Retrieving the constitutive properties of a concrete target from a single instrumented penetration test

Abstract

Penetration into concrete is an important problem that attracted many researches who developed analytical models and numerical solutions. Most of the advanced methods account for the material properties through their constitutive equations, combined from the failure envelope and the equation of state. These properties are rarely available in many cases, and in different experimental reports it is missing. For lack of any other option, analysts borrow available equations that were developed for other concrete types, with the doubt of their suitability. In a limited number of tests, instrumented projectiles are used and the deceleration time history is recorded. Rarely, the constitutive properties are also provided and analysis can be conducted. Otherwise, analysis cannot be carried out and prediction and validation of the recorded deceleration curve cannot be conducted. The present article refers to such instrumented tests where the constitutive properties are not available. It aims at unveiling the relationship between the deceleration curve and the constitutive properties and at solving, for the first time, the inverse problem of a penetration problem in order to retrieve major constitutive equation parameters from the acceleration time-history record. The major new feature of the present article is identification of the failure envelope parameters in the Mohr–Coulomb form as well as the parameters of the Shock Hugoniot equation of state for which procedures have been developed. The present approach allows to retrieve required information regarding the equation of state and failure envelope from a single instrumented experiment that provides the projectile deceleration time-history record. It is shown that the retrieved parameters of the failure envelope and the equation of state are in good agreement with reported constitutive parameters that are based on experimental data. The retrieved data allow to carry out penetration time-history predictions of different projectiles at different impact velocities on the same concrete.

Keywords

Penetration concrete rigid projectile failure envelope equation of state Shock Hugoniot shear strength

Introduction

The interaction of high-velocity projectiles with concrete targets is a rather complex and challenging problem that has gained much interest for many years. A number of procedures have been developed for making such predictions and they generally fall into three categories: empirical equations for depth prediction, analytical models, and numerical solutions. The concrete uniaxial compressive strength is commonly used as the key parameter representing the concrete material in the empirical equations and in some of the analytical models (Chen and Li, 2014; Forrestal et al., 1994, 1996, 2003; Forrestal and Tzou, 1997; Frew et al., 2006; Warren et al., 2014), whereas other analytical models (Feldgun et al., 2017; Yankelevsky, 1982, 1983, 1985; Yankelevsky and Adin, 1980; Yankelevsky et al., 2017; Yankelevsky and Gluck, 1980) and most of the numerical solutions represent the concrete material by its constitutive equations, that is, the equation of state (EOS) and the failure envelope. In recent papers (Yankelevsky, 2016, 2017), it was clarified that the unconfined compressive strength does not identify any specific type of concrete and that different concrete compositions may be prepared with similar unconfined compressive strength and yet behave differently under a variety of triaxial tests. Therefore, the constitutive relationships of a given composition are required to properly define its behavior in penetration analysis. However, the EOS of a given concrete and its deviatoric strength properties depend on the concrete identity information (i.e. the composition, casting, and curing details) of that specific concrete (Yankelevsky, 2017) and are obtained from appropriate triaxial compression tests of that specific concrete. In a penetration event, extremely high pressures are developed at the projectile–concrete interface, and adequate constitutive properties are required to describe the interaction. To obtain the latter, high-pressure tests that require special equipment should be carried out on the target concrete specimens. Presently, only a limited number of tests have been conducted on concrete specimens of specific types at very high pressures and most of the data are not available to the common user. Moreover, the available data are not always specifying the composition identity; thus, its suitability to represent a given type of concrete is questionable. The vast majority of experimental scientific papers and reports on concrete penetration do not provide the concrete identity details as well as the EOS and the deviatoric strength properties of the concrete used in their experiments. The use of experimental data that are related to different concrete compositions may yield predictions that are in poor agreement with penetration test data. Attempts to “tune” the constitutive relationships in order to obtain better agreement between calculated penetration parameters and experimental data are artificial and may lead to wrong conclusions and to pseudo-scientific results. The idea behind the present article is that in absence of the constitutive equations of a specific type of concrete that are required for analysis of instrumented penetration tests, these properties may be retrieved from recorded continuous measurements (i.e. the acceleration time-history record) of a single test that is performed on this same concrete. The present approach is based on the authors’ previous papers (Feldgun et al., 2017; Yankelevsky, 1982, 1983, 1985; Yankelevsky and Adin, 1980; Yankelevsky et al., 2017; Yankelevsky and Gluck, 1980), which relied on a priori given constitutive properties, but present formulation is modified to consider the unknown parameters of these constitutive equations, and solve the penetration problem in terms of these unknown parameters, in order to determine these parameters based on the measured data. This approach of solving the unknown parameters from the penetration analysis and the test data may be denoted as the inverse problem of penetration. The importance of the developed technique is manifested in the fact that commonly, the solution of projectile penetration requires detailed information on the concrete properties, namely, the EOS and the failure envelope, which in many cases is not available, and in different experimental reports it is missing at all. In the case of instrumented tests, the present new approach allows to retrieve the parameters of the failure envelope in the Mohr–Coulomb form as well as the parameters of the Shock Hugoniot EOS. These models have been selected based on the accumulated experience in penetration analysis using the DISCS model, which provided high fidelity predictions of measured signals. These same models are implemented here in their general formulation with the unknown parameters as explained. Undoubtedly, the same suggested procedure may be implemented to other types of equations if desired.

The approach to retrieve the failure envelope is based on the effective simplified solution of the penetration equation using an integral common parameter of the volumetric strain. The approach to retrieve the EOS is based on the Riemann problem analytical solution applied to the initial stage of the penetration process.

It should be clarified that the present approach is not intended to replace the laboratory material triaxial testing, as this is the best direct and detailed way to obtain the material properties. However, these triaxial tests at high pressures are not common, require special and costly equipment, and are difficult and costly to conduct. Therefore, most penetration test series do not provide that information. When instrumented tests are conducted, the present article suggests a technique to retrieve material parameters and enable the analysis.

The proposed method and its novelty

The authors’ recent paper on projectile penetration into concrete (Yankelevsky et al., 2017) considers the following constitutive relationships:

(a) The nonlinear EOS of concrete (i.e. its pressure–density relationship)

\begin{array}{l} p = f_{L} (ρ) f o r t h e a c t i v e l o a d i n g \\ ρ = ρ^{*} f o r u n l o a d i n g \end{array}

(1)

where p is the current hydrostatic pressure, $ρ$ is the concrete current density, and $ρ^{*}$ is the maximum concrete density that is attained in the process of active loading. The concrete is represented by a medium which locks behind the plastic shock wave front (Christescu, 1967; Forrestal et al., 1981; Yankelevsky and Adin, 1980).

(b) The deviatoric plasticity is described by the Drucker–Prager criterion

σ_{r} - σ_{θ} = σ_{Y} (p)

(2)

where $σ_{Y} (p)$ is the yield strength.

It is known that the constitutive functions $f_{L} (ρ)$ and $σ_{Y} (p)$ in equations (1) and (2) depend on the type of concrete and are obtained from appropriate triaxial compression tests. These functions characterize the specific type of concrete into which the projectile is penetrating and, as explained above, are commonly unavailable.

The novelty of the present approach is that it suggests a procedure to retrieve unknown constitutive equation parameters from measured penetration test data and provide the required information for further analysis of projectile penetration with different nose shapes at different impact velocities into that specific type of concrete. The procedure may be formulated as follows. Following the DISCS model formulation, the medium is subdivided into discs that are perpendicular to the projectile axis (Feldgun et al., 2017; Yankelevsky, 1982, 1983, 1985; Yankelevsky and Adin, 1980; Yankelevsky et al., 2017; Yankelevsky and Gluck, 1980). The disc response is formulated with unknown parameters of the constitutive functions and the concrete disc–projectile interaction pressure is determined in terms of the unknown parameters. In order to obtain analytical expressions of the penetration event (depth, velocity, and acceleration), further simplification is required. The DISCS formulation derives the interaction pressure at the disc–nose interface for each disc, and opposed to other models no assumption is needed to fully describe the interaction pressure distribution along the projectile nose at all times. The interaction pressure components contributing to the resisting force at each instant are summed up along the contact zone, and the projectile equation of motion is formulated. To allow this transition from a discrete formulation to an integrated expression, a further assumption is required with regard to the distribution of the locking strains along the discs being in contact with the nose at any instant, and a constant equivalent integrated value of the locking strain is assumed. Its magnitude may be assessed at each instant during the penetration event, and variation with time may then be determined. The failure envelope of a concrete material (equation (2)) is commonly described by the well-known Mohr–Coulomb criterion (Nielsen and Hoang, 2011). This is a rather simple relationship with two unknown parameters, which has been used successfully in the penetration problem solution even in advanced analytical formulations (Forrestal and Tzou, 1997)

σ_{r} - σ_{θ} = σ_{Y} (p) = τ_{0} + λ p

(3)

where $τ_{0}$ is the shear cohesion and $λ$ is the internal friction coefficient. The present approach allows to obtain the shear cohesion and the internal friction coefficient analytically. It should be mentioned that these parameters characterize the material behavior and are identical for all discs at all times. It will be shown that with these constitutive parameters, the resistive force acting on the projectile can be expressed as

F (\overset{\cdot}{W} (t)) = K_{1} (ε_{L} (t), λ, \overset{\cdot}{W}) \cdot {\overset{\cdot}{W}}^{2} (t) + K_{2} (ε_{L} (t), τ_{0}, λ, \overset{\cdot}{W})

(4)

where $\overset{\cdot}{W} (t)$ is the penetration velocity and $ε_{L} (t)$ is an integrated dimensionless parameter of the volumetric strain in the discs at a given instant.

The projectile–concrete interaction problem

The semi-infinite concrete medium is subdivided into discs (Feldgun et al., 2017; Yankelevsky, 1982, 1983, 1985; Yankelevsky and Adin, 1980; Yankelevsky et al., 2017; Yankelevsky and Gluck, 1980) as shown in Figure 1(a). The response properties of each disc are concentrated at the interior expanding cavity, where the interaction with the penetrating nose occurs.

Figure 1.

The problem statement: (a) projectile and discs; (b) local and global coordinates; and (c) ogive geometry.

The following basic assumptions are considered:

The rigid projectile impacts normal to the concrete top surface and remains normal to it at all times, that is, only axisymmetric penetration is analyzed.

Heat transfer and dissipation between the projectile and the concrete mass are neglected.

The principal mechanism governing penetration is assumed to be that of large concrete deformations at the contact zone, and therefore only plastic strains are considered.

The projectile’s nose is slender, that is, there is a high ratio between the nose length to the projectile diameter.

Friction between the concrete and the nose surface is assumed to be negligible.

Only the projectile nose, or part of it, is in contact with the concrete target.

Stresses are assumed to be positive in compression.

In each disc, the stress tensor is

[σ_{i j}] = [\begin{matrix} σ_{r} & 0 & 0 \\ 0 & σ_{θ} & 0 \\ 0 & 0 & σ_{z} \end{matrix}]

(5)

It has already been mentioned that the induced velocity field in the concrete is almost radial; thus, the medium response occurs in the disc’s plane. Therefore, the radial and tangential stresses in the disc are the major and minor principal stresses, and the vertical (axial) stress is the intermediate principal stress. This stress adds an unknown in the direction that is perpendicular to the disc’s plane and to the medium direction of motion. It may be assumed that

σ_{z} = 0.5 (σ_{r} + σ_{θ})

(6)

This condition is formulated by analogy with the theory of elasticity, where in the one-dimensional (1D) cylindrical symmetry case of planar strains, the following exact relationship exists

σ_{z} = ν (σ_{r} + σ_{θ})

(7)

where $ν$ is the Poisson ratio of the elastic material.

A local coordinate system xAy is attached to the nose tip A (Figure 1(b)), and the local coordinate along the nose may be expressed as follows

x = z - W (t)

(8)

where $W (t) \underline{<} 0$ is the penetration depth at the nose tip.

The projectile nose length L is described by the general shape function y(x), which is assumed to have continuous first and second derivatives and to satisfy the boundary conditions

y (0) = 0, y (L) = 0.5 D

(9)

For example, when a projectile with an ogive nose impacts a target, the ogive is commonly defined in terms of its Caliber–Radius–Head (CRH; Figure 1(c))

C R H = \frac{s}{D} = ψ

(10)

At time t, at any distance x, the local radius $R (x, t)$ is identical with the shape function $y (x)$ of the local coordinate system xAy

R (x) = y (x) = \sqrt{s^{2} - {(x - L)}^{2}} - (s - 0.5 D) = \sqrt{s^{2} - {(x - L)}^{2}} - \sqrt{s^{2} - L^{2}}

(11)

In view of equation (11) for the ogive shape, we get

\frac{d y (x)}{d x} = \frac{L - x}{\sqrt{s^{2} - {(x - L)}^{2}}}

(12)

In the global coordinate system rOz (Figure 1(b)), the nose tip penetration depth and the projectile penetration velocity $\overset{\cdot}{W} (t)$ are negative and the deceleration $\overset{\cdot\cdot}{W} (t)$ is positive.

The projectile’s radial velocity at the point with the $t g α$ (Figure 1(b)) is found to be

\overset{\cdot}{R} (x, t) = - \overset{\cdot}{W} (t) t g α = - \overset{\cdot}{W} (t) \frac{d y (x)}{d x}

(13)

The contact stresses and resistive force acting on the projectile surface

The contact radial stress $p_{R} (x, t)$ acting at the nose point $(x_{j}, y (x_{j}))$ that corresponds to a certain disc #j with the coordinate $z_{j} = W + x_{j} (W \leq 0, 0 \leq x_{j} \leq L)$ is (Figure 1(b); Yankelevsky et al., 2017)

\begin{array}{l} p_{R j} (z_{j}, t, τ_{0}, λ, ε_{L j} (t)) = A_{0 j} (ε_{L j} (t), τ_{0}, λ) + ρ_{0} A_{1 j} (ε_{L j} (t), λ) \cdot {\overset{\cdot}{R}}^{2} (z_{j}, t) \\ + ρ_{0} A_{2 j} (ε_{L j} (t), λ) \cdot [{\overset{\cdot}{R}}_{j}^{2} (z_{j}, t) + R_{j} (z_{j}, t) \cdot {\overset{\cdot\cdot}{R}}_{j} (z_{j}, t)] \end{array}

(14)

where

A_{0 j} (ε_{L j} (t), τ_{0}, λ) = \frac{τ_{0}}{λ} [ε_{L j} {(t)}^{\frac{- 3 λ}{2 (3 + λ)}} - 1]

(15)

A_{1 j} (ε_{L j} (t), λ) = ε_{L j} {(t)}^{\frac{- 3 λ}{2 (3 + λ)}} - \frac{1}{1 - ε_{L j} (t)} \frac{3 + λ}{λ - 6} (ε_{L j} {(t)}^{1 + \frac{3 λ}{2 (3 + λ)}} - 1)

(16)

A_{2 j} (ε_{L j} (t), λ) = \frac{3 + λ}{3 λ} \frac{1}{1 - ε_{L j} (t)} (ε_{L j} {(t)}^{\frac{- 3 λ}{2 (3 + λ)}} - 1)

(17)

ε_{L j} (t) = 1 - \frac{ρ_{0}}{ρ_{L j} (t)}

(18)

$ρ_{0}$ is the initial density, $ρ_{L j} (t)$ is the density behind the shock wave front propagating through disc #j, and $R_{j} (z_{j}, t)$ , ${\overset{\cdot}{R}}_{j} (z_{j}, t)$ , and ${\overset{\cdot\cdot}{R}}_{j} (z_{j}, t)$ are the radial values of displacement, velocity, and acceleration of the internal boundary of disc #j.

The total resistive force acting on the projectile at time t is

\begin{array}{l} F (t, τ_{0}, λ) = \sum_{j} p_{S T j} (ε_{L j} (t), τ_{0}, λ) \cdot 2 π y (x_{j}) {\frac{d y (x)}{d x} |}_{x = x_{j}} h_{j} + \\ ρ_{0} \sum_{j} A_{1 j} (ε_{L j} (t), λ) \cdot {\overset{\cdot}{R}}_{j}^{2} (z_{j}, t) 2 π y (x_{j}) {\frac{d y (x)}{d x} |}_{x = x_{j}} h_{j} + \\ ρ_{0} \sum_{j} A_{2 j} (ε_{L j} (t), λ) \cdot [{\overset{\cdot}{R}}_{j}^{2} (z_{j}, t) + R_{j} (z_{j}, t) \cdot {\overset{\cdot\cdot}{R}}_{j} (z_{j}, t)] 2 π y (x_{j}) {\frac{d y (x)}{d x} |}_{x = x_{j}} h_{j} \end{array}

(19)

where $h_{j}$ is the disc thickness.

To allow integration of the force contributions along the nose, a simplifying assumption is introduced, as mentioned above, that an integral common parameter of the volumetric strain $ε_{L} (t)$ represents all the discs in contact with the nose. Thus, an exact integration can be performed in the local coordinate system xAy (Figure 1(b)), and the contact pressure acting on the projectile nose is (Yankelevsky et al., 2017)

\begin{array}{l} p_{R} (x, t, τ_{0}, λ, ε_{L} (t)) = A_{0} (ε_{L} (t), τ_{0}, λ) + ρ_{0} A_{1} (ε_{L} (t), λ) \cdot {\overset{\cdot}{R}}^{2} (x, t) \\ + ρ_{0} A_{2} (ε_{L} (t), λ) \cdot [{\overset{\cdot}{R}}^{2} (x, t) + R (x, t) \cdot \overset{\cdot\cdot}{R} (x, t)] \end{array}

(20)

where the parameters ${\overset{\cdot}{R}}^{2}$ and $({\overset{\cdot}{R}}^{2} + R \cdot \overset{\cdot\cdot}{R})$ are expressed by equations (11) to (13) as follows

{\overset{\cdot}{R}}^{2} (x, t) = {\overset{\cdot}{W}}^{2} (t) \cdot {(\frac{d y (x)}{d x})}^{2} = {\overset{\cdot}{W}}^{2} (t) \cdot \frac{{(L - x)}^{2}}{s^{2} - {(x - L)}^{2}}

(21)

\begin{array}{l} {\overset{\cdot}{R}}^{2} (x, t) + R (x, t) \cdot \overset{\cdot\cdot}{R} (x, t) = {\overset{\cdot}{W}}^{2} (t) \frac{{(L - x)}^{2}}{s^{2} - {(x - L)}^{2}} + \\ [{(s^{2} - {(x - L)}^{2})}^{1 / 2} - (s - 0.5 D)] \cdot {\overset{\cdot}{W}}^{2} (t) \cdot \frac{- s^{2}}{{(s^{2} - {(x - L)}^{2})}^{3 / 2}} \end{array}

(22)

In view of equations (11), (21), and (22), the contact radial stress $p_{R} (x, t)$ becomes

\begin{array}{l} p_{R} (x, \overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t)) = A_{0} (ε_{L} (t), τ_{0}, λ) + \\ ρ_{0} A_{1} (ε_{L} (t), λ) \frac{{(L - x)}^{2}}{s^{2} - {(x - L)}^{2}} {\overset{\cdot}{W}}^{2} (t) + ρ_{0} A_{2} (ε_{L} (t), λ) {\frac{{(L - x)}^{2}}{s^{2} - {(x - L)}^{2}} {\overset{\cdot}{W}}^{2} (t) + \\ [{(s^{2} - {(x - L)}^{2})}^{1 / 2} - (s - 0.5 D)] \cdot [\frac{- s^{2}}{{(s^{2} - {(x - L)}^{2})}^{3 / 2}} {\overset{\cdot}{W}}^{2} (t)]} \end{array}

(23)

where

A_{0} (ε_{L} (t), τ_{0}, λ) = \frac{τ_{0}}{λ} [ε_{L}^{- q / 2} (t) - 1]

(24)

A_{1} (ε_{L} (t), λ) = ε_{L}^{- q / 2} (t) - \frac{1}{1 - ε_{L} (t)} \frac{1}{q - 2} (ε_{L}^{- q / 2 + 1} (t) - 1)

(25)

A_{2} (ε_{L} (t), λ) = \frac{1}{q} \frac{1}{1 - ε_{L} (t)} (ε_{L}^{- q / 2} (t) - 1)

(26)

q = \frac{2 λ}{2 + λ}

(27)

ε_{L} (t) = 1 - \frac{ρ_{0}}{ρ_{L} (t)}

(28)

Therefore, the resistive force is expressed as

F (\overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t)) = \int_{0}^{x^{*}} p_{R} (x, \overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t)) \cdot 2 π y (x) \frac{d y (x)}{d x} d x

(29)

where $x^{*}$ is the length of the contact zone between the projectile nose and the target. It is assumed that during the nose embedment, the contact zone is (Figure 2(a) and (b))

0 \leq x \leq x^{*} (\overset{\cdot}{W} (t)) = | W (t) | \leq L

(30)

otherwise (Figure 2(c))

0 \leq x \leq x^{*} = L < | W (t) |

(31)

Figure 2.

Contact zone and definition of the parameter $x^{*} (t)$ : (a) cratering stage; (b) end of the cratering stage; and (c): tunneling stage.

Thus, using the coordinate $x^{*} (\overset{\cdot}{W} (t))$ allows to simulate the variable (non-stationary) behavior of the contact zone. Thus, the nose embedment stage of the penetration process as well as the tunneling stage can be simulated.

Note that the total resistive force (equation (29)) acting on the projectile at time t should be rewritten as an integral with the variable upper limit of integration $x^{*} (\overset{\cdot}{W} (t))$

F (\overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t)) = \int_{0}^{x^{*} (\overset{\cdot}{W} (t))} p_{R} (x, \overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t)) \cdot 2 π y (x) \frac{d y (x)}{d x} d x

(32)

Substituting equations (23) to (26) into equation (29) and performing the integration yield (see also equation (4))

\begin{array}{l} F (\overset{\cdot}{W} (t), τ_{0}, λ, ε_{L} (t), x^{*} (\overset{\cdot}{W} (t))) = K_{1} (ε_{L} (t), λ, x^{*} (\overset{\cdot}{W} (t))) \cdot {\overset{\cdot}{W}}^{2} (t) \\ + K_{2} (ε_{L} (t), τ_{0}, λ, x^{*} (\overset{\cdot}{W} (t))) \end{array}

(33)

where

\begin{array}{l} K_{1} (ε_{L} (t), λ, x^{*} (\overset{\cdot}{W} (t))) = \\ 2 π ρ_{0} A_{1} (ε_{L} (t), λ) {(2 s^{2} - L^{* 2}) \sqrt{\frac{s^{2} - L^{2}}{s^{2} - L^{* 2}}} + \frac{L^{2} + L^{* 2} - 4 s^{2}}{2} - \frac{s^{2}}{2} \ln \frac{s^{2} - L^{2}}{s^{2} - L^{* 2}}} - \\ 2 π ρ_{0} A_{2} (ε_{L} (t), λ) {s^{2} \sqrt{\frac{s^{2} - L^{2}}{s^{2} - L^{* 2}}} + \frac{s^{2} - L^{2} - L^{* 2}}{2} - \sqrt{(s^{2} - L^{2}) (s^{2} - L^{* 2})} - \frac{s^{2} (s^{2} - L^{2})}{2 (s^{2} - L^{* 2})}} \end{array}

(34)

K_{2} (ε_{L} (t), τ_{0}, λ, x^{*} (\overset{\cdot}{W} (t))) = 2 π A_{0} (ε_{L} (t), τ_{0}, λ) {s^{2} - \sqrt{(s^{2} - L^{2}) (s^{2} - L^{* 2})} - \frac{L^{2} + L^{* 2}}{2}}

(35)

L^{*} = L - x^{*} (\overset{\cdot}{W} (t))

(36)

It should be noted that if the EOS is known, the parameters $ε_{L j} (t)$ can be directly calculated at each disc, as shown in Yankelevsky et al. (2017). Their values and the values of the shear cohesion $τ_{0}$ and of the internal friction coefficient $λ$ allow calculating the contact stresses distribution at each instant of time. If the EOS is unknown, it is not possible to compute the parameters $ε_{L j} (t)$ for all the discs $(j = 1, \dots, N u m b e r o f d i s c s)$ , and in this case, only the average parameter $ε_{L} (t)$ is used and it allows obtaining only average distributions of the contact stresses, but as it will be shown the total resisting force is calculated successfully.

The projectile equation of motion

With the above formulation and assumptions, the equation of motion of a rigid projectile of mass M is described by the following ordinary differential equation that includes the unknown parameters of the EOS and the failure envelope

M \frac{d \overset{\cdot}{W} (t)}{d t} = K_{1} (ε_{L}, λ, x^{*} (\overset{\cdot}{W} (t))) \cdot {\overset{\cdot}{W}}^{2} (t) + K_{2} (ε_{L}, τ_{0}, λ, x^{*} (\overset{\cdot}{W} (t)))

(37)

The initial condition is

\overset{\cdot}{W} (0) = V_{0}

(38)

From equations (34) to (36), it can be seen that the coefficients $K_{1}, K_{2}$ of equation (37) are not constant, and in the general case, this equation can be solved only by a numerical method.

In the special case of $x^{*} = L$ (the entire nose is in contact with the target), the coefficients $K_{1}, K_{2}$ do not depend on $x^{*}$

K_{1} (ε_{L}, λ) = 2 π ρ_{0} A_{1} (ε_{L}, λ) {2 s \sqrt{s^{2} - L^{2}} + \frac{L^{2} - 4 s^{2}}{2} - \frac{s^{2}}{2} \ln \frac{s^{2} - L^{2}}{s^{2}}}

(39)

K_{2} (ε_{L}, τ_{0}, λ) = 2 π A_{0} (ε_{L}, τ_{0}, λ) {s^{2} - s \sqrt{s^{2} - L^{2}} - \frac{L^{2}}{2}}

(40)

and for $t \geq t_{x^{*} = L}$ , equation (37) has the following analytical solution that satisfies the initial condition

\overset{\cdot}{W} (t = t_{x^{*} = L}) = {\overset{\cdot}{W}}_{x^{*} = L}

(41)

\begin{array}{l} W (t) = - L - \frac{M}{K_{1} (ε_{L}, λ)} \times \\ \ln [\frac{\cos {a r c t g ({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} (t - t_{x^{*} = L})}}{\sqrt{1 + {({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}})}^{2}}}] \end{array}

(42)

\begin{array}{l} \overset{\cdot\cdot}{W} (t) = \\ \frac{K_{2} (ε_{L}, τ_{0}, λ)}{M \cos^{2} {a r c t g ({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} (t - t_{x^{*} = L})}} \end{array}

(43)

\begin{array}{l} \overset{\cdot}{W} (t) = \sqrt{\frac{K_{2} (ε_{L}, τ_{0}, λ)}{K_{1} (ε_{L}, λ)}} \times \\ t g {a r c t g ({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} (t - t_{x^{*} = L})} \end{array}

(44)

where $t_{x^{*} = L}$ is the instant at which the parameter $x^{*}$ reaches its maximum value L (the full nose embedment) and ${\overset{\cdot}{W}}_{x^{*} = L}$ is the corresponding projectile velocity.

Thus, the analytical solution (equations (42)–(44)) with the constant coefficients (equations (39) and (40)) is correct for $t \geq t_{x^{*} = L}$ when the penetration depth $W \geq L$ . Deceleration (equation (43)) is schematically shown in Figure 3 by the curve EC, where the curve ABEC corresponds to the numerical solution computed by the approach with the non-constant coefficients $K_{1} (ε_{L}, λ, x^{*} (\overset{\cdot}{W} (t))$ and $K_{2} (ε_{L}, τ_{0}, λ, x^{*} (\overset{\cdot}{W} (t))$ . Note that only the solution with the non-constant coefficients allows modeling the nose embedment stage ABE.

Figure 3.

Numerical solution: the nose embedment stage (ABE) (non-constant coefficients) followed by full nose contact (EC); Analytical solution—full nose contact (DC) (constant coefficients).

The curve CD will be described in the next section.

Retrieving the concrete properties

In the present article, an effective approach to determine the beforehand unknown values of shear cohesion $τ_{0}$ , and internal friction coefficient $λ$ of the failure envelope as well as the unknown parameters of the EOS from a single instrumented penetration test is developed.

Retrieving the parameters of the failure envelope

Figure 4 shows a typical deceleration versus time curve that was measured in an instrumented penetration test into a 23-MPa concrete target (Forrestal et al., 2003). The projectile was characterized by a 7.62-cm diameter, was made of steel (4340), had an ogive nose shape ( $ψ = C R H = 3$ ), and its mass was 13 kg.

Figure 4.

Typical experimental deceleration—time history.

In this specific test, the projectile impacted perpendicular to the concrete surface, at a velocity of 250 m/s.

In this study, we depict termination of the penetration process by the bold point, at which the high deceleration sharply drops to zero as shown in Figure 4.

From equation (37), it may be seen that at the end of the penetration process $(\overset{\cdot}{W} = 0)$ , the final deceleration ${\overset{\cdot\cdot}{W}}_{E N D}$ (related to the bold point in Figure 4) is

{\overset{\cdot\cdot}{W}}_{E N D} = {{\frac{1}{M} [K_{1} (ε_{L}, λ, \overset{\cdot}{W}) \cdot {\overset{\cdot}{W}}^{2} (t) + K_{2} (ε_{L}, τ_{0}, λ, \overset{\cdot}{W})]} |}_{\overset{\cdot}{W} = 0} = \frac{K_{2} (ε_{L}, τ_{0}, λ)}{M}

(45)

It should be noted that this analytical expression of the deceleration at the end of the process is independent on the impact velocity. This is supported by comparisons with various penetration tests that had been conducted on the same concrete type at different impact velocities (Forrestal et al., 2003) as shown in Figure 5. The final deceleration value $D_{E N D}^{E X P} / G$ (G is the gravitational acceleration) from the experiment with V₀ = 250 m/s (Figure 5(c)) is depicted in Figure 5(a), (b), and (d) for different impact velocities.

Figure 5.

Experimental deceleration time histories $(f_{c} = 23 M P a)$ (Forrestal et al., 2003): (a) V₀ = 139 m/s, $ψ = 3$ ; (b) V₀ = 200 m/s, $ψ = 3$ ; (c) V₀ = 250 m/s, $ψ = 3$ ; and (d) V₀ = 379 m/s, $ψ = 6$ .

The analytical solution (equation (43)) obtained for the time interval $t_{x^{*} = L} \leq t \leq t_{E N D}$ also gives the final deceleration value (equation (45)). This may be shown as follows. The time instant $t_{E N D}$ is obtained from equation (44) rewritten as

\begin{array}{l} \overset{\cdot}{W} (t_{E N D}) = \sqrt{\frac{K_{2} (ε_{L}, τ_{0}, λ)}{K_{1} (ε_{L}, λ)}} \times \\ t g {a r c t g ({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} (t_{E N D} - t_{x^{*} = L})} = 0 \end{array}

(46)

Therefore

t_{E N D} = t_{x^{*} = L} - \frac{M}{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}} a r c t g ({\overset{\cdot}{W}}_{x^{*} = L} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}})

(47)

Note that according to the global coordinate system rOz (Figure 1(b)), the penetration velocities $V_{0}$ , $\overset{\cdot}{W} (t)$ , ${\overset{\cdot}{W}}_{x^{*} = L}$ , and the depth $W (t)$ are negative, and therefore the “minus” sign in equation (47) as well as the two first “minus” signs in equation (42) are justified (otherwise, it may seem that $t_{E N D} < t_{x^{*} = L}$ ).

Substituting equation (47) into equation (43) yields $\overset{\cdot\cdot}{W} (t_{E N D}) = (K_{2} (ε_{L}, τ_{0}, λ)) / M$ , and in view of equation (42), the corresponding final penetration depth is

W_{E N D} = W (t_{E N D}) = - L - \frac{M}{K_{1} (ε_{L}, λ)} \ln \sqrt{1 + \frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)} {\overset{\cdot}{W}}_{x^{*} = L}^{2}}

(48)

Based on a typical instrumented test (Figure 4), the following simple and effective approach to obtain the unknown parameters $ε_{L}$ , $τ_{0}$ , and $λ$ of the concrete used in the penetration experiments is suggested.

To determine the three unknowns $ε_{L}$ , $τ_{0}$ , and $λ$ , we refer to the following known experimental values:

$D_{E N D}^{E X P}$ is the deceleration at the end of the penetration process (the bold circle in Figure 4);

$W_{M A X}^{E X P}$ —the final measured penetration depth;

$f_{c}$ —the unconfined compressive strength of target concrete.

Equations (45) and (48) as well as the yield condition (equation (3)) yield the following equations

{\overset{\cdot\cdot}{W}}_{E N D} = D_{E N D}^{E X P}

(49)

W_{E N D} = W_{M A X}^{E X P}

(50)

σ_{Y} (\frac{f_{c}}{3}) = τ_{0} + λ \cdot \frac{f_{c}}{3} = f_{c}

(51)

where the third condition refers to the yield condition (equation (3)) for the case of the standard uniaxial compression test for which $p = f_{c} / 3$ .

It should be noted that the parameter ${\overset{\cdot}{W}}_{x^{*} = L}$ (the velocity at the instant of nose embedment) is unknown. It is clear that the nose embedment duration $t_{x^{*} = L}$ decreases with increase of the impact velocity. Figure 6 shows experimental measured values of the nose penetration duration $t_{x^{*} = L}$ at different impact velocities (Forrestal et al., 2003); it refers to 13 kg mass, 7.62 cm diameter, steel (4340) projectiles, with an ogive nose $(ψ = C R H = 3)$ , impacting a concrete target (with an unconfined compressive strength $f_{c} = 23 M P a$ ) at different velocities.

Figure 6.

Duration of nose embedment (Forrestal et al., 2003).

The projectile velocity at the end of the nose embedment stage, ${\overset{\cdot}{W}}_{x^{*} = L}$ , is somewhat lower than the impact velocity, $V_{0}$ , as shown for the corresponding tests in Figure 7. Thus, the analytical solution for the full nose penetration stage may assume that the initial velocity at this stage, starting at $t_{x^{*} = L}$ , may be approximately equal to the impact velocity $\overset{\cdot}{W} (t = 0) = V_{0}$ , instead of the initial condition $\overset{\cdot}{W} (t = t_{x^{*} = L}) = {\overset{\cdot}{W}}_{x^{*} = L}$ that requires analysis of the nose embedment and evaluation of the velocity at the end of the embedment stage. This approximation is justified (Figure 7) for moderate and high velocities. This yields

\begin{array}{l} W (t) = - \frac{M}{K_{1} (ε_{L}, λ)} \times \\ \ln [\frac{\cos {a r c t g (V_{0} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} t}}{\sqrt{1 + {(V_{0} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}})}^{2}}}] \end{array}

(52)

\overset{\cdot}{W} (t) = \sqrt{\frac{K_{2} (ε_{L}, τ_{0}, λ)}{K_{1} (ε_{L}, λ)}} t g {a r c t g (V_{0} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} t}

(53)

\overset{\cdot\cdot}{W} (t) = \frac{K_{2} (ε_{L}, τ_{0}, λ)}{M \cos^{2} {a r c t g (V_{0} \sqrt{\frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)}}) + \frac{\sqrt{K_{1} (ε_{L}, λ) K_{2} (ε_{L}, τ_{0}, λ)}}{M} t}}

(54)

and the expression of the final penetration depth is

W_{E N D} = - L - \frac{M}{K_{1} (ε_{L}, λ)} \ln \sqrt{1 + \frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)} V_{0}^{2}}

(55)

Figure 7.

Velocity at the end of nose embedment (Forrestal et al., 2003).

A typical deceleration curve (equation (54)) is schematically shown in Figure 3 (curve DC).

Finally, from equations (45) and (49) to (51), we obtain the following system of equations from which the parameters $ε_{L}$ , $τ_{0}$ , and $λ$ are derived

K_{2} (ε_{L}, τ_{0}, λ) = M \cdot D_{E N D}^{E X P}

(56)

- L - \frac{M}{K_{1} (ε_{L}, λ)} \ln \sqrt{1 + \frac{K_{1} (ε_{L}, λ)}{K_{2} (ε_{L}, τ_{0}, λ)} V_{0}^{2}} = W_{M A X}^{E X P}

(57)

f_{c} = τ_{0} + λ \cdot \frac{f_{c}}{3}

(58)

where $D_{E N D}^{E X P}$ and $W_{M A X}^{E X P}$ are known from a single instrumented penetration test.

Thus, the parameters $ε_{L}$ , $τ_{0}$ , and $λ$ that are calculated from equations (56) to (58) enable the analytical solution of the differential equations (37) and (38) in the time interval $t_{x^{*} = L} \leq t \leq t_{E N D}$ and the semi-analytical solution during the nose embedment interval $0 \leq t < t_{x^{*} = L}$ .

Retrieving the parameters of the EOS

Consider a projectile with the initial penetration velocity $V_{0}$ and refer to a small penetration depth of $W_{T I P}^{E X P}$ as shown in Figure 8. The corresponding penetration duration D $t_{T I P}^{E X P}$ can be calculated from the penetration depth time history obtained by a direct double integration of the experimental deceleration time history. Thus, we know the parameters $W_{T I P}^{E X P}$ and ${\overset{\cdot\cdot}{W}}_{T I P}^{E X P}$ at $t = Δ t_{T I P}^{E X P}$ . The value of the corresponding experimental contact stress can be calculated as

σ_{T I P}^{E X P} = {\overset{\cdot\cdot}{W}}_{T I P}^{E X P} M / Ω_{T I P}

(59)

where the contact surface $Ω_{T I P}$ is calculated as

\begin{array}{l} Ω_{T I P} = 2 π \int_{0}^{W_{T I P}} y (x) \frac{d y (x)}{d x} d x = \\ 2 π \int_{0}^{W_{T I P}} [\sqrt{s^{2} - {(W_{T I P} - L)}^{2}} - \sqrt{s^{2} - L^{2}}] \frac{L - x}{\sqrt{s^{2} - {(W_{T I P} - L)}^{2}}} d x = \\ 2 π [- L^{2} / 2 - 0.5 \cdot {(W_{T I P} - L)}^{2} + s^{2} - \sqrt{s^{2} - L^{2}} \sqrt{s^{2} - {(W_{T I P} - L)}^{2}}], s = D ψ \end{array}

(60)

Figure 8.

Initial stage of penetration (small penetration depth).

On the other hand, the contact stress as well as the corresponding volumetric strain on the nose tip can be calculated from the Riemann problem solution (Feldgun et al., 2017; Yankelevsky et al., 2017). If the projectile interacts with an undisturbed concrete, the contact stress S is obtained as

S (P) (\frac{1}{ρ_{0}} - \frac{1}{R o (P)}) = V_{R}^{2} = {(\frac{\overset{\cdot}{W} d y (x)}{d x})}^{2}

(61)

where

R o (P) = f_{L}^{- 1} (P)

(62)

P is the pressure that corresponds to the contact stress S and f⁻¹ is the corresponding inverse function of the EOS (equation (1)).

In view of

E_{V} = 1 - ρ_{0} / R o (P)

(63)

Equation (61) can be rewritten as

S E_{V} = ρ_{0} V_{R}^{2} = ρ_{0} {(\frac{\overset{\cdot}{W} d y (x)}{d x})}^{2}

(64)

where $E_{V}$ is the contact volumetric strain.

For the EOS “pressure-volumetric strain” in the Shock Hugoniot form

P = ρ_{0} A^{2} E_{V} / {(1 - β E_{V})}^{2}

(65)

where the unknown parameters $β$ and A are unknowns that should be found.

In view of the plasticity criterion in the Mohr–Coulomb form (equation (3)) with the parameters $τ_{0}$ and $λ$ , we have

S = n_{1} P + n_{2} τ_{0}

(66)

n_{1} = \frac{1}{2}

(67)

n_{2} = \frac{λ + 2}{2}

(68)

and equation (61) is rewritten as a cubical equation

a_{3} (β) E_{V}^{3} + a_{2} (A, β) E_{V}^{2} + a_{1} (β) E_{V} + a_{0} = 0

(69)

\begin{array}{l} a_{3} (β) = n_{1} τ_{0} β^{3} \\ a_{2} (A, β) = n_{2} ρ_{0} A^{2} - 2 β n_{1} τ_{0} - ρ_{0} β^{2} V_{R}^{2} \\ a_{1} (β) = 2 β ρ_{0} V_{R}^{2} + n_{1} τ_{0} \\ a_{0} = - ρ_{0} V_{R}^{2} \end{array}

(70)

where the parameters of the Shock Hugoniot EOS (equation (65)) are the unknowns that should be found.

Finally, the unknown parameters S, P, $β$ , A, and $E_{V}$ are derived from the system of equations (64), (65), (66), and (69) as well as from the simple condition

S = σ_{T I P}^{E X P}

(71)

where the value of the experimental contact stress $σ_{T I P}^{E X P}$ is known from equations (59) and (60).

The system of equations (64), (65), (66), (69), and (71) is solved numerically.

Examples

Retrieving the failure envelope properties of concrete with f_c = 23 MPa

The developed procedure is applied for a 23-MPa concrete target that had been tested with instrumented projectiles (Forrestal et al., 2003). The failure envelope properties of this concrete are also given in Fossum and Brannon (2004), and although not being used in the proposed procedure, they provide a reference for comparison with the retrieved data. The ogive nose shape 4340 steel projectile of 7.62 cm diameter with a nominal mass of 13 kg impacts the concrete target, perpendicular to the concrete surface, at velocities of 139, 200, and 250 m/s for $ψ = C R H = 3$ , and 379 m/s for $ψ = C R H = 6$ . The density of the concrete is $ρ_{0} = 2040 k g / m^{3}$ .

One of the tests with an impact velocity of $V_{0} = 250 m / s$ was selected to determine the unknown parameters. The corresponding values of the parameters $D_{S T}^{E X P}$ and $W_{M A X}^{E X P}$ (equations (56) and (57)) are

D_{S T}^{E X P} = 5200 \cdot G, W_{M A X}^{E X P} = 57 c m

(72)

The following values of the concrete parameters were obtained

τ_{0} = 130 b a r, λ = 1.299

(73)

To verify the obtained result, let us check the unconfined compression test as a point on the failure envelope. It can be seen that

130 b a r + 1.299 \cdot (f_{c} = 230 b a r) / 3 = 229.59 \approx 230 b a r = f_{c}

(74)

Retrieving the above failure envelope parameters allows their usage in any further penetration analysis in this same concrete, as will be shown soon.

It is interesting to examine the obtained failure envelope parameters, in comparison with available data in the literature. Figure 9(a) shows the magnitude of the shear cohesion term for different values of unconfined compression strength of different concretes (Forrestal and Tzou, 1997; Gran and Frew, 1997; Nielsen and Hoang, 2011; SANDIA GEOMODEL—Fossum and Brannon, 2004, and Warren et al. 2004).

Figure 9.

The coefficients of the Drucker–Prager condition: (a) shear cohesion and (b) internal friction coefficient.

Figure 9(b) shows a comparison with values of the internal friction coefficient. As expected, the scatter of the shear cohesion is small because of its close relation to the given unconfined compression strength, whereas the scatter of the internal friction coefficient is larger, as it represents the failure envelope slope of possibly different concrete compositions. It may be seen that the obtained parameters $τ_{0}$ and $λ$ (equation (73)) are in good agreement with known experimental data presented as a function of the unconfined compressive strength in Figure 9. Note that apart from the concrete with $f_{c} = 23 M P a$ , Figure 9 also shows the parameters obtained for the concrete with $f_{c} = 39 M P a$ for which comparisons with various penetration tests (Forrestal et al., 2003) are also performed (see the next subsection).

Retrieving the failure envelope properties of concrete with f_c = 39 MPa

Comparisons of the present approach predictions with a second series of penetration tests have been carried out, in which instrumented projectiles penetrated into concrete with a different unconfined compression strength (Forrestal et al., 2003). The steel (4340) projectile of 7.62 cm diameter with a mass of 13 kg with an ogive nose shape impacted the concrete target that is defined by its uniaxial compressive strength $f_{c} = 39 M P a$ . Impact was perpendicular to the target surface, at velocities of 238, 276, and 314 m/s for $ψ = C R H = 3$ , and 313 and 449 m/s for $ψ = C R H = 6$ . The concrete density is $ρ_{0} = 2250 k g / m^{3}$ .

The test with an impact velocity of $V_{0} = 276 m / s$ (Forrestal et al., 2003) was selected to determine the unknown parameters.

The following values of the parameters $D_{M A X}^{E X P}$ and $D_{S T}^{E X P}$ (equations (56) and (57)) were taken to determine the parameters

D_{S T}^{E X P} = 9500 \cdot G, W_{M A X}^{E X P} = 35 c m

(75)

The following values of the concrete parameters were obtained

τ_{0} = 180 b a r, λ = 1.6628

(76)

It can be seen that

180 b a r + 1.6628 \cdot (f_{c} = 390 b a r) / 3 = 396.16 \approx 390 b a r = f_{c}

(77)

The obtained parameters $τ_{0}$ and $λ$ (equation (77)) are inserted in Figure 9, similar to the parameters that had been determined above for $f_{c} = 23 M P a$ . A good agreement with known experimental data is shown.

Retrieving the Shock Hugoniot EOS

Consider the same case (f_c = 39 MPa) of a projectile with the initial penetration velocities $V_{0}$ of 238, 276, 314, 370, and 456 m/s (Forrestal et al., 2003) and refer to a small penetration depth of $W_{T I P} = 0.5 c m$ . The corresponding experimental contact stresses calculated by equations (59) and (60) using the experimental deceleration time histories (Forrestal et al., 2003) are, respectively, 4696.7, 5909.4, 7256.1, 9484.5, and 13471 bar.

In absence of experimental data for these specific types of concrete, comparisons will be carried out for the 39 MPa concrete, for which there are experimental data for a very similar concrete small aggregate concrete (SAC)-5 with a compressive strength of 40 MPa (Hall et al., 1999). Figure 10(a) compares the predicted parameter $β$ and Figure 10(b) compares the predicted parameter A with the results of the experiments carried out in Hall et al. (1999; the horizontal lines in Figure 10(a) and (b)), referring to the SAC-5 concrete.

Figure 10.

The parameters of the Shock Hugoniot EOS (equation (65)): (a) the parameter $β$ and (b) the parameter A.

It can be seen that the results computed by the developed approach and the experimental data are in close agreement for the 39 MPa concrete, which has a similar compressive strength.

Figure 11 shows the Shock Hugoniot EOS curves (equation (65)) for various parameters $β$ and A (Figure 10(a) and (b)) obtained by the present approach using different penetration experiments into the 39 MPa concrete (Forrestal et al., 2003).

Figure 11.

The Shock Hugoniot EOS (65) for various parameters $β$ and A obtained by present approach for different penetration experiments (Forrestal et al., 2003) (f_c = 39 MPa). Comparison with SAC-5 (f_c = 40 MPa).

Comparison of the predicted Shock Hugoniot EOS curves with experimental results of SAC-5 concrete shows very good agreement and supports the proposed procedure.

Conclusion

A new approach to retrieve the constitutive properties of concrete targets from instrumented penetration test data of rigid projectiles into concrete media is presented.

The importance of the developed technique is in the fact that commonly, the solution of projectile penetration requires detailed information on the concrete properties, namely, the EOS and the failure envelope (this is the common or “direct” problem). In many cases, that information is not available, and in different experimental reports it is missing.

The present approach is based on the previously developed DISCS model approach in which the concrete medium is subdivided into discs and a convenient mathematical formulation is proposed based on some simplifying assumptions.

The major new feature of this article is the simulation of the rigid projectile penetration into the concrete medium in terms of the unknown parameters of the EOS and strength envelope. Instead of relying on known EOS and strength data, the present approach allows to retrieve required information from a single recorded experiment providing the projectile deceleration time history thus allowing high fidelity penetration time-history predictions at different impact velocities and different values of the CRH on the same concrete. This identification of the unknown parameters and its usage for further predictions of penetration analysis in the same concrete may be denoted as the inverse problem.

A new approach to retrieve the failure envelope and the parameters of the Shock Hugoniot EOS has been developed. The approach to retrieve the failure envelope in the Mohr–Coulomb form is based on the effective simplified solution of the penetration equation using an integral common parameter of the volumetric strain. The approach to retrieve the EOS is based on the Riemann problem analytical solution applied to the initial stage of the penetration process.

It was shown that the failure envelope and the EOS parameters computed by the developed approach are in good agreement with available experimental data.

Footnotes

Appendix 1 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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by a joint grant from the Centre for Absorption in Science of the Ministry of Immigrant Absorption and the Committee for Planning and Budgeting of the Council for Higher Education under the framework of the KAMEA Program.

ORCID iD

Vladimir R Feldgun

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Retrieving the constitutive properties of a concrete target from a single instrumented penetration test

Abstract

Keywords

Introduction

The proposed method and its novelty

The projectile–concrete interaction problem

The contact stresses and resistive force acting on the projectile surface

The projectile equation of motion

Retrieving the concrete properties

Retrieving the parameters of the failure envelope

Retrieving the parameters of the EOS

Examples

Retrieving the failure envelope properties of concrete with fc = 23 MPa

Retrieving the failure envelope properties of concrete with fc = 39 MPa

Retrieving the Shock Hugoniot EOS

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References

Retrieving the failure envelope properties of concrete with f_c = 23 MPa

Retrieving the failure envelope properties of concrete with f_c = 39 MPa