Stress-induced damage evolution in cast AlSi12CuMgNi alloy with one- and two-ceramic reinforcements. Part II: effect of reinforcement orientation

Abstract

While there is a large body of literature on the micromechanical behavior of metal matrix composites (MMCs) under uniaxial applied stress, very little is available on multi-phase MMCs. In order to cast light on the reinforcement mechanisms and damage processes in such multi-phase composites, materials made by an Al-based piston alloy and containing one- and two-ceramic reinforcements (planar-random oriented alumina fibers and SiC particles) were studied. In situ compression tests during neutron diffraction experiments were used to track the load transfer among phases, while X-ray computed tomography on pre-strained samples was used to monitor and quantify damage. We found that damage progresses differently in composites with different orientations of the fiber mat. Because of the presence of intermetallic network, it was observed that the second ceramic reinforcement changed the load transfer scenario only at very high applied load, when also intermetallic particles break. We rationalized the present results combining them with the previous investigations and using a micromechanical model.

Appendix

The components of Wu’s tensor $\varvec{\varGamma}$ for an isolated spheroidal inhomogeneity (with semi-axes $a_{1} = a_{2} = a,\;a_{3}$) of the aspect ratio $\gamma = {a \mathord{\left/ {\vphantom {a {a_{3} }}} \right. \kern-0pt} {a_{3} }}$, with bulk and shear moduli $K_{1}$ and $G_{1}$, embedded in the matrix with bulk and shear moduli $K_{0}$ and $G_{0}$, are as follows:

$$ \begin{aligned} \varGamma_{1111} & = \frac{1}{{2\Delta_{\varGamma } }}\left[ {1 + q_{6} \left( {K + \frac{4}{3}G} \right) + 2q_{4} \left( {K - \frac{2}{3}G} \right)} \right] + \frac{1}{{2\left( {1 + 2q_{2} G} \right)}}; \\ \varGamma_{1212} & = \frac{1}{{2\left( {1 + 2q_{2} G} \right)}} \\ \varGamma_{1133} & = - \frac{1}{{\Delta_{\varGamma } }}\left[ {2q_{1} \left( {K - \frac{2}{3}G} \right) + q_{3} \left( {K + \frac{4}{3}G} \right)} \right]; \\ \varGamma_{3311} & = - \frac{1}{{\Delta_{\varGamma } }}\left[ {2q_{4} \left( {K + \frac{1}{3}G} \right) + q_{6} \left( {K - \frac{2}{3}G} \right)} \right] \\ \varGamma_{3333} & = \frac{2}{{\Delta_{\varGamma } }}\left[ {\frac{1}{2} + 2q_{1} \left( {K + \frac{1}{3}G} \right) + q_{3} \left( {K - \frac{2}{3}G} \right)} \right]; \\ \varGamma_{1313} & = \frac{1}{{2\left( {1 + q_{5} G} \right)}} \\ \end{aligned} $$

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where the functions $f_{0}$ and $f_{1}$ are given by

$$ \begin{aligned} f_{0} & = \frac{1 - g}{{2\left( {1 - \gamma^{2} } \right)^{{}} }},\quad f_{1} = \frac{1}{{4\left( {1 - \gamma^{2} } \right)^{2} }}\left[ {\left( {2 + \gamma^{2} } \right)g - 3\gamma^{2} } \right],\quad \text{with} \\ g & = \left\{ {\begin{array}{*{20}l} {\frac{1}{{\gamma \sqrt {1 - \gamma^{2} } }}\text{arctan}\frac{{\sqrt {1 - \gamma^{2} } }}{\gamma },\quad (\gamma < \text{1})} \hfill \\ {\frac{1}{{\gamma \sqrt {\gamma^{2} - 1} }}\text{ln}\left( {\gamma + \sqrt {\gamma^{2} - 1} } \right),\quad (\gamma > \text{1})} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

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and the following notations are used

$$ \begin{aligned} q_{1} & = G_{0} \left[ {4\kappa - 1 - 2(3\kappa - 1)f_{0} - 2\kappa f_{1} } \right],\quad q_{2} = 2G_{0} \left[ {1 - (2 - \kappa )f_{0} - \kappa f_{1} } \right];\quad q_{5} = 4G_{0} \left( {f_{0} + 4\kappa f_{1} } \right) \\ q_{3} & = q_{4} = 2G_{0} \left[ {(2\kappa - 1)f_{0} + 2\kappa f_{1} } \right],\quad q_{6} = 8G_{0} \left( {\kappa f_{0} - \kappa f_{1} } \right), \\ \end{aligned} $$

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$$ \begin{aligned} K & = {{K_{1} K_{0} } \mathord{\left/ {\vphantom {{K_{1} K_{0} } {\left( {K_{0} - K_{1} } \right)}}} \right. \kern-0pt} {\left( {K_{0} - K_{1} } \right)}};\quad {{G = G_{1} G_{0} } \mathord{\left/ {\vphantom {{G = G_{1} G_{0} } {\left( {G_{0} - G_{1} } \right)}}} \right. \kern-0pt} {\left( {G_{0} - G_{1} } \right)}};\quad \kappa = {{\left( {3K_{0} + G_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {3K_{0} + G_{0} } \right)} {\left( {3K_{0} + 4G_{0} } \right)}}} \right. \kern-0pt} {\left( {3K_{0} + 4G_{0} } \right)}} \\ \Delta_{\varGamma } & = 2\left\{ {\left[ {\frac{1}{2} + 2q_{1} \left( {K + \frac{1}{3}G} \right) + q_{3} \left( {K - \frac{2}{3}G} \right)} \right]} \right.\;\left[ {1 + 2q_{4} \left( {K - \frac{2}{3}G} \right) + q_{6} \left( {K + \frac{4}{3}G} \right)} \right] \\ & \quad - \left. {\left[ {2q_{1} \left( {K - \frac{2}{3}G} \right) + q_{3} \left( {K + \frac{4}{3}G} \right)} \right]\left[ {2q_{4} \left( {K + \frac{1}{3}G} \right) + q_{6} \left( {K - \frac{2}{3}G} \right)} \right]} \right\}. \\ \end{aligned} $$

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The components of the transversely isotropic ($x_{3}$ is the axis of symmetry) tensor $Q^{\varOmega }$ entering (f) are as follows:

$$ \begin{aligned} Q_{1111}^{\varOmega } & = Q_{2222}^{\varOmega } = \mu \kappa \left( {4 - 5f_{0} - 3f_{1} } \right),\quad Q_{1122}^{\varOmega } = \mu \left[ {4\kappa - 2 + (4 - 7\kappa )f_{0} - \kappa f_{1} } \right] \\ Q_{3333}^{\varOmega } & = 8\mu \kappa \left( {f_{0} - f_{1} } \right),\quad Q_{1133}^{\varOmega } = 2\mu \left[ {(2\kappa - 1)f_{0} + 2\kappa f_{1} } \right],\quad Q_{1313}^{\varOmega } = \mu \left( {f_{0} + 4\kappa f_{1} } \right) \\ \end{aligned} $$

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where $f_{0}$ and $f_{1}$ are calculated for the aspect ratio of the domain $\varOmega$. In [42], the Maxwell scheme was completed expressing the aspect ratio of the domain $\varOmega$ in terms of the sums of components of tensors $Q_{ijkl}$ for individual inhomogeneities: for a composite with transversely isotropic microstructure, the effective inclusion is a spheroid with aspect ratio

$$ \gamma_{\varOmega } = {{\sum\limits_{i} {V_{i} Q_{3333}^{(i)} } } \mathord{\left/ {\vphantom {{\sum\limits_{i} {V_{i} Q_{3333}^{(i)} } } {\sum\limits_{i} {V_{i} Q_{1111}^{(i)} } }}} \right. \kern-0pt} {\sum\limits_{i} {V_{i} Q_{1111}^{(i)} } }} $$

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if it is smaller than 1 (case relevant to the present study). This hypothesis was numerically verified in [45].

The procedure of averaging of the transversely isotropic fourth-rank tensor $\varGamma_{ijkl}$ over orientations has been discussed in detail in [34]. In the case of 3-D random orientation, the resulting tensor is isotropic and has only two independent components

$$ \begin{aligned} \bar{\varGamma }_{1111} & = \frac{1}{15}\left( {8\varGamma_{1111} + 2\varGamma_{1133} + 2\varGamma_{3311} + 8\varGamma_{1313} + 3\varGamma_{3333} } \right) \\ \bar{\varGamma }_{1122} & = \frac{1}{15}\left( {\varGamma_{1111} + 5\varGamma_{1122} + 4\varGamma_{1133} + 4\varGamma_{3311} - 4\varGamma_{1313} + \varGamma_{3333} } \right). \\ \end{aligned} $$

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The tensor $\varGamma_{ijkl}$ averaged over 2D random orientations is transversely isotropic with the axis of symmetry normal to the plane of random orientations. The resulting tensor has six independent components (note that stress concentration tensor is not symmetric and $\bar{\varGamma }_{1133} \ne \bar{\varGamma }_{3311}$):

$$ \begin{aligned} \bar{\varGamma }_{1111} & = \frac{1}{8}\left( { - \varGamma_{1111} + 4\varGamma_{1122} + 5\varGamma_{1133} + 5\varGamma_{3311} + 12\varGamma_{1313} + 3\varGamma_{3333} } \right); \\ \bar{\varGamma }_{3333} & = \varGamma_{1111} \\ \bar{\varGamma }_{1212} & = \frac{1}{8}\left( { - 3\varGamma_{1111} + 4\varGamma_{1122} - \varGamma_{1133} - \varGamma_{3311} + 4\varGamma_{1313} + \varGamma_{3333} } \right); \\ \bar{\varGamma }_{1313} & = \varGamma_{1313} \\ \bar{\varGamma }_{1133} & = \frac{1}{4}\left( {\varGamma_{1111} + \varGamma_{1122} + 4\varGamma_{3311} } \right); \\ \bar{\varGamma }_{3311} & = \frac{1}{4}\left( {\varGamma_{1111} + \varGamma_{1122} + 4\varGamma_{1133} } \right). \\ \end{aligned} $$

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