# A Dynamic Cavity Expansion Model for Rigid Projectile Penetration into Concrete considering the Compressibility and Nonlinear Constitutive Relations

Authors: Fei Gao; Zhen Wang; Zhu Wen; Yuguo Ji
Journal: Geofluids
 (2021)

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## Abstract

TheP-α equation of state (EOS) and a nonlinear yield criterion are utilized to characterize the dynamic constitutive behavior of concrete targets subjected to projectile normal penetration. A dynamic cavity expansion model considering the compressibility and nonlinear constitutive relations for concrete material is developed. Then, a theoretical model to calculate the depth of penetration (DOP) for rigid projectile is established. Furthermore, the proposed model is validated based on the available test data as well as the calculation results by the linear compressible EOS and linear yield criterion. This study shows that the proposed model derived using the P-α EOS and nonlinear yield criterion can effectively reflect the plastic mechanical properties of concrete and is also suitable for predicting the DOP of concrete targets. In addition, the influence law of concrete constitutive parameters such as the cohesion strength, shear strength, internal friction coefficient, and elastic limit pressure on the DOP is revealed.


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## Body

## 1. Introduction

The dynamic response and destruction of concrete structures under explosive or impact loading have important theoretical and application value in the field of weapon damage and engineering protection [1]. Li et al. [2] systematically reviewed the empirical formulas, theoretical analyses, and numerical methods related to the concrete penetration problem. To evaluate the damage and failure behavior of concrete, a large number of penetration tests [3–8] have been conducted. The penetration process is driven not only by the mortar strength, aggregate size, aggregate sliding, and redistribution behavior but also by pore breakage and compaction. Therefore, the compressibility and nonlinear effect of concrete must be considered in the process of theoretical analysis.Cavity expansion theory is an accepted analytical method for studying the penetration mechanism of projectiles into a target. Bishop et al. [9] proposed quasi-static cylindrical and spherical cavity expansion equations in a semi-infinite target. Hill [10] modified a spherically symmetric dynamic cavity expansion equation for an incompressible medium. To date, the cavity expansion model is widely used to evaluate the penetration resistance of projectiles into concrete [11–17], rock [18–20], ceramics [21, 22], sand [23, 24], soil [25], and metal [26, 27] targets. When the appropriate EOS and yield criterion are adopted, the numerical and analytical solution for the model will be solved by the similarity transformation method.In the current works that focus on reducing the complexity in the cavity expansion analysis, the simplified EOS is usually used such as incompressible [12], linearly compressible [12, 22], and locked hydrostatic [11, 13] models. Although the incompressibility hypothesis can reduce the difficulty of the analysis step, the DOP obtained by the incompressible EOS is much shallower than the corresponding test data [12]. The linearly compressible model is approximately available for the case of low striking velocities. Therefore, neither the linear model nor the locked model can accurately reflect the compression process of concrete.After reviewing the yield criteria, it was found that the original and revised forms of the Tresca [11], Mohr-Coulomb [12, 14, 19, 22–24], and Drucker-Prager [15] models have been widely used in dynamic cavity expansion analyses for concrete-like material. However, all the above-mentioned yield criteria are in linear form. When considering the triaxial shear data in the present paper, the results show that the relationship between the pressure and shear strength is nonlinear, especially under high pressure. In summary, a model considering pore compaction, a nonlinear EOS, and nonlinear yield criterion are needed to calculate the DOP of projectiles into concrete targets.This paper proposes aP-α EOS and nonlinear yield criterion for rigid projectiles penetrating concrete targets on the basis of a comparative analysis of existing concrete constitutive models. A spherical cavity expansion model considering the compressibility and nonlinear constitutive relations of concrete is established, and the cavity pressure is calculated. Furthermore, the calculation model for the DOP is derived, and the validity of the model is demonstrated by a comparison with the results of the widely recognized model as well as the available penetration test data [12]. Eventually, the influences of the constitutive parameters on the DOP are discussed.


## 2. Constitutive Model

In the following section, the hydrostatic behavior and strength model for concrete are treated separately and described by theP-α EOS and nonlinear yield criterion, respectively.


### 2.1. Equation of State

The concrete exhibits pore compaction and permanent densification effect when subjected to high-pressure loading [28]. The P-α EOS originally proposed by Herrmann [29] can be used to describe the compressibility of concrete. The P-α EOS for concrete can be expressed by Eq. (1).
(1a)α=α0P≤Pe1+α0−1Ps−PPs−Pe3Pe<P≤Ps1P>Ps,(1b)α=ρsρ≥1,where α and α0 are the porosities of the current and initial material, respectively; ρ and ρs are the densities of the current and fully compacted material, respectively; P is the current pressure, Pe is the pressure at which pore compaction occurs, and Ps is the fully compacted pressure.


### 2.2. Nonlinear Yield Criterion

The mechanical properties and yield criterion for rock-like materials are always the concern of researchers in fields as mining science [30], geology [31], and rock engineering [32–34]. In the plastic range, to describe the confining pressure effect on yield strength for rock-like materials, the experimental data are smoothed and approximated according to the method first proposed by Lundborg [35], considering the similarities between concrete and rock in terms of mechanical response. For intact concrete, there is an initial tensile strength ft and a cohesion strength τ0. The local slope of the strength-pressure curve is called the internal friction coefficient μ; the shear strength reaches a constant τm at a sufficiently high-pressure state. The nonlinear yield criterion in a spherically symmetric Euler coordinate system is
(2a)σr−σθ=τ0+P1/μ+1/τm−τ0P,(2b)P=σr+2σθ3,where σr and σθ are the radial and hoop Cauchy stress components, which are taken as positive in compression. In summary, setting τ0=a0, 1/μ=a1, and 1/τm−τ0=a2, and then, Eq. (2a) can be expressed as σr−σθ=a0+P/a1+a2P. a0, a1, and a2 are constants which obtained by a series of triaxial compression test data.


## 2.1. Equation of State

The concrete exhibits pore compaction and permanent densification effect when subjected to high-pressure loading [28]. The P-α EOS originally proposed by Herrmann [29] can be used to describe the compressibility of concrete. The P-α EOS for concrete can be expressed by Eq. (1).
(1a)α=α0P≤Pe1+α0−1Ps−PPs−Pe3Pe<P≤Ps1P>Ps,(1b)α=ρsρ≥1,where α and α0 are the porosities of the current and initial material, respectively; ρ and ρs are the densities of the current and fully compacted material, respectively; P is the current pressure, Pe is the pressure at which pore compaction occurs, and Ps is the fully compacted pressure.


## 2.2. Nonlinear Yield Criterion

The mechanical properties and yield criterion for rock-like materials are always the concern of researchers in fields as mining science [30], geology [31], and rock engineering [32–34]. In the plastic range, to describe the confining pressure effect on yield strength for rock-like materials, the experimental data are smoothed and approximated according to the method first proposed by Lundborg [35], considering the similarities between concrete and rock in terms of mechanical response. For intact concrete, there is an initial tensile strength ft and a cohesion strength τ0. The local slope of the strength-pressure curve is called the internal friction coefficient μ; the shear strength reaches a constant τm at a sufficiently high-pressure state. The nonlinear yield criterion in a spherically symmetric Euler coordinate system is
(2a)σr−σθ=τ0+P1/μ+1/τm−τ0P,(2b)P=σr+2σθ3,where σr and σθ are the radial and hoop Cauchy stress components, which are taken as positive in compression. In summary, setting τ0=a0, 1/μ=a1, and 1/τm−τ0=a2, and then, Eq. (2a) can be expressed as σr−σθ=a0+P/a1+a2P. a0, a1, and a2 are constants which obtained by a series of triaxial compression test data.


## 3. Development of Dynamic Spherical Cavity Expansion Model

The cavity expansion generates an elastic-cracked-compacted response (Figure1(a)) or elastic-compacted response (Figure 1(b)) which depends on the cavity expansion velocity Vr, where r is the radial Eulerian coordinate, c and c1 are the boundary velocities, cd is the dilatational velocity, and t is the time.Figure 1
Two types of target responses: (a) elastic-cracked-compacted response and (b) elastic-compacted response.
(a)(b)


### 3.1. Compacted Region

The mass conservation and momentum conservation equations of the compressible medium in Euler coordinates are(3a)ρ∂v∂r+2vr=−∂ρ∂t+v∂ρ∂r,(3b)∂σr∂r+2σr−σθr=−ρ∂v∂t+v∂v∂r,where r is the radial Eulerian coordinate and v is the particle velocity (outward is positive).By defining the following dimensionless variables and introducing similar transformations,(4)ξ=rct,U=vc,ε=Vrc,F=ftfc,S=σrfc,T=Pfc,β1=c1cp,β=ccp,cp=K/ρ0.Transform Eq. (3) into the following dimensionless forms(5a)dUdξ+2Uξ=fTξ−UdTdξ,(5b)hTfcdTdξ+2gTξ=iTξ−UdUdξ,where fc and ft are the static uniaxial compressive and tensile strength of concrete. fT, gT, hT, and iT can be derived with Eqs. (1) and (2).
(6a)fT=3fcα0−1Ps−fcT2Ps−Pe3+α0−1Ps−fcT3,(6b)gT=a0+fcTa1+af2cT,(6c)hT=1+23a1+af2cT−2af2cT3a1+af2cT2,(6d)iT=α0β2K1+α0−1Ps−fcT3/Ps−Pe3.From Eq. (2), the dimensionless relation between the pressureT and radial stress S obeys
(7)S=23fca0+fcTa1+af2cT+T.Eq. (5) is transformed by the Runge-Kutte method to evaluate it numerically.(8a)dUdξ=2ξ−UfTgT+2UfchTξξ−U2fTiT−fcξhT,(8b)dTdξ=2gT+2Uξ−UiTξξ−U2fTiT−fcξhT.When the Hugoniot jump condition is given, the initial values forU and T in Eq. (8) can be determined by the solutions in the cracked region (Figure 1(a)) or elastic region (Figure 1(b)) at ξ=1. These two types of target responses are presented separately below.


### 3.2. Elastic-Cracked-Compacted Response

In the elastic region, the stress-strain relation is governed by Hooke’s law.(9a)σr=−E1+ν1−2ν1−ν∂u∂r+2νur,(9b)σθ=−E1+ν1−2νν∂u∂r+ur,where u is the outward radial displacement (positive direction).Considering that elastic deformation is very small, ignoring the convective term in Eq. (3b) [14, 22] and substituting Eq. (9) into Eq. (3b) obeys the wave equation
(10a)∂u2∂r2+2r∂u∂r−2ur2=1cd2∂u2∂t2,(10b)cd2=E1−ν1+ν1−2νρ0.By introducing a new dimensionless variableū=u/ct and adopting the similarity transformation, Eq. (10a) can be expressed in another form.
(11a)1−ω2ξ2du¯2dξ2+2ξdu¯dξ−2ξ2u¯=0,(11b)ω=ccd=βK1+ν1−2ν/E1−ν.Integrating Eq. (11a) gives the following general solution.
(12)u¯=A0ωξ−B01−3ω2ξ23ω2ξ2,where A0 and B0 are the unknown integral constants, which can be obtained using the following boundary conditions.
(13a)u¯ξ=1ω=0,(13b)σθξ=β1β=−ft.Combining Eqs. (12) and (13) yields
(14a)B0=−32A0,(14b)A0=2ftE⋅ω2β131+ν1−2ν2ω3β131+ν+β31−2ν−3ω2ββ12.Furthermore, the relation between the particle velocity and radial displacement is(15)v1−∂u∂r=∂u∂t.Substituting Eq. (12) into Eq. (15) and considering the strain in elastic region is small, Eq. (15) can be revised as
(16)U=u¯−ξdudξ.Using Eqs. (9b), (12), (13), and (16), the dimensionless radial stress S1 and particle velocity U1 in the elastic region at ξ=β1/β are derived as
(17a)S1=2F1−2ν+3νγβ12−1+νγβ131−2ν−3γβ12+21+νγβ13,(17b)βU1=3ftβ11+ν1−2νE1−γβ121−2ν−3γβ12+21+νγβ13,(17c)γ2=cpcd2=1+ν31−ν.Using the Hugoniot jump conditions in the cracked region atξ=β1/β, the dimensionless radial stress S2 and particle velocity U2 can be obtained, i.e.,
(18a)S2=S1+2Fβ1−βU123−β1−βU12,(18b)βU2=βU1+2ft/Kβ1−βU13−β1−βU12.Moreover, concrete is considered to be a linear compressible material in the cracked region whenσθ=0, and then, the general solutions in the cracked region can be described as
(19a)Sξ=C0βξ+D01+3βξ2,(19b)Uξ=−fc2KβC03+C0βξ2+4D0βξ,where C0 and D0 are the integral constants. Based on the boundary conditions in Eq. (18), it can be determined as follows:
(20a)C0=−6β122β1fcS2+Kβ12+3βU2fcβ12−32,(20b)D0=β12fcS2β12+3+6β1KβU2fcβ12−32.The above-mentioned nonlinear yield criterion is satisfied in the cracked region atξ=β1/β; thus,
(21a)σr=a0+σr/3a1+a2σr/3,(21b)σrfc=C0β+D01+3β2.By using Eq. (21),β can be expressed as
(22a)β=C0+C02+12D0E0−D02E0−D0,(22b)E0=−3a1+a0a2+1+9a12+a02a2+22a0a2+6a0a1a2−6a1+12a2fc.In the cracked region atξ=1, the dimensionless radial stress S3, pressure T3, and particle velocity U3 can be obtained as
(23)S3=E0,T3=E03,U3=−fcC0β2+36Kβ3−2fcD0Kβ2.


### 3.3. Elastic-Compacted Response

The target response can be simplified to the elastic-compacted type whenc>c1. The solution for ū is consistent with Eq. (12), while the integral constants should be obtained from the following boundary conditions.
(24a)u¯ξ=1ω=0,(24b)σrξ=1−σθξ=1=a0+σrξ=1+2σθξ=13a1+a2σrξ=1+2σθξ=1.Substituting the boundary conditions into the displacement solution in Eq. (12), we have
(25a)B0′=−32A0′,(25b)A0′=−y0+y02−4x0z02x0,where x0, y0, and z0 are constants, which can be written as
(26a)x0=3a2E21−ω2,(26b)y0=3a1E1−2ν1+ω−2Ea0a2+11+νω2,(26c)z0=2a0a1ω21+ν1−2νω−1.In the elastic region at the elastic-compacted interface (ξ=1), using Eqs. (9), (12), (24), (25) and (26), then S3, T3, and U3 are obtained.
(27)S3=EA0′1+νω3−3νω2+2ν−1fc1+ν2ν−1ω2,T3=EA0′ω−1fc2ν−1,U3=321ω2−1A0′.


### 3.4. Hugoniot Jump Conditions

To solve the dimensionless conservation equation in Eq. (8) still requires a dimensionless pressure and particle velocity in the compacted region at the cracked-compacted interface or elastic-compacted interface; it can be obtained with the Hugoniot jump conditions. Noticing that subscript 3 represents the variable in the cracked or elastic region atξ=1, and subscript 4 represents the variable in the compacted region at ξ=1. The concrete is treated as a linear compressible material in the elastic or cracked regions at ξ=1, i.e.,
(28)T3fc=K1−ρ0ρ3.TheP-α EOS is used for the compacted region; therefore,
(29)ρ4=α0ρ01+α0−1Ps−T4fc3/Ps−Pe3.For the dimensionless Hugoniot jump conditions, it follows that(30a)U4=1+ρ3ρ4U3−1,(30b)S4=S3+ρ3c21−U3U4−U3fc.By solving Eqs. (7) and (28)-(30), S4, T4, and U4 in the compacted region at ξ=1 are obtained, and then used as the initial values of the dimensionless conservation equation in Eq. (8).To establish the penetration model, it is necessary to clarify the relationship betweenσr and Vr. The cavity expansion resistance of concrete is generally expressed as the following three components [15], i.e.,
(31)σrfc=A+BVrfc/ρ0+CVrfc/ρ02,where A, B, and C denote the dimensionless resistance constants.Here, we can choose concrete with uniaxial compressive strength of 51 MPa to analyze the proposed dynamic cavity expansion model in detail. For the EOS, referring to the isotropic compression data of 51.2 MPa concrete [36], the fitting results using Eq. (1a) are shown in Figure 2(a). The initial and fully compacted densities of the concrete are 2350 kg/m3 and 2767 kg/m3, respectively. Using the P-α EOS with α0=1.177, Pe=17 MPa, and Ps=6370 MPa gives satisfying agreement with the test data. For the relation between the shear strength and pressure, the existing literature lacks the triaxial data of 51 MPa concrete, so the 48 MPa concrete triaxial data from Hanchak et al. [37] is used instead, and the fitted results by Eq. (2) are presented in Figure 2(b). Obviously, the nonlinear yield criterion presented in this paper can effectively reproduce the test data (adjusted R2=0.975) with a0=26.67 MPa, a1=0.82, and a2=1.84×10−3 MPa-1. In contrast, the Mohr-Coulomb yield criterion shows poor agreement with the experimental data. The constitutive parameters for 51 MPa concrete are summarized as follows:Figure 2
Test data and proposed constitutive laws for 51 MPa concrete: (a) EOS and (b) yield criterion.
(a)(b)ρ0=2350 kg/m3, ρs=2767 kg/m3, α0=1.177, Pe=17 MPa, and Ps=6370MPaa0=26.67 MPa, a1=0.82, and a2=1.84×10−3 MPa-1


## 3.1. Compacted Region

The mass conservation and momentum conservation equations of the compressible medium in Euler coordinates are(3a)ρ∂v∂r+2vr=−∂ρ∂t+v∂ρ∂r,(3b)∂σr∂r+2σr−σθr=−ρ∂v∂t+v∂v∂r,where r is the radial Eulerian coordinate and v is the particle velocity (outward is positive).By defining the following dimensionless variables and introducing similar transformations,(4)ξ=rct,U=vc,ε=Vrc,F=ftfc,S=σrfc,T=Pfc,β1=c1cp,β=ccp,cp=K/ρ0.Transform Eq. (3) into the following dimensionless forms(5a)dUdξ+2Uξ=fTξ−UdTdξ,(5b)hTfcdTdξ+2gTξ=iTξ−UdUdξ,where fc and ft are the static uniaxial compressive and tensile strength of concrete. fT, gT, hT, and iT can be derived with Eqs. (1) and (2).
(6a)fT=3fcα0−1Ps−fcT2Ps−Pe3+α0−1Ps−fcT3,(6b)gT=a0+fcTa1+af2cT,(6c)hT=1+23a1+af2cT−2af2cT3a1+af2cT2,(6d)iT=α0β2K1+α0−1Ps−fcT3/Ps−Pe3.From Eq. (2), the dimensionless relation between the pressureT and radial stress S obeys
(7)S=23fca0+fcTa1+af2cT+T.Eq. (5) is transformed by the Runge-Kutte method to evaluate it numerically.(8a)dUdξ=2ξ−UfTgT+2UfchTξξ−U2fTiT−fcξhT,(8b)dTdξ=2gT+2Uξ−UiTξξ−U2fTiT−fcξhT.When the Hugoniot jump condition is given, the initial values forU and T in Eq. (8) can be determined by the solutions in the cracked region (Figure 1(a)) or elastic region (Figure 1(b)) at ξ=1. These two types of target responses are presented separately below.


## 3.2. Elastic-Cracked-Compacted Response

In the elastic region, the stress-strain relation is governed by Hooke’s law.(9a)σr=−E1+ν1−2ν1−ν∂u∂r+2νur,(9b)σθ=−E1+ν1−2νν∂u∂r+ur,where u is the outward radial displacement (positive direction).Considering that elastic deformation is very small, ignoring the convective term in Eq. (3b) [14, 22] and substituting Eq. (9) into Eq. (3b) obeys the wave equation
(10a)∂u2∂r2+2r∂u∂r−2ur2=1cd2∂u2∂t2,(10b)cd2=E1−ν1+ν1−2νρ0.By introducing a new dimensionless variableū=u/ct and adopting the similarity transformation, Eq. (10a) can be expressed in another form.
(11a)1−ω2ξ2du¯2dξ2+2ξdu¯dξ−2ξ2u¯=0,(11b)ω=ccd=βK1+ν1−2ν/E1−ν.Integrating Eq. (11a) gives the following general solution.
(12)u¯=A0ωξ−B01−3ω2ξ23ω2ξ2,where A0 and B0 are the unknown integral constants, which can be obtained using the following boundary conditions.
(13a)u¯ξ=1ω=0,(13b)σθξ=β1β=−ft.Combining Eqs. (12) and (13) yields
(14a)B0=−32A0,(14b)A0=2ftE⋅ω2β131+ν1−2ν2ω3β131+ν+β31−2ν−3ω2ββ12.Furthermore, the relation between the particle velocity and radial displacement is(15)v1−∂u∂r=∂u∂t.Substituting Eq. (12) into Eq. (15) and considering the strain in elastic region is small, Eq. (15) can be revised as
(16)U=u¯−ξdudξ.Using Eqs. (9b), (12), (13), and (16), the dimensionless radial stress S1 and particle velocity U1 in the elastic region at ξ=β1/β are derived as
(17a)S1=2F1−2ν+3νγβ12−1+νγβ131−2ν−3γβ12+21+νγβ13,(17b)βU1=3ftβ11+ν1−2νE1−γβ121−2ν−3γβ12+21+νγβ13,(17c)γ2=cpcd2=1+ν31−ν.Using the Hugoniot jump conditions in the cracked region atξ=β1/β, the dimensionless radial stress S2 and particle velocity U2 can be obtained, i.e.,
(18a)S2=S1+2Fβ1−βU123−β1−βU12,(18b)βU2=βU1+2ft/Kβ1−βU13−β1−βU12.Moreover, concrete is considered to be a linear compressible material in the cracked region whenσθ=0, and then, the general solutions in the cracked region can be described as
(19a)Sξ=C0βξ+D01+3βξ2,(19b)Uξ=−fc2KβC03+C0βξ2+4D0βξ,where C0 and D0 are the integral constants. Based on the boundary conditions in Eq. (18), it can be determined as follows:
(20a)C0=−6β122β1fcS2+Kβ12+3βU2fcβ12−32,(20b)D0=β12fcS2β12+3+6β1KβU2fcβ12−32.The above-mentioned nonlinear yield criterion is satisfied in the cracked region atξ=β1/β; thus,
(21a)σr=a0+σr/3a1+a2σr/3,(21b)σrfc=C0β+D01+3β2.By using Eq. (21),β can be expressed as
(22a)β=C0+C02+12D0E0−D02E0−D0,(22b)E0=−3a1+a0a2+1+9a12+a02a2+22a0a2+6a0a1a2−6a1+12a2fc.In the cracked region atξ=1, the dimensionless radial stress S3, pressure T3, and particle velocity U3 can be obtained as
(23)S3=E0,T3=E03,U3=−fcC0β2+36Kβ3−2fcD0Kβ2.


## 3.3. Elastic-Compacted Response

The target response can be simplified to the elastic-compacted type whenc>c1. The solution for ū is consistent with Eq. (12), while the integral constants should be obtained from the following boundary conditions.
(24a)u¯ξ=1ω=0,(24b)σrξ=1−σθξ=1=a0+σrξ=1+2σθξ=13a1+a2σrξ=1+2σθξ=1.Substituting the boundary conditions into the displacement solution in Eq. (12), we have
(25a)B0′=−32A0′,(25b)A0′=−y0+y02−4x0z02x0,where x0, y0, and z0 are constants, which can be written as
(26a)x0=3a2E21−ω2,(26b)y0=3a1E1−2ν1+ω−2Ea0a2+11+νω2,(26c)z0=2a0a1ω21+ν1−2νω−1.In the elastic region at the elastic-compacted interface (ξ=1), using Eqs. (9), (12), (24), (25) and (26), then S3, T3, and U3 are obtained.
(27)S3=EA0′1+νω3−3νω2+2ν−1fc1+ν2ν−1ω2,T3=EA0′ω−1fc2ν−1,U3=321ω2−1A0′.


## 3.4. Hugoniot Jump Conditions

To solve the dimensionless conservation equation in Eq. (8) still requires a dimensionless pressure and particle velocity in the compacted region at the cracked-compacted interface or elastic-compacted interface; it can be obtained with the Hugoniot jump conditions. Noticing that subscript 3 represents the variable in the cracked or elastic region atξ=1, and subscript 4 represents the variable in the compacted region at ξ=1. The concrete is treated as a linear compressible material in the elastic or cracked regions at ξ=1, i.e.,
(28)T3fc=K1−ρ0ρ3.TheP-α EOS is used for the compacted region; therefore,
(29)ρ4=α0ρ01+α0−1Ps−T4fc3/Ps−Pe3.For the dimensionless Hugoniot jump conditions, it follows that(30a)U4=1+ρ3ρ4U3−1,(30b)S4=S3+ρ3c21−U3U4−U3fc.By solving Eqs. (7) and (28)-(30), S4, T4, and U4 in the compacted region at ξ=1 are obtained, and then used as the initial values of the dimensionless conservation equation in Eq. (8).To establish the penetration model, it is necessary to clarify the relationship betweenσr and Vr. The cavity expansion resistance of concrete is generally expressed as the following three components [15], i.e.,
(31)σrfc=A+BVrfc/ρ0+CVrfc/ρ02,where A, B, and C denote the dimensionless resistance constants.Here, we can choose concrete with uniaxial compressive strength of 51 MPa to analyze the proposed dynamic cavity expansion model in detail. For the EOS, referring to the isotropic compression data of 51.2 MPa concrete [36], the fitting results using Eq. (1a) are shown in Figure 2(a). The initial and fully compacted densities of the concrete are 2350 kg/m3 and 2767 kg/m3, respectively. Using the P-α EOS with α0=1.177, Pe=17 MPa, and Ps=6370 MPa gives satisfying agreement with the test data. For the relation between the shear strength and pressure, the existing literature lacks the triaxial data of 51 MPa concrete, so the 48 MPa concrete triaxial data from Hanchak et al. [37] is used instead, and the fitted results by Eq. (2) are presented in Figure 2(b). Obviously, the nonlinear yield criterion presented in this paper can effectively reproduce the test data (adjusted R2=0.975) with a0=26.67 MPa, a1=0.82, and a2=1.84×10−3 MPa-1. In contrast, the Mohr-Coulomb yield criterion shows poor agreement with the experimental data. The constitutive parameters for 51 MPa concrete are summarized as follows:Figure 2
Test data and proposed constitutive laws for 51 MPa concrete: (a) EOS and (b) yield criterion.
(a)(b)ρ0=2350 kg/m3, ρs=2767 kg/m3, α0=1.177, Pe=17 MPa, and Ps=6370MPaa0=26.67 MPa, a1=0.82, and a2=1.84×10−3 MPa-1


## 4. Model Validation

### 4.1. Deep Penetration Analysis of an Ogive-Nosed Projectile

The axial resistance acting on the projectile nose is given by the following expression.(32)dFx=σrcosθdS.If the tangential friction stress is neglected [38], the final axial penetration resistance Fx can be expressed as
(33)Fx=mdVdt=πd24AN0fc+BN1ρ0fcV+CN2ρ0V2,where N0, N1, and N2 are the nondimensional quantities related to the projectile nose shape, i.e.,
(34)N0=1,N1=8d2∫0byy′21+y′2dx,N2=8d2∫0byy′31+y′2dx.As Figure3 shows, for ogive-nosed projectile, it follows that
(35)N0=1,N1ψ=4ψ−13/23ψ+2ψ−124ψ−11/22ψ−ψ2ψ−1π−2θ0,N2ψ=8ψ−124ψ2,θ0=sin2ψ−12ψ−1,where ψ=s/d and s is defined in Figure 3. The final DOP H can be derived by integrating Eq. (33).
(36)H=m2πa2CN2ρ0ln1+BN1Aρ0fc1/2V1+CN2ρ0AfcV12+2BN1DarctanBN1D−arctanBN1+2CN2ρ0/fc1/2D+4a,H>4a,where D=4ACN2−BN121/2, and V1 is projectile velocity at penetration depth 4a which was obtained from
(37)m4πa3fc+CN2fc/ρ0⋅V12+BN1fc/ρ01/2⋅V1+A−mVs24πa3fc=0.Figure 3
Diagram of an ogive-nosed projectile.For Eqs. (35)-(37), m is the projectile mass, d=2a is the shank diameter, and Vs is the initial striking velocity.


### 4.2. Comparisons with the Penetration Test Data

First, the proposed model is verified based on the test data of projectile penetration into 51 MPa concrete [4]. The P-α EOS and nonlinear yield criterion shown in Figure 2 are employed. Then, the predicted and experimental DOP are compared in Figure 4(a). It shows that the proposed model gives better agreement than the model in Ref. [12], especially at relatively high-impact velocities. Figure 4(b) illustrates the cavity surface radial stress versus the cavity expansion velocity of the elastic-cracked-compacted and elastic-cracked-plastic [12] models. Correspondingly, the cavity expansion resistance function parameters calibrated as A=7.4, B=2.5, and C=1.1 and A=5.8, B=3.0, and C=1.2, respectively. When Vr/fc/ρ01/2<1.8, the cavity surface radial stress σr calculated by the elastic-cracked-compacted model is slightly larger than that of the elastic-cracked-plastic model. When Vr/fc/ρ01/2>1.8, the σr in the elastic-cracked-compacted model is smaller than the elastic-cracked-plastic model, and the difference between the two models becomes greater with increasing of the cavity expansion velocity. This also explains why the elastic-cracked-plastic model cannot effectively predict the penetration depth under relatively high-impact velocities.Figure 4
Model verification for 51 MPa concrete: (a) comparison of predicted and experimental DOP and (b) cavity surface radial stress calculation results of the elastic-cracked-compacted and elastic-cracked-plastic model.
(a)(b)For 23 MPa concrete, the test results [39] with P-α EOS and nonlinear yield criterion by data fitting are plotted in Figures 5(a) and 5(b), where ρ0=1.97 g/cm3, α0=1.345, Pe=7.6 MPa, Ps=2138 MPa, a0=8.49 MPa, a1=0.79, and a2=2.12×10−4 MPa-1. As Figure 5(c) shows, the predicted DOP by the proposed model is in better agreement with the test data than the classic model in Ref. [12].Figure 5
Comparison of theoretical model and test data for 23 MPa concrete: (a) EOS, (b) yield criterion, and (c) DOP.
(a)(b)(c)For 36 MPa concrete, using the hydrostatic pressure-shear strength test data from Ref. [3], the fitting results with nonlinear least squares are plotted in Figure 6(a). For the EOS, since the pressure-porosity data of 36 MPa concrete are absent, the concrete exhibits similar compaction characteristics when the strength is very close. Therefore, using the EOS parameters from 35 MPa [40] concrete seems reasonable. The constitutive parameters for 36 MPa concrete are summarized as follows: ρ0=2.31 g/cm3, α0=1.19, Pe=23.3 MPa, Ps=6000 MPa, a0=22.89 MPa, a1=0.72, and a2=2.17×10−3 MPa-1. Figure 6(b) demonstrates that the proposed model provides better predictions than the classic model, and it can be inferred that the underestimation DOP of the classic model mainly comes from the use of the linear constitutive relations.Figure 6
Comparison of theoretical model and test data for 36 MPa concrete: (a) yield criterion and (b) DOP.
(a)(b)


### 4.3. Further Discussions

To obtain the parameters of the nonlinear constitutive in Eqs. (1) and (2), i.e.,Pe, Ps, a0, a1, and a2, the pressure-porosity data from flyer-plate impact tests and the hydrostatic pressure-shear strength data from triaxial compression tests are needed. However, most DOP tests lack the necessary data required for analysis and calculation models. Under this circumstance, the parameters of a0, a1, and a2 are taken from the method proposed by Markovich et al. [41] as well as the value of Pe suggested by Holmquist et al. [42]. The default values of Ps for 35 MPa and 140 MPa concrete are both equal to 6 GPa in AUTODYN version 6.1 [43]. Here, the compacted pressure Ps is also recommended to be 6 GPa for fc=58.4 MPa and fc=62.8 MPa concrete. Thus, the constitutive parameters of normal concrete are summarized as follows:
(38a)a0=2.442fc0.4369,a1=1.084fc−0.2463,a1=0.03276fc−0.6416,(38b)Pe=fc3,Ps=6GPa,where fc is measured in MPa.Figure7 gives the experimental data from Refs. [4, 5] and the predicted DOP by Eq. (36), where the parameters of the P-α EOS and nonlinear yield criterion are given in Eq. (38). Most of the predictions show reasonable agreement with the test data except for two points, which are marked by a circle in Figure 7(b). The overestimations of the DOP for these two data points may lie in that if the striking velocity exceeds 987 m/s, the projectile mass loss will aggravate, and the mass abrasive penetration regime occurs.Figure 7
Comparison of predicted and experimental DOP for (a) 58.4 MPa and (b) 62.8 MPa concrete.
(a)(b)


## 4.1. Deep Penetration Analysis of an Ogive-Nosed Projectile

The axial resistance acting on the projectile nose is given by the following expression.(32)dFx=σrcosθdS.If the tangential friction stress is neglected [38], the final axial penetration resistance Fx can be expressed as
(33)Fx=mdVdt=πd24AN0fc+BN1ρ0fcV+CN2ρ0V2,where N0, N1, and N2 are the nondimensional quantities related to the projectile nose shape, i.e.,
(34)N0=1,N1=8d2∫0byy′21+y′2dx,N2=8d2∫0byy′31+y′2dx.As Figure3 shows, for ogive-nosed projectile, it follows that
(35)N0=1,N1ψ=4ψ−13/23ψ+2ψ−124ψ−11/22ψ−ψ2ψ−1π−2θ0,N2ψ=8ψ−124ψ2,θ0=sin2ψ−12ψ−1,where ψ=s/d and s is defined in Figure 3. The final DOP H can be derived by integrating Eq. (33).
(36)H=m2πa2CN2ρ0ln1+BN1Aρ0fc1/2V1+CN2ρ0AfcV12+2BN1DarctanBN1D−arctanBN1+2CN2ρ0/fc1/2D+4a,H>4a,where D=4ACN2−BN121/2, and V1 is projectile velocity at penetration depth 4a which was obtained from
(37)m4πa3fc+CN2fc/ρ0⋅V12+BN1fc/ρ01/2⋅V1+A−mVs24πa3fc=0.Figure 3
Diagram of an ogive-nosed projectile.For Eqs. (35)-(37), m is the projectile mass, d=2a is the shank diameter, and Vs is the initial striking velocity.


## 4.2. Comparisons with the Penetration Test Data

First, the proposed model is verified based on the test data of projectile penetration into 51 MPa concrete [4]. The P-α EOS and nonlinear yield criterion shown in Figure 2 are employed. Then, the predicted and experimental DOP are compared in Figure 4(a). It shows that the proposed model gives better agreement than the model in Ref. [12], especially at relatively high-impact velocities. Figure 4(b) illustrates the cavity surface radial stress versus the cavity expansion velocity of the elastic-cracked-compacted and elastic-cracked-plastic [12] models. Correspondingly, the cavity expansion resistance function parameters calibrated as A=7.4, B=2.5, and C=1.1 and A=5.8, B=3.0, and C=1.2, respectively. When Vr/fc/ρ01/2<1.8, the cavity surface radial stress σr calculated by the elastic-cracked-compacted model is slightly larger than that of the elastic-cracked-plastic model. When Vr/fc/ρ01/2>1.8, the σr in the elastic-cracked-compacted model is smaller than the elastic-cracked-plastic model, and the difference between the two models becomes greater with increasing of the cavity expansion velocity. This also explains why the elastic-cracked-plastic model cannot effectively predict the penetration depth under relatively high-impact velocities.Figure 4
Model verification for 51 MPa concrete: (a) comparison of predicted and experimental DOP and (b) cavity surface radial stress calculation results of the elastic-cracked-compacted and elastic-cracked-plastic model.
(a)(b)For 23 MPa concrete, the test results [39] with P-α EOS and nonlinear yield criterion by data fitting are plotted in Figures 5(a) and 5(b), where ρ0=1.97 g/cm3, α0=1.345, Pe=7.6 MPa, Ps=2138 MPa, a0=8.49 MPa, a1=0.79, and a2=2.12×10−4 MPa-1. As Figure 5(c) shows, the predicted DOP by the proposed model is in better agreement with the test data than the classic model in Ref. [12].Figure 5
Comparison of theoretical model and test data for 23 MPa concrete: (a) EOS, (b) yield criterion, and (c) DOP.
(a)(b)(c)For 36 MPa concrete, using the hydrostatic pressure-shear strength test data from Ref. [3], the fitting results with nonlinear least squares are plotted in Figure 6(a). For the EOS, since the pressure-porosity data of 36 MPa concrete are absent, the concrete exhibits similar compaction characteristics when the strength is very close. Therefore, using the EOS parameters from 35 MPa [40] concrete seems reasonable. The constitutive parameters for 36 MPa concrete are summarized as follows: ρ0=2.31 g/cm3, α0=1.19, Pe=23.3 MPa, Ps=6000 MPa, a0=22.89 MPa, a1=0.72, and a2=2.17×10−3 MPa-1. Figure 6(b) demonstrates that the proposed model provides better predictions than the classic model, and it can be inferred that the underestimation DOP of the classic model mainly comes from the use of the linear constitutive relations.Figure 6
Comparison of theoretical model and test data for 36 MPa concrete: (a) yield criterion and (b) DOP.
(a)(b)


## 4.3. Further Discussions

To obtain the parameters of the nonlinear constitutive in Eqs. (1) and (2), i.e.,Pe, Ps, a0, a1, and a2, the pressure-porosity data from flyer-plate impact tests and the hydrostatic pressure-shear strength data from triaxial compression tests are needed. However, most DOP tests lack the necessary data required for analysis and calculation models. Under this circumstance, the parameters of a0, a1, and a2 are taken from the method proposed by Markovich et al. [41] as well as the value of Pe suggested by Holmquist et al. [42]. The default values of Ps for 35 MPa and 140 MPa concrete are both equal to 6 GPa in AUTODYN version 6.1 [43]. Here, the compacted pressure Ps is also recommended to be 6 GPa for fc=58.4 MPa and fc=62.8 MPa concrete. Thus, the constitutive parameters of normal concrete are summarized as follows:
(38a)a0=2.442fc0.4369,a1=1.084fc−0.2463,a1=0.03276fc−0.6416,(38b)Pe=fc3,Ps=6GPa,where fc is measured in MPa.Figure7 gives the experimental data from Refs. [4, 5] and the predicted DOP by Eq. (36), where the parameters of the P-α EOS and nonlinear yield criterion are given in Eq. (38). Most of the predictions show reasonable agreement with the test data except for two points, which are marked by a circle in Figure 7(b). The overestimations of the DOP for these two data points may lie in that if the striking velocity exceeds 987 m/s, the projectile mass loss will aggravate, and the mass abrasive penetration regime occurs.Figure 7
Comparison of predicted and experimental DOP for (a) 58.4 MPa and (b) 62.8 MPa concrete.
(a)(b)


## 5. The Influence of the Concrete Constitutive Parameters on the DOP

Taking 51 MPa concrete target as an example, the influences of the concrete constitutive parameters on the DOP are discussed below. Figure8 illustrates the effects of one specific variable on the DOP while maintains other variables to be same. During the theoretical analysis, ρ0=2.35 g/cm3 and α0=1.177.Figure 8
Influence of the target material parameters on the DOP (a)a0, (b) a1, (c) a2, (d) Pe, and (e) Ps.
(a)(b)(c)(d)(e)From Figure8(a), it is observed that if a0 increases by 50%, the DOP decreases by 3.8%, 3.4%, and 3.2% when the striking velocity is 400 m/s, 800 m/s, and 1200 m/s, respectively. Obviously, the DOP is relatively less affected by the cohesion in different velocity ranges.As Figure8(b) shows, if a1 increases by 50%, the DOP increases by 6.8%, 7.1%, and 7.2% when the striking velocity is 400 m/s, 800 m/s, and 1200 m/s, respectively. In this respect, the coefficient of internal friction μ has a considerable influence on the DOP.As can be seen in Figure8(c), if a2 increases by 50%, the DOP increases by 7%, 10.2%, and 12.5% when the striking velocity is 400 m/s, 800 m/s, and 1200 m/s, respectively. Therefore, we believe the von Mises plastic limit τm has a substantial influence on the DOP.As is observed in Figure8(d), if Pe increases by 50%, the DOP decreases by 2.5%, 1.9%, and 1.6% when the striking velocity is 400 m/s, 800 m/s, and 1200 m/s, respectively. In general, the influence of the elastic limit on the DOP decreases with increasing striking velocity and is negligibly small at high striking velocities.From Figure8(e), if Ps increases by 50%, the DOP decreases by 6.8%, 12.0%, and 16.6% when the striking velocity is 400 m/s, 800 m/s, and 1200 m/s, respectively. It is observed that the influence of compacted pressure on the DOP increases with increasing striking velocities. For lower impact velocities, the effect of Ps on the DOP is relatively small, and for higher striking velocities, it may become considerably large.


## 6. Conclusions

(1)
TheP-α EOS and nonlinear yield criterion are employed to characterize the plastic mechanical response of concrete under impact loading. Then, an improved dynamic spherical cavity expansion model considering the concrete compressibility and nonlinear constitutive relations is established(2)
A penetration model to predict the DOP is obtained, and the proposed model seems to better predict the available test results compared to the classic model derived from the linear EOS and yield criterion(3)
The parametersa2 and Ps have a great influence on the DOP while the internal friction coefficient μ has a considerable influence on the DOP. In contrast, a1 and Pe have a relatively small influence on the DOP and are weakly dependent on the striking velocities



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*Source: 6664882-2021-01-06.xml*