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Research Papers

# Material Modeling of Concrete for the Numerical Simulation of Steel Plate Reinforced Concrete Panels Subjected to Impacting LoadingPUBLIC ACCESS

[+] Author and Article Information
Huiyun Li

Department of Mechanics,
Tianjin University,
Tianjin 300354, China

Guangyu Shi

Department of Mechanics,
Tianjin University,
Tianjin 300354, China
e-mail: shi_guangyu@163.com

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 6, 2016; final manuscript received September 8, 2016; published online February 7, 2017. Assoc. Editor: Taehyo Park.

J. Eng. Mater. Technol 139(2), 021011 (Feb 07, 2017) (12 pages) Paper No: MATS-16-1170; doi: 10.1115/1.4035487 History: Received June 06, 2016; Revised September 08, 2016

## Abstract

The steel plate reinforced concrete (SC) walls and roofs are effective protective structures in nuclear power plants against aircraft attacks. The mechanical behavior of the concrete in SC panels is very complicated when SC panels are under the action of impacting loading. This paper presents a dynamic material model for concrete subjected to high-velocity impact, in which pressure hardening, strain rate effect, plastic damage, and tensile failure are taken into account. The loading surface of the concrete undergoing plastic deformation is defined based on the extended Drucker–Prager strength criterion and the Johnson–Cook material model. The associated plastic flow rule is utilized to evaluate plastic strains. Two damage parameters are introduced to characterize, respectively, the plastic damage and tensile failure of concrete. The proposed concrete model is implemented into the transient nonlinear dynamic analysis code ls-dyna. The reliability and accuracy of the present concrete material model are verified by the numerical simulations of standard compression and tension tests with different confining pressures and strain rates. The numerical simulation of the impact test of a 1/7.5-scale model of an aircraft penetrating into a half steel plate reinforced concrete (HSC) panel is carried out by using ls-dyna with the present concrete model. The resulting damage pattern of concrete slab and the predicted deformation of steel plate in the HSC panel are in good agreement with the experimental results. The numerical results illustrate that the proposed concrete model is capable of properly charactering the tensile damage and failure of concrete.

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## Introduction

Because concrete is the most commonly used construction material in civil engineering for various structures, the dynamic behavior of concrete has been received extensive study in the past few decades [10,11], and the analysis of high-velocity impact on concrete structures has been an attractive subject of numerical methods in recent years [11]. Many constitutive models have been proposed for the nonlinear response analysis of concrete structures [1215]. Commonly used frameworks for concrete constitutive equations are plasticity, damage mechanics, and their combinations. Plastic model has been used to analyze the constitutive relationship of concrete, but it fails to address the stiffness degradation that is observed in experiments. The plastic-damage model, which combines plasticity with isotropic or anisotropic damage to describe the nonlinear response of concrete due to irreversible deformations and micro cracks, is an appropriate method to model the constitutive equations of concrete. The stochastic model was used also to model the nonlinear behavior of concrete [16]. Anisotropic damage models for concrete are rigorous theoretically but they are often too complex to be applied in engineering analysis [1719]. Therefore, isotropic damage models have been extensively studied [2027] with different types of combinations with plasticity. Moreover, the behavior of concrete is strongly influenced by the confining pressure [2831] and strain rate [10,3243]. The influences of the combination of high confining pressures and high strain rate are not addressed in the above-mentioned models. Consequently, they have limited applicability to account for severe loadings such as impact or explosion.

A large of number of experiments of concrete under static triaxial confining pressure have been conducted [2830]. These experiments show that the peak stresses and ultimate strains of concrete increase with the enhancement of confinement pressure. It has also been observed that dry concrete reaches a limit state characterized by a transition from compaction to dilatancy. This transition can occur around the peak load in the case of low to moderate confinement and during the increment of the axial load in the case of high confinement [31]. Concretelike materials exhibit strain rate sensitivity, and the behavior of concrete under dynamic loads differs from that in static state. The increase in strength under dynamic loads can be expressed through the dynamic increase factor (DIF) that represents the ratio between the dynamic strength for a given strain rate and the static or quasi-static strength. Nevertheless, Cotsovos and Pavlovic [38] as well as Schwer [41] pointed out that the observed DIF is not an actual strength increase. In fact, it occurs exclusively due to the confinement effect produced by the lateral inertia forces [31]. However, as a convenient format for implementation into numerical models, DIF is still widely used to represent the effect of strain rate on the dynamic behavior of concrete. Many empirical relationships are available to estimate strain rate effect on the dynamic behavior of concrete [43]. Aráoz and Luccioni [31] presented a general elastic–viscoplastic model for concretelike materials under high strain rate dynamic loads that produce high confinement pressures, but the degradation of material stiffness was not taken into account.

Computer simulation of structural response under harsh environment such as impacting loading and fluid–structure interaction is nowadays possible by using hydrocodes that are explicit dynamic codes particularly suitable for characterizing highly nonlinear structural response under impulsive loads [31]. A number of commercial hydrocodes such as autodyn and ls-dyna are available for the general simulation of structural nonlinear dynamic responses. Various numerical methods are powerful tools to simulate the dynamic response of concrete structures. Nevertheless, the results obtained from a numerical simulation strongly depend on the constitutive model used for the materials. Tu and Lu [11] presented a detailed review of concrete models used in commercial hydrocodes. The most widely used concrete material models in hydrocodes to simulate concrete structures under severe dynamic loading are the Holmquist–Johnson–Cook (HJC) model [12], continuous surface cap (CSC) model [13], Karagozian and case concrete (KCC) model [14], and Riedel–Thoma–Hiermaier (RHT) model [15]. The CSC, KCC, and RHT models require too many material parameters, some of which are difficult to be determined by simple material tests. The HJC model can account for most of the important material parameters of concrete, such as hydrostatic pressure, strain rate, and compressive damage, to model the compressive damage of concrete under high pressure and high strain rate [43]. It represents a good compromise between the simplicity and the accuracy for computations, and has been widely applied in numerical simulations of the dynamic response of concrete structures [4447]. Gebbeken and Ruppert [44] modified the yield surface of the HJC model with different tensile and compressive meridians and used a shape function to define the octahedral stresses in the deviatoric plane. They also reformulated the strain rate effect. Xiong et al. [46] studied the experimental determinant of the material constants in the HJC model. In a later study, Hartmann et al. [47] combined different approaches for the modeling of the different phenomena of dynamic material behavior from several existing material models of concrete including the HJC and RHT models. However, the tensile damage evolution of concrete and the influence of the third deviatoric stress invariant are not taken into account in the HJC model and the modified HJC models. In general, the major drawback of HJC model is that it cannot capture the brittle failure of concrete [45].

Shi and Sun [45] emphasized that a reliable computational material model of concretelike materials should have the features such as combined hardening of the strain rate and confining pressure, and different damage evolution laws for tensile loading and compressive loading. By considering the above-mentioned features, this work aims to establish a new concrete material model that accounts for high confining pressures and high strain rates, plastic damage, and tensile failure. The proposed constitutive model is programmed in fortran and is implemented in ls-dyna through the user's subroutine UMAT [48]. The proposed concrete model is applied to simulate the damage behavior of concrete specimens under different loading conditions for the model validation. The numerical simulation of a 1/7.5-scale model of an aircraft penetrating into an HSC panel is performed to further validate the practicality and accuracy of the proposed concrete model. The simulation results given by the present concrete material model are compared with the experimental results.

## A Concrete Material Model Accounting for Confining Pressure, Strain Rate, Plastic Damage, and Tensile Failure

As mentioned in Introduction, the dynamic mechanical properties of the steel plate in SC panels can be efficiently modeled by the Johnson–Cook model [9], but the existing concrete material models have certain drawbacks when they are used in the numerical simulation of concrete structures under impact loading. For the concrete in the SC panels under the action of impact loading, it is not only subjected to high strain rate but also high confining pressure resulting from the constraint by the steel plate. Therefore, a reliable material model for the modeling of concrete under impact loading should possess the following main characteristics: (1) strain rate hardening effect; (2) the nonlinear hardening resulting from confining pressure; (3) plastic deformation and the strain softening effect resulting from the plastic strains; (4) the influence of the third deviatoric stress invariant; and (5) different modes for the damage and failure under, respectively, tensile and compressive loading. Since the major drawback in the existing concrete material models used for impact simulation is the modeling of tensile damage and failure, the new concrete material model presented in this section is focused on the modeling of plastic damage and tensile failure of porous concrete.

###### Effective Stress Accounting for the Different Behavior Under Compressive and Tensile Loading.

The proposed material model is an elastoplastic model coupled with isotropic damage, in which the response is decomposed into deviatoric and hydrostatic contributions. The Cauchy stress tensor $σij$ (i, j = 1, 2, 3) is decomposed into deviatoric and hydrostatic parts as Display Formula

(1)$σij=sij−pδij$
where $sij$ is the deviatoric stresses, $p=−σkk/3$ is the hydrostatic stress, and $δij$ is the Kronecker delta.

One basic feature of concrete is that its tensile strength is much smaller than its compressive strength. The smoothness of the loading surface of concrete is very important for the computational efficiency in the numerical simulation. The modified equivalent stress defined in the extended Drucker–Prager criterion is a good choice to compromise the computational efficiency and the result accuracy since it is able to account for the difference of compressive strength and tensile strength, and meanwhile to make the resulting loading surface smooth. Therefore, the modified equivalent stress $τ$ in the extended Drucker–Prager strength criterion defined in Eq. (2) is adopted in this study Display Formula

(2)$τ=12σeq[1+1Ks−(1−1Ks)(rσeq)3]eq$

In the equation above, σeq is the equivalent stress used in the original Drucker–Prager strength criterion which is related to the second invariants $J2$ of deviatoric stress tensor $sij$; $r$ is a parameter related to the third invariants $J3$ of deviatoric stress tensor; and $Ks$, which varies within the range of 0.778 ≤  $Ks$  ≤ 1.0, is a parameter to ensure the convexity of the yield surface. When $Ks$ = 1, the dependence on the third deviatoric stress invariant is vanished and the original Drucker–Prager model is recovered. The equivalent stress σeq and the parameter $r$ take the form, respectively, Display Formula

(3)$σeq=3J2=32sijsij$
Display Formula
(4)$r3=92sijsjkskj=272J3$

The yielding curve in the deviatoric plane given by Eq. (2) is depicted in Fig. 1(a), which illustrates that the equivalent stress $τ$ defined in Eq. (2) can accounts for different responses under tension and compression through parameter $Ks$.

###### Strain Rate Effect.

The Johnson–Cook material model [9] can accurately account for the strain rate effect of metals. Holmquist et al. [12] showed that the Johnson–Cook model can also effectively characterize the strain rate effect of concrete. The framework of the Johnson–Cook model will be used in the present work to establish the concrete material model, but the temperature effect is not taken into account in the present study.

###### Yield Criterion.

By using the Johnson–Cook material model and the extended Drucker–Prager strength criterion defined in Eq. (2), the loading surface of concrete can be expressed as a function of the deviatoric stresses, damage parameter, hydrostatic pressure, and strain rate as Display Formula

(5)$τfc=[A(1−D)+B(p*)N][1+Cln(ε˙*)]$

where $fc$ is the uniaxial static compressive strength; $D$ is the damage parameter with $0≤D≤1$; $p*=p/fc$ is the normalized pressure ($p$ is the actual pressure); and $ε˙*=ε˙/ε˙0$ is the dimensionless equivalent strain rate with respect to reference strain rate $ε˙0$. Parameters A, B, N, and C are the material constants, where A is the normalized cohesive strength, B is the normalized pressure hardening coefficient, N is the pressure hardening exponent, and C is the coefficient of strain rate effect. The loading surface in the meridian plane defined by Eq. (5) is illustrated in Fig. 1(b).

In order to conveniently define the plastic potential which is used later, Eq. (5) can be recast as Display Formula

(6)$f=σeq−2fc1+1Ks−(1−1Ks)(rσeq)3[A(1−D)+B(p*)N]1+Cln(ε˙*)]$

in which Eq. (2) is used. By defining a variable $Y$ as Display Formula

(7)$Y=2fc1+1Ks−(1−1Ks)(rσeq)3[A(1−D)+B(p*)N][1+Cln(ε˙*)]$

then Eq. (6) can be rewritten as Display Formula

(8)$f=σeq−Y$

Consequently, the loading surface is defined by f = 0. The expression of the loading surface of concrete defined in the equation above will be used for the plastic potential in this study.

###### Equation of State for Concrete.

The equation of state defines the hydrostatic pressure–volume relationship. Under the assumption of small deformation, the volumetric strain μ can be expressed in terms of the principal strain $εii$ as Display Formula

(9)$μ=V−V0V0=(1+ε11)(1+ε22)(1+ε33)−1≈ε11+ε22+ε33$

where $V0$ and $V$ are the initial volume and current volume, respectively. Under the action of high hydrostatic pressure, the pressure–volumetric strain response of concrete can be divided into three regions as illustrated in Fig. 2 [12], in which $pcrush$ and $μcrush$ stand for, respectively, the pressure and volumetric strain value that occur in a uniaxial compression test; $plock$ and $μlock$ stand for the pressure and volumetric strain of the material compaction point.

The first stage in the pressure–volumetric strain curve depicted in Fig. 2 is the linear elastic stage (path $OW1$), in which the hydrostatic pressure p is linearly proportional to the volumetric strain μDisplay Formula

(10)${p=KμK=pcrushμcrush$

where K is the bulk modulus of concrete under consideration. After the elastic stage, the second stage ($W1W2$) is the transitional stage. In this stage, the porous concrete gradually turns into compact material, so it produces plastic volumetric strain, and the pressure can be calculated by interpolation method as Display Formula

(11)$p=plock−pcrushμlock−μcrush(μ−μlock)+plock$

The third stage in the equation of stage defines the relationship for fully compacted material followed by nonlinear elastic behavior. The air voids are completely removed from the material when the pressure reaches $plock$, and the relationship is expressed as Display Formula

(12)$p=K1μ¯+K2μ¯2+K3μ¯3$

in which $μ¯=(μ−μlock)/(1+μlock)$ denoted the modified volumetric strain; and $K1$, $K2$, and $K3$ are equivalent to those used for material with no void constants.

Under the action of tension pressure, concretes, which are of typical brittle materials, undergo linear deformation only, and the pressure–volumetric strain is of the form $p=Kμ$, where $K$ is the bulk modulus defined in Eq. (10). The maximum tensile pressure is associated with the ultimate value of the tensile damage parameter $Dt$ defined in Sec. 2.6.1.

The unloading behavior in the pressure–volumetric strain curve has to be considered for the concrete under the action of dynamic loading. Corresponding to the three regions illustrated in Fig. 2, $p=Kμ$ in the elastic region, $p=[(1−F)K+FK1]μ$ in the transition region, where F is the interpolation factor and takes the form $F=(μmax−μcrush)/(μlock−μcrush)$ with $μmax$ being the maximum volumetric strain reached prior to unloading [12], and $p=K1μ$ in the fully compacted region.

###### Plastic Strains.

The associated plastic flow rule is used in this study. Then, it follows from Eq. (8) that the plastic potential g takes the form Display Formula

(13)$g=σeq−Y$

The strain rate in the plastic deformation stage is decomposed into an elastic part $dεije$ and a plastic part $dεijp$ as Display Formula

(14)$dεij=dεije+dεijp$

It follows from plasticity theory that the rate of plastic strains $dεijp$ can be computed from the plastic potential g as Display Formula

(15)$dεijp=λ˙∂g∂σij$

where $λ˙$ is the plastic multiplier. By substituting Eq. (13) into Eq. (15), the equation above becomes Display Formula

(16)$dεijp=λ˙(3sij2σeq−∂Y∂σij)$

The incremental effective plastic strain $dεeffp$ and incremental volumetric strain $dμp$ can be obtained, respectively, from the following relationships: Display Formula

(17)$dεeffp=23dεijpdεijp$
Display Formula
(18)$dμp=λ˙∂Y∂p$

###### Damage Evolution.

A key aspect of the proposed concrete material model is to treat tensile damage mechanism and compressive damage evolution separately. Two internal damage variables $Dt$ and $D$ were introduced to characterize the tensile damage and compressive damage, respectively.

###### Tensile Damage.

Since the concrete, which is a kind of porous material, is extremely sensitive to tensile stress, the damage evolution induced by tensile stress is one of the major concerns in the present study. The tensile damage parameter is based on the work of Yuan et al. [49]. The volumetric expansion strain $μt$ at time t is expressed as Display Formula

(19)$μt=max{μ0,maxs∈[0,t](μ)s}$

in which $μ0$ is the threshold of volumetric strain for the tensile damage initiation when the volume of concrete undergoes expansion. The evolution equation of tensile damage variable $Dt$, 0 ≤  $Dt$  ≥ 1, at time t is defined as Display Formula

(20)$Dt=1−exp[−aV(μt−μ0)]$

where $aV$ is the material constant of the volumetric expansion.

###### Compressive Damage.

The plastic damage in concrete induced by compression is accumulated in a manner similar to that used in the HJC model [12]. The plastic damage is defined in terms of effective plastic strain and plastic volumetric strain. The compressive damage variable D is expressed as Display Formula

(21)$D=∑Δεeffp+Δμpεpf+μpf$

where $Δεeffp$ and $Δμp$ are, respectively, the increment of equivalent plastic strain and the increment of plastic volumetric strain during a cycle of integration, and $εpf+μpf$ is the plastic strain to fracture under a constant pressure $p$. The expression for the fracture plastic strain is given as [12] Display Formula

(22)$εpf+μpf=D1(p*+T*)D2≥εfmin$

in which $D1$ and $D2$ are material constants, $p*$ is as defined previously, and $T*=ft/fc$ is the normalized maximum tensile hydrostatic pressure. $ft$ is the uniaxial tensile strength. $εfmin$ is specified to allow for a finite amount of plastic strain to fracture the material.

It follows from Eqs. (2)(6) that Eq. (8) is a function of the deviatoric stresses $sij$, the hydrostatic stress p, the compressive damage parameter D, and the equivalent strain rate $ε˙$. The consistency condition on the loading surface is Display Formula

(23)$f(sij,p,D,ε˙)=σeq(sij)-Y(sij,p,D,ε˙)=0$

(24)$∂f∂sijdsij+∂f∂pdp+∂f∂DdD+∂f∂ε˙dε˙=0$

Since Eq. (24) holds also for the quasi-static loading, the term associated with the strain rate $ε˙$ can be neglected for the purpose of evaluating plastic strains. By using Eq. (23), Eq. (24) reduces to Display Formula

(25)$∂σ∂sijdsij−(∂Y∂sijdsij+∂Y∂pdp+∂Y∂DdD)=0$

It follows from Eqs. (16), (17), and (21) that the incremental damage dD in the equation above can be expressed in terms of the plastic multiplier $λ˙$ in Eq. (16). Therefore, the plastic multiplier $λ˙$ can be evaluated from Eq. (25) after some standard mathematical manipulation.

## Model Validations: Comparison of Numerical Results With Experimental Data of Standard Tests

The concrete material model proposed in Sec. 2 was implemented into the nonlinear dynamic finite element code ls-dyna via a user's material subroutine [48]. The mechanical behaviors of several standard tests of concrete specimen under uniaxial, biaxial, and triaxial loadings are simulated by ls-dyna with the present concrete material model to evaluate the accuracy of the present concrete material model.

The standard cubic specimen with a dimension of 200 mm × 200 mm × 200 mm is considered in the numerical simulation of uniaxial tensile loading and uniaxial compressive loading. The specimen with a dimension of 200 mm × 200 mm × 50 mm was used for biaxial tensile loading and biaxial compressive loading. Both of these two specimens were modeled by a single solid element only, and one Gauss integration point was employed in the numerical simulation. The boundary condition of fixed displacements was applied to the bottom of the specimens. The displacement-controlled loading was applied to the top of the specimens in order to simulate the whole response of concrete. In order to compare the numerical results with the experimental results reported in Refs. [50] and [51], the following material properties were used. Young's modulus is 31.8 GPa, Poisson's ratio is 0.18, compressive strength is 27.6 MPa, and uniaxial tensile strength is 3.5 MPa. According to the studies of Xiong et al. [46] as well as Guo and Wang [52], the material constants A = 0.3, B = 2.01, and N = 0.73 were used here. The low sensitivity strain rate parameter C = 0.007 can be reasonably used for concrete, and $ε˙0=1.0 s−1$ was used for the reference strain rate based on the HJC model [12]. $Ks$  = 0.8 can be employed according to the suggestion given by Park et al. [53]. $aV$ can be determined through volumetric expansion tests of specimens [49]. When $aV$ is not available, an approximate value of $aV$ can be determined through Eq. (20) together with the numerical simulation of uniaxial tensile test in which the value of $aV$ yields $Dt$  = 1 at the given tensile strength. $av=6500$ was determined by such a way in the present case study. The threshold of volumetric strain $μ0=0$ is adopted in all numerical examples. The elastic limit pressure constants can be determined by $pcrush=fc/3$ and $μcrush=pcrush/K$. The values of $plock$, $μlock$, $K1$, $K2$, and $K3$ are the same as those used in the HJC model [12]. $D1$ and $D2$ were also taken from the HJC model. The material parameters of concrete used in the present simulations are summarized in Table 1.

Figures 3(a) and 3(b) display the comparisons of the present numerical results with the experimental results for the cases under compressive loading [50] and uniaxial tensile loading [51], respectively. Figures 4(a) and 4(b) show the comparisons of the numerical results with the experimental results [54] under biaxial compressive and tensile loading, respectively. The results in Figs. 3 and 4 show that the stress–strain curves obtained from the numerical simulations agree well with those given by the experiments for both uniaxial and biaxial tensile/compressive loadings.

###### Triaxial Compression Tests With Different Confining Pressures.

Triaxial compression tests of concrete have been performed by some researchers using cylindrical specimens with a range of uniaxial compressive strength [2832]. In the present numerical simulation, a cylinder with diameter of 70 mm and height of 140 mm was modeled under a confinement pressure ranged from 50 to 650 MPa. One end of the specimen was fixed and the axial displacement was applied on the other end with small increments. The confinement pressure was applied on the lateral surface of the cylinder. The material properties of the concrete are as follows: Young's modulus of 24 GPa, Poisson's ratio of 0.13, and compressive strength of 42 MPa. Since the material constant of the volumetric expansion the $aV$ only affects the crushing and scabbing failure, it is not needed in the simulation of compression test. A total of 96,000 elements of Solid 164 in LS-DYNA were used to model the cylindrical specimen.

Figure 5 illustrates the evolution of the axial stress–strain curves under different confining pressures. The numerical results given by the present concrete model are compared with the experimental data given by Vu et al. [30]. The peak stresses obtained from the numerical simulation agree well with the experimental results, but the strains corresponding to the peak stresses are smaller than those measured in the experiments. The smaller strains in the simulated stress–strain curves can be attributed to the fixed displacement boundary conditions on the bottom of the specimen used in the simulation. In the lab test, the bottom of the specimen is frictional boundary in the horizontal direction which allows the specimen to expand under the axial compressive load. Nevertheless, the results in Fig. 5 show that the fixed boundary condition used in the simulation does not affect the reliable prediction of the compressive strength for the concrete subjected to high confining pressure.

The plastic theory implies that plastic deformation of concrete must be accompanied by a volumetric increase. This phenomenon is known as dilation. Figure 6 displays the volumetric responses under confining pressures of 200 MPa and 400 MPa obtained from the present numerical simulation and the experiment results reported by Poinard et al. [33]. Once again, the smaller volumetric strain given by the simulation is resulting from the idealized fixed boundary condition specified to the bottom of the specimen.

It can be seen from Fig. 6 that the volumetric strains of concrete are changed from a contraction phase to a dilatancy phase when the strains reach a certain threshold. The point of the contraction–dilatancy transition allows defining a strain limit state for concrete [33].

The dynamic mechanical properties of concrete are markedly different from those exhibited under quasi-static conditions. In the low strain rate range, material strength generally increases slowly with an increase of strain rate. However, when the strain rate exceeds a threshold, the rate of increase in material strength is rapid. In order to further validate the concrete material model proposed in this work, the dynamic compression tests of cylindrical mortar specimens performed by Grote et al. [34] were simulated using the present concrete material model. The cylindrical mortar specimen has the dimension of diameter 76.2 mm and thickness 10.9 mm. The main material properties are as follows: Young's modulus of 20 GPa, Poisson's ratio of 0.2, maximum compressive strength of 46 MPa, and density of 2100 kg/m3. Other material properties are as those given in Table 1. A total of 15,210 elements of Solid 164 in LS-DYNA were used to model the cylindrical specimen. In the simulation, one end of the specimen was totally fixed and the other end was applied an axial displacement load with the given strain rate. The stress–strain curves for different strain rates obtained from the simulations are compared with their experimental counterparts given by Grote et al. [34], and the results are plotted in Fig. 7.

It can be seen from Fig. 7 that the present concrete model can characterize the strain rate effect before the peak stress very well, but it fails to match the experimental stress–strain curves after the peak stresses. The reason for the difference is that a simplified dynamic load and idealized boundary condition were used in the numerical simulation that cannot exactly reproduce the complicated dynamic loading and the interaction between the specimen and the output bar in the split Hopkinson pressure bar (SHPB) tests conducted by Grote et al. [34]. For example, the fixed displacement used in the numerical simulation is equivalent to a rigid wall to the specimen, while the output bar in SHPS behaves like a sort of elastic support which also undergoes some deformation under the action of impact loading. Nevertheless, as illustrated in Fig. 7, the present concrete model can accurately capture the strength of concrete under dynamic loading with high strain rates.

The strain rate effect of concrete under tensile loading is different from that under compression. The dependency of the strength increase on strain rate for concrete under tensile loading was experimentally investigated by Yan and Lin [35]. The specimens were made in dumbbell shape and the tested strain rates were ranged from 10−5/s to 10−0.3/s. The dimension of the specimen with a constant cross section is 70 mm × 70 mm × 100 mm. The specimen group C in Ref. [35] was considered in the present numerical simulation, in which the material properties are as follows: Young's modulus of 18.9 GPa, Poisson's ratio of 0.17, the static tensile strength of 1.3 MPa, and the static compressive strength of 24.46 MPa. A total of 4900 solid elements were used to model the squared prism of the specimen. In the numerical simulation, the fixed boundary conditions were enforced on one end of the specimen, and axial displacement loading with the given strain rate was applied on the other end. The numerical results for different rates of tensile strain are compared with the experimental results given by Yan and Lin as shown in Fig. 8.

Figure 8 illustrates that the present concrete model is able to correctly predict the strain rate effect on the peak stresses under various tensile strain rates. However, the predicted stress–strain curve is different from the experimental results. The different behavior in the simulated stress–strain curves can be attributed to the idealized displacement boundary conditions and dynamic displacement loading on the specimen used in the numerical simulation.

## Application of the Present Concrete Model in the Impact Simulations of Concrete Targets

The performance of the proposed concrete model is demonstrated by the numerical simulations of concrete targets subjected to impact loading in this section.

###### Ogive-Nosed Projectile Penetrating Into Concrete Target.

The ballistic test of ogive-nosed projectile penetrating into concrete target conducted by Hanchak et al. [55] is simulated by using the present concrete model. A smooth-bore powder gun was used to launch 0.05 kg ogive-nosed steel projectiles with a length of 143.7 mm, a diameter of 25.4 mm, and a caliber-radius-head (CRH) of 3.0. The dimensions of the concrete target are 610 mm × 610 mm × 178 mm (thickness).

The rigid material model is used for the steel projectile. The density and Young's modulus of the steel are 8300 kg/m3 and 200 GPa, respectively. The material parameters of concrete used in this simulation are $ρ=2440 kg/m3$, E = 35.664 GPa, $ν=0.2$, $fc=48 MPa$, $ft=4 MPa$, and $av=3700$. The other parameters of the concrete are the same as the data shown in Table 1. The projectile was used to strike at the center of the concrete target at velocities of, respectively, 360 m/s, 381 m/s, 434 m/s, 606 m/s, 746 m/s, 749 m/s, and 1058 m/s. A quarter of the target and projectile is modeled by making use of symmetry as shown in Fig. 9. Some approximate computations using rebar indicate that rebar had little influence on the residual velocity for the reinforced concrete target under consideration [12]. As a result, a plain concrete slab was used as the target for simplification in this work. The Lagrangian coordinate system was employed for the projectile/target system. The element type is Solid 164, and 256,378 elements were used to model a quarter of the target and projectile. Figure 9 depicts the finite element mesh for the simulation. Nonreflective boundaries were exerted on the lateral faces of the target. The “eroding surface-to-surface” algorithm contact interface was adopted. The Flanagan–Belytschko stiffness was applied for the target elements. The strain failure criterion was used as an element erosion criterion in the simulation.

It is assumed that the ogive-nose steel projectile is free from deformation during the entire process of analysis. The tension damage of the concrete target is caused by the tensile volumetric strain. The compressive damage of the concrete target is caused by the plastic deformation. The two different types of damage exist in the concrete slab around the moving projectile. Since the concrete has much higher compressive strength than its tensile strength, the tensile damage appears adjacent to the free surface of the target as the influence of tensile waves is reflected from the free surfaces when the projectile has not reach the back surface of the slab. This reveals the spalling phenomenon of concrete under impact loading.

Figures 10(a) and 10(b) show, respectively, the tensile and compressive damage contours on the target cross section for the case of the steel projectile impacting the concrete slab at an impact velocity of 749 m/s. With the motion of the projectile, the tensile and compressive damage begins to evolve. Thus, cracks gradually grow and expand until the penetration across the target is complete. The damage distributions depicted in Fig. 10 demonstrate that the proposed material model is able to characterize the material failure during the perforation process in a qualitative manner.

Figure 11 displays the residual velocities obtained from the numerical simulation and the experimental data obtained by Hanchak et al. [55]. The results given by the HJC model are also shown in the figure for comparison. When the impact velocity is less than 400 m/s, the residual velocity given by the present concrete model is a little bit larger than the test data; however, the other computed results agree with the experimental data. But on the other hand, all the residual velocities given by the HJC model are lower than the test results as illustrated in Fig. 11, which can be attributed to that the HJC model overestimates the strength of the concrete target as it does not account for the tensile damage of brittle materials [56].

It is worthwhile to examine the predicted material failure on the target surfaces induced by the tensile damage. Figure 12(a) depicts the scabbing failure on the rear surface of the concrete slab in the field test at an impact velocity of 749 m/s reported by Hanchak et al. Figure 12(b) shows the tensile damage contour distribution on the rear surface given by the numerical simulation. It can be seen from Fig. 12 that the radial and circumferential cracks on the rear surface caused by the tensile waves are reproduced well by the numerical simulation using the present concrete model. The diameter of the predicted fracture on the rear surface is approximately 300 mm around the penetration, which matches well with the experimental result [55].

Figure 13 illustrates the compressive damage contour on the target cross section based on the HJC model for the case of impact velocity at 749 m/s. The compressive damage depicted in Fig. 10(b) shows some crabbing failure on the back of the concrete target, while the compressive damage depicted in Fig. 13 given by the HJC model is localized around the hole only. It should be noted that there are no tensile damages in the impact simulation using the HJC material model since the tensile damage is not taken into account in the HJC model. Consequently, the cracking and scabbing failure resulting from the tensile stress wave that are illustrated in Figs. 10(a) and 12 could not be captured by the HJC model. Li and Shi [56] presented the stress–strain curves of rock specimens under various loading conditions given by, respectively, the HJC model and the rock material model based on the framework used in the present study. The comparisons of these stress–strain curves with the experimental results clearly illustrate that the present concretelike material model yields more accurate stress–strain curves than the HJC model.

###### Simulation of Impact Test of the Half Steel Plate Reinforced Concrete Panel.

The analysis presented by Mizuno et al. [3] showed that the steel plate, especially the rear-face plate, has a significant effect in preventing scattering of scabbed concrete debris from the rear face of the panel. The composite concrete panel with only the real steel plate is called as half steel plate reinforced concrete (HSC) panel. Mizuno et al. [4,5] conducted an aircraft impact experiment of 1/7.5-scale model. The dimensions of the HSC panel are 1500 mm × 1500 mm × 80 mm, and the thickness of the steel plate is 1.2 mm. The detailed dimensions of the 1/7.5-scale aircraft model are given in Refs. [46]. The aforementioned 1/7.5-scale model test of HSC panel is simulated by using LS-DYNA with the proposed concrete material mode in this case study. The impact velocity of 1/7.5-scale aircraft is taken at 150 m/s as that used by Mizuno et al. [4]. The HSC panel with a thickness of 80 mm described above is named as HSC-80 for convenience.

###### Computational Model.

The concrete slab of HSC-80 was modeled using solid elements in the numerical simulation. The steel plate in HSC-80 was modeled using shell elements. The scaled aircraft model was simulated with solid and shell elements. The outer skin of the aircraft and the steel casing of the engine were modeled by shell elements while the hexcel core of the aircraft model is modeled by solid element. A total of 159,432 elements were used in the computational model. Fixed boundary conditions along the four outer edges of the HSC-80 panel were applied to constrain the translations and rotations. Figure 14 shows the dimensions of the 1/7.5-scale model test and the finite element mesh used for the impact simulation.

The concrete material model presented in Sec. 2 was applied to the material modeling of concrete in HSC-80. The material parameters of concrete used in the present simulation are $ρ=2300 kg/m3$, E = 19.6 GPa, $ν=0.2$, $fc=35 MPa$, $ft=2.94 MPa$, and $av=1800$. The other material constants of the concrete are the same as the data tabulated in Table 1. The material properties of the mild steel are adopted for the steel plate. The Johnson–Cook model with kinematic hardening is used for the steel plate.

###### Results and Discussion.

The penetration process of aircraft model and the fracture process of HSC-80 panel at three different instants are depicted in Fig. 15, in which 12 ms is the final stage of aircraft impacting test. As shown in Fig. 15(c), the HSC-80 panel escaped the perforation of the projectile, but the concrete slab impacted by the aircraft was severely fractured almost up to the rear surface of the steel plate. The testing result of 1/7.5-scale impact on the HSC-80 panel clearly indicates that the rear steel plate plays the most critical role in preventing projectile perforation and damage for the protective structures used for important facilities, although the ratio of the thickness of the steel plate to the total thickness of the HSC-80 panel is merely 1.2/80.

The velocity–time curve for the engine in the aircraft model is displayed in Fig. 16. A comparison is made between the velocity–time curves obtained from the present numerical simulation and the experimental result [4]. The computed velocity change is small till 5 ms. The engine velocity decelerates rapidly from 5 ms to 8 ms and begins to bounce back around 11 ms.

The distributions of tensile and compressive damaged areas of the concrete slab in the HSC-80 panel are plotted in Figs. 17(a) and 17(b), respectively. The diameter of the tensile damaged area on the front face of the concrete slab is about 450 mm, which is in a good agreement with the experimental result reported by Mizuno et al. [4].

The deformation of the steel plate in the HSC-80 panel, which is located on the rear face of the HSC-80 panel, is shown in Fig. 18. The averaged largest deflection of the steel plate central area at the instant of t = 12 ms is about 62 mm. No cracks were observed on the steel plate even though the steel plate has undergone a large displacement.

The simulation results of the concrete slab damage shown in Fig. 17 and the steel plate deformation depicted in Fig. 18 are consistent with the experimental results illustrated in Fig. 19 reported by Mizuno et al. [4].

## Conclusions

A dynamic material model for concrete is proposed in this paper. The new concrete material model takes account of pressure hardening, strain rate hardening, plastic deformation, and plastic damage. The associated plastic flow rule is employed to evaluate the plastic strains. Besides the plastic-damage parameter to model the strength degeneration, a tensile damage parameter is defined to characterize the tensile damage and tensile failure of the concrete. By implementing the proposed material model to LS-DYNA, the present concrete model was successfully applied to simulate various standard quasi-static and dynamic tests of concrete specimens. The simulation results were found to be in good agreement with the corresponding experimental results for a wide range of loading conditions from uniaxial tension to triaxially confined compression. The numerical simulation of the impact test of a half steel reinforced concrete panel against the scaled aircraft is carried out. The results of fracture process, damage area of concrete, velocity–time histories, and deformations agree well with the experimental results.

The present concrete material model can be easily implemented as the HJC model, but it yields more accurate results because the tensile damage is taken into account in the present concrete model to characterize the crushing and scabbing failure induced by the tensile stress wave. The numerical results of the two impact simulations show that the proposed concrete material model is capable of accurately modeling the cracking, crushing, and scabbing phenomena of porous concrete. The comparison between the experimental data and the simulation results given by, respectively, the present concrete model and the HJC model clearly indicates that the introduction of the tensile damage parameter is very important to model the tensile failure of concrete.

## Acknowledgements

The authors would like to thank the National Basic Research Program of China for the financial support under Grant No. 2013CB035402 to partially support this study.

## References

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## References

Takeuchi, M. , Narikawa, M. , Matsuo, I. , Hara, K. , and Usami, S. , 1998, “ Study on a Concrete Filled Structure for Nuclear Power Plants,” Nucl. Eng. Des., 179(2), pp. 209–223.
Sugano, T. , Tsubota, H. , Kasai, Y. , Koshika, N. , Orui, S. , von Riesemann, W. A. , Bickel, D. C. , and Parks, M. B. , 1993, “ Full-Scale Aircraft Impact Test for Evaluation of Impact Force,” Nucl. Eng. Des., 140(3), pp. 373–385.
Mizuno, J. , Kasai, Y. , Koshika, N. , Kusama, K. , Fujita, T. , and Imamura, A. , 1999, “ Analytical Evaluation of Multiple Barriers Against Full-Scale Aircraft Impact,” 15th International Conference on Structural Mechanics in Reactor Technology (SMiRT-15), Seoul, South Korea, Aug. 15–20, Vol. J04/4, pp. 153–160.
Mizuno, J. , Sawamoto, Y. , Yamashita, S. , Koshika, N. , Niwa, N. , and Suzuki, A. , 2005, “ Investigation on Impact Resistance of Steel Plate Reinforced Concrete Barriers Against Aircraft Impact. Part 1: Test Program and Results,” 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT-18), Beijing, China, Aug. 7–12, pp. 2566–2579.
Mizuno, J. , Morikawa, H. , Koshika, N. , Wakimoto, K. , and Fukuda, R. , 2005, “ Investigation on Impact Resistance of Steel Plate Reinforced Concrete Barriers Against Aircraft Impact. Part 2: Simulation Analyses of Scale Model Impact Tests,” 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT-18), Beijing, China, Aug. 7–12, pp. 2580–2590.
Muhammad, S. , Zhu, X. , and Pan, R. , 2014, “ Simulation Analysis of Impact Tests of Steel Plate Reinforced Concrete and Reinforced Concrete Slabs Against Aircraft Impact and Its Validation With Experimental Results,” Nucl. Eng. Des., 273(1), pp. 653–667.
Yun, S.-H. , and Park, T. , 2013, “ Multi-Physics Blast Analysis for Steel-Plated and GFRPP-Plated Concrete Panels,” Adv. Struct. Eng., 16(3), pp. 529–548.
Yun, S.-H. , Jeon, H.-K. , and Park, T. , 2013, “ Parallel Blast Simulation of Nonlinear Dynamics for Concrete Retrofitted With Steel Plate Using Multi-Solver Coupling,” Int. J. Impact Eng., 60, pp. 10–23.
Johnson, G. J. , and Cook, W. H. , 1983, “ A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures,” 7th International Symposium on Ballistics, Hague, The Netherlands, pp. 541–547.
Fu, H. C. , Erki, M. A. , and Seckin, M. , 1991, “ Review of Effects of Loading Rate on Concrete in Compression,” ASCE J. Struct. Eng., 117(12), pp. 3645–3659.
Tu, Z. , and Lu, Y. , 2009, “ Evaluation of Typical Concrete Material Models Used in Hydrocodes for High Dynamic Response Simulations,” Int. J. Impact Eng., 36(1), pp. 132–146.
Holmquist, T. J. , Johnson, G. R. , and Cook, W. H. , 1993, “ A Computational Constitutive Model for Concrete Subjected to Large Strains, High Strain Rates, and High Pressure,” 14th International Symposium on Ballistics, Quebec City, Canada, Sept. 26–29, pp. 591–600.
Murray, Y. D. , Abu-Odeh, A. , and Bligh, R. , 2007, “ User's Manual for LS-DYNA Concrete Material Model 159,” APTEK, Inc., Colorago Springs, CO, pp. 53–78.
Malvar, L. J. , Crawford, J. E. , Wesevich, J. W. , and Simons, D. , 1997, “ A Plasticity Concrete Material Model for DYNA3D,” Int. J. Impact Eng., 19(9–10), pp. 847–873.
Borrvall, T. , and Riedel, W. , 2011, “ The RHT Concrete Model in LS-DYNA,” 8th European LS-DYNA User's Conference, Strasbourg, France, pp. 1–14.
Fafitis, A. , and Won, Y. H. , 1989, “ Stochastic Nonlinear Constitutive Law for Concrete,” ASME J. Eng. Mater. Technol., 111(4), pp. 443–449.
Simo, J. C. , and Ju, J. W. , 1987, “ Strain and Stress-Based Continuum Damage Model. Part I: Formulation,” Int. J. Solids Struct., 23(7), pp. 821–840.
Yazdani, S. , and Schreyer, H. L. , 1990, “ Combined Plasticity and Damage Mechanics Model for Plain Concrete,” ASCE J. Eng. Mech., 116(7), pp. 1435–1450.
Voyiadjis, G. Z. , Taqieddin, Z. N. , and Kattan, P. I. , 2009, “ Theoretical Formulation of a Coupled Elastic-Plastic Anisotropic Damage Model for Concrete Using the Strain Energy Equivalence Concept,” Int. J. Damage Mech., 18(7), pp. 603–638.
Mazars, J. , and Pijaudier-Cabot, G. , 1989, “ Continuum Damage Theory-Application to Concrete,” ASCE J. Eng. Mech., 115(2), pp. 345–365.
Lubliner, J. , Oliver, J. , Oller, S. , and Oñate, E. , 1989, “ A Plastic-Damage Model for Concrete,” Int. J. Solids Struct., 25(3), pp. 299–326.
Shi, G. , and Voyiadjis, G. Z. , 1997, “ A New Free Energy for Plastic Damage Analysis,” Mech. Res. Commun., 24(4), pp. 377–387.
Salari, M. R. , Saeb, S. , Willam, K. J. , Patchet, S. J. , and Carrasco, R. C. , 2004, “ A Coupled Elastoplastic Damage Model for Geomaterials,” Comput. Methods Appl. Mech. Eng., 193(27–29), pp. 2625–2643.
Wu, J. Y. , Li, J. , and Faria, R. , 2006, “ An Energy Release Rate-Based Plastic-Damage Model for Concrete,” Int. J. Solids Struct., 43(3–4), pp. 583–612.
Grassl, P. , and Jirásek, M. , 2006, “ Damage-Plastic Model for Concrete Failure,” Int. J. Solids Struct., 43(22–23), pp. 7166–7196.
Jason, L. , Huerta, A. , Pijaudier-Cabot, G. , and Ghavamian, S. , 2006, “ An Elastic Plastic Damage Formulation for Concrete: Application to Elementary and Structural Tests,” Comput. Methods Appl. Mech. Eng., 195(52), pp. 7077–7092.
Zheng, F. G. , Wu, Z. R. , Gu, C. S. , Bao, T. F. , and Hu, J. , 2012, “ A Plastic Damage Model for Concrete Structure Cracks With Two Damage Variables,” Sci. China Technol. Sci., 55(11), pp. 2971–2980.
Sfer, D. , Carol, I. , Gettu, R. , and Etse, G. , 2002, “ Study of the Triaxial Behaviour of Concrete,” ASCE J. Eng. Mech., 128(2), pp. 156–163.
Gabet, T. , Malecot, Y. , and Daudeville, L. , 2008, “ Triaxial Behaviour of Concrete Under High Stresses: Influence of the Loading Path on Compaction and Limit States,” Cem. Concr. Res., 38(3), pp. 403–412.
Vu, X. H. , Malecot, Y. , Daudeville, L. , and Buzaud, E. , 2009, “ Experimental Analysis of Concrete Behavior Under High Confinement: Effect of the Saturation Ratio,” Int. J. Solids Struct., 46(5), pp. 1105–1120.
Aráoz, G. , and Luccioni, B. , 2015, “ Modeling Concrete Like Materials Under Sever Dynamic Pressures,” Int. J. Impact Eng., 76, pp. 139–154.
Bischoff, P. H. , and Perry, S. H. , 1991, “ Compressive Behaviour of Concrete at High Strain Rates,” Mater. Struct., 24(6), pp. 425–450.
Poinard, C. , Malecot, Y. , and Daudeville, L. , 2010, “ Damage of Concrete in a Very High Stress State: Experimental Investigation,” Mater. Struct., 43(1–2), pp. 15–29.
Grote, D. L. , Park, S. W. , and Zhou, M. , 2001, “ Dynamic Behavior of Concrete at High Strain Rates and Pressures: I. Experimental Characterization,” Int. J. Impact Eng., 25(9), pp. 869–886.
Yan, D. , and Lin, G. , 2006, “ Dynamic Properties of Concrete in Direct Tension,” Cem. Concr. Res., 36(7), pp. 1371–1378.
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## Figures

Fig. 1

Illustration of the present loading surface: (a) yielding curve on the π-plane and (b) yielding curve on the meridian plane

Fig. 2

The relation of pressure and volumetric strain

Fig. 3

Comparison of stress–strain curves given by numerical simulation and experimental results under uniaxial loading: (a) uniaxial compression [50] and (b) uniaxial tension [51]

Fig. 4

Comparison of stress–strain curves given by numerical simulation and experimental result under biaxial loading: (a) biaxial compression [54] and (b) biaxial tension [54]

Fig. 5

Simulated and measured stress–strain curves of concrete under different confinement pressures

Fig. 6

Comparison of numerical and experimental volumetric responses under triaxial loading: with confining pressures of (a) 200 MPa and (b) 400 MPa

Fig. 14

Finite element analysis model of HSC-80 panel and the 1/7.5-scale aircraft: (a) the front view of HSC-80 panel and (b) the side view of the system

Fig. 13

The distribution of simulated compressive damages based on the HJC material model

Fig. 12

The scabbing on the rear surface of concrete target: (a) field test [55] and (b) simulation result

Fig. 11

The residual velocities using simulation and the test data of Hanchak et al. [55]

Fig. 10

The distributions of predicted damages in the concrete target: (a) tensile damage and (b) compressive damage

Fig. 8

Stress–strain curves of concrete under dynamic tension testing at different strain rates: numerical results and their comparison with the experimental results [35]

Fig. 7

Numerical and experimental [34] stress–strain behaviors of concrete cylinder (the so-called mortar in Ref. [34]) under dynamic compression tests at different strain rates

Fig. 9

Computational model for axial symmetry used in the penetration simulation

Fig. 15

Fracture processes of aircraft model and HSC-80 panel

Fig. 16

Velocity time history curves of engine

Fig. 17

The damage distributions in the concrete slab of HSC-80 panel: (a) tensile damage and (b) compressive damage

Fig. 18

The deformation contours of the steel plate in HSC-80 panel at 12 ms: (a) the side view and (b) the angle view

Fig. 19

The concrete damage and steel plate deformation in HSC-80 panel reported in Ref. [4]

## Tables

Table 1 The material parameters used in the numerical simulation

## Discussions

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