**4. Discussion**

## *4.1. Strengthening Mechanisms*

To further understand the strengthening mechanism of the laser-treated Al–Al2Cu–Si and Al–Al2Cu eutectics, it is essential to consider the roles of interfaces (soft Al phase and hard Al2Cu or Si phase) when the interspacing is refined to nanometer size. In this case, the empirical rule of the mixture is invalid. In this work, the volume fractions of each phase in the studied eutectics are the same even though the length scale has been changed.

A classical constitutive model of the composite structure containing the soft phase and the hard phase could be written as follows:

$$
\sigma\_{\mathfrak{m}} = \mathbb{E}\_{\mathfrak{m}} \cdot \varepsilon\_{\mathfrak{m}'} \; \sigma\_{\mathfrak{m}} < \sigma\_{\mathfrak{m}}^{\text{yield}} \tag{1}
$$

$$
\sigma\_m = \mathbf{E}\_m \cdot \varepsilon\_m + \mathbf{K}\_m \cdot \varepsilon\_{m\prime} \ \sigma\_m \stackrel{\text{yield}}{\sim} \sigma\_m^{\text{yield}} \tag{2}
$$

where the subscript (m) infers to either soft phase (s) or hard phase (h). Here, the uniform stress at the ends of the hard phase, and the soft phase is unaltered with the displacement. It is important to convert the stress and strain into a more fundamental quantity (e.g., energy density). For individual phase, the total work density can be expressed by integrating the area under the stress-strain curve as follows:

$$\mathcal{W}\_{\rm m} = \frac{1}{2} \frac{\left(\sigma\_{\rm m}^{\rm yield}\right)^2}{E\_{\rm m}} + \sigma\_{\rm m}^{\rm yield} \cdot \left(\frac{\sigma\_{\rm m}^{\rm yield}}{E\_{\rm m}} - \varepsilon\_{\rm max}\right) + \int\_{\frac{\sigma\_{\rm m}^{\rm yield}}{E\_{\rm m}}}^{\varepsilon\_{\rm max}} \left[K\_{\rm m}(\varepsilon\_{\rm m})^{u\_{\rm m}}\right] - \sigma\_{\rm i}^{\rm yield} \cdot \left|\cdot d\varepsilon - \frac{1}{2} \frac{P\_{\rm max}}{A} \left(\varepsilon\_{\rm max} - \varepsilon\_{\rm pi}\right)\right| \tag{3}$$

where the subscript (m) infers to either the soft phase (s) or the hard phases (h). *ε*p, *ε*max, and *P*max are the plastic parts of the true-strain and maximum true-strain induced in the experiment and the corresponding maximum indentation load, respectively. We hypothesize that the energies of interaction at the lamellar interfaces (soft Al and hard Al2Cu or Si) play significant roles in the tensile test's deformation process. The total work density of the composites (W) can be divided into a plastic part and an elastic part, with respective interfacial energy terms (denoted by superscript "interface"), as follows:

$$\overline{\mathcal{W}} = \left[ \mathcal{W}\_{\text{e}}^{\text{h}} + \mathcal{W}\_{\text{e}}^{\text{s}} + \mathcal{W}\_{\text{e}}^{\text{interface}} \right] + \left[ \mathcal{W}\_{\text{P}}^{\text{h}} + \mathcal{W}\_{\text{P}}^{\text{s}} + \mathcal{W}\_{\text{P}}^{\text{interface}} \right] \tag{4}$$

The elastic part of the energy of the effective composite equates to the total elastic energy, since there is no other phase combination, therefore a corresponding equation can be written as follows:

$$\frac{1}{2} \cdot \mathbb{E} \cdot \varepsilon^2 (\mathbf{h}\_{\text{s}} + \mathbf{h}\_{\text{h}}) = \frac{1}{2} E\_{\text{m}} \varepsilon^2 h\_{\text{s}} + \frac{1}{2} E\_{\text{f}} \varepsilon^2 h\_{\text{h}} + \mathcal{W}\_{\text{e}}^{\text{interface}}, \ \varepsilon \ll \frac{\sigma\_{\text{i}}^{\text{yield}}}{E\_{\text{i}}} \tag{5}$$

where *h*<sup>s</sup> is the thickness of the soft phase (Al) and *h*<sup>h</sup> is the thickness of the hard phase (Al2Cu or Si), and *W*interface <sup>e</sup> is the elastic energy contribution of the interface. The interlamellar spacing, λ=(*h*1+*h*2)/2, becomes the function of the lamellar geometry. The plastic part of the energy of the composite can be equated with total plastic energy due to the pure phase combination, therefore a corresponding equation can be written as follows:

$$\left[\boldsymbol{\upsigma}^{\text{yield}}\boldsymbol{\varepsilon\_{\text{c}}} + \int\_{\varepsilon\_{\text{f}}}^{\varepsilon} \mathbb{K} \boldsymbol{\varepsilon} \mathbb{H} d\varepsilon \right] \left(h\_{1} + h\_{2}\right) \cong \left[\boldsymbol{\upsigma}^{\text{yield}}\_{\text{m}}\boldsymbol{\varepsilon\_{\text{c}}^{\text{m}}} + \int\_{\varepsilon\_{\text{f}}}^{\varepsilon} \mathbb{K}\_{\text{m}} \left(\boldsymbol{\varepsilon\_{\text{m}}}\right)^{\text{n}^{\text{m}}} d\varepsilon \right] h\_{1} + \left[\boldsymbol{\upsigma}^{\text{yield}}\_{\text{f}}\boldsymbol{\varepsilon\_{\text{c}}^{\text{f}}} + \int\_{\varepsilon\_{\text{f}}}^{\varepsilon} \mathbb{K}\_{\text{f}} \left(\boldsymbol{\varepsilon\_{\text{f}}}\right)^{\text{n}^{\text{f}}} d\varepsilon \right] h\_{2} + \boldsymbol{\upmu}\_{\text{P}}^{\text{interface}} \tag{6}$$

Combining the above two energy equations as described in Equation (4), a total energy balance is obtained. This allows us to evaluate the effect of the length scale of microstructure as expressed through *k* on the mechanical properties. Therefore, with the knowledge of inter-lamellar spacing *k* and the properties of the composite and pure phase, one can estimate the contribution of the interface to the overall elastic energy balance.

In order to verify the above relationships, the Al–Al2Cu+Si eutectic was selected as an example. This eutectic comprises alternate plates of soft phase (Al) and hard phase (Al2Cu and Si) intermetallic phases due to the variation in the lamellar spacing with the cooling rate. The mechanical properties of localized regions were further evaluated from a load-displacement curve obtained by the use of micropillar compression technique. The measured true-stress versus true-strain curves are shown in Figure 2.
