**2.2 Experimental part**

At least six specimens were tested under each condition and then average values (KIC, KI and KII) were considered.

### *2.2.1 Bending test.*

The fracture toughness KIC of the samples were assessed using Semi Circular Bending tests. The samples were positioned on the loading platform by 3-point compressive loading, at a uniform loading speed of 0.075 mm/min (**Figure 2a**). The SCB specimen diameter is equal to 30 mm and 5 mm for thickness. The specimen contains a crack of 4 mm in the semi disc, as shown in **Figure 2b**. The crack-lengthto diameter ratio S/D was 0.13.

Using the SCB specimen with straight crack, the fracture toughness KIC was calculated with the following formula [19]:

$$K\_{IC} = \frac{P\_{\text{max}}\sqrt{\pi a}}{2Rt} Y\_I(a/R, \text{S/R}) \tag{1}$$

Where a is the crack length, Pmax is the maximum load, D is the cylindrical block diameter and YI is the geometry factor. The latter is a function of the ratio of the crack length (a) over the semi-disc radius (R) and the ratio of the half-distance between the two bottom supports (S) over the semi-disc radius (R) (**Figure 2b**). The geometry factor YI is expressed as follows [19]:

**Figure 2.** *Semi-circular bending (SCB): (a) real photo of SCB test (b) illustration of cracked SCB specimen.*

*Combination of Numerical, Experimental and Digital Image Correlation for Mechanical… DOI: http://dx.doi.org/10.5772/intechopen.99357*

$$Y\_I(a/R, S/R) = \frac{S}{R} \left( 2.91 + 54.39 \frac{a}{R} + 391.4 \left( \frac{a}{R} \right)^2 + 1210.6 \left( \frac{a}{R} \right)^3 - 1650 \left( \frac{a}{R} \right)^4 + 875 \left( \frac{a}{R} \right)^5 \right) \tag{2}$$

#### *2.2.2 Cracked straight-through Brazilian disc*

The introduction of the fracture mechanics approach to brittle materials has led to the development of materials fracture mechanics, which refers to the initiation and propagation of a crack or many cracks in materials.

According to the applied stress condition, a crack propagates depending on the three basic failure modes [20]: Mode I loading state is defined as opening mode, the mode II is defined as sliding mode (shear mode) and mode III is defined as tearing.

In this chapter, only mode I and II will be studied and detailed. Bioceramic stress intensity factor under modes I and II was measured using CSTBD specimens for an experimental and analytical investigation [21, 22]. Disc-type specimens are simple in geometry and have many advantages in terms of sample preparation, testing and analysis.

Different combinations of mode I and mode II can be shown by changing the crack angle β: if the direction of compressive applied load is along the crack bi-sector line β = 0, the samples is subjected to pure mode I loading. If β 6¼ 0, the samples are subjected to mixed mode I/II loading. A gradual increase of the loading angle results in an elevation in mode II effects and reduction in mode I effects. Finally, there are a specific loading angle β\_II for which the sample undergoes pure mode II deformation. This angle was found in this research by a series of finite element analyses [11].

The UMTS criterion is a criterion for brittle fracture is proposed by Ayatollahi [11] for prediction the mode II fracture toughness of U notched components and the fracture initiation angle in CSTBD under pure mode II loading.

The International Society for Rock Mechanics (ISRM) proposed many analytical formulas for measuring fracture toughness mode I of brittle materials: the cracked chevron notched Brazilian disc (CCNBD) specimens and the Cracked straight through Brazilian disc (CSTBD) [12, 23] (see **Figure 3a** and **b**). The CSTBD and CCNBD specimens has the same geometry and shape as the conventional Brazilian disc used for measuring the indirect tensile strength, except that the CSTBD specimen has a through notch length of 2a, by means of the straight-through crack assumption (STCA) method.

By comparing these two methods, CSTBD has superiority over CCNBD considering that producing a stream crack is easier than a V-shape crack. [14, 24]

**Figure 3.** *Disc-type specimens: (a) CSTBD and (b) CCNBD.*

The stress intensity factor (SIF) solutions for CSTBD specimens can be met in Cherepanov's book [25] and the handbook [26]. the main formulas to remember are collected as follows:

Cherepanov's book:

$$K\_I = \frac{P}{\sqrt{D} \, t} \, Y \tag{3}$$

$$Y = \sqrt{\frac{2}{\pi}} \sqrt{a} \left[ 1 + \frac{3}{2} a^2 + \frac{3}{4} a^6 + \frac{3}{64} a^8 \right] \tag{4}$$

The handbook:

$$K\_I = \frac{P}{\sqrt{D} \, t} \, \text{Y} \tag{5}$$

$$Y = \sqrt{\frac{2}{\pi}} \sqrt{\frac{a}{1 - a}} [1 - 0.4964 \, a + 1.5582 \, a^2 - 3.1818 \, a^3 + 10.096 \, a^4 - 20.7782 \, a^5]$$

$$+ 20.1342 \, a^6 - 7.5067 \, a^7$$

$$\text{(6)}$$

where *<sup>α</sup>* <sup>¼</sup> *<sup>a</sup> <sup>R</sup>* knowing that KI is the mode-I stress intensity factor, Y is the dimensionless stress intensity factor, P is the concentrated diametral compressive load, D is the diameter, t is the thickness and a is the crack length.

According to [27], the analytical solution of the stress intensity factor SIF for the CSTBD specimen for measuring the fracture toughness of ceramics can be expressed in the following form Shetty et al. [28]:

$$\mathbf{K}\_{I} = \frac{P}{\pi \, R \, t} \sqrt{\pi \, a} \mathbf{N}\_{I} = \frac{P}{\sqrt{\pi \, R} \, t} \sqrt{a} \mathbf{N}\_{I} \tag{7}$$

Where P is the load applied in compression, a is half the notch length and NI is the dimensionless stress intensity factor depending on the dimension less crack length α (a/R) and the notch inclination β.

NI solutions for the CSTBD sample can be determined by several methods:

Starting with Atkinson et al. [29] who has developed NI solutions for determining the fracture toughness and applied the stress intensity factor solutions of the CSTBD.

By small crack approximation (*α*≤0*:*3) and five-term approximation, NI was developed as the following formula:

$$N\_I = 1 - 4\sin^2\theta + 4\sin^2\theta (1 - 4\cos^2\theta)a^2 \tag{8}$$

With

$$a = \frac{a}{R}$$

when β = 0, the problem is reduced to theMode I fracture situation, then according to, the previously equation *NI* becomes [28]:

$$N\_I = 0.991 + 0.141 \,\text{a} + 0.863 \,\text{a}^2 + 0.886 \,\text{a}^3 \tag{9}$$

*Combination of Numerical, Experimental and Digital Image Correlation for Mechanical… DOI: http://dx.doi.org/10.5772/intechopen.99357*

Wherever, Fowell et al. [22] developed the formula on other form 0*:*05≤*α* ≤0*:*95:

$$\begin{aligned} \mathbf{N}\_I &= \sqrt{\frac{\pi}{a}} (0.0354 + 2.0394 \,\mathrm{a} - 7.0356 \,\mathrm{a}^2 + 12.1854 \,\mathrm{a}^3 + 8.4111 \,\mathrm{a}^4 - 30.7418 \,\mathrm{a}^5) \\ &- 29.4959 \,\mathrm{a}^6 + 62.9739 \,\mathrm{a}^7 + 66.5439 \,\mathrm{a}^8 - 82.1339 \,\mathrm{a}^9 - 73.6742 \,\mathrm{a}^{10} \\ &+ 73.8466 \,\mathrm{a}^{11}) \end{aligned} \tag{10}$$

As mentioned, different combinations of mode fracture can be obtained by changing the angle β. For while mode II, we can find the specific loading angle β, for which the specimen undergoes pure mode II deformation, by a series of finite element analyses. The mode II loading angle β was then determined from finite element results for the notch length that is already selected for mode I.

The stress intensity factor, for the CSTBD specimen with a through notch length of 2a, under mode II can be calculated with the following formula [28]:

$$K\_{\rm II} = \frac{P}{\pi \, R \, t} \sqrt{\pi \, a} \mathbf{N}\_{\rm II} = \frac{P}{\sqrt{\pi \, R} \, t} \sqrt{a} \mathbf{N}\_{\rm II} \tag{11}$$

$$\mathbf{N}\_{\rm II} = \left[\mathbf{2} + \left(\mathbf{8}\cos^2\theta - \mathbf{5}\right)a^2\right] \sin 2\theta \tag{12}$$

Where P is the load applied in compression, a is half the crack length and NII is the dimensionless stress intensity factor under mode II, depending on the dimensionless notch length α=a/R and the crack inclination angle with respect to loading direction, β (**Figure 4**).

In this case, NII was developed by [29].

#### *2.2.3 Digital correlation*

Nowadays, various full-field non-contact optical methods have been reported in literature and succeeded in replacing those classical techniques by Digital image correlation for strain and displacement measurements [16, 30]. The principle of DIC analysis is based on the comparison of the different successive digital images acquired during the test.

**Figure 4.** *CSTBD under pure mode II fracture.*

As mentioned in introduction, the experimental displacement was here computed by using Digital Image Correlation (DIC) in order to determinate the crack propagation at different states of loading and different composition. In the present work, DIC calculations have been managed with Correla software.

At each load step and at each composition, a series of images is taken with a CCD camera and digitalized and then compared to the reference image. For this technique, the displacement field analysis was performed inside of a Region of Interest (ROI) divided into discrete subsets. The shape (square or rectangular), the size (number of pixels) and the distribution (vertical and horizontal distances between centres, (Lx,Ly)) of these subsets should be carefully chosen. Those parameters depending to the desired accuracy of measurements (displacement and strain) and to the spatial resolution for map fields [16].

For each subset, a correlation function is used to estimate the degree of similarity between the reference image state and the current one (for each given load) [31].

Increasing the sub-set size allows decreasing the uncertainty because DIC error mainly depends on the number of pixels in thesubset [17]. The dedicated subsets were voluntary chosen with different scale factor (as shown on **Table 1**).

### **2.3 CDM model**

In this study, the FE simulations were performed by a constitutive model describing the mechanical behavior of brittle material is based on the CDM approach.

At first, The CDM approach was introduced by Kachanov [32] and generalized later by Le maitre and Chaboche [33]. It is a concept which provides a mathematical description of the effect of micro-defects and micro-cracks, at a macro-scale, on the macroscopic properties of the material. After that, the works of J. Ismail [15] show that even the mathematical formulation of this damage mechanics model that allows it to predict the cracking damage patterns in brittle materials.


#### **Table 1.** *Calibration scalor factor.*

*Combination of Numerical, Experimental and Digital Image Correlation for Mechanical… DOI: http://dx.doi.org/10.5772/intechopen.99357*

For an exact estimation of damage patterns, there is a clear need of constitutive equation of brittle material that is defined by:

$$
\sigma = k \ast \varepsilon \tag{13}
$$

where K is the fourth-order stiffness tensor which is written as

$$k = k^{\epsilon} + k^{d} \tag{14}$$

in which *k<sup>e</sup>* denotes the fourth-order stiffness tensor for the isotropic virgin material. *kd* is a fourth-order tensor which represents the added damage influence and the final expression is given by

$$\mathcal{M}\_{ijkl}^{d} = \mathbf{C1} \left( \delta\_{\vec{\eta}} D\_{kl} + \delta\_{kl} D\_{\vec{\eta}} \right) + \mathbf{C2} \left( \delta\_{jk} D\_{il} + \delta\_{il} D\_{jk} \right) \tag{15}$$

where *δ* is the Kronecker-delta symbol and C1 and C2 are the damage parameters.

The damage variables that expressed as functions of stress state is introduced into an anisotropic damage tensor Dij. Their values vary between 0 for virgin state and 1 for fully damaged (cracking) state.

Both damage patterns (mode I/mode II) are modeled by taking into consideration the effects of tensile principal stresses as well as compressive and shear stresses.

For the functioning of damage tensor, account should be taken of the effect of normal principal stress and those shear stress components in the damage mechanisms in brittle materials and the damage for both modes I and II is modeled.

• The first types of components (mode I)

This mode is involved by normal tensile principal stresses. The damage components representing are the diagonal terms of tensor K and their values are expressed in term of critical and threshold stress limits by:

$$\begin{aligned} \text{.1} \\\\ D\_{ii} = \left\{ \begin{aligned} 0 & \text{if } \sigma\_i \le \sigma\_t \\\\ \frac{\sigma\_i - \sigma\_t}{\sigma\_c - \sigma\_t} & \text{if } \sigma\_t < \sigma\_i < \sigma\_c \end{aligned} \right. \\\\ \text{-} \quad \text{1} \quad \text{if } \sigma\_i \ge \sigma\_c \end{aligned} \tag{16}$$

Where *σ<sup>c</sup>* and *σ<sup>t</sup>* are the critical and threshold stresses that corresponds to the stress below which no damage occurs.

• The second types of components (mode II)

In some cases, the shear mode (mode II) could be activated. Those components are formulated as a function of shear stress in the symmetry plane. The general form is:

$$D\_{\vec{\eta}} = \left\{ \begin{array}{c} \text{0} \quad \text{if } \sigma\_{\vec{\eta}} \le \tau\_t \text{ and } \max\left(\sigma\_i\right) > \mathbf{0} \\\\ \frac{\sigma\_{\vec{\eta}} - \tau\_t}{\tau\_c - \tau\_t} \quad \text{if } \tau\_t < \sigma\_{\vec{\eta}} < \tau\_c \text{ and } \max\left(\sigma\_i\right) < \mathbf{0} \\\\ \mathbf{1} \quad \text{if } \sigma\_{\vec{\eta}} \ge \tau\_c \text{ and } \max\left(\sigma\_i\right) > \mathbf{0} \end{array} \right. \tag{17}$$
 
$$i = 1, 2, 3 \text{ and } i \ne j$$

Where *τ<sup>c</sup>* and *τ<sup>t</sup>* are the critical and threshold shear parameters, respectively.

The damaged constitutive equations were coded in the Fortran programming language and implemented in the commercial FE code MSC. Marc to simulate the behavior and damage evolution in our materials.

The micro-cracks and micro-defects are an irreversible phenomenon. For this reason, the damage does not decrease during the loading and the Dij is taken as a monotonic increasing function of time increment:

$$D\_{\vec{\eta}} = \max\left(D\_{\vec{\eta}}^n, D\_{\vec{\eta}}^{n-1}\right) \tag{18}$$

Where *Dn ij* is the damage value at the current time step n and *D<sup>n</sup>*�<sup>1</sup> *ij* is the damage value at the previous time increment n-1.

A procedure to identify the model parameters must now be defined. The input parameters required are:


The material properties such as the ultimate tensile strength (*σt*Þ, the elastic modulus (E) and the Poisson's ratio (*μ*) for Al2O3-TCP were determined experimentally using the standard test techniques at room temperature.

#### **2.4 Finite element analysis**

In this section, the commercial FE code MSC. Marc was used to perform the simulations. A two-dimensional calculation has been performed using the finite element program MARC. A plane stress FE model with a total number of 8000 Quad 4 elements was created for simulating the specimen by moving two plates to effect compression on the disk. **Figure 5a** shows a sample FE grid pattern used for simulating a CSTBD specimen. The finest elements were located near the notch tip due to its high stress gradient (**Figure 5b**).

In order to determine the angle βII for which the sample undergoes pure mode II deformation, this angle was found in this research by a series of finite element analyses. Then, the values of the tangential and the shear stresses (*σ*12, *σ*22) along the notch bi-sector line could be obtained from the FE results In a Cartesian coordinate. In an auxiliary system of curvilinear coordinates, when *σθθ*ð Þ¼ *r*0, 0 0, the mode I is zero, and hence the specimen is subjected to pure mode II deformation. Therefore, the mode II loading angle *βII* is the angle for which *σθθ*ð Þ¼ *r*0, 0 0 [11].

*Combination of Numerical, Experimental and Digital Image Correlation for Mechanical… DOI: http://dx.doi.org/10.5772/intechopen.99357*

**Figure 5.** *A FE grid used for the simulations: (a) FE-mesh for the whole sample, (b) FE-mesh near the notch tip.*

After having used the matrix for passing from a Cartesian coordinate system to a cylindrical coordinate system, *σθθ* can write Eq. (19),

$$
\sigma\_{\theta\theta} = \sigma\_{22} \ast \cos\left(\theta\right) - \sin\left(\theta\right) \ast \sigma\_{12} \tag{19}
$$

In order to obtain the pure mode II loading angle *βII*, the angle b was gradually increased from zero and the value of tangential stress *σθθ*ð Þ *r*0, 0 at the notch tip was calculated for each loading angle, under a compressive load already found by the mechanical tests. As the loading angle increased, the value of *σθθ*ð Þ *r*0, 0 decreased until it was equal to zero.

In a second step, a CDM criterion for brittle materials has been introduced in the MARC-2005 to predict the mechanical behavior of our biomaterials subjected to a mechanicals test and this modeling was used to simulate the damage process. This combination has allowed to detect crack initiation and to analyze fracture process.

The mechanical properties were chosen to represent the composite specimens, for which elastic modulus and Poisson's ratio are (47.03; 75.96; 55.75; 46.86 and 33.51 GPa) and (0.283; 0.318; 0.361; 0.363 and 0.28), respectively for the variation of TCP.
