### Materials

The materials used in this work are glass/epoxy unidirectional laminates. Our test specimen is a rectangular plate following proportioning: fiber 50%, epoxy resin 37.5% and 12.5% of hardener. The “Technique of Elaboration” consists of applying successively into a mould surface, a layer of resin (epoxy) a layer of reinforcement (glass fiber) and to impregnate the reinforcement by hand with the aid of a roller [15]. The stacking sequence consisted of 12 layers. Polymerization is achieved at room temperature during approximately 12 h period under pressure. After that, this plate is cut using a diamond wheel saw. The specimens obtained are 95 mm in length, and 30 mm in width with 4 mm of thickness. The type of samples is a 0°off-axis unidirectional. To evaluate the influence of the size of the body incorporated on the mechanical behavior of material, one metal patch in the form of disc is integrated within each test-beam, so we obtained tow test specimens: one simple laminate beam (SLB) and other incorporated laminate beam (ILB) with patch.

### Optical set-up

The experimental set-up adapted to the investigation of the three point flexural loading, is described in Fig. 1. It uses three continuous diode-pumped solid-state lasers (red line R at *λ*
_{
R
} = 671 nm, green line G at *λ*
_{
G
} = 532 nm, and blue line B at *λ*
_{
B
} = 457 nm). The color sensor consists of a color camera made up of three stacked layers of photodiodes with a single 8 bit per channel digital output, including (M, N) = (1060, 1420) pixels with pitches *p*
_{
x
} = *p*
_{
y
} = 5 μm. Each collimated laser beam illuminates the object under interest with *θ*
_{
R
}
*, θ*
_{
G
}, and *θ*
_{
B
} angles, respectively for the red, green, and blue lines leading to the three sensitivities. The R and G beams are included in the {x, z} plane, whereas the B beam is included in the {y, z} plane. The three beams are separated in reference and object beams by cubes PBS1, PBS2 and PBS3. The R and B reference beams are combined into an unique beam thanks to the use of a dichroic plate, which it reflects the B beam and transmits the R beam, while G is adjusted independently of the two other beams.

The smooth plane reference wave G is produced through the SF1, which includes an achromatic lens although the RB reference waves are, produced the SF2. Thus, the two reference beams RB impact the sensing area with the same incidence angle to produce spatial frequencies (*u*
_{
R
} = −28.54; *υ*
_{
R
} = −51.34) mm^{−1} of the red hologram and (*u*
_{
B
} = −41.91; *υ*
_{
B
} = −75.39) mm^{−1} of the Blue hologram. The green reference beam is regulated with an incidence angle leading to spatial frequencies (*u*
_{
G
} = 45.82; *υ*
_{
G
} = −30.73) mm^{−1} of green hologram.

The reconstructing horizon is chosen equal K × L = 1024 × 1024 data points.

The reconstruction algorithm follows the method based on convolution method with an adjustable magnification [16]. The method is based on the image locations and magnification relations of holography when the illuminating beam is a spherical wave front. The curvature radius of the spherical wave front is *R*
_{
c
} with the transversal magnification between the reconstructed object and the real one is given by:

$$ \gamma =-\frac{d_r}{d_0} $$

(1)

Thus the reconstruction distance *d*
_{
r
} depends on *R*
_{
c
} and the curvature of the spherical wave, as shown in this relation:

$$ {d}_r=-{\left(\frac{1}{d_0}+\frac{1}{R_c}+\frac{1}{d_s}\right)}^{-1} $$

(2)

The transversal magnification and the number of points of the algorithm are linked by this expression.

$$ \left\{K,L\right\}=\left|\gamma \right|\left\{\frac{\varDelta {A}_x}{p_x},\frac{\varDelta {A}_x}{p_x}\right\} $$

(3)

According to the potentialities of this strategy, the transverse magnification that must be put into the algorithm is *γ* = 0.16, the curvature radius of the numerical spherical reconstructing wave is *R*
_{
c
} *= γd*
_{
0
}
*/ (γ-1)* = −157.3481 mm; and the effective reconstruction distance is *d*
_{
r
} *= −γd*
_{
0
} = −132.27 mm, for each wavelength.

The object is illuminated from three directions, which produces three sensitivity vectors and, subsequently, a three-sensitivity measurement. The relation between the displacement vector **U** = *u*
_{
x
}
**i** + *u*
_{
y
}
**j** + *u*
_{
z
}
**k** and the illuminating geometry is \( \varDelta {\varphi}_{\lambda }=2\pi \mathbf{U}.\left({\mathbf{K}}_e^{\lambda }-{\mathbf{K}}_{\mathbf{o}}\right)/\lambda \) for each wavelength, where \( {\mathbf{K}}_e^{\lambda }=-\mathit{\cos}{\theta}_{yz}^{\lambda}\mathit{\sin}{\theta}_{xz}^{\lambda}\mathbf{i}-\mathit{\sin}{\theta}_{yz}^{\lambda}\mathbf{j}-\mathit{\cos}{\theta}_{yz}^{\lambda}\mathit{\cos}{\theta}_{xz}^{\lambda}\mathbf{k} \) is the illumination vector and **K**
_{
o
} ≅ **k** is the observation vector.

Using three wavelengths with three different lighting directions, we obtain a matrix relationship between the monochrome phase changes measured from the holograms and each component of the 3D displacement field.

$$ \left(\begin{array}{l}{\lambda}_1\varDelta {\phi}_{\lambda_1}\\ {}{\lambda}_2\varDelta {\phi}_{\lambda_2}\\ {}{\lambda}_3\varDelta {\phi}_{\lambda_3}\end{array}\right)=2\pi\;\mathbf{A}\left(\begin{array}{l}{u}_x\\ {}{u}_y\\ {}{u}_z\end{array}\right) $$

(4)

The inversion of the matrix leads to the determination of the 3D displacement vector.

$$ \left(\begin{array}{l}{u}_x\\ {}{u}_y\\ {}{u}_z\end{array}\right)=\frac{1}{2\pi }{A}^{-1}\left(\begin{array}{l}{\lambda}_1\varDelta {\phi}_{\lambda_1}\\ {}{\lambda}_2\varDelta {\phi}_{\lambda_2}\\ {}{\lambda}_3\varDelta {\phi}_{\lambda_3}\end{array}\right) $$

(5)

To simplify the notation, we note: *θ*
_{
R
} *= θ*
^{R}
_{
xz
}
*, θ*
_{
G
} *= θ*
^{G}
_{
xz
} et *θ*
_{
B
} *= θ*
^{B}
_{
yz
}
*.* In the setup, we have:

$$ {\displaystyle \begin{array}{l}{K}_e^R=\sin {\theta}_Ri-\cos {\theta}_Rk\\ {}{K}_e^G=-\sin {\theta}_Gi-\cos {\theta}_Gk\\ {}{K}_e^B=-\sin {\theta}_Bi-\cos {\theta}_Bk\end{array}} $$

(6)

According to Eq. (5) the calculation of the three components of the displacements field is given by the following relation:

$$ \left(\begin{array}{c}\hfill {u}_x\hfill \\ {}\hfill {u}_y\hfill \\ {}\hfill {u}_z\hfill \end{array}\right)=\frac{1}{2\pi \alpha}\left(\begin{array}{ccc}\hfill 1+\cos \left({\theta}_G\right)\hfill & \hfill -1-\cos \left({\theta}_G\right)\hfill & \hfill 0\hfill \\ {}\hfill \sin \left({\theta}_G\right)\left(1+\cos \left({\theta}_B\right)\right)/\sin \left({\theta}_B\right)\hfill & \hfill \sin \left({\theta}_R\right)\left(1+\cos \left({\theta}_B\right)\right)/\sin \left({\theta}_B\right)\hfill & \hfill -\alpha /\sin \left({\theta}_R\right)\hfill \\ {}\hfill -\sin \left({\theta}_G\right)\hfill & \hfill -\sin \left({\theta}_R\right)\hfill & \hfill 0\hfill \end{array}\right)\times \left[\begin{array}{c}\hfill {\lambda}_R\varDelta {\varphi}_R\hfill \\ {}\hfill {\lambda}_G\varDelta {\varphi}_G\hfill \\ {}\hfill {\lambda}_B\varDelta {\varphi}_B\hfill \end{array}\right] $$

(7)

where *α* = sin (*θ*
_{
R
}) (1 + cos (*θ*
_{
G
})) + sin (*θ*
_{
G
}) (1 + cos (*θ*
_{
R
})) and *Δφ*
_{
R
}
*, Δφ*
_{
G
}
*,* and *Δφ*
_{
B
} are the optical phase changes between two deformation states, which are obtained from the reconstructed holograms, respectively, for the R, G, and B beams. In the setup, the angles are adjusted to *θ*
_{
R
} = 30.17, *θ*
_{
G
} = 11.80, and *θ*
_{
B
} = 46.27. Figure 2a indicates the illuminating geometry and Fig. 2b is a photograph illustrates the “white” illumination of the sample by the three laser wavelengths.