3. Concrete microstructure damage by projectile impact

Penetration of projectiles into concrete target was investigated in this research by using both experimental and simulation methods. The over-all objectives of the project included: (a) Building up an experimental facility to conduct the penetration test; (b) Developing of a rational constitutive model to incorporate distributed damage effects; (c) Enhancing of the model implementation by combining the Finite Element Method (FEM) and the Discrete Element Method (DEM) so that post-fracture behavior can be simulated; and (d) Developing of methods to backcalculate model constants from comparing experimental with simulation results.

Figure 5. Gas operated facility for testing concrete samples.

#### Compressive Strength of Concrete

The equipment assembled for testing concrete specimens of different sizes and strengths is showed in Figure 5. It contains a gas tank that can be filled at various pressures, a launch tube, a gas expansion tank, and specimen housing chamber.

A three-dimensional penetration model was constructed using ABAQUS as shown in Figure 6. The concrete target was tentatively modeled as 40 � 40 � 100 cm blocks and penetrated by high speed projectiles made of rigid materials. No deformation is assumed for the projectile when penetrating the target. The concrete is assumed to be an elasto-plastic material with damage property. 8-node linear brick elements are used for the FEM mesh. An unbounded boundary domain is defined by using 8-node linear infinite elements, which are connected with concrete specimen at the periphery (Figure 6).

The microstructure of concrete specimen is considered by assigning different material properties to the three components of the mixture, aggregate, mortar (hydration products plus fine aggregate particles) and air void. These components are discriminated by utilizing image analysis techniques shown in Figure 7(a) and (b). Pixel intensity value determines what component each pixel belongs to. When meshing the domain to be modeled, the properties of the material between two scanned images are assumed to be the same as the front image.

#### 3.1 Concrete damage

A reduction in the elastic stiffness of concrete is the result of damage typically associated with the failure mechanisms of the concrete (cracking and crushing). According to the scalar-damage theory, the isotropic stiffness degradation is characterized by a single degradation variable, d. Based on continuum damage mechanics notions, the effective stress is defined as Eq. (1).

$$\overline{\sigma} = \mathbf{D}\_0 \prescript{el}{}{\cdot} : \left( \mathbf{e} - \mathbf{e}^{pl} \right) \tag{1}$$

D0

Figure 7.

fully damaged material.

marized below.

according to Eq. (4).

37

3.2 Strain rate decomposition

el is the elastic stiffness due to damage; and d is the scalar stiffness variable due to degradation. A d value of zero indicates undamaged material while one shows a

The constitutive behavior of concrete was illustrated using the concrete damaged plasticity model. The model describes the inelastic behavior of concrete based on the concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity. Moreover, the scalar damaged elasticity combined with the non-associated multi-hardening plasticity describe the irreversible damage that occurs during the fracturing process. The main ingredients of the model are sum-

Additive strain rate decomposition is assumed for the rate-independent model

For any given cross-section of the material, the ratio of the effective loadcarrying area (i.e., the total area minus the damaged area) to the overall section area is represented by the (1 � d) factor. Thus, the effective stress is equivalent to the Cauchy stress, σ, if there is no damage, d = 0. When damage occurs, however, the effective stress is larger than the Cauchy stress because the external loads are supported by the effective stress area. Therefore, the plasticity problem can be

<sup>έ</sup> <sup>¼</sup> <sup>έ</sup>el ð Þþ elastic <sup>έ</sup>pl ð Þ elastic (4)

el : <sup>ε</sup> � <sup>ε</sup>pl (3)

σ ¼ ð Þ 1 � d D0

3D microstructure of concrete specimen (a) and the image-based reconstruction (b).

Concrete Microstructure Characterization and Performance

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

The Cauchy stress is related to the effective stress through the scalar degradation relation per Eq. (2).

$$
\sigma = (1 - d)\overline{\sigma} \tag{2}
$$

The stress-strain relations are governed by scalar damaged elasticity given in Eq. (3), where D0 el is the elastic stiffness of the undamaged material; Del = (1 � <sup>d</sup>) Concrete Microstructure Characterization and Performance DOI: http://dx.doi.org/10.5772/intechopen.90500

The equipment assembled for testing concrete specimens of different sizes and strengths is showed in Figure 5. It contains a gas tank that can be filled at various pressures, a launch tube, a gas expansion tank, and specimen housing chamber. A three-dimensional penetration model was constructed using ABAQUS as shown in Figure 6. The concrete target was tentatively modeled as 40 � 40 � 100 cm blocks and penetrated by high speed projectiles made of rigid materials. No deformation is assumed for the projectile when penetrating the target. The concrete is assumed to be an elasto-plastic material with damage property. 8-node linear brick elements are used for the FEM mesh. An unbounded boundary domain is defined by using 8-node linear infinite elements, which are connected with concrete

The microstructure of concrete specimen is considered by assigning different material properties to the three components of the mixture, aggregate, mortar

A reduction in the elastic stiffness of concrete is the result of damage typically associated with the failure mechanisms of the concrete (cracking and crushing). According to the scalar-damage theory, the isotropic stiffness degradation is characterized by a single degradation variable, d. Based on continuum damage mechan-

The Cauchy stress is related to the effective stress through the scalar degradation

The stress-strain relations are governed by scalar damaged elasticity given in

el is the elastic stiffness of the undamaged material; Del = (1 � <sup>d</sup>)

el : <sup>ε</sup> � <sup>ε</sup>pl (1)

σ ¼ ð Þ 1 � d σ (2)

components are discriminated by utilizing image analysis techniques shown in Figure 7(a) and (b). Pixel intensity value determines what component each pixel belongs to. When meshing the domain to be modeled, the properties of the material between two scanned images are assumed to be the same as the front image.

(hydration products plus fine aggregate particles) and air void. These

σ ¼ D0

ics notions, the effective stress is defined as Eq. (1).

specimen at the periphery (Figure 6).

Compressive Strength of Concrete

FEM modeling of concrete target impacted by projectile.

3.1 Concrete damage

Figure 6.

relation per Eq. (2).

Eq. (3), where D0

36

D0 el is the elastic stiffness due to damage; and d is the scalar stiffness variable due to degradation. A d value of zero indicates undamaged material while one shows a fully damaged material.

$$\boldsymbol{\sigma} = (\mathbf{1} - d)\mathbf{D}\_0^{\;el} : \left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{pl}\right) \tag{3}$$

The constitutive behavior of concrete was illustrated using the concrete damaged plasticity model. The model describes the inelastic behavior of concrete based on the concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity. Moreover, the scalar damaged elasticity combined with the non-associated multi-hardening plasticity describe the irreversible damage that occurs during the fracturing process. The main ingredients of the model are summarized below.

#### 3.2 Strain rate decomposition

Additive strain rate decomposition is assumed for the rate-independent model according to Eq. (4).

$$
\dot{\varepsilon} = \dot{\varepsilon}^{el} \left( \text{elastic} \right) + \dot{\varepsilon}^{pl} \left( \text{elastic} \right) \tag{4}
$$

For any given cross-section of the material, the ratio of the effective loadcarrying area (i.e., the total area minus the damaged area) to the overall section area is represented by the (1 � d) factor. Thus, the effective stress is equivalent to the Cauchy stress, σ, if there is no damage, d = 0. When damage occurs, however, the effective stress is larger than the Cauchy stress because the external loads are supported by the effective stress area. Therefore, the plasticity problem can be

conveniently formulated using the effective stress component. The development of the degradation variable is governed by a set of hardening variables, ~ε pl (plastic strains), and the effective stress, d = d(σ, ~ε pl).

### 3.3 Hardening variables

Two hardening variables, ~ε<sup>t</sup> pl and ~ε<sup>c</sup> pl, defined as equivalent plastic strains in 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 to generate microcracks.

Also, a yield function, F(σ, ~ε pl), that represents a surface in effective stress space, will determines the states of failure or damage. For the inviscid plasticdamage model, it is represented by Eq. (5).

$$\mathbf{F}(\overline{\sigma}, \tilde{\varepsilon}^{pl}) < \mathbf{0} \tag{5}$$

4.1 Projectile model

Concrete Microstructure Characterization and Performance

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

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].

<sup>f</sup> <sup>¼</sup> <sup>f</sup>inf <sup>þ</sup> <sup>f</sup>stat � <sup>f</sup>inf <sup>e</sup>

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.

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 vari-

ables to minimize the iterations for parallel bond.

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

tension.

39

Figure 8.

not presented in this paper.

1.Matching the material's Young's modulus by varying E<sup>c</sup> and Ēc.

3.Varying the mean normal and shear strength, σ<sup>c</sup> and τc, as well as their

standard deviation, to obtain the strength envelope for both compression and

4.Properties, such as post-peak behavior or crack-initiation stress, can also be obtained by adjusting related variables, such as friction coefficient to match with those from the real samples; for conciseness purpose however, they are

2.Matching the Poisson's ratio by varying kn/k<sup>s</sup> and kn/ks.

<sup>γ</sup> <sup>∗</sup> vel (6)

### 4. DEM modeling

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 deformation conveniently.

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 particle translational and rotational motion.
