**2. Model of porous ceramics under shock wave compression**

Dynamic response of porous ceramics under rapid impulsive loading relates to evolution of a crack network following the shock wave. Although some pioneer works have been conducted on modeling ceramic shock fracture via mesh-based computational methods [10–13], such methods encounter significant difficulties when dealing with fracture and fragmentation induced by shock wave compression. The reason is that partial derivatives are used in mesh-based methods to represent the relative displacement and force between any two neighboring particles [14]. But, the necessary partial derivatives with respect to the spatial coordinates are undefined along the cracks and need to be redefined. However, the redefinition requires us to know where the discontinuity is located. This limits the usefulness of these methods in addressing problems involving the spontaneous formation of cracks, in which one might not know their location in advance [14]. In contrast, as a particle method, the lattice-spring model (LSM, also known as discrete-element method) [15–20] could avoid various numerical difficulties caused by displacement discontinuity. In this section, details of the LSM model (lattice interactions, spring mapping procedure, fracture criterion, microstructures, loading) and its validation are introduced.

particles, when the sum exceeds a certain threshold corresponding to the fracture energy. The deformation energy induced by compression in the normal spring will not be counted in this criterion, because it is assumed that hydrostatic compression would not cause fracture in the homogeneous media. When the microcrack forms between two particles, tension and shear interactions are removed; however, repulsion and friction interactions exist, when the broken particles come

**Figure 1.** Particle interaction in the LSM model and schematic of the parameter mapping procedure.

The parameters used in the interaction formulae of LSM were usually given empirically, resulting in a qualitative representation of mechanical properties of target materials. Several outstanding studies have been done to overcome this shortcoming [14–16, 22–25]. Gusev proposed a parameter mapping procedure between finite-element method (FEM) and LSM [26]: consider a network that is both a LSM lattice and a FEM mesh; first, elastic constants of the target material are transformed into stiffness matrix of the FEM mesh; next, using the same network, the interaction-

To obtain the deformation state for the FEM mesh, the force-displacement equations assem-

{*F*} = [*K*]{*δ*} (1)

where {*F*} and {*δ*} are the respective column vectors formed from the external forces and displacements of all nodes. The so-called global stiffness matrix [*K*] is a sparse symmetric matrix, which is determined by elastic constants of the material and geometrical structure of the mesh. Under

Since motion of translation would not change the strain energy of the whole system, Eq. (3)

*<sup>i</sup>* = −*Fi* = −(*Ki*<sup>1</sup> × *δ*<sup>1</sup> + *Ki*<sup>2</sup> × *δ*<sup>2</sup> + ⋯ +*Kii* × *δ<sup>i</sup>* + ⋯ +*Kij* × *δ<sup>j</sup>* + ⋯ +*KiN* × *δN*) (2)

acting on node *i* can be written, according to Eq. (1), as

Shock Compression of Porous Ceramics http://dx.doi.org/10.5772/intechopen.72246 203

*i*

parameter conversion between FEM and LSM is performed (**Figure 1**).

bled from all elements need to be solved, that is,

the equilibrium state, the internal force *f*

holds between elements of the matrix [*K*] [26],

into collision.

*f*

**2.2. Parameter mapping procedure**

#### **2.1. Lattice-spring model**

A two-dimensional LSM was established to explore the shock behavior of porous ceramics. In the LSM, continuum medium is described as discrete material particles. The nearest neighboring particles are interconnected and interact through springs. Evolution of this network can represent the global response of macroscopic materials, if the interactions of material particles are described accurately. Through simplifications of real materials and the model's discrete nature, LSM has the advantage in treating fracture, fragmentation, and other dynamic damage processes of brittle materials subjected to tension, compression, shear, and other complex loading [17].

The model established here has an elastic-brittle interaction, which ignores the small plasticity contribution to the response that possibly exists in brittle materials; only a linear elastic interaction is used. Particle interaction is shown in **Figure 1**. Between pairs of nearest-neighbor particles, indexed by *i* and *j*, there are the central potential forces *f ij n* and the shear resistance forces *f ij τ* . They could be visualized as forces provided by a normal spring that lies along the normal direction and a shear spring that lies along the tangent direction.

An energy threshold based on Griffith's energy balance principle [21] has been used as the fracture criterion. The summation of the deformation energy induced by tension in the normal spring and shear in the shear spring is calculated when the relative position between two neighboring particles changes. And the two springs break irreversibly to create a microcrack between the two

**Figure 1.** Particle interaction in the LSM model and schematic of the parameter mapping procedure.

particles, when the sum exceeds a certain threshold corresponding to the fracture energy. The deformation energy induced by compression in the normal spring will not be counted in this criterion, because it is assumed that hydrostatic compression would not cause fracture in the homogeneous media. When the microcrack forms between two particles, tension and shear interactions are removed; however, repulsion and friction interactions exist, when the broken particles come into collision.

#### **2.2. Parameter mapping procedure**

Hence, a good understanding of the dynamic response of porous ceramics under rapid impulsive loading is vital to the design, manufacture, and usage of these materials. To this objective, a two-dimensional lattice-spring model (LSM) has been newly established, and the shock compression behavior of porous ceramics is explored and the mechanisms and strategies for

Dynamic response of porous ceramics under rapid impulsive loading relates to evolution of a crack network following the shock wave. Although some pioneer works have been conducted on modeling ceramic shock fracture via mesh-based computational methods [10–13], such methods encounter significant difficulties when dealing with fracture and fragmentation induced by shock wave compression. The reason is that partial derivatives are used in mesh-based methods to represent the relative displacement and force between any two neighboring particles [14]. But, the necessary partial derivatives with respect to the spatial coordinates are undefined along the cracks and need to be redefined. However, the redefinition requires us to know where the discontinuity is located. This limits the usefulness of these methods in addressing problems involving the spontaneous formation of cracks, in which one might not know their location in advance [14]. In contrast, as a particle method, the lattice-spring model (LSM, also known as discrete-element method) [15–20] could avoid various numerical difficulties caused by displacement discontinuity. In this section, details of the LSM model (lattice interactions, spring mapping procedure, fracture criterion, microstructures, loading) and its validation are introduced.

A two-dimensional LSM was established to explore the shock behavior of porous ceramics. In the LSM, continuum medium is described as discrete material particles. The nearest neighboring particles are interconnected and interact through springs. Evolution of this network can represent the global response of macroscopic materials, if the interactions of material particles are described accurately. Through simplifications of real materials and the model's discrete nature, LSM has the advantage in treating fracture, fragmentation, and other dynamic damage processes of brittle materials subjected to tension, compression, shear, and other complex loading [17].

The model established here has an elastic-brittle interaction, which ignores the small plasticity contribution to the response that possibly exists in brittle materials; only a linear elastic interaction is used. Particle interaction is shown in **Figure 1**. Between pairs of nearest-neighbor

An energy threshold based on Griffith's energy balance principle [21] has been used as the fracture criterion. The summation of the deformation energy induced by tension in the normal spring and shear in the shear spring is calculated when the relative position between two neighboring particles changes. And the two springs break irreversibly to create a microcrack between the two

. They could be visualized as forces provided by a normal spring that lies along the

*ij n*

and the shear resistance

particles, indexed by *i* and *j*, there are the central potential forces *f*

normal direction and a shear spring that lies along the tangent direction.

**2. Model of porous ceramics under shock wave compression**

improving robustness are discussed.

202 Recent Advances in Porous Ceramics

**2.1. Lattice-spring model**

forces *f ij τ* The parameters used in the interaction formulae of LSM were usually given empirically, resulting in a qualitative representation of mechanical properties of target materials. Several outstanding studies have been done to overcome this shortcoming [14–16, 22–25]. Gusev proposed a parameter mapping procedure between finite-element method (FEM) and LSM [26]: consider a network that is both a LSM lattice and a FEM mesh; first, elastic constants of the target material are transformed into stiffness matrix of the FEM mesh; next, using the same network, the interactionparameter conversion between FEM and LSM is performed (**Figure 1**).

To obtain the deformation state for the FEM mesh, the force-displacement equations assembled from all elements need to be solved, that is,

$$\{F\} \quad = \{K\} |\delta\rangle\tag{1}$$

where {*F*} and {*δ*} are the respective column vectors formed from the external forces and displacements of all nodes. The so-called global stiffness matrix [*K*] is a sparse symmetric matrix, which is determined by elastic constants of the material and geometrical structure of the mesh. Under the equilibrium state, the internal force *f i* acting on node *i* can be written, according to Eq. (1), as

$$f\_{\parallel} = -F\_{\parallel} = -\left(\mathbf{K}\_{\parallel} \times \delta\_{\mathbf{i}} + \mathbf{K}\_{\oplus} \times \delta\_{\mathbf{2}} + \cdots \ + \mathbf{K}\_{\parallel} \times \delta\_{\mathbf{i}} + \cdots \ + \mathbf{K}\_{\parallel} \times \delta\_{\mathbf{j}} + \cdots \ + \mathbf{K}\_{\mathbf{N}} \times \delta\_{\mathbf{N}}\right) \tag{2}$$

Since motion of translation would not change the strain energy of the whole system, Eq. (3) holds between elements of the matrix [*K*] [26],

$$\mathbf{K}\_{\rm il} = -\sum\_{\substack{\mathbf{x}=\mathbf{l} \\ \mathbf{x} \neq \mathbf{l}}}^{\rm N} \mathbf{K}\_{\rm ir} \tag{3}$$

*<sup>G</sup>*(*η*) <sup>=</sup> <sup>3</sup>*E*(*η*) \_\_\_\_\_\_\_\_\_\_\_ (<sup>9</sup> <sup>−</sup> *βeff*(*η*)*E*(*η*)) (7)

together with *βeff*(*η*) estimated from Eq. (6) and Young's moduli *E*(*η*) obtained from the simula-

the parameter-mapping procedure is verified as having the capability of representing elastic

To capture the influence of grain boundaries (GBs) on porous ceramics, polycrystalline sketching has been randomly produced using Voronoi tessellation [10]. As shown in **Figure 2(a)**, particles (small circles) in the model are assigned into grains (large polygons). If two particles connected by springs belong to different grains, then the springs are assumed to be a small segment of a GB. Given that media on GBs have higher energy state than media in grains, the deformation energy required for creating a pair of new crack surfaces on GBs is smaller

and *EGB* are the threshold in a grain and the additional energy that exists on GBs, respectively.

lines), which are much weaker than grains (blue media). A few GBs are low-angle GBs (green, yellow, and brown lines), which have various thresholds according to their relative angles.

Voids are set by removing portions of the model particles (**Figure 2(c)**). In the model, the balance distance between nearest neighbor particles is 1 μm, characteristic size of the grains is 10 μm, and the diameter of a round void is 50 μm. The length of the model along the shock direction is 1.6 mm. The model is illustrated schematically in **Figure 3**. A piston composed of two columns of particles is set on the left-hand side of the model; it moves with piston veloc-

) towards the right and produces a shock wave, which propagates from the left to the right. In order to reduce computational cost, periodic boundary conditions are applied on the upper and lower boundaries. Free boundary condition has been applied on the right side. At appointed simulation steps, evolution information such as particles' coordinates, velocities,

**Figure 2.** (a) Sketch of polycrystalline model. (b) Fracture energy set in the polycrystalline model. (c) Sketch of porous

extracted directly

205

estimated via Eq. (7) [17]. Thus,

Shock Compression of Porous Ceramics http://dx.doi.org/10.5772/intechopen.72246

*grain* − *EGB*, where *U<sup>S</sup>*

*grain*

*GB* = *U<sup>S</sup>*

in grains and GBs. Most GBs are high-angle GBs (red

tion, the shear moduli *G*(*η*) of the porous samples could be worked out. *G*/*G*<sup>0</sup>

from acoustic velocity tests are in good agreement with *G*/*G*<sup>0</sup>

**2.4. Microstructures and shock wave loading**

**Figure 2(b)** shows the distribution of *Us*

ity (*vp*

properties of both dense and porous brittle medium quantitatively.

than that in grains. The energy threshold on GBs is given as *U<sup>S</sup>*

stresses, springs' forces and connection states will be recorded.

ceramics. White circles are randomly distributed voids and small colored dots are grains.

Using *Kii*, Eq. (3) could be rearranged as

$$\begin{aligned} f\_i &= -\left[ \mathbf{K}\_{\text{in}} \times \boldsymbol{\delta}\_1 + \mathbf{K}\_{\text{in}} \times \boldsymbol{\delta}\_2 + \cdots \quad + \left( -\sum\_{\substack{\text{s}=1\\\text{s}\neq i}}^N \mathbf{K}\_{\text{in}} \times \boldsymbol{\delta}\_i \right) + \cdots \quad + \mathbf{K}\_{\text{ij}} \times \boldsymbol{\delta}\_j + \cdots \quad + \mathbf{K}\_{\text{in}} \times \boldsymbol{\delta}\_N \right] \\ &= -\left\{ \left[ \mathbf{K}\_{\text{in}} \times (\boldsymbol{\delta}\_1 - \boldsymbol{\delta}\_j) \right] + \left[ \mathbf{K}\_{\text{in}} (\boldsymbol{\delta}\_2 - \boldsymbol{\delta}\_j) \right] + \left[ \mathbf{K}\_{\text{ij}} (\boldsymbol{\delta}\_j - \boldsymbol{\delta}\_j) \right] + \left[ \mathbf{K}\_{\text{in}} (\boldsymbol{\delta}\_N - \boldsymbol{\delta}\_j) \right] \right\} \\ &= \sum\_{\substack{\mathbf{x}=1\\\mathbf{x} \neq i}}^N \mathbf{K}\_{\text{in}} (\boldsymbol{\delta}\_i - \boldsymbol{\delta}\_{\text{x}})^m \sum\_{\substack{\mathbf{x} \neq i}}^N f\_{i\mathbf{x}} \end{aligned} \tag{4}$$

The resultant internal force *f i* is the sum of the forces from all the neighbor particles (1, 2,…, *j*,…, *N*; *i* excluded). Hence, the internal force acting on particle *i* by particle *j* is

$$f\_{\parallel} = K\_{\parallel}(\delta\_{i} - \delta\_{\parallel}) = K\_{\parallel}\delta\_{\parallel} \tag{5}$$

where *δij* = *δ<sup>i</sup>* − *δ<sup>j</sup>* . Eq. (5) has the form of Hooke's law. The *Kij* could be taken as the stiffness coefficients of the springs of the LSM.

#### **2.3. Model validation**

In order to validate the parameter mapping procedure, dense and porous samples have been built and tested. Young's modulus, *E*<sup>0</sup> = 250 GPa; shear modulus, *G*<sup>0</sup> = 104 GPa; and density *ρ* = 5 × 103 kg/m3 are set into the lattice-spring networks of those samples. Samples with porosity 0, 2, 4, 6, 8, and 10% are subjected to quasi-static compression and tension. The maximum and minimum strains are 0.1 and −0.1%, respectively. Young's modulus of the dense sample is 251 GPa, which is in good agreement with the preset *E*<sup>0</sup> [17]. In porous samples, Young's modulus decreases with the porosity increasing.

Shear wave speeds (*Cs* ) of dense and porous samples have been obtained via acoustic velocity tests. Then, the shear modulus *G* = *ρCs* 2 could be calculated. For the dense sample, shear modulus is 105 GPa, which is almost the same with the preset *G*<sup>0</sup> [17]. As the porosity increases, shear modulus decreases. In rock physics, the elastic property of rock with spherical pores could be estimated from [27]

$$\mathcal{J}\_{\rm eff}(\eta) = \beta\_s \left( 1 + \frac{3(1 - \nu\_s)}{2(1 - 2\,\nu\_s)} \frac{\eta}{1 - \eta} \right) \tag{6}$$

where *β<sup>s</sup>* is the compression coefficient (the inverse of bulk modulus) of the dense medium, *βeff* the effective compression coefficient of the porous medium, *ν s* Poisson's ratio of the dense medium, and *η* porosity. With

$$\mathbf{G}(\eta) = \frac{3E(\eta)}{(\theta - \beta\_{\rm gf}\eta\eta E(\eta))}\tag{7}$$

together with *βeff*(*η*) estimated from Eq. (6) and Young's moduli *E*(*η*) obtained from the simulation, the shear moduli *G*(*η*) of the porous samples could be worked out. *G*/*G*<sup>0</sup> extracted directly from acoustic velocity tests are in good agreement with *G*/*G*<sup>0</sup> estimated via Eq. (7) [17]. Thus, the parameter-mapping procedure is verified as having the capability of representing elastic properties of both dense and porous brittle medium quantitatively.

#### **2.4. Microstructures and shock wave loading**

*Kii* = −∑

, Eq. (3) could be rearranged as

(*δ<sup>i</sup>* − *δ<sup>x</sup>* )=∑ *x*=1 *x*≠*i*

*Ki*<sup>1</sup> × *δ*<sup>1</sup> + *Ki*<sup>2</sup> × *δ*<sup>2</sup> + ⋯ +

*N f ix*

*i*

Using *Kii*

*f <sup>i</sup>* = − [

204 Recent Advances in Porous Ceramics

= ∑ *x*=1 *x*≠*i*

The resultant internal force *f*

where *δij* = *δ<sup>i</sup>* − *δ<sup>j</sup>*

*ρ* = 5 × 103 kg/m3

Shear wave speeds (*Cs*

estimated from [27]

and *η* porosity. With

where *β<sup>s</sup>*

**2.3. Model validation**

*f*

coefficients of the springs of the LSM.

built and tested. Young's modulus, *E*<sup>0</sup>

lus decreases with the porosity increasing.

*βeff* (*η*) <sup>=</sup> *<sup>β</sup>s*(<sup>1</sup> <sup>+</sup>

tests. Then, the shear modulus *G* = *ρCs*

251 GPa, which is in good agreement with the preset *E*<sup>0</sup>

lus is 105 GPa, which is almost the same with the preset *G*<sup>0</sup>

effective compression coefficient of the porous medium, *ν*

*N Kix*

*x*=1 *x*≠*i*

*Kix* × *δ<sup>i</sup>*

<sup>=</sup> <sup>−</sup> {[*Ki*<sup>1</sup> <sup>×</sup> (*δ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup><sup>i</sup>* )] <sup>+</sup> [*Ki*<sup>2</sup>

(−<sup>∑</sup> *x*=1 *x*≠*i*

*N*

(*δ*<sup>2</sup> − *δ<sup>i</sup>* )] + [*Kij*

*ij* = *Kij*(*δ<sup>i</sup>* − *δj*) = *Kij*

In order to validate the parameter mapping procedure, dense and porous samples have been

ity 0, 2, 4, 6, 8, and 10% are subjected to quasi-static compression and tension. The maximum and minimum strains are 0.1 and −0.1%, respectively. Young's modulus of the dense sample is

modulus decreases. In rock physics, the elastic property of rock with spherical pores could be

<sup>3</sup>(<sup>1</sup> <sup>−</sup> *<sup>ν</sup>s*) \_\_\_\_\_\_\_ 2(1 − 2 *νs*)

is the compression coefficient (the inverse of bulk modulus) of the dense medium, *βeff* the

= 250 GPa; shear modulus, *G*<sup>0</sup>

are set into the lattice-spring networks of those samples. Samples with poros-

) of dense and porous samples have been obtained via acoustic velocity

*<sup>η</sup>* \_\_\_

*s*

could be calculated. For the dense sample, shear modu-

*j*,…, *N*; *i* excluded). Hence, the internal force acting on particle *i* by particle *j* is

. Eq. (5) has the form of Hooke's law. The *Kij*

2

*Kix* (3)

]

(4)

*δij* (5)

[17]. In porous samples, Young's modu-

[17]. As the porosity increases, shear

<sup>1</sup> <sup>−</sup> *<sup>η</sup>*) (6)

Poisson's ratio of the dense medium,

could be taken as the stiffness

= 104 GPa; and density

) <sup>+</sup> <sup>⋯</sup> <sup>+</sup>*Kij* <sup>×</sup> *<sup>δ</sup><sup>j</sup>* <sup>+</sup> <sup>⋯</sup> <sup>+</sup>*KiN* <sup>×</sup> *<sup>δ</sup><sup>N</sup>*

(*δ<sup>j</sup>* − *δ<sup>i</sup>* )] + [*KiN*(*δ<sup>N</sup>* − *δ<sup>i</sup>* )]}

is the sum of the forces from all the neighbor particles (1, 2,…,

*N*

To capture the influence of grain boundaries (GBs) on porous ceramics, polycrystalline sketching has been randomly produced using Voronoi tessellation [10]. As shown in **Figure 2(a)**, particles (small circles) in the model are assigned into grains (large polygons). If two particles connected by springs belong to different grains, then the springs are assumed to be a small segment of a GB. Given that media on GBs have higher energy state than media in grains, the deformation energy required for creating a pair of new crack surfaces on GBs is smaller than that in grains. The energy threshold on GBs is given as *U<sup>S</sup> GB* = *U<sup>S</sup> grain* − *EGB*, where *U<sup>S</sup> grain* and *EGB* are the threshold in a grain and the additional energy that exists on GBs, respectively. **Figure 2(b)** shows the distribution of *Us* in grains and GBs. Most GBs are high-angle GBs (red lines), which are much weaker than grains (blue media). A few GBs are low-angle GBs (green, yellow, and brown lines), which have various thresholds according to their relative angles.

Voids are set by removing portions of the model particles (**Figure 2(c)**). In the model, the balance distance between nearest neighbor particles is 1 μm, characteristic size of the grains is 10 μm, and the diameter of a round void is 50 μm. The length of the model along the shock direction is 1.6 mm. The model is illustrated schematically in **Figure 3**. A piston composed of two columns of particles is set on the left-hand side of the model; it moves with piston velocity (*vp* ) towards the right and produces a shock wave, which propagates from the left to the right. In order to reduce computational cost, periodic boundary conditions are applied on the upper and lower boundaries. Free boundary condition has been applied on the right side. At appointed simulation steps, evolution information such as particles' coordinates, velocities, stresses, springs' forces and connection states will be recorded.

**Figure 2.** (a) Sketch of polycrystalline model. (b) Fracture energy set in the polycrystalline model. (c) Sketch of porous ceramics. White circles are randomly distributed voids and small colored dots are grains.

**Figure 3.** Schematic of the shock wave compression model for porous ceramics.
