## 3 Methodology for Measuring Damage in a Metal Matrix Composite and Discussion of Results
### Damage Measurement Technique
The reduced elastic modulus \(E^{*}\) is calculated from the results of instrumented indentation, which is related to the elastic modulus \(E\) by the following formula (ISO 14577-1-2002):  
\[E=\frac{1-\left(\mathrm{v_{s}}\right)^{2}}{\frac{1}{E^{*}}-\frac{1-\left( \mathrm{v_{s}}\right)^{2}}{E_{i}}} \tag{2}\]  
where \(\mathrm{v_{s}}\) is [PERSON]'s ratio for the test specimen; \(\mathrm{v_{i}}\) is [PERSON]'s ratio for the indenter, which is equal to 0.07 for diamond (ISO 14577-1-2002); \(E^{*}\) is the reduced elastic modulus of the test specimen; \(E_{i}\) is the elastic modulus of the indenter, which is equal to \(1.14\cdot 10^{6}\) N/mm  
Figure 11: The strain (\(\varepsilon\)) dependence of the reduced elastic modulus \(E^{*}\) for the V95\(\backslash\)10% SiC MMC specimen at 300, 400 and 500 \({}^{\circ}\)C, the dependence being valid for the zone near the side surface of the specimen (see Figure 2). for diamond (ISO 14577-1-2002). Using the relation between the elastic modulus and the reduced elastic modulus represented by Equation (2) and then substituting it into Equation (1), we obtain the following formula for calculating material damage:  
\[D_{E}=\frac{E_{i}\cdot\left(E_{i\mu}^{*}-E_{cur}^{*}\right)}{E_{i\mu}^{*}\cdot \left(E_{i}+\left(E_{cur}^{*}\cdot\mathbf{v}_{i}^{2}-E_{cur}^{*}\right)\right)} \tag{3}\]  
According to the technique described above, the \(D_{E}-\varepsilon\) dependencies for the V95\(\backslash\)10% SiC metal matrix composite deformed at temperatures of 300, 400, and 500 \({}^{\circ}\)C were obtained for the zone near the side surface of the specimen (see Figure 12). The last point on each \(D_{E}-\varepsilon\) curve is marked with a cross; it corresponds to the state of the specimen with a macrocrack detected on the side surface, and this corresponds to damage equal to 1. Figure 12 shows that the beginning of damage accumulation in the composite belongs to the strain range from 0.2 to 0.3 and increases with deformation temperature. The results of damage accumulation indicate that the slowest damage accumulation occurs at 300 \({}^{\circ}\)C, that it is more rapid at 400 \({}^{\circ}\)C and the most rapid at 500 \({}^{\circ}\)C. The history of changes in the stress state is one of the factors affecting damage accumulation. In this paper, the stress state coefficient \(k\) and the Lode-Nadai coefficient \(\upmu_{\sigma}\) are used as the stress state indices. They are calculated by the following formulas:  
\[k=\frac{\sigma}{T}\text{ and }\upmu_{\sigma}=2\frac{\upsigma_{22}-\upsigma_{33} }{\upsigma_{11}-\upsigma_{33}}-1,\]  
where \(\sigma=\frac{1}{3}(\upsigma_{11}+\upsigma_{22}+\upsigma_{33})\) is the average normal stress; \(T=\sqrt{0.5\upsigma_{ij}^{\prime}\upsigma_{ij}^{\prime}}\) is the intensity of tangential stresses; \(\upsigma_{ij}^{\prime}\) are the stress deviator components; \(\upsigma_{11}\), \(\upsigma_{22}\), \(\upsigma_{33}\) are principal stresses. The stress state indices are calculated from finite element method computer simulation of specimen compression under conditions of the axisymmetric stress-strain state (see Section 2). Figure 13a,b show changes in the stress state indices \(k\) and \(\upmu_{\sigma}\) on the side surface of the specimens during testing at three test temperatures. The difference from the values \(k=-\frac{1}{\sqrt{3}}\) and \(\upmu_{\sigma}=+1\), which are typical for uniaxial compression conditions, is caused by the curvature of the side surface of the specimens under compression due to the action of friction forces on the contact surfaces. It is obvious from these figures that, even though the most favorable conditions of the stress state at the time of macrocracking on the lateral surface of the specimen exist at 500 \({}^{\circ}\)C, the composite demonstrates significantly lower plasticity at this temperature. At this temperature, the strain to fracture on the side surface of the specimen is the lowest and equal to 0.63, and it is 0.78 for test temperatures of 300 and 400 \({}^{\circ}\)C (see Figure 13c).