Numerical predictions of the behaviour of plain concrete targets subjected to impact

Abstract

One widely used model implemented in the hydrocode ANSYS Autodyn is the Riedel–Hiermaier–Thoma material model, which is used for the prediction of concrete behaviour under severe loading, such as blast and impact. In the Riedel–Hiermaier–Thoma model, parameters are adjusted to be a function of the compressive strength of specific concrete. This advantage enables occasional users (and those who cannot conduct experiments to determine all the required concrete parameters) to input only this parameter, while the rest is automatically calculated by default settings. This study is an attempt to calibrate (not modify) this model, with the least number of possible changes, to better predict the performance of concrete targets, particularly spalling and scabbing phenomena, impacted by hard projectiles. It is shown that the present calibration of the Riedel–Hiermaier–Thoma material model performed well when compared to the default settings currently available in the model.

Keywords

Impact projectile concrete scabbing Riedel–Hiermaier–Thoma model

Introduction

Concrete, one of the major construction materials, has been extensively used in both civil and defence engineering. An understanding of the response of concrete structures subjected to projectile impact or blast loading is of great significance for the design and assessment of weapons as well as for protective structures (Xu and Wen, 2016). However, in spite of the extensive use of concrete as a construction material, our knowledge about its exact mechanical properties and physical behaviour under complex stress states is still limited (Cui et al., 2017). The analysis of concrete structures against dynamic loading scenarios is an especial challenge, since difficulties arise from the heterogeneity and influence of microstructure on the general behaviour of concrete, which until today is not understood in every detail (Grunwald et al., 2017). Concrete response to severe loading such as projectile impact is an interesting subject that has attracted researchers for many years, in spite of its great complexity. For many decades, much research has been conducted in an attempt to enhance our understanding of this problem and develop calculation tools to analyse the projectile impact of concrete target and to estimate the target response (Yankelevsky, 2017). However, the actual behaviour of concrete under severe loading cannot always be precisely reflected in any model dedicated to the constitutive modelling of the material (Nguyen, 2005).

When a concrete target (e.g. slab or wall) is subjected to projectile impact, one or more of the damage forms (e.g. penetration, spalling, scabbing, perforation, shear failure or flexural failure) may occur (Abdel-Kader and Fouda, 2014; Corbett et al., 1996; Hanchak et al., 1992; Li et al., 2005; Kennedy, 1976). Spalling and scabbing phenomena are frequently observed on its front (impact) and rear faces, respectively, Figure 1. Spalling arises from the compressive stress wave propagation in the target, the reflected tensile stress wave on the front face of the target and the associated (crushing) material failure, while scabbing is caused by the reflected tensile stress wave on the rear free face of the concrete target (Kong et al., 2017; Wang et al., 2007).

Figure 1.

Impact effects on concrete target caused by hard projectile (Li et al., 2005).

The most important concrete parameters that influence the response to projectile impact are the compressive strength, the tensile strength, the fracture energy, and the strain rate dependency for both compression and tension. Other factors that influence the response to projectile impact are the geometry of the target and the properties of the projectile (Chen et al., 2007, 2008). In addition, there are other parameters related to numerical solution such as the size of meshing and the criteria of erosion (when using Lagrangian method), which also influence the results (e.g. Leppanen, 2002).

Hydrocodes (computer programs used to simulate numerically highly dynamic events) are nowadays an accepted tool to evaluate impact or blast. They often accompany experimental research and development of engineering methods, and in some cases they may be the only available option to assess specific loading situations in engineering applications (Grunwald et al., 2017). Pioneer work to model concrete in hydrocodes over a large range of dynamic loading situations was undertaken by Holmquist and Johnson (1993). Using hydrocodes to predict efficiently the concrete response (e.g. penetration depth, cracks and crater size) to severe loading such as projectile impact, material models that take into account the large deformations, triaxial stress states and strain rate effect are essential. The models Taylor–Chen–Kuszmaul (TCK) (Taylor et al., 1986), Holmquist–Johnson–Cook (HJC) (Holmquist and Johnson, 1993), Karagozian & Case (K&C) (Malvar et al., 1997) and Riedel–Hiermaier–Thoma (RHT) (Riedel et al., 1999) are the most widely used models for concrete and concrete-like brittle materials, knowing that their capabilities in describing the actual non-linear behaviour of the material under different loading conditions vary (Tu and Lu, 2009). Many publications deal with assessment and comparison of these material models (Borrvall and Riedel, 2011; Brannon and Leelavanichkul, 2009; Kong et al., 2016; Leppanen, 2004, 2006; Riedel, 2009; Riedel et al., 2009; Tu and Lu, 2009, 2010; Wu et al., 2012, 2014).

The RHT model developed by Riedel et al. (1999, 2009) from Ernst Mach Institute was implemented in the hydrocode ANSYS Autodyn (2007) and it became a standard model in the year 2000. Since then it has been widely used in modelling concrete impact analysis. It was found reliable in predicting major dynamic failure phenomena of concrete structures (Grunwald et al., 2017). The original RHT model is well documented in many previous publications (ANSYS Autodyn, 2007; Riedel et al., 1999, 2009). For reviews of applications and validation cases, see for example, Leppanen (2002), Riedel et al. (1999), Tu and Lu (2009), Brannon and Leelavanichkul (2009), Riedel (2009), Leppanen (2004), Ansys Autodyn (2011), Hansson and Skoglund (2002) and Unosson and Nilsson (2006).

Because of the general complexity of the models, determination of the model parameters plays an important role in the actual performance of these models. This requires sufficient understanding of the modelling formulation and the associated considerations (Tu and Lu, 2009). A review and evaluation study of the RHT model (Tu and Lu, 2009, 2010) identified a few issues regarding its capability of describing the concrete behaviour under certain loading conditions, particularly concerning the tension response and softening behaviour. For example, the need to modify the default parameters of the tensile-to-compressive meridian ratio to produce results more consistent with experimental observations, and the need to update the rate-dependency of concrete in tension to keep in line with generally accepted macroscopic dynamic enhancement functions. In addition, using Autodyn, when the crack-softening model is employed in the model for tension softening, the softening law should be carefully formulated to ensure an anticipated softening process while maintaining the specified fracture energy. Furthermore, under the triaxial hydrostatic tension, the model does not exhibit a descending branch, instead it shows plastic behaviour that is apparently unreasonable for a brittle material.

Numerical analyses of concrete targets subjected to projectile impact (Huang et al., 2005; Kong et al., 2016; Leppanen, 2006; Tu and Lu, 2010; Wang et al., 2007) have shown that the tensile strength, strain softening, fracture energy and strain-rate effect on the tensile strength in a concrete model can greatly influence the accurate modelling of cratering and scabbing phenomena (Kong et al., 2017).

Parametric studies to improve the numerical predictions of concrete response are important for adjustment of the model parameter settings to achieve better predictions of the observed responses from relevant experimental studies. This is despite the fact that such a treatment could become problem-specific if the material model itself is not robust enough for representing the underlying physical mechanisms (Tu and Lu, 2010), due to the lack of experimental data. In general, even if sufficient experimental data are available, considerable effort is needed to determine sound parameter settings (Grunwald et al., 2017).

Numerous parametric studies and modifications have been performed on the standard RHT model to improve the numerical predictions of concrete response, particularly the crater size. For example, Leppanen (2006) conducted a parametric study using the RHT model with a bilinear tensile softening model based on fracture energy and strain-rate effect in tension; the accuracy of the RHT model results was verified using numerical simulations of three series of experiments on concrete impacted by steel projectiles. Tu and Lu (2010) made several modifications to the RHT model, including inclusion of the third stress invariant in residual strength surface, tensile softening law and the dynamic tensile strength function; the improved performance of the modified RHT model was verified using numerical simulations of two series of experiments on concrete impacted by steel projectiles.

This study is an attempt to calibrate (not modify) the RHT material model, with the least number of possible changes, to better predict the performance of concrete targets, particularly spalling and scabbing phenomena, impacted by hard projectiles. After a brief description of the original RHT concrete model, comparison analyses of some parameters are carried out to know the influence of each individual parameter on the prediction results of the model. These parameters are uniaxial compressive strength, tensile strength of concrete (tensile-to-compressive meridian ratio) and the residual strength surface constant and exponent. Comparisons are made among the predictions using the original RHT model, the RHT model with modified settings and experimental results with regard to such characteristic responses as depth of penetration, radial cracks and spalling and/or scabbing crater size. The improved performance of the RHT model with modified settings is demonstrated using the numerical simulation results of a series of experiments on concrete targets subjected to impact by hard projectiles.

RHT concrete model in Autodyn

The RHT material model is an advanced plasticity model for brittle materials. It consists of three pressure-dependent surfaces in stress space: the elastic limit (yield) surface, the failure surface and the residual strength surface. The three surfaces are implemented to model material hardening (the post-yield–pre-failure behaviour) and damage or softening (the post-failure behaviour), that is, it takes into account the strain hardening, strain softening, third invariant dependence and strain rate. The pressure, in this model, is governed by a polynomial equation of state (EOS), which can be derived from Mie–Gruneisen, together with a P-α model to describe the pore compaction effects and, thus, give a realistic response in the high-pressure regime (Borrvall and Riedel, 2011). A brief description of the three stress surfaces, strain hardening, strain softening, strain rate effect and damage model, is discussed in the following sections; a more detailed description of the RHT model can be found in Riedel (2009) and Riedel et al. (1999).

Elastic limit surface and strain hardening

The elastic limit surface Y_elastic is scaled down from the failure surface Y_failure along the tensile and compressive meridian by defined ratios (given as user input ratios, see Table 1, the default for tension: Elastic strength/f_t = 0.70 and the default for compression: Elastic strength/f_c = 0.53). Y_elastic additionally includes a cap that closes at the current pore crush pressure (see Table 1, default: Use cap on elastic surface = yes). The elastic limit surface Y_elastic can be expressed as

Y_{e l a s t i c} = Y_{f a i l u r e} \times F_{e l a s t i c} \times F_{c a p}

(1)

where the scaling factor F_elastic is the ratio of elastic tensile or compressive strength over the corresponding ultimate strength (f_t for ultimate tensile strength or f_c for ultimate compressive strength). The parabolic cap function F_cap is used to close consistently the elastic surface with the porous EOS towards higher pressures involving pore compaction (Riedel, 2009; Riedel et al., 1999, 2009).

Table 1.

Employed material data in the RHT model for concrete 35 MPa.

Name	Value	Units	Notes
Strength
Shear modulus	16.700	GPa
Compressive strength (f_c)	35.000	MPa	20.8, 26.0 and 41.6
Tensile strength (f_t/f_c)	0.1000	None	0.06, 0.08, 0.10 and 0.12
Shear strength (f_s/f_c)	0.1800	None
Intact failure surface constant (A_FAIL)	1.6000	None
Intact failure surface exponent (N_Fail)	0.6100	None
Tens./Comp. meridian ratio (Q_2.0)	0.6805	None
Brittle to ductile transition (BQ)	0.0105	None
Hardening slope G_el/(G_el – G_pl)	2.0000	None
Elastic strength/f_t	0.7000	None
Elastic strength/f_c	0.5300	None
Residual strength surface constant (B)	1.6000	None	1.4, 1.6, 1.8 and 2.0
Residual strength surface exponent (m)	0.6100	None	0.30 and 0.61
Compressive strain rate exponent (α)	0.0320	None	0.00909 for concrete 140 MPa
Tensile strain rate exponent (δ)	0.0360	None	0.01250 for concrete 140 MPa
Maximum fracture strength ratio (SFMAX)	1.0E+20	None
Use cap on elastic surface	Yes
Failure
Damage constant (D₁)	0.0400	None
Damage constant (D₂)	1.0000	None
Minimum strain to failure	0.0100	None
Residual shear modulus fraction	0.1300	None
Tensile failure	Hydro (P_min)
Erosion strain
Geometric strain (instantaneous)	1.5000	None
Equation of state	P alpha
Reference density	2.7500	g/cm³
Porous density	2.3140	g/cm³	2.5200 for concrete 140 MPa
Porous sound speed	2.9200E+3	m/s	3.2420E+3 for concrete 140 MPa
Initial compaction pressure, P_e	2.3300E+4	kPa	9.3300E+4 for concrete 140 MPa
Solid compaction pressure, P_s	6.0000E+6	kPa
Compaction exponent, n	3.0000	None
Solid EOS	Polynomial
Bulk modulus, A1	3.5270E+7	kPa
Parameter A2	3.9580E+7	kPa
Parameter A3	9.0400E+6	kPa
Parameter B0 = Γ	1.2200	None
Parameter B1 = Γ	1.2200	None
Parameter T1	3.5270E+7	kPa
Parameter T2	0.0000	kPa
Reference temperature	3.0000E+2	K
Specific heat	6.5400E+2	J/(kg K)
Compaction curve	Standard

RHT: Riedel–Hiermaier–Thoma.

The model is elastic until the stress reaches the elastic limit surface Y_elastic (P₁ in Figure 2). With the increase in stress, plastic strains evolve and strain hardening takes place until the stress reaches failure surface Y_failure (P₂ in Figure 2).

Figure 2.

The three strength surfaces of RHT model and loading scenario (Riedel et al., 1999).

The pre-peak loading surface Y_pre-peak is subsequently defined through interpolation between the elastic Y_elastic and the failure surface Y_failure using the hardening slope. This can be seen in Figure 3(a) in the case of uniaxial compression, from which the equation can be expressed as

Y_{p r e - p e a k} = Y_{e l a s t i c} + \frac{ε_{p l}}{ε_{p l - p r e s o f t e n i n g}} (Y_{f a i l u r e} - Y_{e l a s t i c})

(2)

where the definitions of ε_pl and ε_{pl-presoftening} are shown in Figure 3(a). ε_pl is the current plastic strain and ε_{pl-presoftening} is the plastic strain at failure, which can be determined by the secant modulus between the elastic limit surface and the failure surface.

Figure 3.

(a) Bilinear strain hardening between the elastic limit and the failure surface (Riedel et al., 1999) and (b) typical deviatoric cross section of strength surface (Tu and Lu, 2009).

Failure surface and damage model

When the stress reaches the failure surface Y_failure a parameterized damage model governs the evolution of damage, driven by plastic strain, which in turn represents the post-failure surface by interpolating between the failure surface and the residual friction surface. The failure surface Y_failure is expressed as a function of hydrostatic pressure P, Lode angle θ and strain rate $\overset{\cdot}{ε}$ . The failure surface Y_failure can be expressed as

Y_{f a i l u r e} = Y_{T X C} (P) \times R_{3} (θ) \times F_{r a t e} (\overset{\cdot}{ε})

(3)

where Y_TXC (P) represents the compressive meridian and it is formed from material parameters including the compressive, tensile and shear strength of the concrete.

Equation (3) indicates that the failure surface Y_failure is established from the compressive meridian to form the shape of a rotational body around the hydrostatic axis (see Figure 2). Multiplying lode angle dependence R₃ (θ) with the compressive meridian, the shear and tension meridians in the stress space can be taken into account in the failure surface.

The compressive meridian normalized by the uniaxial compressive strength of the material, $Y_{T X C}^{*}$ , is defined as

Y_{T X C}^{*} = {\begin{cases} A_{F a i l} {(P^{*} - P_{s p a l l}^{*} F_{R a t e})}^{N_{F a i l}} for P^{*} \geq \frac{1}{3} \\ 0 for P^{*} < P_{s p a l l}^{*} F_{R a t e} \end{cases}

(4)

where A_Fail and N_Fail are two constant model parameters which can be determined from curve fitting of the experimental data, defined by the user (default values: A_Fail = 1.6 and N_Fail = 0.61). $P^{*}$ is pressure normalized by the uniaxial compressive strength of the material. $P_{s p a l l}^{*}$ is the hydrodynamic tensile (hydro tensile) stress limit normalized by the uniaxial compressive strength of the material. F_Rate is a rate-dependent enhancement factor representing the dynamic increase factor as a function of the strain rate $\overset{\cdot}{ε}$ .

The third invariant dependence term R₃ (θ) is formulated using the expression (Willam and Warnke, 1975)

R_{3} (θ) = \frac{2 (1 - Q_{2}^{2}) \cos θ + (2 Q_{2} - 1) \sqrt{4 (1 - Q_{2}^{2}) \cos^{2} θ + 5 Q_{2}^{2} - 4 Q_{2}}}{4 (1 - Q_{2}^{2}) \cos^{2} θ + {(1 - 2 Q_{2})}^{2}}

(5)

where Q₂ = Q_2.0 + BQ.P*, BQ = 0.0105 and 0.5 < Q₂ < 1

The input parameter Q_2.0 defines the ratio of strength at zero pressure and the coefficient BQ defines the rate at which the fracture surface transitions from an approximately triangular form to a circular form with increasing pressure. Figure 3(b) illustrates a typical shape of the deviatoric section plane of a strength surface. It should be noticed that for the concrete material, the deviatoric section typically transits from a triangular shape (the case of Q₂ in equation (5) equal to 0.5) at a low pressure to a circular shape (the case of Q₂ equal to 1) at a high pressure. The Lode angle θ is a function of the second and third deviatoric stress invariants, and it can be obtained as

\cos 3 θ = \frac{3 \sqrt{3}}{2} \frac{J_{3}}{J_{2}^{3 / 2}}

(6)

After failure is initiated, a damage model is used for strain softening, which considers the gradual loss of the load-carrying capacity of material after reaching its ultimate tensile or compressive strength. This post-failure surface Y_fracture can be achieved by linear interpolation from the failure surface Y_failure to the residual surface Y_residual (as P₂ to P₃ in Figure 2) and incorporation of a damage factor D, as follows

Y_{f r a c t u r e d}^{*} = (1 - D) Y_{f a i l u r e}^{*} + D Y_{r e s i d u a l}^{*}

(7)

The damage parameter D is defined as

D = {\begin{cases} \sum \frac{Δ ε_{p}}{ε_{p}^{f a i l u r e}} for ε_{p}^{f a i l u r e} > ε_{f \min} \\ \sum \frac{Δ ε_{p}}{ε_{f \min}} for ε_{p}^{f a i l u r e} \leq ε_{f \min} \end{cases}

(8a)

ε_{p}^{f a i l u r e} = D_{1} {(P^{*} - P_{s p a l l}^{*})}^{D_{2}}

(8b)

where Δε_p is the plastic strain increment. From equation (8), it can be observed that the damage factor D is dependent on the normalized pressure P^*, the shape parameters D₁ and D₂ and minimum failure strain ε_f _min. A proper selection of parameters D₁, D₂ and ε_f _min is crucial in order to obtain a reasonable post-failure softening behaviour of the material (see Table 1, default values: D₁ = 0.04, D₂ = 1.0 and ε_f _min = 0.01).

Residual surface

In the case of uniaxial stress state, the residual strength of the crushed concrete is always zero. However, in the case of multi-axial stress states as in real structures, the crushed concrete will contribute to the resistance, that is, if confining pressure exists, the concrete retains a certain level of shear strength due to friction among crushed particles (Tu and Lu, 2009). During the projectile penetration into concrete, the crushed concrete will be pushed in both the longitudinal and lateral directions, producing confining effects. As implemented in Autodyn, the residual strength surface in the standard RHT model exhibits a circular deviatoric cross-sectional plane in the principal stress space. This simplified treatment creates some difficulties in replicating the material behaviour under certain stress conditions, in that the model tends to exhibit a hardening response after the peak failure strength is reached instead of softening, as would normally be expected, and it needs to be modified (Tu and Lu, 2010).

The residual surface, which describes the strength of the completely crushed material, is expressed as

Y_{r e s i d u a l}^{*} = M i n [B \times P^{* m}, Y_{X T C}^{*} \times S F M A X]

(9)

where the term Y_XTC×SFMAX is used to limit the maximum residual shear strength for completely damaged material to be a fraction (SFMAX) of the current fracture strength (see Table 1, default value: SFMAX = 1.0E+20). The two parameters B and m, which are the residual strength constant and exponent, respectively, can be determined from curve fitting of the experimental data. The default values of these parameters in the RHT model are B = 1.6 and m = 0.61 (see Table 1).

The two residual strength parameters B and m have been given much attention in several studies (Hu et al., 2016; Leppanen, 2002; Nystrom and Gylltoft, 2011; Tu and Lu, 2009, 2010; Xu and Wen, 2016). The modification of these two parameters can have an effect to improve the descending branch in the stress–strain curve. When Leppanen (2002) carried out a parametric study with Autodyn, on experiments of concrete targets impacted by steel projectile, he made a comparison of the residual strengths with the parameter B equal to 0.9, 1.1 and 1.5, while a constant value was assumed for m (m = 0.7). The study concluded that the residual strength influences the depth of penetration; by increasing the level of residual strength, the depth of penetration is decreased. Tu and Lu (2009) proposed two different values (B = 0.7 and m = 0.8) in simulation of a concrete slab under explosive loading; they noted that, using these values for B and m led to considerable improvement in the simulation results. After a few trials, Hu et al. (2016) proposed other values for B and m (1.1 and 0.9, respectively). With such modification, the residual yield stress becomes nearly linearly related to the pressure, due to increasing the exponent m near to 1.0, see equation (9). When Xu and Wen (2016) compared their developed constitutive model with the HJC, RHT and K&C models, they assumed two values for the residual surface parameters in the RHT model, B = 1.78 and m = 0.8, showing that their developed constitutive model for concrete is advantageous over existing models. Nystrom and Gylltoft (2011) adopted the values determined by Leppanen (2004) (B = 1.5 and m = 0.7) when they carried out comparative numerical studies of projectile impacts on plain and steel-fibre-reinforced concrete.

It can be noticed from the above previous studies that all the assumed values for m (0.7, 0.8 and 0.9) were higher than the default value (0.61). In this study, a smaller value of m (0.3) was used (i.e. reducing the increase in the residual strength level with pressure increase).

Compressive strength

Accurate constitutive concrete models require a long set of input parameters. In order to obtain the values for these parameters, several (e.g. uniaxial, biaxial and triaxial in tension and compression) tests should be performed on many concrete samples. Therefore, there is a need for a single parameter that can characterize the majority of common concretes of various strengths, which is the uniaxial unconfined compressive strength f_c. The RHT model is formulated such that the required input parameters can be scaled with f_c. Consequently, a question arises about the value that should be used to simulate the uniaxial unconfined compressive strength of concrete f_c to get better numerical results: is it the uniaxial compressive strength of 150 mm cube ‘f_cu’ or the uniaxial cylindrical compressive strength of 150 × 300 mm cylinder ‘ $f_{c} ’$ ’? This question arises because of the lack of clarity in determining the value of f_c, which should be used in the RHT model. For example, Riedel (2009) used the symbol f_c for uniaxial cubic compressive strength f_cu (of 68.9 MPa which was tested on 150 mm cubes, see Leppanen, 2006). While in their recent paper, Grunwald et al. (2017) used f_c of 35 MPa for standard concrete C30/37 (cylinder/cube) and 73.9 MPa for high-performance concrete C70/85, which are not of cylinder or cubic compressive strength.

Therefore, in this numerical study, comparative simulations using f_cu and $f_{c} ’$ were performed to demonstrate their effect on the simulation results.

Tensile strength

In the standard RHT model, tensile failure is achieved using a hydrodynamic tensile limit (HTL), also referred to as P_min, which is the minimum pressure to which the material can sustain continuous expansion. The maximum tensile pressure in the material is limited to

P = \max [D \times P_{\min}, P (ρ, θ)]

(10)

where D is the damage parameter, see equation (8). Using this option, no additional user input is required, since the value of P_min is received by linear extrapolation of the shear and the tensile strength projected on the pressure meridian, see Grunwald et al. (2017).

The crack-softening failure may be used in conjunction with the standard RHT model. However, the model has a limited capability of representing this tension-cracking behaviour (Leppanen, 2006); it is not possible to describe different post-crack behaviours (different stress-crack opening relations, for example, bilinear crack-softening relation) in the standard RHT material model (Nystrom and Gylltoft, 2011). In addition, it has been observed that under the triaxial hydrostatic tension, the RHT model does not exhibit a descending branch, which is apparently unreasonable for a brittle material, and instead it shows plastic behaviour (Tu and Lu, 2009).

Strain rate effect

As mentioned above in equation (3), the failure surface Y_failure is affected by strain rate $\overset{\cdot}{ε}$ via F_rate $(\overset{\cdot}{ε})$ , which represents the dynamic increase factor (differential item functioning (DIF), defined as the dynamic strength normalized by its counterpart under quasi-static loading (Cui et al., 2017)) as a function of the strain rate $\overset{\cdot}{ε}$ . In the standard RHT model as implemented in Autodyn, the DIF for both tension and compression is expressed as a linear function of the strain rate in the logarithmic scale, as

F_{r a t e} (\overset{\cdot}{ε}) = D I F = {\begin{matrix} {(\frac{\overset{\cdot}{ε}}{{\overset{\cdot}{ε}}_{o}})}^{α} for compression (P^{*} > \frac{1}{3}) \\ {(\frac{\overset{\cdot}{ε}}{{\overset{\cdot}{ε}}_{o}})}^{δ} for tension (P < \frac{1}{3} f_{t}) \end{matrix}

(11)

where α and δ are defined user constants (see Table 1, default values: Compressive strain rate exponent (α) = 0.032 and tensile strain rate exponent (δ) = 0.036). The quasi-static strain rate ${\overset{\cdot}{ε}}_{o}$ is taken as 3.0 × 10⁻⁵ s⁻¹ for compression and 3.0 × 10⁻⁶ s⁻¹ for tension.

The above expressions of DIF are simple, however, they are criticized for being inconsistent with experimental observations, which tend to support a bilinear DIF expression (Tu and Lu, 2010), particularly for tensile DIF, as the transition strain rate limit is low (around 1 s⁻¹) and is easily reached.

It should be noted that the RHT model does not consider the strain rate effect on Young’s modulus, in spite of the Young’s modulus increase with strain rate, but with a less pronounced increment ratio, compared to that of compressive and tensile strength (Cui et al., 2017). It should also be noted that the strain rate is dependent on the numerical mesh, and therefore the increase in dynamic strength is mesh-dependent (Leppanen, 2002).

Numerical simulations

The calibration (adjustment of the parameters) of the RHT model was done according to tests conducted on 100-mm-thick plain (unreinforced) concrete panel specimens (Abdel-Kader and Fouda, 2014). The specimens were unreinforced concrete panels with dimensions of 500 × 500 × 100 mm³. The test specimen was clamped around the periphery, such that a square of dimensions 400 × 400 mm² was exposed (Figure 4). In the experiment, a projectile (made of hard-steel alloy) was shot at different impact velocities (201, 240, 270, 299 and 354 m/s). The blunt-nosed projectile used has a diameter of 23 mm, length 64 mm and weight 175 g. The concrete compressive strength obtained from tests of 150-mm cubic specimens is about 26 MPa. The post-test measurement from 270 m/s shot gave a size of spalling crater of diameter 112 mm and depth 35 mm, a size of scabbing crater of diameter 295 mm and depth 65 mm and radial cracks starting from the impact centre and passing through the concrete panel at both the front and the rear faces. Figure 4 shows the damage after an impact velocity of 270 m/s at the front and rear faces of half of the specimen. In addition, experiment of the 270 m/s shot showed a rebound of the projectile (the projectile was found near the front of specimen after impact).

Figure 4.

Damage after impact velocity of 270 m/s at the front (left) and rear faces (right) (Abdel-Kader and Fouda, 2014).

The calculations were carried out using a three-dimensional (one-quarter) model, with applying the appropriate symmetry boundary conditions along the two planes of symmetry (planes normal to x and z; Figure 5). The 100-mm-thick concrete panel was supported at the perimeter by the front and back structural steel fixation as shown in Figure 5. The figure also shows modelling of the panel using a Lagrangian mesh of 22,500 elements (24,986 nodes) to represent one-quarter of the panel (with 25 elements through the concrete panel thickness). To improve the accuracy of the analysis, smaller cells were used in the region of the projectile and a coarser mesh was used in the region of the panel less affected by the projectile impact. The size of the mesh in the Lagrange processor was selected, after an elementary study (see Appendix 1), in such a way that their further refinement does not measurably affect the computed results and, at the same time, does not consume much time. The projectile was also modelled using a Lagrangian mesh of 402 elements (562 nodes) to represent one-quarter of the projectile (with eight elements across the projectile diameter), Figure 5. The front and back steel fixation along the perimeter of the concrete panel was represented by 18 elements for one-quarter as shown in Figure 5. Frictionless contact between the concrete panel and the steel fixation surfaces is assumed.

Figure 5.

A schematic representation and finite element mesh of the projectile and concrete panel (one-quarter) used in this study.

The standard Lagrangian finite element method was used in the simulation for modelling both the projectile and the concrete panel. In order to deal with the numerical difficulty that could arise when elements suffer severe distortion, element erosion technique was employed.

For impact simulations, a sensitivity study was performed by several authors (e.g. Leppanen, 2002; Tu and Lu, 2010) with erosion strains of the range [50%–350%], suggesting that 150% or around that level is a good choice. Therefore, in the present simulations, the strain limit of 150% (the default in ANSYS Autodyn) is taken for initiating the element erosion.

The steel present in the projectile was modelled using the Johnson–Cook model, which is suitable for high rate deformation; it can capture the main features of penetration and perforation (Dean et al., 2009; Johnson and Cook, 1983). The main material parameters for steel 4340, which were chosen from the programme material library, were used for modelling the projectile, except that the value of the initial yield stress was taken as 1726 MPa. Projectile–concrete panel interaction was achieved using the gap interaction logic.

Concrete present in the panels was modelled first using the standard RHT model implemented in ANSYS Autodyn (2007) and Riedel et al. (1999, 2009). Then, some of the default parameters were modified as will be shown later.

Numerical results and discussion

Some factors that are thought to have a significant effect on the prediction of concrete response under impact, such as compressive strength of concrete, tensile strength and residual strength surface constants, are discussed in the following sections. It should be stated that four characteristic response parameters, which were observed in the experiment from Abdel-Kader and Fouda (2014), are used for verification/comparison purposes. These response parameters are (a) rebound of the projectile, (b) spalling crater size at the front of the specimen of diameter 112 mm and depth 35 mm, (c) scabbing crater size at the rear of diameter 295 mm and depth 65 mm and (d) radial cracks starting from the impact centre and passing through the specimen at both the faces. It is worth mentioning that since the crater is not circular in the experiments, the crater diameter is represented by an equivalent diameter, which is the average of the horizontal, vertical and two diagonal diameters (Abdel-Kader and Fouda, 2014). In addition, since none of the cohesive zone (CZ) methods or eXtended finite element methods (XFEM) are employed in the simulations, the radial cracks are indicated by the concentration of contours of damage (or equivalent plastic strain contours) in spite of the fact that this may not exactly indicate evident cracks.

Effect of the compressive strength

First, the simulations were performed using the default computational settings in the RHT material model for concrete 35 MPa (ANSYS Autodyn, 2007; Riedel, 2009), see Table 1. The RHT model is formulated such that the input can be scaled with the concrete compressive strength, that is, the remaining terms will automatically scale proportionately. However, there was a question about the value that should be used to simulate the compressive strength of concrete, whether it is the uniaxial cube compressive strength of 150 mm cube ‘f_cu’ (here, f_cu = 26 MPa) or the uniaxial cylinder compressive strength of 150×300 mm cylinder ‘ $f_{c} ’$ ’ (here, $f_{c} ’ = 20.8 MPa$ ). Therefore, two values of the concrete compressive strength were used; two comparative simulations (Nos 1 and 2) were performed to demonstrate the effect of the compressive strength in the RHT material model on the simulation results.

Comparisons among the predictions of the two simulations (No. 1: using f_cu and No. 2: using $f_{c} ’$ ) and experimental results, for impact velocity of 270 m/s, with regard to the four characteristic response parameters mentioned above, are shown in Table 2 and Figure 6.

Table 2.

Comparisons among the predictions of the two simulations and experimental results.

Response parameter	Experiment	Simulation no. 1f_cu = 26 MPa	Simulation no. 2 $f_{c} ’ = 20.8 MPa$
Projectile travel distance	Rebound	Perforation exit velocity of ~25 m/s	Perforation exit velocity of ~44 m/s
Size of spalling crater: diameter (mm) – depth (mm)	122 – 35	110 – 23	115 – 28
Size of scabbing crater: diameter (mm) – depth (mm)	295 – 65	170 – 77	170 – 72
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks passing through the panel	No radial cracks	No radial cracks

Figure 6.

Comparison of damage patterns at time of 0.003 s, when using f_cu (left) and f_c (right).

It can be seen from Table 2 and Figure 6 that, in the two simulations, neither the rebound of the projectile nor the radial cracks are predicted. In both the cases, the simulation predicted perforation of the projectile into the concrete target (with exit or residual velocity of ~25 m/s when using f_cu to represent the compressive strength of concrete in the RHT model and with exit velocity of ~44 m/s when using $f_{c} ’$ ), which disagrees with the experiment. Besides, it can also be observed that damage in the form of a scabbing crater is significantly under-predicted (occurs with a diameter of ~170 mm in the two simulations), which is much less than that in the experiment (295 mm). The damage in the form of spalling, at the front side of the specimen, occurs with the diameter (~110 when using f_cu and 115 mm when using $f_{c} ’$ ; see Figure 6) to some extent less than in the experiment (122 mm). With regard to the perforation hole, the hole diameter in the case of f_cu is smaller (and closer to the experiment) than that in the case of $f_{c} ’$ .

It is to be observed that the size of craters is ambiguous when the damage plot is used to measure the crater size as shown in Figure 6. It is difficult to distinguish between the depth of spalling crater, which occurs at the front of the specimen, and the (depth of) the scabbing crater, which occurs at the rear (in the direction of impact). This is because both spalling and scabbing are represented, after complete damage, by the same colour (as shown in Figure 6). It was found in the study that using the directional deformation plot allows to distinguish between the spalling and scabbing craters more easily. Not only this, but using this plot allows to measure the depth of the spalling and scabbing craters more accurately. Therefore, the directional deformation plot is used to measure the crater size, see Figure 7. The spalling crater radius is estimated as the distance from the impact axis to the outermost nodal directional deformations of 1 mm (i.e. where the nodes are moving out of the concrete panel in the opposite direction of impact, with deformations greater than 1 mm, Figure 7). While the scabbing crater radius is estimated as the distance from the impact axis to the outermost 1-mm nodal directional deformations (where the nodes are moving out of the concrete panel at the rear face of the concrete panel, in the direction of impact, with deformations greater than 1 mm, darkest area in Figure 7).

Figure 7.

Comparison of directional deformations (in impact direction) at time of 0.003 s, when using f_cu (left) and f_c (right).

From Figure 7, it can be noticed that the depth of spalling crater (23 mm) when using f_cu is smaller than that (28 mm) when using $f_{c} ’$ , and this is attributed to the higher value of compressive strength (f_cu = 1.25 $f_{c} ’$ ); as mentioned above, spalling arises from the compressive stress wave propagation in the target and the associated (crushing) material failure.

It was also observed that there was no need for the simulation to be so long (to let the projectile out of the specimen) to verify perforation. Careful observation of the projectile velocity against the time plot (e.g. that shown in Figure 8) clarifies whether there is still perforation resistance of the specimen (concrete contact with the projectile or part of it) or not. Constant velocity of a significant value for a certain period indicates the projectile perforation.

Figure 8.

Projectile velocity and position versus time, when using f_cu.

As can be seen from the above simulations (Nos 1 and 2), prediction of the concrete response, using either f_cu or $f_{c} ’$ was not satisfactory. One of the possible ways, without changing the default parameters setting in the model, to improve the prediction result of the simulations is by increasing the compressive strength of concrete. It is well known that the depth of penetration depends mainly on the compressive strength of concrete (f_c appears in most of the empirical equations for penetration depth; see, for example, Li et al., 2005). Therefore, the simulation was repeated using a remarkably higher value (41.6 MPa, which is double of $f_{c} ’$ ) to represent the compressive strength of concrete. The simulation results (Simulation No. 3, using twice $f_{c} ’$ ) show that the concrete specimen could catch the projectile after a penetration depth of 40 mm, Figure 9. However, this improvement in the prediction of penetration did not accompany any noticeable improvement in the prediction of scabbing diameter or even redial cracks. The predicted scabbing diameter was just increased by 10 mm (from 170 mm when using $f_{c} ’$ to 180 mm when using ${2 f}_{c}^{’}$ ). Figure 9 shows the directional deformations at time 0.002 and 0.0045 s. It can be concluded from this that the compressive strength of concrete has little effect on the scabbing crater size, which has been numerically concluded in other works (e.g. Leppanen, 2002).

Figure 9.

Directional deformations at time 0.002 s (left) and 0.0045 s (right), when using ${2 f}_{c}^{’}$ .

Generally speaking and based on the above simulations, using f_cu to represent the compressive strength of concrete f_c in the RHT model gives better prediction results (lesser exit velocity and closer hole diameter to experiment) than using $f_{c}^{'}$ , in spite of not predicting the projectile rebound or radial cracks. Therefore, it will be fixed in the rest of the simulations.

The under-prediction of scabbing size and radial cracks, which was observed in the above simulations, may attribute to a strong tensile strength or fracture energy of concrete assumed in the RHT material model. Previous parametrical studies show that the tensile strength, fracture energy and the strain rate law influence the cracking and scabbing of concrete. The tensile strength of the material as represented by the RHT model can be rectified by adjusting the normalized tensile strength (tensile strength to compressive strength ratio f_t/f_c) and the fracture strength hardening can be rectified by adjusting the residual fracture strength surface (e.g. by modifying the residual strength constant B or/and the residual strength exponent m). Therefore, in order to improve the prediction result of simulations, not only for better prediction of projectile penetration of concrete but also for better prediction of scabbing size of concrete, the effects of the normalized tensile strength and the residual fracture strength were studied and the simulation results are presented below.

Effect of the normalized tensile strength

It is well known that the ratio (f_t/f_c) is influenced by the level of concrete strength. At low compressive strengths, the tensile strengths are as high as 10% of the cylinder compressive strength, but at extremely higher compressive strengths, this ratio reduces to about 5% (Arιoglu et al., 2006).

Three comparative simulations (Nos 4–6) are performed to demonstrate the effect of the normalized tensile strength f_t/f_c on the simulation result. The computational model settings are kept the same among the three simulations, except for the ratio f_t/f_c in the RHT material model, which takes three values (0.06, 0.08 and 0.12), besides the default value (0.1; Simulation No. 2).

Comparisons among the predictions of the four simulations and experimental results, for impact velocity of 270 m/s, with regard to the four characteristic response parameters mentioned above are shown in Table 3.

Table 3.

Comparisons among simulation results using different values of normalized tensile strength.

Response parameter	Simulation no. 4f_t/f_c = 0.06	Simulation no. 5f_t/f_c = 0.08	Simulation no. 2f_t/f_c = 0.10	Simulation no. 6f_t/f_c = 0.12
Projectile travel distance	Perforation exit velocity of ~27 m/s	Perforation exit velocity of ~18 m/s	Perforation exit velocity of ~25 m/s	Rebound after penetration depth of ~46 mm
Size of spalling crater: diameter (mm) – depth (mm)	120 – 25	128 – 25	110 – 23	74 – 23
Size of scabbing crater: diameter (mm) – depth (mm)	370 – 75	175 – 75	170 – 77	115 – 77
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks	Radial cracks	No radial cracks	No radial cracks
Directional deformations (in impact direction) at time of 3 ms

It can be seen from Table 3 that the simulation Nos 3 and 4 (f_t/f_c = 0.06 and 0.08) could predict the radial cracks. However, the rebound of the projectile could not be predicted, the two simulations predicted perforation of the projectile into the concrete target with exit velocities of ~27 m/s and ~18 m/s, which disagrees with the experiment. In simulation No. 3 (f_t/f_c = 0.06), more damage in the form of concrete part splitting is observed, which overestimates the reported in the experiment.

For simulation No. 5 (f_t/f_c = 0.12), the radial cracks could not be predicted, however, the rebound (or one can say embedded because of the small rebound velocity of 1.0 m/s) of the projectile is predicted (after penetration of ~46 mm inside the specimen), which nearly agrees with the experiment. The damage in the form of scabbing occurs, at the rear side of the target, with diameter (115) much less than that in the experiment (295 mm). The damage in the form of spalling occurs with the diameter (74 mm) significantly less than that in the experiment (122 mm).

With regard to the exit velocities in the four simulations, it can be observed that there is no gradual decrease in the exit velocity, as was expected, by increasing f_t/f_c. It seems that this is the case when the impact velocity is far from the perforation velocity, not quite near the perforation velocity as in these simulations.

Generally speaking, prediction of simulation No. 4 (f_t/f_c = 0.08) is closer to the experiment than that of other simulations. However, the prediction is still not satisfactory.

Effect of the residual strength

It is worth mentioning first that the residual strength of concrete should be determined from the triaxial compression tests, where the concrete is subjected to high confinement pressure as in the case of projectile impact. Furthermore, the tests should be under dynamic loading. Therefore, the experiments performed under static loading with low confinement pressure (Attard and Setunge, 1996; Xie et al., 1995) should be taken just as an indication of the level of the residual strength. For dynamic loading at high confinement pressure, no experimental results have been reported, to the author’s knowledge. This makes performing reverse fitting of some parameters helpful.

To rectify the fracture strength hardening in the RHT material model, one possible way, without changing the shape of the residual fracture strength surface, is to rectify (B and m in equation (9)) the residual fracture strength in the model. In this section, in conjunction with the above modifications (using f_cu to represent the concrete compressive strength f_c and taking the normalized tensile strength f_t/f_c = 0.08), different values of the residual strength constant B (while m is kept as default = 0.61) will be implemented. Three comparative simulations (Nos 7–9) are performed to demonstrate the effect of B in the RHT model on the simulation results. The computational model settings are kept the same among the four simulations, except for the value of B, which was taken as 1.4, 1.8 and 2.0, besides the default value (1.6).

Comparisons among the predictions of the four simulations using different B, and experimental results for impact velocity of 270 m/s, with regard to the four characteristic response parameters mentioned above are shown in Table 4.

Table 4.

Comparisons among simulation results using different values of residual surface constant B.

Response parameter	Simulation no. 7B = 1.4	Simulation no. 5B = 1.6	Simulation no. 8B = 1.8	Simulation no. 9B = 2.0
Projectile travel distance	Perforation exit velocity of ~45 m/s	Perforation exit velocity of ~18 m/s	Perforation exit velocity of ~3 m/s	Perforation exit velocity of ~12 m/s
Size of spalling crater: diameter (mm) – depth (mm)	125 – 23	125 – 25	122 – 23	116 – 20
Size of scabbing crater: diameter (mm) – depth (mm)	200 – 77	175 – 75	215 – 77	166 – 80
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at rear only	Radial cracks at both faces	Radial cracks at both faces	Radial cracks at both faces
Directional deformations (in impact direction) at time of 3 ms

It can be seen from Table 4 that simulation No. 7 (B = 1.4) could predict the radial cracks at the rear face only; however, the rebound of the projectile could not be predicted. Damage in the form of scabbing occurs, at the rear side of the target, with diameter (200 mm) much less than that in the experiment (295 mm). Damage in the form of spalling occurs with the diameter 122 mm, which agrees with the experiment (122 mm).

For simulation No. 8 (B = 1.8), the radial cracks could be predicted, and the projectile was almost embedded (after penetration of ~68 mm inside the specimen, with very small residual velocity of 3 m/s). Damage in the form of scabbing occurs at the rear side of the target, with the diameter (215 mm) still less than that in the experiment (295 mm). Damage in the form of spalling occurs with diameter 123 mm, which agrees well with the experiment (122 mm). The prediction of simulation No. 9 (B = 2) is less satisfactory than that of simulation No. 8 (B = 1.8), as shown in Table 4.

Generally speaking, prediction of simulation No. 8 (B = 1.8) is closer to the experiment than that of other simulations (Nos 7, 5, 9) of B = 1.4, 1.6 and 2.0, respectively. However, the prediction is still not satisfactory, especially for the scabbing diameter.

Finally, the above four simulations (Nos 5 and 7–9) are repeated after modifying the residual strength exponent m from 0.61 to 0.3 (it should be mentioned that greater decrease in m leads to a little change in the residual strength with increasing pressure). This modification lets the residual strength surface move closer to the hydrostatic axis at a pressure higher (and farther at lower pressure) than that corresponding to the default value (0.61), see Figure 10.

Figure 10.

Residual strength surface with different B and m.

Comparisons among the predictions of the four simulations (Nos 10–13) using residual surface exponent m = 0.3 (rather than the default value, 0.61) with different values of B, and experimental results, for impact velocity of 270 m/s, with regard to the four characteristic response parameters mentioned above, are shown in Table 5.

Table 5.

Comparisons among simulation results using m = 0.3 with different values of B.

Response parameter	Simulation no. 10B = 1.4	Simulation no. 11B = 1.6	Simulation no. 12B = 1.8	Simulation no. 13B = 2.0
Projectile travel distance	Perforation exit velocity of ~28 m/s	Perforation exit velocity of ~4.0 m/s	Rebound penetration of 59 mm	Embedded penetration of 52 mm
Size of spalling crater: diameter (mm) – depth (mm)	125 – 35	125 – 35	122 – 35	122 – 30
Size of scabbing crater: diameter (mm) – depth (mm)	250 – 65	256 – 65	285 – 65	283 – 70
Radial cracks: damages at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at both faces	Radial cracks at both faces	Radial cracks at both faces	Radial cracks at both faces
Directional deformations (in impact direction) at time of 3 ms

It can be seen from Table 5 that simulation Nos 10 and 11 (B = 1.4 and B = 1.6) could still not predict the rebound of the projectile; however, an improvement in the prediction of scabbing crater size is observed. This improvement in the prediction of scabbing crater size is also observed for the other simulations. Damage in the form of scabbing occurs with diameters in the range of 250–285 mm and depths in the range of 65–70 mm, which is close to the experiment (diameter of 295 mm and depth of 65 mm). Damage in the form of spalling diameter is not affected by changing m and remains in the range (122–125 mm) of the experiment, while prediction of the spalling depth improved. The overall improvement in the simulation results is evident. The prediction of simulation No. 11 (B = 1.8 and m = 0.3) is closer to the experiment than that of other simulations, particularly in predicting the spalling and scabbing crater size. The diameter and depth of spalling crater are predicted to be 122 and 35 mm, which agree with the measurements (122 and 35 mm); in addition, the simulated scabbing crater diameter and depth are 285 and 65 mm, which also agree well with the experimental results (295 and 65 mm). It can be seen that the prediction is satisfactory, especially for the scabbing crater size.

To sum, the calibration was performed via a total of 13 simulations, as shown in Table 6. The modified parameters are as follows:

Using the uniaxial compressive strength of 150 mm cube ‘f_cu’ instead of the uniaxial compressive strength of 150 × 300 mm² cylinder ‘ $f_{c}^{’}$ to simulate the compressive strength of concrete.

The normalized tensile strength is simulated by 0.08 (the default is 0.10).

The residual strength surface parameters B and m are taken as 1.8 and 0.3 (the default are 1.6 and 0.61).

Table 6.

Calibration of parameters.

Parameter	Simulation nos 1 and 3	Simulation nos 4 and 6	Simulation nos 7–13	Final modifications
Compressive strength	$f_{c}^{'}$ f_cu → ${2 f}_{c}^{'}$	f_cu	f_cu	f_cu
Tensile strength/compressive strength	0.10	0.060.08 →0.100.12	0.08	0.08
Residual surface parameters B and m	1.6 and 0.61	1.6 and 0.61	1.4 and 0.611.6 and 0.611.8 and 0.612.0 and 0.611.4 and 0.301.6 and 0.301.8 and 0.302.0 and 0.30	1.8 and 0.3

Shading shows the variable studieded in these simulations.

It is shown that the present calibration of RHT model performed well for the experiment conducted by Abdel-Kader and Fouda (2014) when compared with the default settings currently available in the model.

Verification for other specimens

In order to examine the accuracy of the RHT model with modified settings, for predicting penetration depth, cratering and scabbing phenomena and exit velocity (in the case of perforation), projectile penetration/perforation experiments of Abdel-Kader and Fouda (2014), Hansson (1998), Erkander and Pettersson (1985) and Unosson and Nilsson (2006) are chosen for simulation and comparison in this section. The corresponding simulation results of the original RHT model are also presented for comparison purposes.

Simulation of experiments by Abdel-Kader and Fouda (2014)

The adjustment of the parameters of the RHT model performed in the previous sections was done according to a test conducted on 100-mm-thick plain concrete specimen under the projectile impact velocity of 270 m/s (Abdel-Kader and Fouda, 2014). As mentioned above, in the experiment of Abdel-Kader and Fouda (2014), a projectile was shot at different impact velocities (201, 240, 270, 299 and 354 m/s). The post-test measurements (size of spalling and scabbing crater) for impact velocities of 201, 240, 299 and 354 m/s are shown in Tables 6 –9. The tables also show the simulation results of the RHT model, with modified settings and the original RHT model.

Concerning the exit velocities, determination of the V50 (the projectile velocity at which the probability for the target perforation is 50%; there are various semi-empirical and theoretical models to predict V50) and a Lambert–Jonas–Fit (correlation between the impact, exit and ballistic limit velocities) of the experiments would help to estimate the difference between simulation and experiment more clearly. However, the exit velocities were not recorded in the experiments from Abdel-Kader and Fouda (2014).

It can be seen from Tables 7 and 8 that for projectile impact velocities of 201 and 240 m/s, the RHT model with default settings could not predict the projectile rebound that occurred in the experiment, whereas the RHT model with the modified settings predicted the projectile rebound after penetrations of about 30 and 41 mm (26 and 30 mm in the experiment), respectively. In addition, simulation using the RHT model with modified settings could predict the radial cracks that occurred in the experiment, while that using the default settings could not predict these radial cracks.

Table 7.

Comparisons among simulation results using default settings, modified settings and experiment for projectile impact velocity of 201 m/s.

Response parameter	Experiment	Default settings	Modified settings
Projectile travel distance	Rebound	Perforation with exit velocity of ~10 m/s	Rebound after penetration of ~30 mm
Size of spalling crater: diameter (mm) – depth (mm)	99 – 26	100 – 20	96 – 27
Size of scabbing crater: diameter (mm) – depth (mm)	No scabbing	Scabbing	No scabbing
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at rear face only	Cracks at rear face only	Radial cracks at rear face only
Directional deformations (in impact direction) at time of 3 ms

Table 8.

Comparisons among simulation results using default settings, modified settings and experiment for projectile impact velocity of 240 m/s.

Response parameter	Experiment	Default settings	Modified settings
Projectile travel distance	Rebound	Perforation with exit velocity of ~11 m/s	Rebound after penetration of ~41 mm
Size of spalling crater: diameter (mm) – depth (mm)	116 – 30	106 – 25	121 – 30
Size of scabbing crater: diameter (mm) – depth (mm)	200 – 70	200 – 65	264 – 70
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at both faces	Cracks at rear face only	Radial cracks at both faces
Directional deformations (in impact direction) at time of 3 ms

For the specimens impacted by projectile velocities of 299 and 354 m/s (see Tables 9 and 10), all the simulations predicted the perforation that occurred in the experiment; however, with regard to crater size, the simulation results using the modified settings were better than those using the default settings. In addition, as for projectile velocities of 201 and 240 m/s, the simulations using the modified settings could predict the radial cracks that occurred in the experiment, while those using the default settings could not predict these radial cracks.

Table 9.

Comparisons among simulation results using default settings, modified settings and experiment for projectile impact velocity of 299 m/s.

Response parameter	Experiment	Default settings	Modified settings
Projectile travel distance	Perforation exit velocity not recorded	Perforation with exit velocity of ~59 m/s	Perforation with exit velocity of ~20 m/s
Size of spalling crater: diameter (mm) – depth (mm)	120 – 40	120 – 32	120 – 32
Size of scabbing crater: diameter (mm) – depth (mm)	246 – 60	192 – 68	280 – 68
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at both faces	Cracks at rear face only	Radial cracks at both faces
Directional deformations (in impact direction) at time of 3 ms

Table 10.

Comparisons among simulation results using default settings, modified settings and experiment for projectile impact velocity of 354 m/s.

Response parameter	Experiment	Default settings	Modified settings
Projectile travel distance	Perforationexit velocity not recorded	Perforationwith exit velocity of ~121 m/s	Perforationwith exit velocity of ~104 m/s
Size of spalling crater: diameter (mm) – depth (mm)	144 – 35	137 – 32	142 – 34
Size of scabbing crater: diameter (mm) – depth (mm)	305 – 65	188 – 68	220 – 66
Radial cracks: damage at front (top) and rear (bottom) faces at time of 3 ms	Radial cracks at both faces	Cracks at rear face only	Radial cracks at both faces
Directional deformations (in impact direction) at time of 3 ms

Simulation of experiment by Hansson (1998)

In the experiment conducted by Hansson (1998), two shots of steel projectile were fired into cylindrical plain concrete targets. The steel projectile had an ogive nose with diameter 75 mm, length 225 mm and a total mass of 6.28 kg. The steel material of the projectile had a density of 7830 kg/m³, bulk modulus 159 GPa and shear modulus 81.8 GPa. The same projectile impact velocity of 485 m/s was measured in the two shots. The cylindrical concrete target had a length (thickness) of 2.0 m and diameter of 1.6 m cast in a steel culvert. In one shot, a support was used at the rear face of the target. The concrete compressive strength obtained from tests of 150 mm cubic specimens was about 40 MPa.

The post-test measurement from the two shots gave a penetration depth of 655 for the target with support at the rear face, and 660 mm for the target without support at the rear face. The diameter of the front-face crater was also measured; it was about 800 mm in both the shots.

The numerical simulations of the experiment (without support at the rear face of the target) conducted by Hansson (1998), made in two dimensions with axial symmetry and a uniform mesh of quadratic Lagrangian elements (64,019 elements) of length 5 mm were used for the concrete target. In addition, the projectile was modelled with an element size of average 5 mm (319 elements), as shown in Figure 11.

Figure 11.

Numerical mesh for simulation of Hansson (1998), region of impact.

In the case of a target without support at the rear face, the numerical simulation results using the RHT model with default settings gave a depth of penetration of 550 mm (with rebound velocity of 32 m/s) and spalling crater diameter of about 600 mm (see Figure 12). While when using the RHT model with modified settings, the depth of penetration and spalling crater diameter were about 573 mm (with rebound velocity of 27 m/s) and 980 mm, respectively, as shown in Figure 12. Also in Figure 12, the measured penetration depth and crater diameter in the experiment (660 mm and 800 mm, respectively) are shown.

Figure 12.

Damage (left) and directional deformations (right) at time of 0.003 s using default RHT model (above) and modified RHT model (below) for the experiment of Hansson (1998).

Figure 12 shows that in the case of default settings, both the predicted penetration depth (550 mm, 83% of experiment) and the spalling crater diameter (600 mm, 75% of experiment) underestimate the experimental results. However, when the modified settings are used an improvement of the predicted penetration depth (573 mm, 87% of experiment) occurred, and a closer spalling crater diameter (980 mm, 111% of experiment) to the experiment was predicted.

From the above, it can be concluded that use of the RHT model with modified settings instead of default settings lead to better results for both the depth of penetration and the spalling crater size.

Simulation of experiments by Erkander and Pettersson (1985)

In the experiment conducted by Erkander and Pettersson (1985) and Leppanen (2006), single fragments were shot against concrete targets with velocities of 1024, 1163 and 1238 m/s. The fragments were spherical with a radius of 10.3 mm and a weight of 35.9 g. The dimensions of the concrete targets were 1000 × 1000 × 140 mm³. The concrete had unconfined compressive strength (of 150 mm cubes in uniaxial stress) of 68.9 MPa. In the first shot of impact velocity of 1024 m/s, the depth of penetration was 50 mm, spalling diameter was 270 mm and the diameter of scabbing was 380 mm. The fragment in the second shot had a velocity of 1163 m/s, a depth of penetration of 66 mm, spalling diameter of 240 mm and a scabbing diameter of 310 mm. Finally, in the third shot of impact velocity of 1283 m/s, the depth of penetration was 50 mm, spalling diameter was 230 mm and the diameter of scabbing was 360 mm.

The simulations of the experiment conducted by Erkander and Pettersson (1985) made in two-dimensional (2D) model with axial symmetry and a uniform mesh of quadratic Lagrangian elements of length 5 mm were used for the concrete target (mesh of 17,500 elements). In addition, the projectile was modelled with element size of an average of 5 mm (mesh of 52 elements), as shown in Figure 13.

Figure 13.

Numerical mesh for simulation of Erkander and Pettersson (1985), region of impact.

For the first shot of impact velocity of 1024 m/s, the simulation results using the RHT model with the default settings (concrete 35 MPa: f_t/f_c = 0.1, B = 1.6, m = 0.61 and $f_{c}^{’}$ = ) and modified settings (concrete 35 MPa: f_t/f_c = 0.08, B = 1.8, m = 0.3 and f_c = f_cu = 68.9 MPa) are shown in Figure 14. The simulation results show that there is no big difference in the predicted penetration depth; it is 54.6 mm (with rebound velocity of 2.6 m/s) in the case of using default settings and 53.9 mm (with rebound velocity of 23.3 m/s) in the case of using modified settings. However, there is a significant difference in the spalling diameter results (about 130 mm using default settings and about 180 mm using the modified settings; compared to 270 mm in the experiment). Also, in the scabbing diameter results (about 200 mm using default settings and about 400 mm using the modified settings; compared to 380 mm in the experiment), as shown in Figure 14. A significant improvement in the prediction results, especially in the scabbing diameter, is evident. Both the spalling and the scabbing diameters are in good agreement with the experimental result in the case of modified settings.

Figure 14.

Damage (left) and directional deformations (right) at time of 0.5 ms using default RHT model (above) and modified RHT model (below) for the experiment of Erkander and Pettersson (1985).

Simulation of experiments by Unosson and Nilsson (2006)

Unosson and Nilsson (2006) conducted two series of experiments (namely, perforation and perforation tests). The armour piercing steel projectile was shot to a plain concrete cylindrical target. The projectile with an ogive nose had a radius of 127 mm, a total length of 225, diameter of 75 mm and a mass of 6.3 kg. In the perforation experiments, the thickness of the concrete target was 0.4 m and the diameter was 1.4 m. The concrete target used in the penetration experiments had a thickness (length) of 0.8 m and a diameter of 1.4 m. The concrete targets were all made of high strength concrete, with unconfined compressive strength (of 150 mm cubes in uniaxial stress) of 153 MPa. The tensile strength of the concrete was determined to be 8.2 MPa. The density of the concrete was 2770 kg/m³. For each series of the tests, three shots were performed in order to observe the consistency of the test results. The projectile impact velocity was about 616 m/s. For the perforation tests, the mean exit velocity (mean of 276, 303 and 293 m/s) was about 291 m/s. While for the penetration tests, the mean penetration depth (mean of 0.45, 0.54 and 0.51 m) was about 0.5 m. The variation in the individual test results from the mean is within 5% for the perforation tests and 10% for the penetration tests.

Similar to the simulations for the experiment conducted by Hansson (1998), numerical simulations of the experiment conducted by Unosson and Nilsson (2006) made in 2D with axial symmetry and a uniform mesh of quadratic Lagrangian elements of length 5 mm (11,214 and 22,402 elements for 400 and 800 mm thick targets, respectively) were used for the concrete target. In addition, the projectile was modelled with an element size of average 5 mm (313 elements), as shown in Figure 15. Here, the simulations were performed using the default computational settings in the RHT material model for concrete 140 MPa. The default settings in the RHT material model for concrete 35 MPa presented in Table 1 are the same for concrete of 140 MPa, except for compressive strain rate exponent α and tensile strain rate exponent δ, see Table 1. The exponent α is equal to 0.0320 and 0.00909 for concrete 35 and 140 MPa and the exponent δ is equal to 0.0360 and 0.0125 for concrete 35 and 140 MPa, respectively.

Figure 15.

Numerical mesh for simulation of Unosson and Nilsson (2006), region of impact.

For the perforation simulations of 400-mm-thick target, the predicted projectile exit velocity from the simulation using the RHT model with default settings was about 380 m/s, which is much larger than the average test result of 291 m/s. With the modified RHT model, the simulation results are markedly improved, and the predicted exit velocity is about 330 m/s. Figure 16 shows a comparison of the damage and directional deformations, before the projectile exits from the target (at time of 0.96 ms) using the RHT model with both the default and modified settings. It should be mentioned that no data for crater sizes were recorded in the experiment.

Figure 16.

Damage (left) and directional deformations (right) at time of 0.96 ms using default RHT model (above) and modified RHT model (below) for the experiment of Unosson and Nilsson (2006).

In the case of penetration simulations of 800-mm-thick specimen, the numerical simulation results using the RHT model with default settings gave a depth of penetration of 446 mm (with rebound velocity of 49 m/s). While in the case of using the RHT model with modified settings, the depth of penetration was 400 mm (with rebound velocity of 51 m/s), compared to 500 mm in the experiment. It can be observed that the RHT model with the modified settings gave unsatisfactory prediction. This may be attributed to the high value of f_t/f_c (= 0.08) assumed for high strength concrete of 153 MPa. The simulation, using the RHT model with modified settings, was repeated using f_t/f_c equal to 0.06 instead of 0.08; the numerical simulation results gave a depth of penetration of 473 mm (and the projectile embedded into the target), which is very close to the average test result of 500 mm, see Figure 17. Figure 18 shows a comparison of the damage and directional deformations, at time of 1.5 ms using the RHT model with both the default and modified settings. It should be mentioned that no data for crater sizes were recorded in the experiment of the 800-mm-thick specimen for comparison.

Figure 17.

Projectile position versus time (left) and directional deformations at time of 1.26 ms (right), using the modified RHT model for the experiment of 800-mm-thick specimen of Unosson and Nilsson (2006).

Figure 18.

Damage (left) and directional deformations (right) at time of 1.5 ms using default RHT model (above) and modified RHT model (below) for the experiment of 800-mm-thick specimen of Unosson and Nilsson (2006).

Finally, to sum up, the modified parameters are: a) using f_cu instead of $f_{c}^{’}$ to simulate the compressive strength of concrete f_c, b) f_t/f_c is simulated by 0.08 for concrete 35 MPa and 0.06 for 140 MPa (instead of 0.10 in both) and c) the parameters B and m are taken as 1.8 and 0.3 (instead of 1.6 and 0.61).

Conclusion

In the RHT model implemented in the hydrocode ANSYS Autodyn, parameters are adjusted as a function of the uniaxial unconfined compressive strength of specific concrete f_c. This advantage enables occasional users (and those who cannot conduct experiments to determine all the required concrete parameters) to input only f_c, which can characterize the majority of common concretes of various strengths, while the rest is automatically calculated by default settings. This study is an attempt to calibrate this model, with the least number of possible changes, to better predict the performance of concrete targets, particularly spalling and scabbing phenomena, impacted by hard projectiles.

The recommended modified parameters are as follows:

The cube compressive strength of 150 mm cube f_cu is used instead of the cylinder compressive strength $f_{c}^{’}$ to simulate the compressive strength of concrete f_c.

The default value of 0.10 to simulate the normalized tensile strength f_t/f_c in both concrete 35 and 140 MPa is replaced by 0.08 for concrete 35 MPa and 0.06 for concrete 140 MPa.

The residual strength surface parameters (constant B and exponent m) are taken as 1.8 and 0.3 (instead of the default 1.6 and 0.61).

It is shown that the RHT model with modified parameter settings gives better prediction results than the RHT model with default settings, from the corresponding experiment results including penetration depth, damage pattern, crater size (diameter and depth) and exit projectile velocity (in the case of perforation). In addition, it is shown that the method used in this study (using the directional deformation plot) is an appropriate way to measure and distinguish between spalling and scabbing crater sizes after penetration/perforation of a projectile.

It should be borne in mind that although the results of the modified settings reproduce the chosen experiments better than the default settings, particularly in crater size, they cannot be taken to be a better choice.

An incorporation of both the tensile strength and the residual strength surface based on dynamic loading at high confinement pressure in the experimental results is a right step in improving the numerical simulation results, particularly for spalling and scabbing phenomena in concrete targets impacted by hard projectiles.

Modelling both tensile strength and residual strength surface based on dynamic loading at high confinement pressure experimental results is a right step in improving the numerical simulation results, particularly for spalling and scabbing phenomena in concrete targets impacted by hard projectiles.

Footnotes

Appendix 1

Based on previous experience with a similar type of impact problem, four different meshes were chosen for simulation of the concrete panel of dimensions 500 × 500 × 100 mm³ (one-quarter of dimensions 250 × 250 × 100 mm³ is used) in the case of using the RHT default settings (and $f_{c}^{’} = 20.8 Mpa$ ). A comparison of the projectile travel distance and the damage to the four meshes is shown in Table 11. The table shows that in all cases, the projectile perforated the concrete panel (in experiment, the projectile was rebounded) but with different exit velocities (8, 14, 44 and 31 m/s for meshes a, b, c and d, respectively, as shown in Table 11. The damages are also shown in Table 1, which nearly have the same trend. Mesh c was chosen and kept constant for all the simulations.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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