Lε p σdendrite = σref ⎜ d + εdp⎟ + d D ⎠ ⎝ Ed σdy ,
(4)
εdp
where σref , and are the reference stress, the yielding stresses and the plastic strain of the dendrites, respectively. n is the hardening coefficient of the dendrites, L is the intrinsic material length of ductile phase, and D is the average diameter of the dendrites. From the Eq. (1) and Eq. (2), it can be obtained that under the same plastic strain, the dendrite size and the stress of the composites are in a negative relationship, which is in agreement with the previous study [16]. Fig. 6(a) plots the dependence of the dimensionless yielding stress on the strain rate, and the equation can be obtained as follows [40]:
⎛ ε⋅ ⎞m σd = k⎜⎜ ⋅d ⎟⎟ σq ⎝ εq ⎠
(5)
where σd and σq are the yielding stress under dynamic and quasistatic loadings, respectively, k is the strain rate sensitivity coefficient, ε⋅d and ε⋅q are the strain rates upon dynamic and quasi-static compressions, and m is the strain rate sensitivity exponent. An approximately nonlinear relationship is established, which can be simply expressed as follows:
⎛ ε⋅ ⎞0.08 σd = 0.32⎜⎜ ⋅d ⎟⎟ σq ⎝ εq ⎠
(6)
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Fig. 5. The TEM images of the Ti60 composites after quasi-static compression as shown in (a)–(c). (a) BF TEM image of the composite at low magnification. (b) HRTEM image taken near the interface. (Inset) Corresponding SEAD patterns. (c) IFFT image in of the dendrites marked by the red square in (b). (d)–(f) show the images after dynamic compression. (d) BF TEM image of the composite at low magnification. (e) HRTEM image near the interface of the dendrites and glass matrix. (f) IFFT pattern from the area of the dendrites. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the current study, the temperature effect is ignored, due to that the dynamic compression occurs at room temperature. Based on the Johnson-Cook (J-C) plastic model [41], which is widely employed in metals and alloys, the flow stress of ductile materials can be effectively quantified as:
σ = (A + Bεn)(1 + C ln ε⋅*)
(7)
where σ is the von Mises flow stress, A, B, and C are constants, n is the strain hardening exponent, and ε⋅* is the normalized strain rate, i.e., ε⋅* = ε⋅ /ε⋅ 0 ( ε⋅0 is the reference strain rate and equals to 10 3 s 1). According to the experimental results at strain rates of 2250 and 4050 s 1, the parameters of the J-C constitutive model are respectively obtained by fitting experimental flow stress. Further, the corresponding average values of the parameters are calculated and adopted. In order to validate the fitted J-C constitutive model, the intermediate strain rate of 2550 s 1 is used in fitted J-C constitutive model to obtain the predicted plastic flow. Actually, the experimental curve at the strain rate of 2550 s 1 coincides well with the predicted one by the J-C model, which faithfully validates the fitted J-C constitutive model. Using this model and fitting the flow curves in Fig. 3(b) based on the J-C model, the constitutive relationship for the present in-situ MGMCs is obtained as σ = (1120 + 654.5ε0.0065)(1 − 0.0161 ln ε⋅*). For the sake of the consistency between the fitting curves and experimental data, as shown in Fig. 6(b), it further confirms the validity of the constitutive relationship. It can be distinctly seen that the experimental curves are basically consistent with those based on the J-C equations, which signifies, for the current Ti60 MGMCs, the J-C model legitimately forecasts the plastic flows under dynamic compression.
During quasi-static and dynamic compression, the plastic deformation energy may be transferred to heat and led to an adiabatic temperature rise, which can be estimated using the heat diffusion equation [42]:
⎛ ⎞ 1 ⎛ −x 2 ⎞ H ⎟⎟ exp⎜ ⎟ ΔT = ⎜⎜ ⎝ 4α t ⎠ ⎝ 2ρCp πα ⎠ t
(8)
where ρ is the material density, Cp is the specific heat capacity, α is the material thermal diffusivity, and H = βσyδ ( β is a constant of value about 0.3 [43,44], σy is the uniaxial yielding stress, and δ is the shear displacement [44]). For the present Ti-based MGMCs, it is assumed that σy≈1300 MPa at a strain rate of 2,550 s 1 and 1120 MPa at a strain rate of 5 10 4 s 1, ρ ¼ 5,445 kg m 3 [45], Cp ¼520 J kg 1 K 1 [46], α ≈ 10−6 m2 s 1 [47], and δ ≈ 495 nm [44]. The profile of the temperature rise along the x is schematically illustrated in Fig. 7. It should be noted that at a short time after shear band growth, the temperature increase is relatively high. At the same time, the increasement of the temperature under dynamic loading is higher than that under quasi-static loading, which is consistent with the SEM in Fig. 4(c) and Fig. 4(f). Furthermore, the shorter the time, the wider the temperature between quasi-static and dynamic compression, which is shown in Fig. 7(a) and (b), respectively. In addition, at a given strain rate, from the above Eqs. (7) and (8), it can be seen that the adiabatic temperature rise and the strain rate are in a positive relationship. Actually, upon dynamic compression, the dendrite-dominated mechanism associated with dislocation movement within the ductile secondary phase is
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Fig. 6. (a) The dependence of the dimensionless yielding stress on the strain rate. (b) The experimental flow stresses and their corresponding J-C equation for the Ti60 composite.
mainly responsible for work-hardening behavior during plastic flow, and matrix-dominated deformation related to structural and thermal softening leads to strain-softening effect for the MGMC [35]. For the present composite upon dynamic compression, the flow behaviors during deformation mainly depend on the competition between two different deformation mechanisms. Indeed, the work-hardening behavior prevails upon quasi-static compression, indicating that the dislocation movement within the crystalline dendrites assumes more dominance on the plastic flow. Compared to work-hardening behavior upon quasi-static compression, the flow stress is almost the same upon dynamic compression, demonstrating that two different deformation mechanisms basically keep the balance during deformation.
5. . Conclusion In summary, a plastic Ti-based MGMC with the composition of Ti60Zr14V12Cu4Be10, containing homogeneously distributed β-Ti dendrites embedded in the glass matrix, was fabricated by coppermould suction casting. Distinguishing plastic behaviors are observed upon both quasi-static and dynamic loadings. Upon the quasi-static loading, the considerably large plasticity up to 40% was observed due to the abundant dislocations and the multiple shear bands. However, upon dynamic loading, excellent plasticity of 16% was obtained, which was superior to the previous results. The equation the dependence of the dimensionless yielding stress on the strain rate σd/σq = 0.32(ε⋅d /ε⋅q )0.08. The Johnson-Cook (J-C) plasticity model is employed to model the dynamic flow behavior,
Fig. 7. The profile of the adiabatic temperature rise along x upon quasi-static and dynamic compressions are shown in (a) and (b), respectively.
and the constitutive relationship is obtained as σ = (1120 + 654.5ε0.0065)(1 − 0.0161 ln ε⋅*). According to theory of the adiabat, the adiabatic temperature rise increases with increasing strain rates, resulting in that the softening effect within the glass matrix is obviously enhanced during deformation.
Acknowledgements The authors would like to acknowledge the financial support of National Natural Science Foundation of China (Nos. 51371122 and 51501123), the Program for the Innovative Talents of Higher Learning Institutions of Shanxi (2013), the Youth Natural Science Foundation of Shanxi Province, China (Nos. 2015021005 and 2015021006), and the project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology), and the project number is KFJJ1519M. The authors declare that they have no conflict of interest.
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