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# First of all two assumptions just as

First of all, two assumptions just as in Ref. [30] are need to simplify the penetration process: (1) The crater phases is short and the motilin receptor of EFP is consumed little; thus, the influence of crater on penetration can be ignored; (2) the initial stagnation radial pressure is equal to axial pressure, and radial pressure changes as the crater radius.
The process diagram for the typical EFP penetrating RHA normally is drawn in Fig. 1.
The axial force of the EFP on the RHA can be obtained from Ref. [30] thatand the crater velocity isand the penetration depth of EFP when the plug is formed iswhere R is the radius of crater (mushroom), r is the cross section radius of EFP, u is the penetration velocity, v is the velocity of undeformed part of EFP, t is the penetration time and the penetration starts at t = 0. And ρt and Rt are the density and strength factor of RHA respectively, ρp and Yp are the density and strength factor of EFP respectively. And u0 is the penetration velocity at t = 0, v0 is the impact velocity of EFP. And the subscript c represents the parameters when the plug is just formed, for example, tc is the forming time of plug.
In Ref. [30], an assumption was made that the shape of plug is frustum of a cone and the angle between generatrix and bottom is 45°, furthermore the axial length of mushroom is considered. Now this assumption is ignored and the shape of plug is frustum of a cone and the angle between generatrix and bottom is 45° or cylinder is assumed, what's more the axial length of mushroom could be considered or ignored.
If the shape of plug is frustum of a cone and the angle between generatrix and bottom is 45°, then parameters when the plug is formed are shown in Fig. 2, and in which the meshes are the internal fragments of the plug.
The axial shear force of the plug is equal to the axial force of EFP on RHA, as equation (4) shows.where τmax is the maximum shear strength of RHA and the calculation method was introduced in details in Ref. [30].
If the axial length of mushroom is ignored, then,where H0 is the thickness of RHA. And the mass of BAD generated by RHA satisfies the following relation,
The residual mass of EFP is the mass of undeformed part of EFP, so mass of BAD generated by EFP satisfies the following relation,
If the axial length of mushroom is considered, then,
Then the mass of BAD generated by the RHA satisfies the following relation,
The residual mass of EFP is the sum mass of undeformed part of EFP and mushroom, so the mass of BAD generated by EFP satisfies the following relation,
If the shape of plug is cylinder, Fig. 3 shows parameters when the plug is formed.
And the axial shear force of the plug is equal to the axial force of EFP on RHA, as
If the axial length of mushroom is ignored, the mass of BAD generated by EFP satisfies equation (7), and the mass of BAD generated by RHA satisfies the following relation,
If the axial length of mushroom is considered, the mass of BAD generated by EFP satisfies equation (10), and the mass of BAD generated by RHA satisfies the following relation,
Since the crater radius is not considered after the formation of the plug, the time range is 0 ≤ t ≤ tc.

Numerical simulation model verification
The main parameters of the numerical simulation and the main parameters of experimental setting about the formation of EFP had already been introduced in details in our previous work [8,9] and the credibility of numerical simulation method had already been proved in Refs. [8,9].
Dalzell [4] used AUTODYN-3D™ to simulate the formation of BAD clouds subjected to penetration of EFP, and pointed out that the SPH algorithm had more advantageous than the Lagrange and Euler algorithms in the study of BAD. Yarin [25] found that the mass loss range of target is nearly 30% in the experimental study of tungsten alloy long rod penetrating RHA. And budding was often found that the BAD generated by RHA and EFP would be bonded together [30].