Boron carbide (B₄C) has wide industrial and military applications due to its excellent properties, including high hardness, low density and good chemical stability.^[1–3] With a low density and high Hugoniot elastic limit (HEL) of about 20 GPa,^[4,5] B₄C is expected to be a promising material for protective armor. However, when impact pressure is above HEL, an abrupt drop in strength occurs, leading to much poorer dynamics performance than expected.^[5]

Since the 1990s, a number of groups have studied the mechanical behavior and related deformation mechanism experimentally.^[4–10] Chen et al.^[6] studied the formation of nanoscale intragranular amorphous bands in a ballistic experiment, which might be responsible for the lower dynamics response of B₄C. Amorphization of B₄C was also observed in scratching and indentation experiments.^[7–10] By contrast, Zhang et al.^[4] performed shock-wave experiments on B₄C, and found a sudden drop in the pressure-density plot above 38 GPa. They pointed out that it is related to a phase transition. Similarly, using plate impact experiments including reshock and release configurations, Vogler et al.^[5] found that a possible structural change occurs at about 40 GPa. However, it is still difficult to analyze the failure mechanism in B₄C experimentally because of its complex crystal structure and the nearly identical scattering cross sections for neutron and electron of B and C.

Over the past few decades, ab initio methods based on the density functional theory have been widely used to study mechanical behaviors, deformation mechanisms and various properties in crystalline materials.^[11–14] In particular, several theoretical studies in B₄C have been performed.^[14–20] An et al.^[14,15] considered the structure deformation mechanism under shear stress. They found that the amorphization band formation is associated with the breaking of B–C bond between neighboring icosahedra and the bond formation between the B atoms in the chain center and the C atoms in the nearby icosahedra.^[14] They also concluded that the brittle failure of B₄C arises from the formation of higher density amorphous bands due to the fracture of the icosahedra through reactive molecular dynamics (RMD) simulations.^[15] Besides, Aryal et al.,^[16] Yan et al.,^[17] and Taylor et al.^[18,19] investigated the nature of structural deformation in B₄C under c-axis (the direction of three-atom chains) compression. They concluded that the deformation mechanism of B₄C is the abrupt bending of three-atom chains. Betranhandy et al.^[20] suggested that the formation of C–C bonds in the C–B–C chains results in the dynamic failure of B₄C instead of the abrupt bending of C–B–C chains under the c-axis compression. Most of the above studies focused on the deformation mechanism of B₄C under the c-axis compression. The deformation behaviors for B₄C loading along other directions have not been understood.

Furthermore, Clellan et al.^[21] showed experimentally that B₄C is more strongly anisotropic in elasticity and interatomic bonding than most solids. Thus, it needs to be considered whether the deformation mechanism for B₄C has strong anisotropy. Korotaev et al.^[22] studied the structural changes in B₄C during compression along different directions (x, y, and z axes). The z axis is the direction of three-atom chains, and the x axis is chosen so that the edge of the unit cell containing the carbon atom lays in the plane. They found three types of structural changes: abrupt and continuous bending of the three-atom chain, and the disordering of the structure. They concluded that B₄C has a strongly anisotropic deformation behavior, because sudden chain bending occurs along the x axis and in y axis and z axis of deformation only continuous bending of the chain occurs. Actually, the sudden bending of three-atom chains was also observed along the z-axis compression (the three-atom direction).^[16–19] Thus, the reason Korotaev et al.^[22] proposed for the anisotropy of B₄C might not be the real anisotropic deformation behaviors. Thus, the anisotropic mechanical behaviors and associated deformation mechanisms still need investigating.

In order to investigate the anisotropic deformation mechanism in B₄C, ab initio methods are used to study the uniaxial compression deformation along a axis and c axis in this work. The rest of this paper is organized as follows. In Section 2, we give the details about crystal structure and computation. In Section 3, we present the results and discussion, containing stress–strain relation and the analyses of deformation mechanisms under a-axis and c-axis compressions. In Section 4, we draw some conclusions from the present study.

As illustrated in Fig. 1, the crystal structure of B₄C consists of 12-atom icosahedra located at vertices of a rhombohedral lattice and the three-atom chain (C–B–C) lying on the main diagonal location. This structure can also be described as a hexagonal lattice with its c axis along the body diagonal of the primitive rhombohedral cell. Within an individual icosahedron, there are two types of distinct sites: polar and equatorial. Atoms at the polar sites connect the icosahedra together and atoms at the equatorial sites form bonds between the icosahedra and three-atom chains. In the present study, the hexagonal unit cell of B₁₁C –CBC, in which B₁₁C is the 12-atom icosahedron at polar sites and C–B–C is the 3-atom chain, is considered. This structure is generally accepted as the most stable form of B₄C.^[23–25]

Fig. 1.

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	Fig. 1. (color online) Atomic structure of polar boron carbide B11C –CBC in the hexagonal unit cells. The red balls in the figure denote the C atoms, and the blue balls refer to the B atoms.

In the current study, all periodic calculations were performed with an ab initio method at a local density approximation (LDA) functional level, and Vienna ab initio Simulation Package (VSA)^[26–28] was used. To ensure accuracy and efficiency, tests were first made by using experimental unit cell parameters^[29] of B₁₁C –CBC (Table 1) to determine the number of k-points and the cutoff energy required. According to the test results, 75 k-points and a high cutoff energy of 1200 eV were adopted in the present calculations. The convergence criteria were set to be 1 10 eV energy difference for solving the electronic wave function and 1 10 eV/Å force for geometry optimization. The lattice parameters for B₄C were first optimized in VASP to ensure that the calculations were based on equilibrium lattice parameters. The experimental lattice parameters (table 1) were taken as the basis for the optimization by energy minimization. Due to the substitution of carbon into the B₁₂ icosahedron, the B₁₁C –CBC structure rather deviates from hexagonal symmetry. Our optimized lattice parameters are listed in Table 1, which are in agreement with previous results.^[16,30]

Table 1.

Table 1.

Table 1. Lattice parameters of B11C –CBC. . Lattice parameters Present work Aryal et al.[16] Ivashchenko et al.[30] Morosin et al.[29] a/Å 5.532 5.581 5.548 5.589 b/Å 5.532 5.581 5.548 5.589 c/Å 11.907 11.907 11.865 12.054 92.034 90.00 91.74 90 87.966 92.075 88.26 90 119.887 120.055 119.89 120

Table 1. Lattice parameters of B11C –CBC. .

In order to investigate the deformation mechanism of B₄C, the uniaxial compression along different directions were performed. The uniaxial compressive strains were imposed along a axis and c axis respectively. At each strain level, the strain components of the other directions were fully relaxed, and the residual stresses for relaxation of these other strain directions were controlled to be less than 0.1 GPa. To obtain stress–strain curves, a small uniaxial compression strain was applied sequentially to the B₄C structure relaxed in the previous step. At each deformation step, we defined a 1% level of strain as a small compression strain increment. The volume of B₄C remained unchanged during relaxation. This stress–strain relationship was used to obtain mechanical behaviors and to analyze the related deformation mechanisms for B₄C.

In this work, ab initio methods are used to study the intrinsic mechanical properties and deformation mechanisms of B₄C under a-axis and c-axis compression. Strong anisotropy is observed. The maximum stresses are 89.0 GPa and 172.2 GPa along a axis and c axis, respectively. Under the a-axis compression, there is a slight decrease in the stress–strain curve, which is caused by the sudden bending of the C–B–C chains. However, the structure is still in elastic deformation at this time. Then the drop in stress takes place, which is associated with the distortion of B₄C structure, while the destruction of icosahedra is the main mechanism for causing the unrecoverable deformation. Under the c-axis compression, the icosahedra remain intact, and the C–B–C chains are almost straight before a drop in stress occurs. The formation of B–B bonds, which leads to the sudden bending of the C–B–C chains, is the main mechanism for causing the stress to abruptly drop.