4.1 Projectile model

conveniently formulated using the effective stress component. The development of

tension and compression, respectively, can be used to independently characterized a material damaged states in tension and compression. Generally, increasing values of the hardening variables may lead to microcracking and crushing in concrete. These variables also control the degradation of the elastic stiffness and the progression of the yield surface, as well as affecting the dissipated fracture energy required

space, will determines the states of failure or damage. For the inviscid plastic-

Penetration test is also modeled using the Discrete Element Method (DEM). DEM was first introduced by Cundall [19] in the early 1970s. It was originally applied on rocks, then extended to granular material, which triggered much wider uses in different kinds of material like fluid, soil, and composites. DEM has not received much attention in penetration simulation before 1990. Before 1990, Heuze's overview [20, 21] indicated that only 3 computer programs based their theory on DEM. However, DEM has its intrinsic advantages, especially related to penetration simulation, when compared to other numerical simulation methods, such as FEM-based on continuum meshing. DEM allows transitioning from continuum to discontinuum to be easily simulated, while handling fracturing and large

The geometry of a projectile is one of the key factors affecting the penetration process. A number of studies have addressed the shape effects including those on flat nose [22–24], ogive [25], and spherical ball [26]. Zhu and Zhang [27] compared the effects on penetration using projectiles of ogive and flat nose shape. While most researchers consider projectiles as rigid, others investigated the effects due to a deformable projectile. As for the impact velocity, Nishida [26] studied the penetration at a low velocity of 16 m/s while most others focused on velocities larger than 100 m/s. DEM is also used in the theoretical formulation of PFC3D (a particle modeling software) known as particle-flow model. Particles of arbitrary shapes that displace independent of each other and occupy a finite amount of space constitute the basic element of the model. The model uses a finite normal stiffness to represent the contact stiffness, while the interaction between the particles, which are assumed rigid, is defined using a soft contact approach. Force-Displacement Law and Motion Law are the two primary rules to define the mechanical computation. The former law is used to calculate the contact force and momentum between two entities based on their relative displacement. It should be noted that the momentum part could only be modeled in the parallel bond model for contacts. The second law, also referred as Newton's second law, governs how force and momentum determine the

pl).

pl, defined as equivalent plastic strains in

pl), that represents a surface in effective stress

F σ, ~εpl <0 (5)

pl (plastic

the degradation variable is governed by a set of hardening variables, ~ε

pl and ~ε<sup>c</sup>

strains), and the effective stress, d = d(σ, ~ε

3.3 Hardening variables

Compressive Strength of Concrete

to generate microcracks.

4. DEM modeling

deformation conveniently.

particle translational and rotational motion.

38

Two hardening variables, ~ε<sup>t</sup>

Also, a yield function, F(σ, ~ε

damage model, it is represented by Eq. (5).

In order to build the required cone shape mono-size balls are decreased in size from tail to tip. To keep a compact status inside the projectile the overlap of balls and large stiffness were purposely assigned. The balls forming the projectile were clumped into one object using the PFC3Dclump function. The created object does not allow any relative movements for the balls constituting the projectile. The friction between projectile and the target varies with their relative velocity and is defined by Eq. (6), where the static friction was determined by using the idealized infinite velocity Chen [28].

$$f = f\_{\text{inf}} + (f\_{\text{stat}} - f\_{\text{inf}})e^{r \ast \text{vel}} \tag{6}$$

where finf is the friction with idealized infinite velocity and fstat is the static friction.

Figure 8 illustrates a projectile model used to simulate penetration velocity versus depth relationship established by Forrestal et al. and their corresponding microscopic scale parameters. A model of a projectile created in PFC3D is showed in Figure 9. Most experiments use the cylindrical projectile shape which allows the convenient monitoring of symmetric damage. However, cubic specimens are used in simulations due to their simple geometry. By using large dimensions, the corner or boundary effects can be minimized. Although a semi-infinite target can be used in the classic penetration theory, a DEM simulation only accepts finite size targets, with its specific dimension needing to be determined to eliminate the size effect.

#### Figure 8.

Geometry of projectile used in test (Forrestal et al., 1994).

For both projectile and target, there are several major parameters contributing to the entire penetration process significantly. The major variables for projectile are mass (m), diameter (dia), nose shape, and impact velocity (vel). The former three are set in the projectile geometric and mechanical property file, while the last variable is input in the main code for penetration simulation. Key variables for target are macro Young's modulus (E), Poisson's ratio (ν), compressive strength (σc), and tensile strength (σt). They together represent the mechanical characteristics of the material.

PFC3D provides an optimized calibration sequence for some major control variables to minimize the iterations for parallel bond.


As illustrated in Figure 10, the macro compressive strength depends on both normal and shear strength of the contact balls, while normal strength contributes a little more. An important feature for this case is that it is the smaller one of these two micro-parameters controls the upper limit of the macro-strength, i.e. the compressive strength cannot increase when either one of the micro strengths stay at a

For Young's modulus and Poisson's ratio, neither the normal nor the shear strength of particles has much contribution in the normal range. However, when both these two micro strengths decrease to very small values, the Young's modulus

The empirical discoveries found in the above calibration test can be used to form concrete target with the required mechanical property, although the calibration still needs to be conducted step by step. This is because different micro-variable changes may result in similar macro-property, and the changing magnitudes prob-

Two types of concrete targets were made for penetration testing, i.e., the frusta

of either a pyramid or a cone (Figure 11). The pyramid and cone shapes were intended to save materials in the rear end of the samples. The larger-area side was subjected to the projectile penetration. The frustum of pyramid had dimensions of <sup>12</sup>″ <sup>12</sup>″ in the larger-end side and 13″ in depth, while the cylinders were 6″ and 11″ in diameter and 11″ in height. Both types of concrete targets were cast in 5000 psi and 8000 psi uniaxial compressive strengths. For verification purposes of the gas operated facility the first two shots were on two 2500 psi concrete cylinders (6″ <sup>12</sup>″). Limestone aggregate (#67) was used for the 5000 psi samples while (#78) was used

Projectiles of three different diameters (12, 20, and 30 mm) were used for penetration into the concrete targets and were launched using the same pressures

Figure 12 shows an example of the projectile after impacting the concrete targets, the damaged concrete targets and location of projectiles after the impact. Some target specimens were shattered by the projectile and the penetration of the projectile were not observed. This was primarily due to the size of concrete specimen relative to that of the projectile. However, as projectile size decreases (or concrete specimen size increases), the phenomena of projectile penetrating through

(1200 psi) to assess their speeds, penetration depths, and target damage.

concrete target become more likely to occur.

Types of concrete targets used for penetration testing.

and Poisson's ratio have a little more influence.

Concrete Microstructure Characterization and Performance

DOI: http://dx.doi.org/10.5772/intechopen.90500

ably vary widely as other parameters vary.

5. Experimental results

for the 8000 psi samples.

Figure 11.

41

constant level.

Figure 9. Projectile model created in DEM.
