# Downscaled Finite Element Modeling of Metal Targets for Surface Roughness Level under Pulsed Laser Irradiation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methodology

## 3. Numerical Modeling

_{p}, k are the mass density, specific heat at constant pressure, and the thermal conductivity of the target material, respectively, and L

_{i}is the latent heat of melting, which is equal to zero in the thermoelastic regime. The source term, Q(x, y, z, t), has an elliptic spatial form and represents the absorbed laser energy per unit volume per unit time by the sample:

_{0}is the incident laser intensity on target, R is the optical reflectivity of the sample, α

_{b}is the optical absorption coefficient, t

_{0}is the laser pulse duration at full width at half maximum (FWHM), r

_{α}is the semi-major axis and r

_{b}is the semi-minor axis of the laser spot, x

_{o}and y

_{o}are the center coordinates, and θ is the rotation angle. The conservative equations of mass, momentum, and energy are also solved:

_{Τ}is the thermal expansion coefficient, E is the energy, and σ

_{ij}is the stress tensor. The strain tensor is given by:

_{ij}is the deviatoric stress component. The hydrostatic component of stress is associated to the pressure in the material, which is equal to the trace of the complete stress tensor

_{0}is the sound speed, γ

_{0}the unitless, Grüneisen parameter that defines the effect on the atom’s vibration due to the change in energy and a is the unitless, first-order volume correction to γ

_{0}; while ${\mu}_{v}$ is a volumetric parameter that holds ${\mu}_{v}$ = ρ/ρ

_{0}−1. For most of the materials, shock velocity U

_{s}varies linearly, in relation to the particle velocity, U

_{p}, as U

_{s}= C

_{0}+ S

_{1}U

_{p}, where S

_{1}is the unitless coefficient of the slope of the U

_{s}-U

_{p}curve. For expanded materials the following holds:

^{−1}is a reference strain rate used to normalize the strain rate, T

_{m}is the melting temperature of the workpiece, T

_{r}is the room temperature, and m is the thermal softening exponent. The J–C material model also adopts a fracture model that considers the nucleation, growth, and coalescence of voids in a ductile material at high-strain rates. The equivalent plastic strain at the onset of damage is defined as:

_{1}is the initial failure strain, D

_{2}is the exponential factor, D

_{3}is the triaxility factor, D

_{4}is the strain rate factor, D

_{5}is the temperature factor, σ

_{VM}is the Von Mises stress, and ε

_{f}is the plastic strain at fracture. The D

_{1}–D

_{5}constitute unitless failure parameters. Material fracture occurs when the damage parameter D reaches the value of 1 and the concerned elements are removed from the computation.

^{2}, which was computed to be above the melting threshold of the sample, two SAWs could be depicted having an amplitude of 3 nm and 6 nm and 550 μm and 350 μm, far away the epicenter, respectively. However, this non-uniform grid generation is not efficient to properly model the microscale characteristics of the surface roughness. Any geometrical modification of the non-uniform element distribution is affecting the entire solution domain and disturbs the smooth nodal distribution.

## 4. Results and Discussion

^{2}, above the melting threshold of the Al alloy, which was computed to be 1.5 J/cm

^{2}. For the simulated laser fluence, no ablation phenomena occurred and the structural changes of the target due to laser irradiation were restricted in the plastic and melting regimes. Thus, structural changes related to plastic and melting effects occurred in the submicrometer scale. Figure 3 presents results for five models where sequential downscaling of the surface roughness level was performed. Roughness was neglected in the flat surface model. To perform the downscaling, the average roughness profile, presented in Figure 1e, was processed in MATLAB using a Gaussian smoothing filter that resulted in the cyan, dashed curve, presented in Figure 3a. Based on the geometrical characteristics of this curve, a linear approximation was performed and the flat surface model was firstly downscaled to describe a step of 3 μm. Then, this model was downscaled to describe two steps of 1.5 μm each, following the geometry of the cyan, dashed curve. Following this procedure, two more downscaled models were built and resulted in a total of four downscaled models having a geometrical step accuracy from 3 to 0.5 μm. Each one of the sequentially developed multiscale models could simulate the thermal-mechanical laser–solid interactions, according to the irradiated area of interest and the surface roughness profile therein, with a predefined geometrical accuracy. The identification of these positions, where changes in surface roughness took place after their interaction with a single laser pulse in this energy regime, is of high importance, especially for research works that concern changes of matter of micro- or nanoscale order, which affect the material structure, since this is where different types of cracks initialize [26], ripples may be formed [27,28], and mechanical and optical materials’ properties change [29,30].

^{3,}while for the rest of the models this value was gradually reduced to ~310 μm

^{3}(model neglecting roughness). It was obvious that as the geometrical accuracy of the roughness approximation was downscaled to the maximum of 0.5 μm, the submicron regions, where fracture occurs, permanent deformations are developed, and the material properties change, were more precisely identified.

^{2}. The heated sample at 10 ns was in the thermoelastic regime. Plastic effects started to develop 2 ns later and, at 16 ns, the maximum surface temperature exceeded the melting point. At 20 ns the heated material was in the melting regime and from this temporal moment and afterwards the maximum surface temperature started to decrease. Furthermore, the maximum developed Von Mises stresses of 380 MPa were observed at 18 ns. The advantage of such a high geometrical accuracy for surface roughness level was clearly visible at 50 ns, where the residual stresses that existed due to the plastic deformation of the heated samples were apparent and were above the yield stress of the Al alloy, which is 324 MPa [25].

^{2}. Experiments performed with pulse energies below 0.6 mJ showed that no detectable effect of the laser pulse shot was measurable on the surface of the target. Figure 5 shows two-dimensional images of the area of interest and cropped images, at a position of the target with the maximum roughness variation (~3 μm depth), on which the laser elliptical spot was focused. The cropped images cover a length of x = 220 μm. The two-dimensional graphs of Figure 5 show the surface roughness (height) plotted against the x dimension, passing through the center of the irradiated region as determined by the experiment. For every x position, the roughness values at every point along the y direction were averaged, for a length y = 4 μm within the yellow rectangular area. The graphs produced exhibit the average surface roughness (z dimension, height) plotted along the x direction. The images refer to the same area of interest before and after the laser pulse interacts with the surface.

^{3}. It is worth noting that, through the averaging performed on the roughness values, large fluctuations arising from local surface discontinuities were also smoothened, while information on the location and the type of the changes experienced by the roughness peaks was less influenced, compared to the case where a single line was chosen for plotting the roughness profile, by its exact position relative to the pulse spot center. The complicated pattern of the structural changes in surface roughness due to the interaction with a single laser pulse, as well as the agreement between the experimental and computational outcomes regarding qualitative and quantitative characteristics of the modified roughness pattern, established that downscaling for surface roughness level is of great importance in order to fully characterize the microstructures and new features that evidently arise, even at low energies, and very importantly determine the scale (micro- and/or nanoscale) and the nature of the changes that take place.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Optical setup and the periodic surface roughness profile. (

**a**) Experimental setup for the target–laser interaction measurements: M, metallic mirror; F, converging lens with focal length f = 100 cm; SA, aluminum sample. Profilometry setup: WLS, white light source; BS, beam splitter; RB, reference beam; OB, object beam; PZT, piezoelectric transducer; OL, objective lens. (

**b**) The 2D grayscale surface image of the SA as extracted from the profilometry measurements, of ~3 mm of length. (

**c**) The periodic roughness profiles and the measured values of R

_{a}, R

_{t}, R

_{z}, as extracted from the Roughness gauge. (

**d**) The corresponding roughness line-out plot of image (

**b**) measured by the profilometry setup; the line-out refers to the marked yellow rectangular region of the image. (

**e**) Average surface roughness profile of the heated area of interest of (

**b**).

**Figure 2.**The FEM model. (

**a**) Geometry and mesh. (

**b**) Representative result of simulated ultrasound propagation 60 ns after the beginning of laser irradiation.

**Figure 3.**Downscaling for surface roughness from 0 to 0.5 μm. The computed nodal positions and the plastic strains 100 ns after the laser beam irradiation are presented. (

**a**) A step-like linear approximation of the roughness profile is shown and the original flat surface was gradually downscaled to reach the resulting roughness curve (dashed, cyan line) after the Gaussian filtering. (

**b**) The permanent nodal positions on the top surface along the central horizontal x-axis 100 ns after the laser beam irradiation.

**Figure 4.**Representative FEM results of temperature and Von Mises stress distribution for the model with 0.5-μm step accuracy at 10 ns, 20 ns, and 50 ns and for laser fluence of 2 J/cm

^{2}.

**Figure 5.**The 2D images of the area of interest, cropped images, and line-out graphs of the Al surface before and after irradiation. The images highlight the vicinity where the elliptical laser spot hits the surface of the workpiece.

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**MDPI and ACS Style**

Kaselouris, E.; Kosma, K.; Orphanos, Y.; Skoulakis, A.; Fitilis, I.; Markopoulos, A.P.; Bakarezos, M.; Tatarakis, M.; Papadogiannis, N.A.; Dimitriou, V. Downscaled Finite Element Modeling of Metal Targets for Surface Roughness Level under Pulsed Laser Irradiation. *Appl. Sci.* **2021**, *11*, 1253.
https://doi.org/10.3390/app11031253

**AMA Style**

Kaselouris E, Kosma K, Orphanos Y, Skoulakis A, Fitilis I, Markopoulos AP, Bakarezos M, Tatarakis M, Papadogiannis NA, Dimitriou V. Downscaled Finite Element Modeling of Metal Targets for Surface Roughness Level under Pulsed Laser Irradiation. *Applied Sciences*. 2021; 11(3):1253.
https://doi.org/10.3390/app11031253

**Chicago/Turabian Style**

Kaselouris, Evaggelos, Kyriaki Kosma, Yannis Orphanos, Alexandros Skoulakis, Ioannis Fitilis, Angelos P. Markopoulos, Makis Bakarezos, Michael Tatarakis, Nektarios A. Papadogiannis, and Vasilis Dimitriou. 2021. "Downscaled Finite Element Modeling of Metal Targets for Surface Roughness Level under Pulsed Laser Irradiation" *Applied Sciences* 11, no. 3: 1253.
https://doi.org/10.3390/app11031253