On the mechanical properties of particle reinforced metallic glass matrix composites

Accepted Manuscript On the mechanical properties of particle reinforced metallic glass matrix composites J.C. Li, X.W. Chen, F.L. Huang PII:

S0925-8388(17)34200-7

DOI:

10.1016/j.jallcom.2017.12.024

Reference:

JALCOM 44110

To appear in:

Journal of Alloys and Compounds

Received Date: 19 September 2017 Revised Date:

14 November 2017

Accepted Date: 4 December 2017

Please cite this article as: J.C. Li, X.W. Chen, F.L. Huang, On the mechanical properties of particle reinforced metallic glass matrix composites, Journal of Alloys and Compounds (2018), doi: 10.1016/ j.jallcom.2017.12.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT On the mechanical properties of particle reinforced metallic glass matrix composites J.C. Li a, b, c, X.W. Chen a, c, *, F.L. Huang a, *

State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology,

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a

Beijing, 100081, China b

Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, Sichuan, 621999, China

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c Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province, Mianyang, Sichuan, 621999, China

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* Corresponding authors. E-mail: [email protected], [email protected]

Abstract: In the present manuscript a modified coupled thermo-mechanical constitutive model is employed to describe the mechanical property of metallic glass (MG) matrix, and the geometrical models of particle reinforced MG matrix composites are established based on the inner structures.

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Then finite element method (FEM) numerical simulations on the mechanical properties of composites are conducted systemically by integrating with related experimental investigations, and the effects of different factors, including the volume fraction of particle (which determines the

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amount of particle and the interspace between particles), the mechanical property of particle

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material and the strain rate, on the deformation and failure characteristics of composites are analyzed. Related analysis demonstrates that the composite mainly displays as a shear manner, however, the reinforced particles can lead to an improvement for its plasticity. Furthermore, several internal and external factors have significant influences on the deformation and failure characteristics of composites. The corresponding mechanical properties depend firmly on the initiation and propagation of shear band and shear crack within the composite. Keywords: Particle reinforced metallic glass matrix composite, Mechanical behavior, Shear band, Crack, Numerical analysis

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Introduction Metallic glass (MG), or called amorphous alloy, has many excellent mechanical and physical

properties [1-7]. One of its important performances is that highly localized shear bands are very easily to be motivated during the deformation process of material, and thus failure is usually localized in a dominant shear band. Consequently, MG material usually displays as catastrophic

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fracture and shows little macroscopic plasticity [7-9]. This high shear sensitivity of MG usually impedes its application as a structural material.

Therefore, MG matrix composites were stimulated widely. By introducing the ductile

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reinforced phase into the MG substrate, the corresponding composite could integrate the excellent properties of MG matrix and that of the reinforced phase [3, 10-52]. The popular kinds of

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composites are that reinforced with fibers and with particles. Composites reinforced with fibers may show an anisotropic feature, and comparatively, that reinforced with particles usually displays the isotropic characteristic. Among various ductile reinforced phases, the tungsten (W) material has good comprehensive properties and a good wettability on the MG substrate [53], and thus tungsten particle reinforced MG matrix (WP/MG) composites are very popular and there are

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abundant research focusing on this kind of composite [11-14, 16, 17, 21, 22, 24, 28-35]. Besides, the steel particle reinforced MG matrix (FeP/MG) composite has remarkable cost effectiveness in the practical application, and it has great potentiality in the engineering field [45]. Titanium

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particle reinforced (TiP/MG) composite could be used as implant material due to its good mechanical properties as well as good biocompatibility, and it gains more and more investigations

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[26, 46-48].

Regarding to the structural material, one of its important factors is the mechanical property. For the mechanical performances of particle reinforced MG matrix composites, abundant experimental investigations and theoretical analysis have been conducted, including quasi-static experiments [12-15, 17-29, 32, 33, 35-52], dynamic experiments [20, 21, 24, 30, 31, 34] and high-speed impact experiments [13], etc. The corresponding investigations showed that the deformation and failure forms of composite are similar to that of the monolithic MG material, i.e., it still behaves as a shear manner. However, its plasticity achieves a significant improvement compared with that of MG. 2

ACCEPTED MANUSCRIPT Along with the progress of computer technology, numerical simulation, especially the finite element method (FEM) simulation, has become an effective approach for investigating the mechanical properties of materials [19, 20, 23, 27, 33, 49, 54-64]. The detailed deformation process and variations of stress and strain within the material are usually difficult to investigate in the real experiment, and thus FEM simulation could be an effective supplement. It is favorable to

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conduct deeper analysis on the experiment.

Based on our related work [61-64], the present manuscript further investigates the mechanical properties of particle reinforced MG matrix composites. A modified coupled thermo-mechanical

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constitutive model is employed to describe the special mechanical behaviors of MG matrix, and the geometrical models of composites are established based on their inner structures. Then

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systemic FEM simulations for various deformation conditions, including the quasi-static and dynamic compression and tension, the dynamic compression and tension and the high-speed impact, are conducted by integrating with related experiments. The detailed deformation and failure processes are analyzed. Based on the analysis on the initiation and propagation of shear band as well as the shear banding-induced failure of material, the mechanical performances of

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composites are investigated, and the influences of different internal and external factors on the mechanical properties of material are discussed in detail.

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2 Geometrical models and constitutive models 2.1 Geometrical models

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LS-DYNA software is employed to implement the FEM simulations. As the authors’ previous work [62-64] demonstrated, though 3-D FEM simulation could represent the actual structure of composite as well as the corresponding deformation and failure forms better, 2-D simulation can also capture the main deformation and failure characteristics of material, besides, it is convenient for investigating the initiation and propagation processes of shear bands and cracks within the material, and it is also beneficial to save the solution time. Thus, in the following analysis 2-D simulation will be mainly focused, and 3-D simulation is regarded as an assist. The quasi-static compressive test is usually based on the Materials Test System (MTS). Integrated with related experimental investigations [12-15, 17, 18, 20, 22-29, 32, 33, 35-51], a 3

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compressive specimen with a cylindrical shape of φ 3mm×6mm is selected as the sample. According to the axis-symmetry of experiment condition, the corresponding sketch of axis-symmetrical model is shown in Fig. 1, wherein the bottom surface of the base is fixed, and the top surface of the indenter moves downwards at a constant velocity v (sketched as solid arrows).

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Contact is defined between the two ends of specimen and the base as well as the indenter. Regarding to the tensile experiment, the gauge zone of tensile specimen is also designed as a cylindrical structure [15, 18, 31]. For convenience of comparative analysis on the deformation behaviors under the compressive and tensile conditions, the gauge zone of tensile specimen here is

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set as the same structure with the compressive one (Fig. 1). Correspondingly, the tensile specimen is fixed with the two other parts, i.e., the indenter and the base in the compressive experiment

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become fixtures under the tensile condition. The bottom surface of the lower fixture is still fixed and the top surface of the upper fixture moves upwards at a constant velocity (dashed arrows in Fig. 1).

Investigations on the dynamic behavior of material is usually based on the Hopkinson test or the instrumented anvil-on-rod impact test [20, 21, 24, 30, 31, 34], and the corresponding specimen

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is almost the same with that in the quasi-static experiment. In the same way, for comparative analysis on the mechanical behaviors of composites under different strain rates, specimens in the dynamic conditions are also set as the same structure with that in the quasi-static cases (see Fig. 1),

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and correspondingly the loading with different strain rates is obtained from changing the value of the indenter/fixture’s velocity.

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The composite specimen in Fig. 1 mainly contains two parts, i.e., the MG matrix and the reinforced particles, and particles distribute randomly in the matrix. Moreover, there are also some shear-band zones with inhomogeneous distribution within the MG matrix [65, 66]. To describe the mechanical behavior of composite well in the FEM simulations, the corresponding geometrical model should reflect the reinforced particles as well as the shear-band zones in the MG matrix. Regarding the reinforced particles, they usually distribute randomly in the MG matrix, and their sizes are usually not uniform. Consequently, it will be onerous to establish the geometrical model of composite specimen in the pre-processing program by following a normal procedure. Also, it will be difficult to achieve a fine quality of meshes. The meshes will affect the accuracy of 4

ACCEPTED MANUSCRIPT simulation result and the solution time of calculation [59]. In the present manuscript, based on the format of keyword file in the LS-DYNA software and the FORTRAN program, the reinforced particles can be defined conveniently in the FEM geometrical model and their random distribution is achieved at the same time. LS-DYNA keyword file contains all related information for the calculation, wherein the element information section lists the detailed messages for each element

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in the geometrical model, including element ID number, part ID number to which the element belongs and node ID number that belongs to the element, etc. Similarly, the part information section includes part ID number and material ID number, and the node information section

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contains node ID number and the details of spatial coordinate for the node. Besides, these ID numbers correspond to each other uniquely. Thus, by changing the part ID number of an element

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one can change its material dependence correspondingly.

The approach for establishing the geometrical models of reinforced particles is as the following: Firstly, define the spatial range for particles in the FORTRAN program according to the dimension of composite specimen, and then randomly generate the spherical particles (circular particles for the 2-D model) within the spatial range based on Monte Carlo algorithm. The corresponding

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information includes parameters about the position as well as the diameter of the spherical (circular for 2-D model) particle. Secondly, establish the geometrical model and the corresponding meshes of specimen without reinforced particles in the pre-processing program of LS-DYNA, and

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in this procedure relatively fine meshes of specimen can be obtained. Finally, integrated with related information about the particle and the corresponding meshes of the geometrical model, and

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by employing the mapping algorithm, the geometrical model of reinforced particles can be achieved. The corresponding implementation steps are listed below: (1) Generation of reinforced particles Step 1 Input the dimension parameters of the composite specimen and define the spatial range of reinforced particles; Step 2 Calculate the amount of reinforced particles; Step 3 Randomly generate the position of a particle within the range of specimen, and define the corresponding diameter of particle simultaneously; Step 4 Check whether the non-superposition boundary condition is satisfied to avoid overlapping among particles (touch is allowed); 5

ACCEPTED MANUSCRIPT Step 5 If the generated particle satisfies the non-superposition boundary condition, record its corresponding parameters; otherwise delete the particle and perform a new generation until the generated particle satisfies the boundary condition; Step 6 Repeat steps 3-5 until the calculated amount of reinforced particles is reached. (2) Mapping of meshes for reinforced particles in the FEM geometrical model

Step 2 Calculate the central coordinate of each mesh element;

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Step 1 Generate regular meshes of specimen without reinforced particles;

Step 3 Check the position of each element with respect to that of each particle generated via approach (1) above;

Step 4 If the element center locates in one of particles, assign the element with particle material;

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otherwise assign it with MG matrix material.

According to related experiments [12-15, 17, 18, 20, 22-29, 32, 33, 35-51], two typical volume

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fractions of particle, V p =10% and V p =50%, are adopted for comparative analysis, and an average value of 50µm is set for the diameter of reinforced particles. Fig. 2 lists the 2-D geometrical models of the two composite specimens, and the zones labeled with “A′-C′” in the specimen with V p =50% (Fig. 2(b)) are the key points for the following analysis, and their distances to the top

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surface of specimen are, respectively, 0.5mm, 3.0mm and 5.5mm. The geometrical models are meshed with 4-node axis-symmetrical elements and the mesh size is 5µm, and Fig. 2(b) shows the magnifications of local meshes in the right bottom of specimen. Fig. 2(c) further shows the SEM

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micrograph of a real composite reinforced with tungsten and rhenium particles which occupy a volume fraction of V p =50% [13]. The amount of rhenium particles in the specimen is small (with

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a mass ratio of 6.25% among the total particles), and they will be regarded as tungsten material in the FEM simulation for convenience of analysis. By comparing Figs. 2(a-b) with Fig. 2(c) and with other similar experimental observations on the microstructure of composites (e.g., Refs. [37, 38, 47, 49, 50]), it can be seen that the distribution of reinforced particles in FEM models is similar to that in the real composites, i.e., the geometrical models represent the structures of composites well. It is seen from Figs. 2(a-b) that particles distribute randomly, and there are four sizes for their diameters, i.e., 70µm, 50µm, 30µm and 15µm, respectively. It is worth noting that in the composite shear bands from the metallic glass matrix cannot travel freely, whereas they will be arrested by 6

ACCEPTED MANUSCRIPT the reinforced particles. Consequently, the traveling distance (or usually called as the mean free path) of shear bands within the composite will dominate the deformation and failure characteristics of the material [38, 42-44, 47, 51]. This mean free path of shear band is an important parameter to evaluate the inner structure as well as the mechanical properties of particle reinforced metallic glass matrix composites, and it usually depends on the interspace between reinforced particles (or called the confinement zone size), which is determined by the particle size and the volume fraction

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of particle simultaneously. The confinement zone size is an effective index for the mean free path of shear band. Generally speaking, for a given particle size, the larger volume fraction of particle would lead to smaller confinement zone size and thus shorter mean free path. Additionally, the increment of particle amount will also induce more particle / matrix interfaces. Similarly, for a

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given volume fraction of particle, the smaller particle size would induce smaller mean free path of shear band and more interfaces. Figs. 2 (a-b) indeed show such a structure characteristic. It is seen

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that because the particle sizes are comparable, the amount of particles and the corresponding particle / matrix interfaces in the specimen with V p =50% is much larger than that in the specimen with V p =10%, whereas the confinement zone size in the former is much smaller than that in the latter.

Besides, in the FEM geometrical models the matrix and particles are fixed together by the

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shared nodes, and it indicates that these two phases are assumed to be ideally bonded, i.e., the adhesion of interface is strong. The interface between the tungsten reinforced phase and the MG matrix is usually well bonded [53]. Regarding to other reinforced material, a strong adhesion for

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the particle / matrix interface can also be achieved by improving related manufacture technology (e.g., Ref. [51]). Hence, the corresponding approximation of ideal particle / matrix interface with

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strong adhesion could reflect the microstructure feature for most composites, and it has a relatively good representativeness.

Moreover, shear-band zones in the MG matrix are also established. According to related analysis [61-63, 67-69], five kinds of shear-band zones are defined and their volume fractions in the matrix is 1%, the corresponding establishing method for the shear-band zones can be seen in our previous work [62]. From Fig. 2 it can be seen that all the shear-band zones indeed distribute randomly. The corresponding 3-D geometrical models are listed in Fig. 3. Similarly, due to the axis-symmetry feature, only a quarter of the geometrical model is established. The models are 7

ACCEPTED MANUSCRIPT meshed with 8-node hexahedral elements. Fig. 4 further shows the elements of shear-band zones in the specimens, and it is seen that the shear-band zones also distribute randomly in the MG matrix. The high-speed impact experiment is based on the penetrating test. The simulation here is mainly based on Choi-Yim et al [13]’s test, in which a WP/MG composite rod with a size of

φ 6.35mm×38.1mm was launched to penetrate a 102mm-thick 6061-T651 aluminum target, and

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the corresponding sketch is listed in Fig. 5. Similarly, related 2-D and 3-D simulations will be conducted to investigate the mechanical behaviors of composites under the high-speed impact condition. The corresponding geometrical models of composite rod are listed in Fig. 6. The zones

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labeled as “A-C” in Fig. 6(a) are also the key points in the following analysis, and the corresponding distances from the rod nose are, respectively, 2.5mm, 19mm and 37mm. Besides,

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related analysis demonstrated that in the impact case, the difference between the deformation in the shear-band zones and that in the matrix is little [61-63]. Thus, for convenience of establishing the model, the corresponding shear-band zones are neglected in the simulations.

2.2.1 Metallic glass (MG)

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2.2 Constitutive models

The authors have developed a modified coupled thermo-mechanical model and a failure criterion of critical free volume concentration for the MG material [61, 62], and they can describe

Appendix A.

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the mechanical behaviors of MG well. The corresponding constitutive equations are listed in

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In the current study, this constitutive model is also employed to describe the deformation and failure of MG matrix. For comparative analysis with the corresponding experimental results, the MG matrix in the simulations is selected as Zr41.2Ti13.8Cu12.5Ni10.0Be22.5 (Vit1), and related material parameters are listed in Table A1 in Appendix A. The detailed analysis on the parameters can be seen in our previous work [61, 62]. 2.2.2 Metals Regarding to the metal material, e.g., WPs and FePs in the composites and the 6061-T651 aluminum target in the impact test, similar to the authors’ previous analysis [63, 64], their mechanical properties are also described through the Johnson-Cook constitutive model [70] 8

ACCEPTED MANUSCRIPT combined with the accumulative damage failure criterion [71] as well as Gruneisen equation of state [72]. The corresponding equations are listed, respectively, in Appendixes B and C. Integrated with related analysis on the tungsten and aluminum materials [63, 64, 73-75], and based on the mechanical properties of metals in the above experiments [12, 13, 21, 24, 34], the corresponding parameters for tungsten (W) and 6061-T651 aluminum are listed in Table 1,

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wherein the letter “C” represents the compressive condition, “T” the tensile condition and “P” the penetrating condition. Additionally, for comparative analysis on the influence of mechanical property of particle material, another kind of particle material, STS304 stainless steel, is also

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selected as a sample in the simulations. This steel material displays a property of low strength and high plasticity, and its corresponding parameters are also listed in Table 1. More detailed analysis

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on the mechanical property of STS304 stainless steel can be seen in Ref. [76] and our previous work [63]. Besides, discussions about the plasticity of metals and the corresponding deformation and failure characteristics under different conditions can also be referred in the authors’ previous work [63, 64].

In addition, the other parts, e.g., the indenter and the base in the compressive experiment, and

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the fixtures in tensile experiment, etc., are all defined as elastic material.

3 Model validation and related analysis

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Based on the constitutive and geometrical models in section 2, FEM simulations and related analysis will be conducted in this section. The macroscopic mechanical behaviors of composites

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and the development of strain within the material under different deformation conditions will be emphasized. For simulations of quasi-static experiments, the load is defined as a constant velocity of the experimental equipment (see Fig. 1), and to save the solution time, the strain rate is taken as 10-2s-1. Our previous work demonstrated that such an approximation can still represent well the mechanical behaviors of MG and its composites [62, 63]. Regarding to the dynamic tests, integrated with the authors’ previous work [63] and related experiments [20, 21, 24, 30, 31, 34], the strain rate is adopted as 103s-1. Comparatively, regarding to the penetrating experiment, the load is taken as the actual impact velocity of the composite rod in Choi-Yim et al. [13]’s test, i.e., v0 =998m/s. 9

ACCEPTED MANUSCRIPT 3.1 Compressive experiment 3.1.1 Quasi-static compressive experiment The numerical stress-strain curves under the quasi-static compressive experiment for the two kinds of composite specimens are shown in Fig. 7, wherein Choi-Yim et al. [12, 13]’s test data are

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also listed for comparative analysis. It is seen that the numerical results agree relatively well with the corresponding test data, especially for the specimen with V p =10% the numerical curve almost superposes the test result during both the initial elastic response and the latter plastic deformation

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stage. Regarding to the specimen with V p =50%, there exists a deviation between the numerical result and the test datum in the initial deformation stage. The experimental curve doesn’t show the

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feature that an obvious initial elastic response is combined with latter plastic deformation. Besides, the modulus of composite begins to decrease significantly from the initial deformation stage, and it is even smaller than that of the MG matrix. These characteristics may be derived from the defects existing in the real composite, e.g., some particle / matrix interfaces with weak adhesion, voids

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between different particles, etc. These factors may lead to an earlier yielding of composite. In the specimen with V p =50% the amount of particles and their corresponding interfaces is relatively large, thus the amount of interfaces with weak adhesion and that of voids may increase, and

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correspondingly the deviation of numerical results from the test data is remarkable. Nevertheless, for some other similar composites with high volume fraction of particle and relatively strong

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interfaces, there exist a significant initial elastic deformation stage and a latter plastic deformation stage in the stress-strain curves [20, 21], being similar to the numerical result for the case of V p =50% in Fig. 7. Thus, it can be inferred that the simulation results are reasonable. Besides,

From Fig. 7 it is also found that the 2-D numerical results have relatively good agreements with the corresponding 3-D results. It is seen from Fig. 7 that the elastic modulus of the composite increases gradually with increasing the volume fraction of particle, and it almost follows the rule of mixtures for the elastic modulus and locates within the corresponding Hashin-Shtrikman bound [77]. When the material begins to yield, the yielding of MG matrix will induce a precipitous decrease of stress [61-69], and 10

ACCEPTED MANUSCRIPT curves in Fig. 7 show a certain softening characteristic. After the MG matrix yields, the initiation and propagation of shear bands in the material start immediately. During the propagation, shear bands will encounter the reinforced WPs and their propagation will be hindered by the particles. Consequently, it will lead to a complex deformation process within the composite. The shear band or shear crack is difficult to run through the composite specimen speedily, and thus the composite

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does not fails catastrophically like the case of monolithic MG, whereas behaves as a manner of plastic deformation. Correspondingly, the stress rises up again after a short duration of softening, and the stress-strain curves display a platform phenomenon during the latter stage. Besides, a

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certain serrated flow is seen in the curves. Moreover, it can be found that during the initial yielding stage the softening of stress in the specimen with V p =50% is less significant compared with that

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in the specimen with V p =10%, and this is just due to that the confinement zone size in the former is relatively small, and the hindering effect of particles becomes remarkable, thus the influence of the softening of MG matrix is weakened. In addition, the plastic strain of specimen with V p =50% is a little higher, and it is also derived from the decrease of the mean free path of shear band.

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Fig. 8 and Fig. 9 list the corresponding deformation and failure patterns of specimens, wherein the 2-D results in Fig. 8 are presented as the macroscopic form, and Figs. 8(c-d) further list the magnified images of the zones located in the dashed square frames in Figs. 8(a-b); comparatively,

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the 3-D simulation results are shown as the effective strain within the specimen. Furthermore, Fig. 10 lists the deformation and failure pattern of real composite specimen with V p =50% after the

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quasi-static compression [13]. By comparison among Figs. 8-10 it is seen that both the 2-D and 3-D simulations depict relatively well the main deformation and failure characteristics of composites. Both the composite specimens behave as a shear manner. Additionally, it can be seen that shear bands initiate from the MG matrix and then start to propagate speedily. During the propagation process, shear bands encounter the surrounding WPs, and at the same time this leads to an impact on the particle. However, it is relatively difficult for the shear bands to cut WPs off. Thus, failure mainly occurs in the MG matrix, and finally shear crack runs through the whole specimen. During this process only a few particles experience shear deformation, especially, for the specimen with V p =10%, because the confinement zone size is relatively large, the hindering 11

ACCEPTED MANUSCRIPT effect of particles on the propagation of shear bands is much less significant, and thus few particles are cut off (see Figs. 8(c-d)). Regarding to the real composite, Li et al. [24]’s investigation also demonstrated such a deformation feature. 3.1.2 Dynamic compressive experiment The dynamic compressive stress-strain curves for the two specimens are shown in Fig. 11. It is

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found that in the dynamic case, there is a certain increase for the yielding stress of composite compared with that under the quasi-static condition (see Fig. 7). It is demonstrated that the increase of yielding stress derives mainly from the positive response of tungsten material to the

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strain rate (see Appendix B, Table 1 and Ref. [63]). Besides, after the MG matrix yields, a softening feature also occurs in the stress-strain curve, and the two specimens also experience

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certain duration of plastic deformation. However, the stress-strain curves display remarkable vibration, implying that the deformation is unstable. Moreover, it is found that the stress softening in the specimen with V p =10% is more significant whereas its plasticity is lower compared with that of specimen with V p =50%. Additionally, by comparison between Fig. 11 and Fig. 7 it can be

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seen that the compressive engineering strain ε C that the specimen can sustain decreases when under the dynamic condition, especially for the case of V p =10% the decline is much significant, and it is also due to the relatively large mean free path of shear band in the specimen. Li et al.

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[24]’s experimental investigation on the similar composites also demonstrated the same tendency. Under the dynamic condition, the composite indeed displays a higher yielding stress and a lower

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plasticity compared with that in the quasi-static case. Similarly, the dynamic deformation and failure patterns of specimens are listed in Figs. 12-13, and the magnified images of the zones within the dashed square frames in Figs. 12(a-b) are also listed in Figs. 12(c-d). It is seen that under the dynamic condition, the deformation and failure of specimen transform to a local response from the global response in the quasi-static case, i.e., the deformation and failure zones are mainly localized at the top end which is compressed directly by the indenter, and the deformation in the parts is very little. Besides, as shown in Figs. 12(c-d), for the two specimens, after the top end is compressed, shear bands initiate and propagate speedily from the MG matrix and further impact onto the 12

ACCEPTED MANUSCRIPT reinforced WPs. However, it is still difficult to cut the particle off completely, and failure also mainly propagates along the MG matrix, especially in the specimen with V p =10%. During the further compression process, WPs at the top end also experience certain plastic deformation and become as elliptic shape from the initial round shape, and thus it induces a certain blunting

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deformation at the top end of specimen. When failure occurs in the dominate shear plane, the top part of specimen slips along the shear fracture area and separates from the main body, being just like the case of the monolithic MG material [3, 62]. Finally, by comparing with the quasi-static case (see Figs. 8 and 9) it can be found that under the dynamic condition, the impact of the shear

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bands on WPs is much more intense, and thus the amount of WPs with shear deformation or shear damage increases.

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Fig. 14 shows the dynamic deformation and failure patterns of real WP/MG specimens, wherein Fig. 14(a) is that of a composite rod with V p =70% under the instrumented anvil-on-rod impact in which the local strain rate achieves a value of about 3×103s-1 [3, 34], and Fig. 14(b) is that of a composite specimen with V p =60% in the SHPB test with a strain rate of 1×103s-1 [24].

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By comparison among Figs. 12-14 it can be seen that the simulation results have captured the main deformation and failure characteristics of composites well. In special, in the dynamic case, the deformation at the bottom end of specimen is little, whereas remarkable shear deformation and

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failure occur at the top end which is impacted directly, and cracks also mainly propagate along the MG matrix. Additionally, shear deformation in WPs within the deformation zone is much more

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remarkable than that in the quasi-static condition, and shear failure is easier to occur in WPs. Hence, it could be inferred that the impact of shear bands on the particle will become more intense under higher strain rate.

3.1.3 High-speed impact experiment From section 3.1.2 it can be known that the deformation of composites transforms to the local response gradually with increasing the strain rate, and the shear feature of reinforced particles becomes more significant. The deformation and failure characteristics under the high-speed impact condition will further be investigated in this section. Fig. 15 lists the simulation results for Choi-Yim et al. [13]’s penetration of composite rod with V p =50% into 6061-T651 aluminum 13

ACCEPTED MANUSCRIPT target at v0 =998m/s, and it is seen that the numerical results accord with the test data well. It can be found that in the case of extremely high strain rate (a high value of about 106s-1 for the local strain rate in the deformed material [13]), the localization of deformation and failure in the long rod becomes much more significant. The rod nose behaves as significant shear manner,

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and it becomes as a sharp shape, i.e., a “self-sharpening” behavior occurs [13, 63, 64]. Thus, it is known that in the case of extremely high strain rate, the composite material still behaves as shear deformation, and the local response feature is much more remarkable.

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3.2 Tensile experiment 3.2.1 Quasi-static tensile experiment

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Similarly, the numerical stress-strain curves under the quasi-static tensile experiment are shown in Fig. 16, wherein Choi-Yim et al. [12]’s experimental result for the composite with V p =5% is also included. It can be seen that the composite behaves as brittle fracture under tension,

and the tensile engineering strain ε T that the specimen can sustain is relatively small, even less

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than that of the monolithic MG specimen. This is because the tensile plasticity of tungsten is remarkably less than its compressive plasticity (see Table 1), and it is also less than the tensile plasticity of MG material (see Refs. [61, 62]). Moreover, with increasing the value of V p , the

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elastic modulus of composite increases gradually, however, the fracture strain further decreases. This implies that the influences of yielding and failure of reinforced particles on the mechanical

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properties of composite become more considerable with increasing the volume fraction of WPs. The corresponding 2-D and 3-D forms of specimens are listed in Figs. 17 and 18, respectively, and Fig. 19 shows that of composite specimen reinforced with tungsten balls (with particle diameter of 1.5mm and V p =52%) [29]. From Figs. 17-19 it is also found that the simulations depict relatively well the test result. All the specimens display brittle fracture, and the fracture surfaces are almost perpendicular to the loading direction. Choi-Yim et al. [12] and Conner et al. [17]’s corresponding tension tests also demonstrated that corresponding to different values of V p , fracture surfaces of specimens are inclined to be perpendicular to the loading direction. During the tension process, the specimen fails speedily along the dominant fracture surface, and shear bands 14

ACCEPTED MANUSCRIPT in the MG matrix are unable to achieve a sufficient propagation in this duration. Because the tensile plasticity of tungsten material is lower compared with that of MG matrix, failure occurs in some WPs located near the fracture surface. However, WPs are still well bonded with the MG matrix. In addition, from Figs. 17 and 18 it can also be found that the propagation path of crack in

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the specimen with V p =50% is more sinuous compared that for the case of V p =10%, and the reason is that the smaller confinement zone size in the former leads to more significant hindering effect of particles on the propagations of shear bands and cracks, and thus the deviation of propagation path is more remarkable.

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3.2.2 Dynamic tensile experiment

The corresponding simulations show that under the dynamic tension, the stress-strain curves as

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well as the deformation and fracture forms of specimens are similar to that under the quasi-static tensile condition (see Figs. 16-18), and only the deformation and failure are further localized into the top end of specimen, besides, the critical engineering strain at fracture decreases a little compared with that in the quasi-static case (see Fig. 16), being consistent with Xue et al. [31]’s experimental investigations.

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From the above FEM simulations it can be concluded that related simulations capture the main deformation and failure characteristics of composites well, and thus it has reference significance for practical application and the corresponding analysis based on the simulations is physically

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meaningful. Besides, it can be seen that the 2-D simulation is also able to represent the main mechanical properties of composites relatively well. For investigating directly the deformation and

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failure process of material and reducing the amount of calculation, in the following related analysis will mainly base on 2-D simulations.

4 Influences of internal and external factors on the mechanical properties of composites In this section the influences of different factors on the mechanical properties of composites, especially on the shear banding behaviors within the material, will be further analysed. From Section 3 it is known that several factors have relatively significant effects, e.g., the volume fraction of particle and the strain rate. Regarding to the volume fraction of particle, one of its influence is the amount of particles; Besides, different volume fraction of particle will lead to 15

ACCEPTED MANUSCRIPT different amounts of particle / matrix interfaces and different confinement zone sizes, and thus the mean free paths of shear bands and cracks within the composites differ from each other. For the present two kinds of composites, the particle sizes are comparable, and thus corresponding to the composite with higher volume fraction of particle, the mean free path of shear band is shorter, whereas the amount of particle / matrix interfaces is larger. Furthermore, related investigations

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show that the mechanical property of particle material also plays an important role [49], and the authors’ previous analysis on the fiber reinforced composites also demonstrated such a feature [63]. These three factors will be analyzed in detail in the following. For the convenience of comparative

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analysis, the corresponding discussion will mainly regard to the compressive condition.

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4.1 Volume fraction of particle (the amount of particle and the inter-particle spacing) Considering the shear band can achieve more sufficient propagation under the quasi-static condition, the corresponding discussion will be based on this deformation case. The deformation process as well as the corresponding distributions of strain within specimens during the quasi-static compression is shown in Figs. 20 -21.

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From Fig. 20 it is seen that the introduction of WPs leads to stress concentration in the MG matrix located around the particle. During the early elastic deformation stage, the deformation in MG material near the particle is obviously larger than that in other zones, as seen in Fig. 20(a)

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which shows the morphology at the engineering strain of ε C =1%. Comparatively, because the elastic modulus of tungsten particle is about 4 times of that of MG matrix (see Table A1 and Table

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1), and its yielding stress is also relatively high (see Table 1), the deformation in WPs is significantly smaller than that in MG matrix. After certain duration, the MG matrix begins to yield, and shear bands initiate from the shear-band zones located near particles, and then they begin to propagate speedily. It is found from Fig. 20(b) that there are a large number of dominant and secondary shear bands in the MG matrix. For the composite with V p =10%, because the amount of particles is small and their distributions are relatively sparse, the confinement zone size is large, and correspondingly the mean free path of shear band is relatively long. Consequently, the hindrance for the propagation of shear bands is 16

ACCEPTED MANUSCRIPT relatively weak, and thus the change of their propagation directions is not significant, and shear bands can achieve a relatively long propagation. During the propagation process, shear bands intersect with each other, and thus more secondary shear bands are promoted. Besides, it can also be found that due to the intersections of shear bands, the deformation within several main shear

deformation is usually controlled by one dominant shear band.

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bands is almost the same, being different from the case of monolithic MG in which the

With continuing the loading, crack begins to initiate in someone dominant shear band, and correspondingly the deformation is localized into this shear band. As aforementioned, the

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deformation within many main shear bands is almost the same with each other, and thus failure may occur simultaneously in several shear bands, as seen in the dashed circle frame in Fig. 20(c)).

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In the early stage shear crack propagates along the direction of shear band. When the crack encounters WPs, it runs along the edge of particle and can’t cut the particle off. Finally, cracks in different zones in the specimen coalesce together and run through the whole specimen. During this process the engineering strain ε C only increases by a little value of 0.3%. Besides, as shown in Fig. 20(d), the fracture surface may display a step feature, according with

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Kim et al. [22]’s test results. At the same time, there are also several micro-cracks in other zones. Additionally, it can be found that due to the high elastic modulus and relatively high yielding stress, the deformation in WPs is always significantly smaller than that in the MG matrix, and only a few

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particles located near the crack show a certain plastic deformation.

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For the specimen with V p =50%, due to that the amount of WPs and their corresponding interfaces is large, the confinement zone size decreases remarkably, and thus the stress concentration in the MG matrix becomes more significant. As seen in Fig. 21(a), during the early elastic response, the deformation in the MG matrix located around particles is significant and a little larger than that in the specimen with V p =10% (see Fig. 20(a)). When the MG matrix yields, shear bands also initiate from the shear-band zones. Due to that the confinement zone size is small, the corresponding confinement on the MG matrix is relatively strong, and thus the hindrance for the propagation of shear band is remarkable. From Fig. 21(b) it is found that shear bands also mainly propagate along the edge of particle, and the propagation 17

ACCEPTED MANUSCRIPT direction is swerved. Besides, because of the hindrance effect of particle and the intersection among shear bands, their propagation distances are relatively short, and the deformation within shear bands is larger compared with the case in the specimen with V p =10% (see Fig. 20(b)). Furthermore, the amount of promoted secondary shear bands also increases significantly.

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As the deformation continues, the MG material within shear bands begins to fail. Similarly, failure initiates from several shear-band zones simultaneously, and then cracks begin to propagate in the specimen, as shown in the dashed circle frame in Fig. 21(c).

Finally, cracks coalesce together and further propagate along the edges of particles. Because of

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the hindrance effect of dense particles, the mean free path of crack is relatively short, and thus the corresponding propagation becomes a little slower, and when the dominant crack runs through the

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specimen the increase of engineering strain ε C achieves a value of about 1%, as seen in Fig. 21(d). Besides, it is seen that the amount of micro-cracks in other zones is also larger than that in the specimen with V p =10% (see Fig. 20(d)). Moreover, the deformation of WPs is also small during the whole compression process, and the amount of WPs with plastic deformation is still

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relatively small.

Related tests also demonstrated the above deformation and failure characteristics. Fig. 22 lists the SEM micrographs of a WP/MG composite specimen with V p =7% after quasi-static

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compression [22] and that of a specimen reinforced with tungsten balls with V p =52% [28]. From Fig. 22(a) it is seen that there are a large amount of shear bands which intersect with each other. In

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the intersections propagation directions of shear bands change and even the propagation is stopped, and thus secondary shear bands are further promoted. Such intersections among shear bands significantly delay the occurrence of failure in the MG matrix, and correspondingly the plasticity of composite is improved. In Fig. 22(b), for the case that the space for MG matrix is narrow, the propagation of shear band is usually only within the region between two adjacent particles, i.e., the propagation distance decreases significantly. Accordingly, much more secondary shear bands are promoted in the matrix after the propagation is hindered, and in the zone with shorter confinement zone size there are more promoted shear bands, and thus the plasticity of composite is significantly improved. For convenience of comparative analysis, Figs. 22(c-d) further list the magnified images 18

ACCEPTED MANUSCRIPT of zones within the dashed square frames in Fig. 20(b) and in Fig. 21(b), and it can be found that the simulation results also demonstrate the similar deformation features. Consequently, it can be known that corresponding to different volume fractions of particles, composites mainly behave as a shear manner. Besides, the introduction of reinforced particles can effectively delay the occurrence of failure in composites, and regarding to both the low and the

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high volume fractions of particle, the compressive plasticity of composite can be improved significantly. Especially, for the given particle size, within the composite with high volume fraction of particle the amount of particle / matrix interfaces is large, thus the confinement zone

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size is small, and the mean free path of shear band is short. Consequently, the improvement for the plasticity of composite with higher volume fraction of particle is more significant in the case of

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strong interface adhesion.

4.2 Mechanical property of particle material

Integrated with Lee et al. [76]’s investigation and the authors’ previous work [63], another kind of particle material, STS304 steel, is introduced in this section, and comparative analysis will

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be conducted by comparing with the composite reinforced with WPs to discuss the effect of particle material. Related analysis will mainly consider two factors, i.e., the strength and the plasticity of particle material. For convenience of comparative analysis, the deformation as well as

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the development of effective strain in the corresponding FeP/MG composite specimens during the quasi-static compression is listed in Figs. 23 and 24, respectively.

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Different from the case of WPs, because the yielding stress of steel is far less than that of MG material (see Table 1 and Table A1), FePs yield speedily in the early deformation stage. For the specimen with V p =10%, almost all the FePs yield when the engineering strain ε C increases to a value of 1% and the deformation is remarkably larger than that in the MG matrix, as shown in Fig. 23(a). Due to the plastic deformation of particles, interaction between the MG matrix and the reinforced particle is much more significant than the case in the WP/MG composite. Consequently, the stress state within the specimen is significantly changed, and thus much more nucleation points of shear band are promoted in the MG matrix. When the MG matrix begins to yield, shear bands also initiate and propagate in the matrix. It is 19

ACCEPTED MANUSCRIPT seen that the deformation within the shear bands located between two adjacent particles is also considerable compared with that in FePs, and these shear bands string particles which with significant deformation together, as shown in Fig. 23(b). Besides, it is also found that because of the further plastic deformation, the deformation degree in FePs is still higher than that in shear bands in MG matrix.

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In the latter deformation process, particles continue their plastic deformation and keep on interacting with the MG matrix, and thus they are further compressed into certain oblate shape. Consequently, the interaction between the MG matrix and particles becomes more severe, and

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much more secondary shear bands are promoted in the MG matrix. At the same time, secondary shear bands also intersect with each other. During these complex deformation processes, the

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specimen experiences a relatively long duration of plastic deformation and doesn’t fail. Besides, the deformation localization feature in the specimen further becomes less remarkable. As shown in Fig. 23(c), except zones located near particles, deformation in other regions of MG matrix is relatively homogeneous. Moreover, it is seen that the deformation within FePs is still larger than that in the MG matrix.

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Finally, failure occurs in some main shear bands in the MG matrix, and then cracks begin to propagate and further impact on the surrounding FePs. Due to that FePs have experienced significant plastic deformation, they are easily to behave as shear manner after the impact of

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cracks from the MG matrix, i.e., shear cracks are easily to run through FePs (see Fig. 23(d)), being significantly different from the case of WPs (see Fig. 20 (d)). Besides, it can also be found that

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after the plastic deformation, most of FePs display as an oblate shape. Correspondingly, the plasticity of FeP/MG composite is further improved, and the critical strain at failure achieves a value of 16.5% (see Fig. 23(d)), being remarkably higher than that of the WP/MG composite (7.3%, see Fig. 20(d)).

For the specimen with V p =50%, because the amount of FePs with plastic deformation increases obviously, and the confinement zone size remarkably decreases, deformation within the specimen is more homogeneous than that in the specimen with V p =10%, and deformation in particles is a little lower (see Figs. 23(a) and 24(a)). During the latter stage, because there are lots 20

ACCEPTED MANUSCRIPT of FePs with plastic deformation and they squeeze the MG matrix intensely, deformation localization in the specimen further becomes less remarkable, and the occurrence of failure is further delayed. Thus, the plasticity of specimen is improved compared with the WP/MG composite, achieving a value of about 27% (see Figs. 21(d) and 24(d)). Similarly, during the failure process shear cracks can cut FePs off and leads to a shear manner in particles.

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The corresponding deformation characteristics are also demonstrated in real tests. Lee et al. [23]’s investigation on the tantalum particle (with a strength of about 800MPa) reinforced composite in the quasi-static compression found that severe plastic deformation occurs in particles,

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and some particles are cut off after the impact of shear bands, as shown in Fig. 25(a), wherein the insert is a magnified image. Similarly, for convenience of comparative analysis, Fig. 25(b) further

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lists the magnified image of zones located near the crack in the specimen (dashed frame) in Fig. 23(d), and it can be found that the simulation result shows the same feature with the experimental result. Just because of the remarkable plastic deformation of reinforced particles, the plasticity of composite is significantly improved.

From the above analysis it can be seen that the mechanical property of particle material also

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plays an important role. Generally speaking, when the strength of the metal particle is larger than that of the MG matrix, the deformation in particles is relatively small during the compression process, and the propagation of cracks in composites is mainly along the MG matrix.

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Comparatively, regarding to the case that the strength of the particle is lower than that of MG matrix, particles will yield early in the deformation process and further experience significant

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plastic deformation. This will further weaken deformation localization in the composite, and it is beneficial to improve the plasticity. When the composite begins to fail, particles with extreme deformation are also easy to be cut off by the shear crack from MG matrix, i.e., cracks will mainly propagate along reinforced particles. Moreover, corresponding to the higher particle plasticity, it is harder to be cut the particle off, and thus it is more favorable for improving the plasticity of composites.

4.3 Strain rate It is found from the above analysis that under the dynamic compression, the composite also 21

ACCEPTED MANUSCRIPT behaves as shear manner, however, the deformation is localized gradually at the end which is impacted directly, and shear deformation in reinforced particles also becomes more remarkable. The effect of strain rate will be analyzed in this section. For convenience of comparison with related test data, the WP/MG composite with V p =50% is selected as the sample. The deformation

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process of the cylindrical specimen under dynamic compression is listed in Fig. 26, and that of the composite long rod and aluminum target during the penetration is shown in Fig. 27, wherein Fig. 26(d) is the magnification of zones within the dashed frame in Fig. 26(b), and Fig. 27(d) is the zoom of rod nose in Fig. 27(b).

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From Fig. 26 it is seen that in the early stage, significant plastic deformation has occurred at the top end of specimen, and it leads to certain blunting deformation, as shown in Fig. 26(a).

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Besides, a cone-shaped deformation area is formed at this end, being similar to the case of monolithic MG rod under impact condition [3, 62]. As the compression continues, the deformation zone expands towards the back end, and at the same time shear bands from the MG matrix continue to impact on the surrounding WPs. It is found that under the dynamic condition, shear deformation or damage occurs in WPs due to the impact of shear bands, as seen in particles

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labeled with dashed frames in Fig. 26(d). Finally, cracks initiate in some shear bands and run through the specimen speedily, and this leads to the decrease of carrying capacity of the specimen, as shown in Fig. 26(c). Furthermore, it can also be found that the top end of specimen further

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becomes blunter, and remarkable plastic deformation occurs in WPs within the top end. Especially, some particles fail along previous shear deformation zones, indicating that the impact of shear

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band on the particle becomes more intense under the dynamic condition. Comparatively, there is no obvious deformation in the middle and bottom parts of specimen, i.e., significant deformation localization occurs, being different from the case under the quasi-static condition (see Fig. 21). As shown in Fig. 27, in the early stage of penetration, the MG matrix in the rod nose fails speedily, and extreme plastic deformation occurs in the surrounding WPs (see Fig. 27(a)). Then materials with shear deformation in the rod nose are squeezed by the target material and further separate from the rod nose. Thus, the rod nose will become as a sharp shape (see Fig. 27(b)). In the subsequent penetration process, materials in the sharp nose continue to peel out and flow backwards, and the rod nose keeps on the “self-sharpening” behavior (see Fig. 27(c)). 22

ACCEPTED MANUSCRIPT Besides, it can be seen that during the penetration process shear cracks initiate speedily in the rod nose and run through the composite rod immediately. This leads to that materials within the front part separate fleetly from the rod and further results in a “self-sharpening” performance, as shown in Fig. 27(d). Similar to the case of a tungsten fiber reinforced (WF/MG) composite rod [63, 64], this separated part in the front of composite rod can be called “edge layer”. From Fig. 27(d) it

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is seen that within the “edge layer” materials break up due to the squeeze of target material. Comparatively, there is almost no obvious deformation in materials located behind the edge layer. Furthermore, the amount of WPs with shear deformation and failure further increases considerably.

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Therefore, it can be known that in the case of high-speed impact, shear feature of composites becomes more remarkable, and the deformation localization characteristic is also more significant.

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In the following the variation of stress within composites will be further investigated, and the analysis is based on the corresponding key points in the above composite specimen (Fig. 2(a)) and in the composite rod (Fig. 6(a)). Fig. 28 lists variations of effective stress in the key points, respectively, under the dynamic compression and in the penetration process. From Fig. 28(a) it can be seen that due to the high shear sensitivity of MG material [61, 62],

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the stress within the MG matrix at the top end of specimen increases significantly after the compression, and then the matrix begins to yield. After the yielding, the MG material softens speedily and fails immediately. Besides, the yielding stress of MG doesn’t achieve considerable

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increase compared with that in the quasi-static case. This is consistent with related experimental investigation [78]. For the surrounding WP, the stress also increases quickly to the yielding point.

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After the failure of MG matrix, the WP doesn’t fail at once, whereas still experiences a duration of plastic deformation until about 67µs after the compression. The main reason is that the impact of shear band or shear crack from the MG matrix cannot immediately cut the WP off. Regarding to the middle and bottom regions of specimen, when the impact stress wave arrives, the stress within the WP also increases to the yielding value, and then the particle further behaves as plastic deformation. Comparatively, for the MG matrix, due to the dependence of its strength to the strain rate (see Appendix A and Table A1), the increase of stress is slow at the beginning, and then a sharp drop occurs immediately after the yielding. Due to that the high-speed impact induces an extremely high strain rate (about 106s-1), the 23

ACCEPTED MANUSCRIPT strength of MG matrix located in the rod nose increases remarkably to a high value of about 7.5GPa, as shown in Fig. 28(b). For the surrounding WP, because of the strong impact of shear band and the squeeze effect of the target material, the particle also fails speedily after the failure of MG matrix. For the middle part of rod, the stress in the WP also increases to its yielding value at the arrival of impact stress wave, and then the WP undergoes the plastic deformation.

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Comparatively, regarding to the MG matrix, the increase of stress is also slow when the stress wave arrives, and before the material is going to yield, the stress increases quickly to a high value, at the end it further drops rapidly. Similarly, the surrounding WP is cut off immediately after the

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failure of MG matrix. At the back end, for the back surface is free, the corresponding stress is always low. Consequently, under the high-speed impact, the MG matrix can achieve a high

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strength before failure, and thus it is able to prevent the deformation of the surrounding WPs. Comparatively, after the failure of MG matrix, materials within the rod nose also fail speedily as a shear manner and separate from the rod. Hence, the rod displays a “self-sharpening” behavior and its penetrating capability achieves an improvement [13].

Hence, it is seen that the strain rate also has a significant influence. The impact of shear band

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derived from the MG matrix onto the reinforced particle will become more intense with increasing the strain rate, and thus shear deformation or damage in the particle is more remarkable. Consequently, failure is more easily to propagate along the pre-damage zone in the particle during

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the subsequent deformation process, and thus the shear characteristic of composite becomes more significant. Especially, for the ultrahigh strain rate, the MG matrix further achieves a remarkable

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improvement of its strength, and it is also favorable for the strength of whole composite.

5 Discussions

The above simulations depict the main mechanical properties of composites relatively well. It is worth noting that for convenience of investigating the deformation and failure process in the material and for saving the solution time in the software, 2-D models are employed, and for the 3-D simulation, only a quarter of the geometrical model is established. This doesn’t mean that the inner structures and the deformation and failure morphologies of composites have the corresponding symmetrical characteristics. For comparative analysis, the whole model of a smaller 24

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WP/MG specimen ( φ 1.5mm × 3mm) with V p =50% is established, and the corresponding simulation results for quasi-static compression at ε C =8.2% is listed in Fig. 29. It is seen that the fracture surface displays as a single plane without symmetrical feature, and the morphology is more consistent with the test result (see Fig. 10). This validates again the applicability of the

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constitutive and geometrical models. Besides, for the real composites, there exists an interface layer with a thickness of about 250nm between the reinforced particle and the MG matrix, and the adhesion of interface usually depends on the physical properties of the particle and matrix materials as well as the manufacture

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technology [11, 12, 14-16, 51, 53, 79]. In addition, because of the difference between the physical properties of the particle and matrix materials, e.g., the thermal expansion coefficient and the

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elastic modulus, etc., there may be some residual stress within the composite, especially in the interface layer [14, 15, 80]. These factors will affect the mechanical behaviors of composite at a certain extent. Moreover, the particle size also contributes to the deformation and failure characteristics of composites, Choi-Yim et al. [15] and Jang et al. [44]’s experimental

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investigations demonstrated that when the diameter of reinforced particle is larger than the width of shear band in MG matrix, the particle can hinder the propagation of shear bands effectively, and thus regarding to any volume fraction of particle, the plasticity of composite can be improved. The corresponding simulations above also show such a feather. Especially, for a given particle size,

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increasing the volume fraction of particle would lead to more interfacial areas; similarly, regarding

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to a given volume fraction of particle, decreasing the particle size will also induce more interfaces. Along with increasing the amount of particles and their corresponding interfaces, the plasticity of composite may be improved gradually when the interface adhesion is strong [38, 42-44, 47, 51]; comparatively, for the weaker interface adhesion, the interface layer may become the main site where the material failure initiates, and thus the variation of plasticity with the increment of interfacial areas does not show a monotonous tendency [50]. In the present numerical simulations an approximation of ideal interface with strong adhesion is adopted for all cases, and the residual stress in the interface is neglected, thus there is a little deviation between the simulation result and the test data for the minutia of deformation and failure 25

ACCEPTED MANUSCRIPT forms. To discuss the influences of these factors, it needs to establish a much finer geometrical model, e.g., the particle / matrix interface layers should be included in the model, and the mesh size needs to be set smaller than its thickness (~250nm). Besides, various adhesion levels and residual stress state in the interface layer should be considered at the same time. This will lead to a remarkable increment for the loading of calculation, and maybe it needs to conduct multi-scale

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simulations. At present related numerical simulations [19, 20, 23, 27, 33, 49, 54-58, 63, 64] seldom focus on the interface layer, and the deeper and more detailed investigations on the particle

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/ matrix interface will be conducted in our future work, and it will be arranged as a special subject.

6 Conclusions

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In the present manuscript, a modified coupled thermo-mechanical constitutive model is employed to describe the mechanical behaviors of the MG matrix, and the geometrical models of particle reinforced MG matrix composites are established based on their inner structures. Integrated with the corresponding experimental investigations, FEM simulations and related analysis on the mechanical properties of composites are conducted systemically. Especially, shear

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banding behaviors within composites are discussed in detail. Additionally, the influences of several internal and external factors on the deformation and failure characteristics of composites are analyzed.

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Related analysis demonstrates that under the compressive condition, the composite behaves as a shear manner, and its plasticity is significantly improved compared with that of monolithic MG.

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Comparatively, in the tensile case, the composite will show a feature of brittle fracture and the fracture surface is nearly perpendicular to the loading direction, and thus its tensile plasticity is relatively hard to be improved. The mechanical properties of composites depend firmly on the shear banding behaviors within the material, including the initiation and propagation of shear band, and the shear banding-induced failure, etc. In the deformation process, shear bands initiate and propagate from the MG matrix, and then they impact on the surrounding reinforced particles. At the same time, particles will hinder the propagation of shear bands, and this will also promote the nucleation and propagation of secondary shear bands. Regarding to a given particle size, for the composite with low volume 26

ACCEPTED MANUSCRIPT fraction of particle, the interspace between particles (confinement zone size) is large, and thus the mean free path of shear band in the material is relatively long, however, shear bands intersect with each other severely; For the case of high volume fraction of particle, the confinement zone size decreases, and the mean free path of shear band becomes much shorter, the shear band is hard to propagate sufficiently due to the significant hindrance of particles. Thus, corresponding to various

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volume fractions of particle, the plasticity of composites can be improved effectively. Especially, in the case of strong adhesion of particle / matrix interface, the plasticity of composite increases with increasing the amount of reinforced particles as well as their corresponding interfaces.

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Besides, the higher particle plasticity will be more beneficial to the plasticity of composites. Moreover, when the particle strength is high, failure of composites mainly propagates along the

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MG matrix, and comparatively, for the low particle strength, the reinforced particle will yield early and then undergo the plastic deformation. This is beneficial to weaken the deformation localization in composites, and particles with low strength are also easily to be cut off by the shear band from the MG matrix. Additionally, with increasing the strain rate, the impact of shear band on the

behave as a shear manner.

Acknowledgements

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reinforced particle will become more intense, and thus it induces that the particle is more easily to

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The project is supported by the Science and Technology Development Foundation (2015B0201025) as well as the key subject “Computational Solid Mechanics” of China Academy

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of Engineering Physics, the National Outstanding Young Scientists Foundation of China (11225213) and the National Natural Science Foundation of China (11521062, 11602258). The authors are grateful to Mr. T.H. Lv, Mr. Z.P. Shen and Prof. G. Chen for their assistance and discussion in the FEM simulations.

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Appendix A: Modified coupled thermo-mechanical constitutive model

By considering all the effects of free volume, temperature and hydrostatic stress on the

constitutive model as [61, 62]

ε&ij = ε&ij e + ε&ij p 1 +ν E

 & ν &  σ kk δ ij   σ ij − 1 +ν  

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ε&ij e =

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deformation and failure of MG material, it can be obtained a modified coupled thermo-mechanical

 ∆G m   σ e' Ω  1 − exp sinh      ξ  K BT   2 K BT

ε&ij p = f exp  −

 Sij  σe

σ e = 3J 2

σ e' = Λσ m + J 2  ∆G m f exp  − α  K BT 1

  1   2 K BT exp   − ξ   ξ v* s    

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ξ&=

  σ e' Ω cosh    2 K BT 

  1   − 1 −    nD 

EP

β p T&= TQ σ e' ε& ij ρ Cv

(A1a) (A1b) (A1c) (A1d) (A1e) (A2)

(A3)

g

In Eqs. (A1-A3), a dot over a quantity

( )

denotes the differentiation with time.

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In Eq. (A1) E and ν are the Young’s modulus and Poisson’s ratio, respectively; f is the frequency of atomic vibration; ∆G m is the activation energy; K B is the Boltzmann constant and T the absolute temperature; ξ = v f

(α v ) *

is the free volume concentration in MG (here v f ,

v* and α are, respectively, the average free volume per atom, critical volume and a geometrical

factor); Ω is the average atomic volume; σ e is the von-Mises effective stress, in which J 2 = Sij Sij 2 is the J 2 invariant of stress, Sij = σ ij − δ ijσ m is the deviatoric stress tensor and

σ m = σ kk 3 the mean stress; σ e' is an effective stress which considers the contribution of 32

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hydrostatic stress, in which Λ is the hydrostatic stress sensitivity factor, and its value varies with the stress state, i.e., the value of Λ C under compression is different from that of ΛT under tension. Eq. (A2) is the free volume evolution equation, in which s = E 3 (1 −ν )  is the Eshelby

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modulus and nD the number of diffusive jumps necessary to annihilate a free volume as v* . Eq. (A3) is the temperature evolution equation, where ρ and Cv are the density and the specific heat at constant volume, respectively; ε ep = 2ε ijpε ijp 3 is the effective plastic strain. βTQ

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is the Taylor-Quinney coefficient and considered to be a function of the effective plastic strain rate. Furthermore, when the net free volumes exceed a critical value, failure occurs in the MG

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material, and thus the corresponding failure criterion is defined as that the free volume concentration ξ exceeds the critical value ξc :

ξ ≥ ξc

(A4)

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Appendix B: Johnson-Cook constitutive model and cumulative damage failure criterion The corresponding expression of Johnson-Cook model is [70] m σ = ( A + Bε p n ) 1 + C ln ε&*  1 − (T * ) 





(B1)

EP

where A , B , C , n and m are the material constants. A represents the yield stress at the reference strain rate ( ε&* =1), B and n represent the effect of strain hardening, C is the strain

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rate constant and m characterizes the thermal softening. ε&* = ε&p ε& 0 is the dimensionless plastic strain, in which ε& is the reference strain rate and usually takes a value of 1s-1. 0

T * = (T − Tr ) (Tm − Tr ) is the homologous temperature, wherein Tm is the melting temperature, Tr is the reference temperature and usually taken as the room temperature. The temperature rise in the material is derived from the deformation energy, and the corresponding equation is dT =

βTQ σ e ⋅ d ε ep ρ Cv 33

(B2)

ACCEPTED MANUSCRIPT In the calculation it is usually assumed as the adiabatic deformation, and the value of Taylor-Quinney coefficient βTQ in Eq. (B2) is taken as 0.9. In the cumulative damage failure criterion a degree of damage, D , is defined to represent the failure of material, and the corresponding expression is [71] ∆ε p

(B3)

εf

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D=∑

The value of D ranges from 0 to 1. When there is no deformation in the material, D =0. Comparatively, when the damage parameter increases up to its critical value, i.e., D =1, fracture

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occurs in the material. In Eq. (B3) ∆ε p is the increase of effective plastic strain in a time step, and ε f is the critical strain at failure of material with an expression of

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ε f = [ D1 + D2 exp( D3σ * )][1 + D4 ln ε&* ][1 + D5T * ]

(B4)

In Eq. (B4) σ * = P σ e is the ratio of pressure divided by the effective stress, in which P is the pressure. The parameters of D1 - D5 are the material constants.

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Appendix C: Gruneisen equation of state

The Gruneisen equation of state defines the pressure for the compressed material as [72]





γ0 

a  µ − µ2   2 2 

µ2 µ3 [1 − ( S1 − 1) µ − S 2 − S3 ] ( µ + 1) 2 µ +1

+ (γ 0 + a µ ) ⋅ e0

(C1)

EP

P=

 

ρ 0C0 2 µ 1 +  1 −

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where e0 is the initial internal energy in the material; C0 is the intercept of the vs − v p curve; S1 - S3 are the coefficients of slope of the vs − v p curve; vs is the shock wave velocity and v p the particle velocity; γ 0 is the Gruneisen coefficient; a is the first order volume correction to

γ 0 ; µ = ρ ρ 0 − 1 , in which ρ is the current density and ρ 0 the initial density.

34

ACCEPTED MANUSCRIPT Lists of Tables and Figures Table 1 Johnson-Cook model parameters of the metals Table A1 Parameters of Zr41.2Ti13.8Cu12.5Ni10.0Be22.5 (Vit1) metallic glass, Ref. [61, 62] Sketch of FEM geometrical model for the quasi-static and dynamic compressive and tensile experiments Fig. 2 2-D models of composite specimens with different volume fractions of particle: (a) Specimens, (b) Magnifications of local meshes in the right bottom of specimens, (c) SEM micrograph of real composite with V p =50%, Ref. [13]

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Fig. 1

3-D models of composite specimens with different volume fractions of particle: (a) V p =10%, (b) V p =50%

Fig. 4

Shear-band zones in 3-D composite specimens with different volume fractions of particle: (a) V p =10%, (b) V p =50%

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Fig. 3

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Fig. 5 Sketch of penetrating experiment for the composite rod Fig. 6 Geometrical models of the composite rod: (a) 2-D model, (b) 3-D model Fig. 7 Stress-strain curves corresponding to composite specimens with different volume fractions of tungsten particle under the quasi-static compressive condition, the test data are from Refs. [12, 13] Fig. 8 2-D deformation and failure patterns of composite specimens with different volume fractions of tungsten particle under the quasi-static compressive condition: (a) V p =10%, ε C =7.3%, (b) V p =50%, ε C =8.0%, (c) V p =10%, magnification of local zone, (d) V p =50%, magnification of local zone

3-D deformation and failure patterns as well as distributions of effective strain for composite specimens with different volume fractions of tungsten particle under the quasi-static compressive condition: (a) V p =10%, ε C =7.6%, (b) V p =50%, ε C =8.2%

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Fig. 9

Fig. 10 Experimental deformation and failure patterns of tungsten particle reinforced composite specimen with V p =50% under the quasi-static compressive condition,

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from Ref. [13] Fig. 11 Stress-strain curves corresponding to composite specimens with different volume fractions of tungsten particle under the dynamic compressive condition Fig. 12 2-D deformation and failure patterns of composite specimens with different volume fractions of tungsten particle under the dynamic compressive condition: (a) V p =10%, ε C =5.8%, (b) V p =50%, ε C =6.8%, (c) V p =10%, magnification of local zone, (d) V p =50%, magnification of local zone

Fig. 13 3-D deformation and failure patterns as well as distributions of effective strain for composite specimens with different volume fractions of tungsten particle under the dynamic compressive condition: (a) V p =10%, ε C =5.8%, (b) V p =50%, ε C =7.0% Fig. 14 Experimental deformation and failure patterns of real WP/MG composite specimens under the dynamic compressive condition: (a) V p =70%, macroscopic form, Ref. [34], (b) V p =60%, SEM micrograph of vertical section, Ref. [24] Fig. 15 Deformation and failure patterns of the WP/MG composite rod with V p =50% after penetration at v0 =998m/s: (a) 2-D simulation result, (b) 3-D simulation result, (c) Experimental result, Ref. [13] 35

ACCEPTED MANUSCRIPT Fig. 16 Stress-strain curves corresponding to composite specimens with different volume fractions of tungsten particle under the quasi-static tensile condition, the test datum is from Ref. [12] Fig. 17 2-D deformation and failure patterns of composite specimens with different volume fractions of tungsten particle under the quasi-static tensile condition: (a) V p =10%,

ε T =3.0%, (b) Vp =50%, ε T =2.7%

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Fig. 18 3-D deformation and failure patterns as well as distributions of effective strain for composite specimens with different volume fractions of tungsten particle under the quasi-static tensile condition: (a) V p =10%, ε T =3.0%, (b) V p =50%, ε T =2.7% Fig. 19 Experimental deformation and failure patterns of the composite specimen reinforced with tungsten balls with V p =52% under the quasi-static tensile condition, Ref. [29] Fig. 20 Development of effective strain within the WP/MG composite specimen with V p =10% under the quasi-static compressive condition: (a) ε C =1.0%, (b) ε C =2.3%,

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(c) ε C =7.0%, (d) ε C =7.3% Fig. 21 Development of effective strain within the WP/MG composite specimen with V p =50% under the quasi-static compressive condition: (a) ε C =1.0%, (b) ε C =2.3%, (c) ε C =7.0%, (d) ε C =8.0% Fig. 22 Local deformation and failure patterns of composites with different volume fractions of tungsten particle under the quasi-static compressive condition: (a) V p =7%, SEM micrograph [22], (b) V p =52%, SEM micrograph [28], (c) V p =10%, magnification of local zone within the dashed frame in Fig. 20(b), (d) V p =50%, magnification of local

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zone within the dashed frame in Fig. 21(b) Fig. 23 Development of effective strain within the FeP/MG composite specimen with V p =10% under the quasi-static compressive condition: (a) ε C =1.0%, (b) ε C =2.3%, (c) ε C =12.0%, (d) ε C =16.5% Fig. 24 Development of effective strain within the FeP/MG composite specimen with V p =50% under the quasi-static compressive condition: (a) ε C =1.0%, (b) ε C =2.3%,

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(c) ε C =20.0%, (d) ε C =27.0% Fig. 25 Local deformation and failure patterns of composites reinforced with particles with low strength under the quasi-static compressive condition: (a) Tantalum particle, V p =5%, SEM micrograph, Ref. [23], (b) Steel particle, V p =10%, simulation result Fig. 26 Development of effective strain within the WP/MG composite specimen with V p =50% under the quasi-static compressive condition: (a) t=9.5ms, (b) t=25.0ms, (c) t=68.0ms, (d) Magnification of local zone within the dashed frame in (b) Fig. 27 Development of effective strain within the WP/MG composite rod with V p =50% and the aluminum target during penetration: (a) t=4µs, (b) t=120µs, (c) t=200µs, (d) Magnification of the rod nose in (b) Fig. 28 Variations of effective stress within composites under different deformation conditions: (a) Dynamic compression, (b) High-speed impact Fig. 29 Numerical deformation and failure patterns as well as distributions of effective strain for the whole model of WP/MG specimen with V p =50% under the quasi-static compressive condition 36

ACCEPTED MANUSCRIPT Table 1

ν

ε0

(kg/m3)

E (GPa)

(-)

Cv J/(kg·K)

Tr (K)

Tm (K)

Tungsten Aluminum STS304 steel

19300 2700 7900

410 68.9 190

0.28 0.33 0.29

134 910 477

300 300 300

1752 925 1455

Material

A (MPa)

B (MPa)

n (-)

C (-)

m (-)

Tungsten

1700

650

0.12

0.016

Aluminum

270

154

0.22

0.13

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ρ

Material

STS304 steel

463

426

0.25

0.015

Tungsten Aluminum STS304 steel

D2 (-)

1.34

D1 (-) 1.320 (C) 0.019 (T) 3.0 (P) 2.50 (P)

1.03

2.00(C)

0

1.00

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D4 (-) 0 0 0

D5 (-) 0 0 0

C0 (m/s) 3850 5200 4578

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D3 (-) 0 0 0

Material

S1 (-) 1.44 1.40 1.36

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Table A1

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Parameter Young’s modulus Density Poisson ratio Glass transition temperature

Notation

Unit GPa kg/m3 K

Value 96 6125 0.36 625

ξ0

K J/(kg·K) s-1 A3 A3 K -

993 400 1×1013 25 20 300 0.05-0.0525

E

ρ ν

Tg

Tm

ξc

-

0.065

α

0.05

Activation energy

∆G m

eV

Number of diffusive jumps

nD

-

Hydrostatic factor

Λ

-

AC C

Melting temperature Specific heat at constant volume Frequency of atomic vibration Average atomic volume Critical volume Initial temperature Initial free volume concentration Critical free volume concentration Geometrical factor

stress

sensitivity

Cv f Ω v* T0

37

(s-1) 1 1 1

g

∆G m ( ε ) 3 Λ C =0.05

ΛT =0.35

γ0 (-) 1.58 1.97 1.65

0 0

a (-) 0 0.48 0.45

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Fig.1

38

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V p =10%

V p =50%

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(a)

V p =10%

V p =50%

(b)

(c) Fig. 2

39

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Whole specimen

Reinforced particles

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(a)

Whole specimen

Reinforced particles

Fig. 3

AC C

EP

(b)

(a)

(b) Fig. 4 40

TE D

SC

M AN U

Fig. 5

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

(a)

(b) Fig. 6

41

ACCEPTED MANUSCRIPT 2000

1200 800 400

10%-test

50%-test

10%-2D

50%-2D

10%-3D

50%-3D

0 0

3

6

9

AC C

EP

TE D

M AN U

Fig. 7

12

SC

Engineering strain,%

RI PT

Stress (MPa)

1600

(a)

(b)

(c) Fig. 8 42

(d)

(a)

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

(b)

AC C

EP

TE D

Fig. 9

Fig. 10

43

ACCEPTED MANUSCRIPT 2100

1500 1200 900

RI PT

Stress (MPa)

1800

600 300

10%-2D

50%-2D

10%-3D

50%-3D

0 2

4

6

SC

0

8

M AN U

Engineering strain,%

AC C

EP

TE D

Fig. 11

(a)

(b)

(c) Fig. 12 44

(d)

(a)

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

(b)

(a)

AC C

EP

TE D

Fig. 13

(b) Fig. 14 45

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

(a)

(b)

(c)

TE D

Fig. 15

1600

EP

1400

1000

AC C

Stress (MPa)

1200

800 600 400 200

10%-2D

50%-2D

10%-3D

50%-3D

5%-test

0 0

1

2

3

Engineering strain,%

Fig. 16

46

4

(a)

SC

RI PT

ACCEPTED MANUSCRIPT

(b)

AC C

EP

TE D

M AN U

Fig. 17

(a)

(b) Fig. 18

Fig. 19 47

(a)

(b)

SC

RI PT

ACCEPTED MANUSCRIPT

(c)

EP

TE D

M AN U

Fig. 20

(d)

AC C

(a)

(a)

(b)

(c)

(d)

Fig. 21

(b)

(c) Fig. 22 48

(d)

(a)

(b)

(c)

(d)

AC C

(a)

EP

TE D

M AN U

Fig. 23

SC

RI PT

ACCEPTED MANUSCRIPT

(b)

(c)

(d)

Fig. 24

(a)

(b) Fig. 25 49

(a)

(b)

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

(c)

AC C

EP

TE D

Fig. 26

(a)

(b)

(c)

(d) Fig. 27 50

(d)

ACCEPTED MANUSCRIPT A'-MG B'-MG C'-MG

2500

A'-WP B'-WP C'-WP

2000 1500 1000

RI PT

Effective stress (MPa)

3000

500 0 0

30

60

90

Time (µs)

SC

(a) 8000

A-WP

6000

B-MG

M AN U

Effective stress (MPa)

A-MG

B-WP

4000

C-MG C-WP

2000

0

30

60 Time (µs)

TE D

0

(b)

AC C

EP

Fig. 28

Fig. 29 51

90

120

ACCEPTED MANUSCRIPT Highlights

FEM geometrical models of composites are established based on the inner structures.

•

A new constitutive model is employed to describe the properties of MG matrix.

•

Systemic FEM simulations on the mechanical behaviors of composites are conducted.

•

Effects of various factors on the mechanical behaviors of composites are analyzed.

AC C

EP

TE D

M AN U

SC

RI PT

•