## 3 The Plastic Constitutive Equation of 6061 Al  
### Constitutive Model of 6061 Al Matrix  
Particle reinforced SiCP/6061 Al composites are not homogeneous materials from a microscopic point of view. Due to the introduction of the particle-reinforced phase, the microstructure and mechanical properties of the 6061 Al matrix material change greatly. The strengthening mechanism mainly includes fine grain strengthening, dislocation strengthening, and dispersion strengthening. Therefore, it is of great significance to study the deformation and fracture process of the SiC/6061 Al composites by the finite element method. [PERSON] et al. [23] proposed a reverse analysis algorithm for obtaining the elasticity and plasticity from the load-displacement curve obtained from the nanoindentation experiments. The inverse analysis algorithm assumes that the plastic behavior of the metal can be described by Equation (1):  
\[\sigma=\sigma_{y}\bigg{(}1+\frac{E}{\sigma_{y}}\varepsilon_{p}\bigg{)}^{n}, \tag{1}\]  
where \(\sigma_{y}\) is the yield stress, \(\varepsilon_{p}\) is the plastic strain, and \(n\) is the strain hardening exponent. \(E\) is the Young's modulus. The load-depth (_P-It_) response of elastoplastic materials to sharp indentations follows [PERSON]'s law:  
\[P_{ave}=\frac{P_{m}}{A_{m}}, \tag{2}\]  
where \(A_{m}\) is the indentation projection area corresponding to the maximum load, and \(P_{m}\) is the maximum pressure applied by the indenter. The hardness is defined as the average contact pressure:  
\[P=Cl^{2} \tag{3}\]  
where \(C\) is the loading curvature, that is, the slope of the depth load curve in the loading stage in Figure 1a. \(h\) is the indentation depth, and \(P\) is the pressure. The characteristic Young's modulus \(E^{*}\) used in the calculation of the mechanical properties of the tested sample is defined as:  
\[E^{*}=\bigg{[}\frac{1-v^{2}}{E}+\frac{1-v_{i}^{2}}{E_{i}}\bigg{]}^{-1}, \tag{4}\]  
where \(E_{i}\) is the [PERSON]'s modulus of the indenter, and \(v_{i}\) is the [PERSON]'s ratio of the indenter. For the diamond indenter, \(E_{i}\) = 1000 GPa, and \(v_{i}\) = 0.07 [23]. \(E\) and \(v\) are the [PERSON]'s modulus and [PERSON]'s ratio of the measured material, respectively. Figure 1a shows the loading versus displacement curves of the matrix 6061 aluminium alloy with volume fractions of 5%, 10%, and 15%, and they were basically coincident with each other (the difference of the feature points was within 0.02 \(\upmu\)m). In order to reduce the difficulty of calculation, the loading versus displacement curve with a volume fraction of 10% was used to establish the plasticity constitutive model of the matrix 6061 aluminium alloy. At the same time, in order to further improve the accuracy of the measurement results, the values of C (loading curvature), \(h_{r}/h_{m}\) (ratio of residual indentation depth to maximum indentation depth), \(P_{ave}\) (hardness), \(h_{m}\) (Maximum depth of indentation), and other parameters were obtained from the loading-displacement curves of different indentation points of the SiCp/6061 Al composite matrix with a volume fraction of 10%, and the average value was obtained. The final values of each parameter are shown in Table 1.  
The dimensionless function (\(\pi\) function) in this paper is listed in Equations (5)-(8):  
\[C=\sigma_{0.033}\Pi_{1}\bigg{(}\frac{E^{*}}{\sigma_{0.033}}\bigg{)}, \tag{5}\]  
\[\frac{1\,dp}{E^{*}h_{m}dh_{h_{m}}}=\Pi_{2}\bigg{(}\frac{E^{*}}{\sigma_{0.033}}, n\bigg{)}, \tag{6}\]  
\[\frac{P_{ave}}{E^{*}}=\Pi_{4}\bigg{(}\frac{h_{r}}{h_{m}}\bigg{)}, \tag{7}\]  
\[\Pi_{6}=\frac{1\,dp}{E^{*}\sqrt{A_{m}}dh_{h_{m}}}=C^{*}, \tag{8}\]  
\[\sigma_{0.033}=\sigma_{y}\bigg{(}1+\frac{E}{\sigma_{y}}0.033\bigg{)}^{n}. \tag{9}\]  
The detailed expression of each \(\pi\) function is given in the appendix of Dao [23]. For the Berkovich indenter, \(C^{*}\) is 1.2370 [21].