*3.3. Finite Element Analysis (FEA) Simulation of Bending of Thin Metal Plate*

The bending of the thin metal plate was modeled by finite element analysis (FEA), using the Ansys workbench software package, version 16.0. A numerical model of the bending test was created identical to the experimental bend test device; see sub-chapter 3.2. The general view of the numerical model is shown in Figure 3a. The geometry and dimensions of the FEA models of the plate were identical to the plates used in the experimental investigation; see Figure 1. The shape and depth of the laser-processed metal (0.35 mm) were identical for all FEA models of the plates (see Figure 3 and Table 6). The distance between supports (76 mm), punch width (10 mm) and fixation conditions in the FEA simulation were the same as those used during the real experiments. The plates in FEA modeling were supported simply, as it is shown in Figure 2b. For all three variants, the load was imposed in the middle of the span of the stand; see Figure 3a. The laser processing cases and loading variants used in the present FEA are summarized in Table 6. As in the case of experimental investigation in the numerical modeling in general, 10 different simulations were conducted; see Table 6.

The three-dimensional solid brick and tetrahedral elements were used for the discretization of the complex geometry of the modelled plate with laser-processed layers [20]. Large-scale finite elements with a maximum size of 0.7 mm were used to mesh the laser-unprocessed parts of the plate, while a finer mesh, with finite element sizes up to 0.12 mm, was adopted for the discretization of the laser-processed layer.

**Figure 3.** Part (**a**) shows: geometry, dimensions, stand and the loading scheme of the laser-processed plate of the numerical model of the bending simulation; and in (**b**): σ*<sup>b</sup>* denotes the base metal σ − curve; σˆ *<sup>b</sup>* denotes the bilinear approximations of σ*<sup>b</sup>* ; σˆ*<sup>l</sup>* denotes the bilinear approximation of the σ − curve of unknown laser-processed metal; yield strengths, ultimate strength, and fracture strains of the base metal and the laser-processed metal are denoted by σ0.2,*<sup>b</sup>* , σ0.2,*<sup>l</sup>* , σ*B*,*<sup>b</sup>* , σ*B*,*<sup>l</sup>* , *ul*,*<sup>b</sup>* and *ul*,*<sup>l</sup>* respectively.


**Table 6.** The laser processing and variants of geometry of the laser-processed area and bending loading.

The different mechanical properties of the base metal and laser-processed layer were used in the FEA models (treatment cases IIA—IIC, IIIA—IIIC, IVA—IVC). The yield strength, ultimate strength and modulus of elasticity of the steel were determined by mechanical tensile and bending testing of the specimens. The approximate yield strength and ultimate strength of the metal of the laser-processed layers were obtained by applying the relations between hardness and strength that are explained in [21–23]. The modulus of elasticity, Poisson's ratio and other required properties of materials are taken from reference literature [5]. The bilinear isotropic hardening plasticity model was used for the bending case numerical investigation [8].

The so-called bilinear constitutive laws of the laser-unprocessed and laser-processed metals of the plates were adopted for the finite element analysis of the bending of the plates; see σˆ *<sup>b</sup>* and σˆ*<sup>l</sup>* curves depicted in Figure 3b. These constitutive laws are approximations of the real physical stress-strain (σ − ) curves of the laser-processed and unprocessed metals. The direct investigation of the σ − curve of the laser-processed metal is very complicated; therefore, the simplified bilinear σ − diagrams already mentioned were adopted for the finite element analysis. Below, explanations are given of the obtained and adopted mechanical parameters of the laser-unprocessed and processed metals of the plates.

The σ− curve of the base, i.e., laser-unprocessed, metal was obtained experimentally, as described above in Sub-Chapter 3.2. From this σ − curve, the yield strength σ0.2,*<sup>b</sup>* = 256 MPa, the ultimate strength σ*B*,*<sup>b</sup>* = 410 MPa and the ultimate strains *ul*,*<sup>b</sup>* = 0.302 of the base metal were obtained directly. The σ − curve, denoted as σ*<sup>b</sup>* , is depicted as a smooth curve in Figure 3b.

As can be seen from Figure 3b, the σ − curve does not exhibit a clear yield plateau. After reaching the yield strength, so-called work hardening is specific to the base metal. Due to the difficulty of experimental investigation of the mechanical properties of the laser-processed metal, the σ − curve of this metal is unknown. Only indirect estimations of the yield strength, σ0.2,*<sup>l</sup>* = 412 MPa, and the ultimate strength, σ*B*,*<sup>l</sup>* = 665 MPa, were obtained, using the J. R. Cahoon equations, [21,22], which establish the relations between the hardness and strength of metals. However, the ultimate strains, *ul*,*<sup>l</sup>* of the laser-processed metal are still unknown. Therefore, it is assumed that the ultimate strains of the laser-processed metal are the same as for the laser-unprocessed metal, i.e., it is assumed that *ul*,*<sup>l</sup>* = *ul*,*<sup>b</sup>* = 0.302.

The moduli of elasticity, *E<sup>b</sup>* = 200 GPa and *E<sup>l</sup>* = 210 GPa, Poisson's ratios *v<sup>b</sup>* = *v<sup>l</sup>* = 0.28 and the shear moduli, *G<sup>b</sup>* = 78.1 GPa and *G<sup>l</sup>* = 82 GPa of the base metal and the laser-processed metal respectively, were taken from the reference literature [23]. Having the yield strength, σ0.2,*<sup>i</sup>* , and the moduli of elasticity, *E<sup>i</sup>* , *<sup>i</sup>* <sup>∈</sup> {*b*, *<sup>l</sup>*}, the strains 1,*<sup>b</sup>* = 1.28 · <sup>10</sup>−<sup>3</sup> and 1,*<sup>l</sup>* = 1.962 · <sup>10</sup>−<sup>3</sup> that correspond to the yield strength σ0.2,*<sup>i</sup>* were calculated by the formula 1,*<sup>i</sup>* = σ0.2,*i*/*E<sup>i</sup>* .

The strength coefficients of the laser-processed and unprocessed metals, *E*1,*<sup>b</sup>* = 512 MPa and *E*1,*<sup>l</sup>* = 843 MPa, respectively, were calculated by the formula *E*1,*<sup>i</sup>* = (σ*B*,*<sup>i</sup>* − σ0.2,*i*)/ *ul*,*<sup>i</sup>* − 1,*<sup>i</sup>* , *i* ∈ {*b*, *l*}, where σ0.2,*<sup>b</sup>* = 256 MPa and σ0.2,*<sup>l</sup>* = 412 MPa are the yield strengths of the base metal and the laser-processed metal, respectively, while σ*B*,*<sup>b</sup>* = 410 MPa and σ*B*,*<sup>l</sup>* = 665 MPa are the ultimate strengths of the same metals.

On the basis of the obtained values of the mechanical properties of the metals, the bilinear approximations of σ − curves of the laser-unprocessed and processed metals σˆ *<sup>b</sup>* . and σˆ*<sup>l</sup>* , respectively, are depicted in Figure 3b. All the values of the mechanical properties of the two metals discussed above are summarized in Table 7.


**Table 7.** Parameters of base metal and laser-processed layer used to simulate elastoplastic deformation of samples [24,25].

#### *3.4. Analytical Analysis of the Sti*ff*ness Properties of the Laser-Unprocessed and Processed Plates*

The geometry and dimensions of the modeled plate are given in Figure 1. The material properties are given in Table 7. The main assumptions are as follows: (1) the laser-processed metal has a perfect bond with the base metal; that is, no slip occurs between the laser-processed layer and the layer of the base metal; (2) the hypotheses of the plane sections are valid; (3) the influence of Poisson's ratio is ignored; (4) only the normal stresses σ acting along the laser tracks (see Figure 1a) are taken into account; (5) the stress-strain diagram for the laser-processed layer and the base metal is bilinear, as shown in Figure 3b.

Then, under the accepted assumptions, we can write that the total axial force acting on the cross-section of the plate, *N*() = *N<sup>b</sup>* () + *N<sup>l</sup>* (); where *N<sup>b</sup>* () = *Ab*σˆ *<sup>b</sup>* () and *N<sup>l</sup>* () = *Al*σˆ*<sup>l</sup>* () are the axial forces acting in the base metal and laser-processed metal of the plate, respectively; *A<sup>b</sup>* and *A<sup>l</sup>* are the cross-sectional areas of the base metal and the laser-processed metal, respectively; and σˆ*<sup>l</sup>* are the stress functions of the base and the laser-processed metals, respectively, depending on the strains, .

When the bilinear stress–strain diagrams are applied (see Figure 3b), then these functions σˆ *<sup>b</sup>* and σˆ*<sup>l</sup>* can be expressed as follows:

$$\partial\_{i}(\boldsymbol{\varepsilon}) = \begin{cases} \boldsymbol{E}\_{i}\boldsymbol{\varepsilon}\_{\prime} \text{ as } \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{1,i\prime} \\\ \boldsymbol{E}\_{i}\boldsymbol{\varepsilon}\_{1,i} + \boldsymbol{E}\_{1,i}(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{1,i}), \text{ as } \boldsymbol{\varepsilon}\_{1,i} < \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{\text{ul},i\prime} \text{ } i \in \{b, l\} \\\ \boldsymbol{0}, \text{ as } \boldsymbol{\varepsilon} > \boldsymbol{\varepsilon}\_{\text{ul},i}. \end{cases} \tag{1}$$

where *E<sup>b</sup>* , *E*1,*<sup>b</sup>* , *E<sup>l</sup>* , *E*1,*<sup>l</sup>* , *ul*,*<sup>b</sup>* and *ul*,*<sup>l</sup>* are the moduli of elasticity, the stiffness moduli and the ultimate strains of the base and the laser-processed metals, respectively; see Table 7 and Figure 3b.

Since 1,*<sup>b</sup>* ≤ 1,*<sup>l</sup>* and *ul*,*<sup>b</sup>* ≤ *ul*,*<sup>l</sup>* , then by putting functions σˆ *<sup>b</sup>* and σˆ*<sup>l</sup>* (see Equation (1)), in *N*() = P *<sup>i</sup>*∈{*b*,*l*} *Ai*σˆ*i*(), we obtain an explicit function for the axial force, *N*:

$$N(\boldsymbol{\varepsilon}) = \begin{cases} \begin{array}{c} \varepsilon \sum\_{i \in \{l,b\}} A\_i E\_{i\prime} \text{ as } \boldsymbol{\varepsilon} \le \varepsilon\_{1,b\prime} \\\ A\_b \big( E\_b \varepsilon\_{1,b} + E\_{1,b} \big( \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{1,b} \big) \big) + A\_l E\_l \boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}} \text{ as } \boldsymbol{\varepsilon}\_{1,b} < \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{1,l\prime} \\\ \sum\_{i \in \{l,b\}} A\_i \big( E\_l \varepsilon\_{1,i} + E\_{1,i} (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{1,i}) \big), \text{ as } \boldsymbol{\varepsilon}\_{1,l} < \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{ul,b\prime} \\\ A\_l \big( E\_l \varepsilon\_{1,l} + E\_{1,l} (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{1,l}) \big), \text{ as } \boldsymbol{\varepsilon}\_{ul,b} < \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{ul,l}. \end{cases} \tag{2}$$

#### *3.5. Evaluation of the Cross-Sectional Area of Laser-Processed Metal Track*

The geometry of the laser-processed layer is not rectangular, but of more difficult geometry, as shown in Figure 1. Therefore, it is worth discussing further the evaluation of the area of the cross-section of the laser-processed metal, *A<sup>l</sup>* . As can be seen from Figures 1d and 4, the cross-sections of the laser-processed layer and a single laser-processed track are not rectangular. If the laser tracks do not overlap, i.e., *d<sup>c</sup>* ≥ *d<sup>t</sup>* , where *d<sup>t</sup>* is the laser-processed track width and *d<sup>c</sup>* is the distance between the tracks of the laser-processed metal (see Figure 1d, case IV), and the number of laser tracks, *ntr*, is known, then the cross-section of the laser-processed metal can be calculated easily.

$$A\_l = n\_{\rm s} \eta\_{\rm tr} A\_{\rm tr, 1\prime} \text{ if } d\_{\rm c} - d\_l \ge 0, \text{ and } n\_{\rm tr} \text{ is known} \tag{3}$$

where A*tr*,1 = π*d* 2 *t* /8 is the cross-sectional area of one laser-processed track, assumed to be a semicircle; *n<sup>s</sup>* ∈ {1, 2} is the number of laser-processed sides of the plate. It should be noted that, in Equation (3) and hereafter in the article, the number of laser tracks, *ntr*, may not be an integer; i.e., in general, *ntr* ∈ R.

However, if *ntr* is not known or the laser-processed tracks overlap, i.e., *d<sup>c</sup>* < *d<sup>t</sup>* , then the exact evaluation of *A<sup>l</sup>* can be complicated. Some suggestions are made below on the following assumptions: the cross-section of a laser track is a semicircle whose area A*tr*,1 = π*d* 2 *t* /8, where *d<sup>t</sup>* is the width of the laser track (see Figure 1d); the width of the tracks, *d<sup>t</sup>* , is constant for all laser tracks and for the entire length of all laser tracks; the distance between centers of the laser tracks, *d<sup>c</sup>* (see Figure 1d) is also constant for all pairs of the vicinal laser tracks.

The calculation of *A<sup>l</sup>* depends on the distances between the tracks, *dc*. If the difference *d<sup>c</sup>* − *d<sup>t</sup>* > 0 then the laser tracks do not overlap each other and, in general, the exact evaluation of *A<sup>l</sup>* is impossible without knowledge of the exact position of the laser tracks. However, the lower and upper bounds *Al*,*in f* and *Al*,*sup* can be suggested to estimate *A<sup>l</sup>* , *Al*,*in f* ≤ *A<sup>l</sup>* ≤ *Al*,*sup*:

$$A\_{l, \inf} = n\_{\text{s}} \mathbf{A}\_{l\text{tr},1} floor(b\_l/d\_\text{c}), \text{ if } d\_\text{c} - d\_l > 0,\tag{4}$$

$$A\_{l,sup} = n\_{\rm s} \mathbf{A}\_{\rm tr,1} \text{ceil}(b\_l/d\_c), \text{ if } d\_c - d\_t > 0. \tag{5}$$

where *floor*(*x*) = *max y* ∈ Z, *y* ≤ *x* and *ceil*(*x*) = *min y* ∈ Z, *y* ≥ *x* are ceiling and floor functions; Z is the set of the integer numbers; *b<sup>l</sup>* is the width of the laser-processed area. When *bl*/*d<sup>c</sup>* increases, then the relative differences of the estimations *<sup>A</sup>l*,*sup* <sup>−</sup> *<sup>A</sup>l*,*in f* /*Al*,*sup* and *<sup>A</sup>l*,*in f* <sup>−</sup> *<sup>A</sup>l*,*sup* /*Al*,*sup* decreases. For the present case, when *d<sup>t</sup>* = 0.7 mm, and *d<sup>c</sup>* ∈ {1.05, 1.4} mm (see Figure 1d), Equations (3) and (4) give:

*Al*,*in f* = 14A*tr*,1 and *Al*,*sup* = 15A*tr*,1 when *d<sup>c</sup>* = 1.4 mm, and *Al*,*in f* = 19A*tr*,1 and *Al*,*sup* = 20A*tr*,1 when *d<sup>c</sup>* = 1.05 mm.

**Figure 4.** Cross-sections of a laser-processed track and the base metal of a FEA and a real specimen respectively (**a**) and the general view of the laser-processed tracks of case III of the real specimen and its FEA model respectively (**b**).

If *ntr* is unknown, then it can be estimated as follows:

$$\mathfrak{h}\_{\rm tr} = \mathfrak{b}\_{\rm l} / d\_{\rm c} \tag{6}$$

It is clear that Equation (6) can be used to evaluate the distance between the laser-processed tracks *d<sup>c</sup>* = *bl*/*n*ˆ*tr*.

When *d<sup>c</sup>* − *d<sup>t</sup>* = 0 and *ntr* is known, then *A<sup>l</sup>* can be calculated by Equation (3). Otherwise, when the number of laser-processed tracks, *ntr* is not known, then *A<sup>l</sup>* can be calculated by assuming that there are no laser-unprocessed bands between the laser-processed tracks by the following formulae:

$$A\_l = n\_\text{s} \mathbf{A}\_{\text{tr},1} b\_l / d\_\text{c} \text{ if } d\_\text{c} - d\_l = \mathbf{0}.\tag{7}$$

When *d<sup>c</sup>* − *d<sup>t</sup>* < 0, then the laser tracks overlap each other and we have to take into account the overlapped areas of the tracks, see Figure 1d, case II. Therefore, Equations (3)–(5) and (7) are not valid. On the basis of the above assumptions, the following bounds infimum *Al*,*in f* and supremum *Al*,*sup*, *Al*,*in f* ≤ *A<sup>l</sup>* ≤ *Al*,*sup*, of the cross-section area *A<sup>l</sup>* of the laser-processed metal are derived when *ntr* is not known

$$A\_{l, \inf} = n\_{\rm s} \hbar\_{\rm lr} (A\_{\rm tr, 1} - A\_{\rm av})\_{\prime} \text{ if } d\_{\rm c} - d\_{\rm l} < 0,\tag{8}$$

$$A\_{l, \text{sup}} = \min \left\{ n\_{\text{s}} (\hat{n}\_{\text{tr}} \mathbf{A}\_{\text{tr},1} - (\hat{n}\_{\text{tr}} - 1) A\_{\text{ov}}), b\_l \mathbf{1}/2d\_l \right\}, \text{ if } d\_{\text{c}} - d\_l < \mathbf{0}, \tag{9}$$

where *n*ˆ*tr* is an estimation of the number of laser-processed tracks:

$$\mathfrak{H}\_{\rm tr} = f \text{hor} \left( \frac{b\_l}{d\_c} \right) - 1 + \left( 1 + \frac{f}{\pi} \text{arc} \left( \frac{b\_l}{d\_c} \right) \right) \frac{d\_c}{d\_t} \tag{10}$$

where *f rac*(*x*) = *x* − *floor*(*x*) is the fractional part of a number *x*.

When the number of the tracks *ntr* is known:

$$A\_l = n\_s(n\_{lr}A\_{lr,1} - (n\_{lr} - 1)A\_{\text{ov}}), \text{ if } d\_c - d\_l < 0 \text{ and } n\_{lr} \text{ is known} \tag{11}$$

In Equations (8), (9) and (11), *Aov* is the overlapping area of two adjacent laser tracks; see Figure 1d, case II. Under the accepted assumptions, *Aov* can be calculated as the area of the circle segment *Aov* = *d* 2 *t* (<sup>α</sup> <sup>−</sup> sin(α))/8, where <sup>α</sup> = <sup>2</sup> tan−<sup>1</sup> q *d* 2 *t* − *d* 2 *<sup>c</sup>*/*d<sup>c</sup>* is the sector angle, tan−<sup>1</sup> is the inverse tangent function. It should be noted that *Al*,*in f* ≤ *Al*,*sup* ≤ *Al*,*upp* = 1/2 *bld<sup>t</sup>* , where *Al*,*upp* is the upper bound of the cross-section of the laser-processed layer.

#### **4. Results**

The results of the structural analysis and hardness of the laser-processed surface layer are given in Section 4.1. The comparative analyses of the experimental and numerical FEA results of the bending of the laser-unprocessed (case I) and laser-processed plates are given in the Sections 4.2 and 4.3. In Section 4.2 the numerically obtained von Mises stresses and forces *Fexp* at the different vertical displacements *w* ∈ {0.5, 1.0, 1.5, 2.0} mm imposed at midpoint B are presented, see Figure 2b. The relation between the experimentally determined vertical forces *Fexp*, imposed at the midpoint B, and the deflection *w* is analyzed in Section 4.3. Furthermore, a comparison of the experimental and calculated-by-FEA vertical forces *Fexp* and *Fcalc* is given in Section 4.3. In Section 4.4.1, the results of the analytical analysis of the cross-section areas *A<sup>l</sup>* of the laser-processed metal depending on the track width of the laser-processed metal *d<sup>t</sup>* and the distances between these tracks centers *d<sup>c</sup>* are given. In Section 4.4.2, the results of the analytical analysis of the influence of the laser processing of the plate metal on the axial stiffness of the cross-section and the force-strain behavior of the plates under investigation are given. The obtained results definitely showed that the axial and flexural stiffnesses of the laser-processed cross-sections are bigger than the stiffnesses of the laser-unprocessed cross-sections.

#### *4.1. Results of the Structural Analysis of the Laser-Processed Surface Layer*

The thickness of the laser-processed layer, which is established by metallographic examination of cross-sections of samples of metal processed by laser, was about 0.35 mm (see Figure 4). There are no unacceptable inclusions, porosity or internal defects in the remelted area and transition area. The microstructure of the base metal consists of 70% ferrite and 30% pearlite (see Figure 5a), according to GOST 8233 [26]. Granularity in the processed zone decreases from G8 (average grain diameter about 18 µm) to G10 (average grain diameter about 10 µm), according to ISO 643 [27]. According to the XRD data analysis (see Figure 6), the X-ray diffraction pattern of the laser-processed layer is typical for the ferrite microstructure family (including sorbite and troostite) with BCC crystal structure). The same crystal structure has martensite and bainite. The Fe3C peak and other carbide peaks were not observed. This is typical, so carbon content in the steel is low or peak sensitivity is below the limit of detection sensitivity. Therefore, XRD analysis confirms only that there is no unstable, retained austenite in the laser-processed area, because the austenite has the FCC crystal structure and other XRD diffraction peaks. In low-carbon steel, high-retained austenite contents are usually found together with the martensite or bainite phase in the quenched steel after cooling. The measurement of microhardness shows that the hardness of the laser-processed layer increases up to 200 HV (by 60%) compared to the laser-unprocessed base metal hardness (see Figure 7). According to the hardness measurement results, there are no hard and brittle bainite and martensite microstructures in the laser-processed areas or their proportion is very low, because such quenching microstructures have the highest hardness: bainite—about 400 HV and martensite—above 450 HV [28,29]. Consequently, to achieve a hardness of about 200 HV, it is necessary to have a minimum of about 25% of bainite or martensite structure in the present low-carbon steel after rapid cooling of the microstructure of the melted area, or the steel's

microstructure must be strengthened by refining the grain size and creating a finely dispersed perlite (sorbite, troostite) structure.

**Figure 5.** SEM images of microstructure: (**a**) ferrite-pearlite structure of base metal (magnification ×2000); (**b**) distance between lamella in the sorbite structure of laser-processed layer (magnification ×10,000).

**Figure 7.** Distribution of hardness through the cross-section of the laser-processed surface of sample IIA: (**a**) and (**b**) show the distributions of three- and one-laser-processed tracks respectively.

The microstructure located at the laser-processed area of the low-carbon steel sample demonstrated a typical sorbite structure (Figure 5b). The distance between lamella, which is measured by SEM, is less than 0.3–0.4 µm. This distance is typical for sorbite, because perlite has a distance between lamella of about 0.6–0.7 µm, troostite—about 0.1 µm, martensite and bainite—about 0.2 µm and thickness of retained austenite lamellae—0.05–0.2 µm [30]. The hardness of the laser-processed layer also corresponds to the typical hardness of sorbite, which must be in the range 200–300 HV [31].

The sorbite structure of the laser-processed layer was formed due to the applied high overlap coefficient of laser spots, with inevitable additional heating and partial remelting of the crystallized pool during the next laser pulse. This effect allows the cooling rate of the melted pool to be reduced and prevents the formation of more brittle, quenched structures in the laser-processed layer.

The sorbite structure has many advantages in this case, because sorbite has a finer texture, with higher dispersity and stiffness than pearlite, which increases the strength and wear resistance of the laser-processed metal parts, without loss in plasticity, that is typical for hard and brittle quenching (martensite or bainite) microstructures [32]. SEM-EDS element mapping, line scan and point analysis of the distribution and concentration of chemical elements (C, Mn, Si) in the remelted area show that there is no significant chemical heterogeneity in the laser-processed layer (see Figures 8 and 9, Table 8). This uniform distribution of the chemical elements and the homogeneous structure of the remelted layer can positively influence the mechanical properties of the laser-processed layers and the entire plate.

**Figure 8.** Line scan element analysis along the surface of laser-processed layer: (**a**) place of line scan in cross-section of layer; (**b**) C distribution; (**c**) Fe distribution; (**d**) Si distribution; (**e**) Mn distribution.

**Figure 9.** Line scan element analysis normal to the laser-processed surface: (**a**) place of line scan in cross-section of layer; (**b**) C distribution; (**c**) Fe distribution; (**d**) Si distribution; (**e**) Mn distribution.


**Table 8.** Chemical compositions (by energy-dispersive X-ray spectroscopy analysis) of surface regions.

#### *4.2. Results of the FEA Simulations of the Bending of the Laser-processed Plates*

The increase in the bending force varied, depending on the plate treatment case and deflection level: for 0.5 mm deflection, the force was 11–27%; for 1.0 mm deflection about 43–49%; for 1.5 mm deflection—32–55%; for 2.0 mm deflection—30–52%. The increase in the maximum equivalent stresses varied according to the plate treatment case and deflection level: for 0.5 mm deflection—13–30%; for 1.0 mm deflection—13–32%; for 1.5 mm deflection—8–26%; for 2.0 mm deflection—8–25%.

The maximum equivalent stress at the bent plates depends on the volume and position of the laser-processed layer. The greatest increase in the required bending load and available stresses, according to the modeling results, was in the double-sided laser-processed samples, while the volume of treatment was greatest from 15.4% to 19.8% of the volume of the bent plate part. The increased bending load required for similar deflection depends also on the position of the laser-processed layer. Better results were obtained in the bent samples where the zone of the laser-processed layer was subjected to tension, rather than compression.

It was determined that double-side laser-processed metal plates with 30% overlay of laser tracks had the highest resistance to bending, compared to the other treatment modes; see Table 9. However, modeling results (see Table 9 and Figure 10) show that it is possible to use a non-continuous laser with either one-side or double-side processing with a certain distance between tracks, because the difference in the efficiency of such surface treatments is small. The difference of the required bending loads for one deflection level of the plate, applied to different positions of the laser-processed layer (cases A, B, C) did not reach 20%.


**Table 9.** The maximum Von Mises equivalent stresses (in MPa), and corresponding forces *Fcalc* (in N) of the finite element analysis (FEA) simulation of the bending of the differently laser-processed plates at different imposed vertical displacements *w* ∈ {0.5, 1.0, 1.5, 2.0} mm of the middle point B.

**Figure 10.** FEA results of the bending modeling of the laser-processed plates when the imposed vertical displacement of the plate middle point B, see Figure 2b, equals 1 mm; von Mises stresses are shown in (**a**–**c**): (**a**) for case IIIA, (**b**) for case IVB and (**c**) for case IIC; while (**d**) shows the deflections or vertical displacements of the plate of case IIIA.
