4.4.1. Area *A<sup>l</sup>* of the Laser-Processed Metal

Figure 12 shows the dependencies of the bounds of the cross-sectional area of the laser-processed metal, *Al*,*in f* and *Al*,*sup*, as well as their ratios *Al*/(1/2*blbt*), where *A<sup>l</sup>* ∈ n *<sup>A</sup>l*,*in f* , *<sup>A</sup>l*,*sup*<sup>o</sup> on the estimated number of the laser-processed tracks, *n*ˆ*tr*, calculated by Equations (6) or (10), and on the ratio *dc*/*d<sup>t</sup>* . The areas *Al*,*in f* and *Al*,*sup* were calculated by Equations (4), (5), (7)–(10) of the plate, shown in Figure 1, when the width of the laser-processed track *d<sup>t</sup>* = 0.7 mm and the width of the laser-processed area of the plate under consideration, *b<sup>l</sup>* = 0.02 m. The vertical dotted lines correspond to the minimum and maximum of *n*ˆ*tr* or *dc*/*d<sup>t</sup>* and laser-processing cases: case II when *dc*/*d<sup>t</sup>* = 2/3; case III when *dc*/*d<sup>t</sup>* = 1.5; and case IV when *dc*/*d<sup>t</sup>* = 2; see Table 5.

**Figure 12.** The dependencies of the bounds of the cross-section of the laser-processed metal *Al*,*in f* and *Al*,*sup* and the ratios *Al*,*in f* /(1/2*bldt*) and *Al*,*sup*/(0.5*bldt*) on the number of tracks *n*ˆ*tr* and ratio *dc*/*d<sup>t</sup>* of the considered plate; see Figure 1: (**a**) when *n*ˆ*tr* ∈ [14.29, 142.86], corresponding to *d<sup>c</sup>* ∈ h 1.4 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , 1.4 <sup>×</sup> <sup>10</sup>−<sup>3</sup> i or *dc*/*d<sup>t</sup>* ∈ [0.2, 2]; and (**b**) when *dc*/*d<sup>t</sup>* ∈ [0.02, 2], corresponding to *d<sup>c</sup>* ∈ h 1.4 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , 1.4 <sup>×</sup> <sup>10</sup>−<sup>3</sup> i or *n*ˆ*tr* ∈ [14.286, 1428.571].

Please note: in the present calculations, the numbers of tracks, *ntr*, for cases II–IV were evaluated by Equations (6) or (10), i.e., *ntr* was not taken from Table 5.

As can be seen from Figure 12, *Al*,*in f* approaches *Al*,*sup* with increasing numbers of laser-processed tracks, *n*ˆ*tr*, or with a decrease in the ratio, *dc*/*d<sup>t</sup>* or *dc*. Also from Figure 12, it can be seen that the rate of increase of the areas, *A<sup>l</sup>* ∈ n *<sup>A</sup>l*,*in f* , *<sup>A</sup>l*,*sup*<sup>o</sup> , is practically constant with respect to increasing *ntr* within the interval *ntr* ∈ [min{*ntr*}, 28, 57], corresponding to the interval, *dc*/*d<sup>t</sup>* ∈ [1, *max*{*dc*\*dt*}], or when *dc*/*d<sup>t</sup>* ≥ 1.

More remarkably, the rate of increase of *A<sup>l</sup>* ∈ n *<sup>A</sup>l*,*in f* , *<sup>A</sup>l*,*sup*<sup>o</sup> even increases with respect to decreasing *dc*/*d<sup>t</sup>* within the interval *dc*/*d<sup>t</sup>* ∈ [*max*{*dc*\*dt*}, 1]. However, as shown in Figure 12, the increase in the areas *A<sup>l</sup>* ∈ n *<sup>A</sup>l*,*in f* , *<sup>A</sup>l*,*sup*<sup>o</sup> becomes slower with increasing *ntr* or decreasing *dc*/*d<sup>t</sup>* or *d<sup>c</sup>* at *dc*/*d<sup>t</sup>* < 1.

Finally, the increase of *A<sup>l</sup>* ∈ n *<sup>A</sup>l*,*in f* , *<sup>A</sup>l*,*sup*<sup>o</sup> practically does not change at a very large *n*ˆ*tr*. It should be noted that the biggest difference *max*<sup>n</sup> *<sup>A</sup>l*,*sup* <sup>−</sup> *<sup>A</sup>l*,*in f*<sup>o</sup> = 0.1924 mm<sup>2</sup> and the (*max*n *<sup>A</sup>l*,*sup* <sup>−</sup> *<sup>A</sup>l*,*in f* /*Al*,*sup*<sup>o</sup> · 100 = 6.66%) occurs at the largest value of ratio *dc*/*d<sup>t</sup>* = 2, or at the smallest *n*ˆ*tr* = 14.29.

Figure 13 shows the dependency of the ratio of the area of the remelted metal to the supremum bound of the area of the laser-processed metal *Al*,*rem*/*Al*,*sup* (Figure 13a), and the dependency of the ratio of the overlapping area on the cross-sectional area of the single laser-processed track *Aov*/*Atr*,1 on the ratio *dc*/*d<sup>t</sup>* of the plate under consideration; see Figure 1, where the total area of the remelted metal is calculated as follows: *Al*,*rem* = (*ntr* − 1)*Aov* = (*bl*/*dc*)*Aov*.

**Figure 13.** The dependencies of the ratios *Al*,*rem*/*Al*,*sup* and *Aov*/*Atr*,1 on the ratio *dc*/*d<sup>t</sup>* of the considered plate, see Figure 1: (**a**) *Al*,*rem*/*Al*,*sup* versus *dc*/*d<sup>t</sup>* ∈ [0.02, 1]; and (**b**) *Aov*/*Atr*,1 versus *dc*/*d<sup>t</sup>* ∈ [0.02, 2].

It is clear that *Al*,*rem* = *Aov* = 0 as *d<sup>c</sup>* > *d<sup>t</sup>* or *dc*/*d<sup>t</sup>* > 1. As we can see from Figure 13a,b, the rate of the increase of the ratio *Aov*/*Atr*,1 on *dc*/*d<sup>t</sup>* jumps sharply at *dc*/*d<sup>t</sup>* = 1, and this rate is almost constant within the interval *dc*/*d<sup>t</sup>* ∈ [0.02, 1], since *Aov*/*Atr*,1 on *dc*/*d<sup>t</sup>* depends very similarly to the line. However, the dependence of the ratio of the total area of the remelted metal to the supremum bound of the laser-processed metal *Al*,*rem*/*Al*,*sup* increases very slowly at *dc*/*d<sup>t</sup>* = 1, but becomes very steep with a higher ratio of *dc*/*d<sup>t</sup>* .

The behavior of *Al*,*rem*/*Al*,*sup* with respect to *dc*/*d<sup>t</sup>* differs from *Aov*/*Atr*,1 due to the influence of the number of laser-processed tracks, *ntr*. At relatively large *dc*/*d<sup>t</sup>* , the number *ntr* is small. Therefore, the contribution of *Aov* to *Al*,*rem* is also small. With decreasing *dc*/*d<sup>t</sup>* the number *ntr* increases and, hence the contribution of increased *Aov* to *Al*,*rem* increases also.

As we can see from Figure 13a, the ratio *Al*,*rem*/*Al*,*sup* = 1 as *dc*/*d<sup>t</sup>* = 0.405. Therefore *Al*,*rem* > *Al*,*sup* as *dc*/*d<sup>t</sup>* < 0.405, and the difference *Al*,*rem* − *Al*,*sup* or the ratio *Al*,*rem*/*Al*,*sup* increases very quickly with increasing *dc*/*d<sup>t</sup>* ≥ 1. It should be noted that the cross-sectional area of the remelted metal, *Al*,*rem* can be many times, ten- or even twenty-times, bigger than the supremum bound of the cross-sectional area of the laser-processed metal *Al*,*sup* at small values of *dc*/*d<sup>t</sup>* . Throughout this subchapter, a decrease in *dc*/*d<sup>t</sup>* corresponds to increasing the number of laser-processed tracks, *ntr* and vice versa, while increasing *dc*/*d<sup>t</sup>* corresponds to a decrease in *ntr*.

The analysis above allows us to conclude that making a laser-processed layer is very inefficient at small values of *dc*/*d<sup>t</sup>* or large values of *ntr*, since a very large volume of the metal is melted repeatedly. For practical applications, it is reasonable to take the ratio, *dc*/*d<sup>t</sup>* = 1. In this case, as we can see from Figure 12b, *Al*/(1/2*blbt*) = 0.785. This means that the increase in the number of laser-processed tracks *ntr* or decrease of the distance between the tracks, *d<sup>c</sup>* or the ratio *dc*/*d<sup>t</sup>* can increase the cross-sectional area of the laser-processed metal only up to (1 − 0.785) · 100 = 21.5%. However, as shown above, the cross-section of the remelted metal, or the ratio *Al*,*rem*/*Al*,*sup*, increases very quickly with decreasing *d<sup>c</sup>* or increasing *ntr*. Hence, the efficiency of laser processing decreases very quickly.

For practical applications, the optimal limit of ratio *dc*/*d<sup>t</sup>* can be 0.405, since at this point *Al*,*rem*/*Al*,*sup* = 1 and ratios *Al*.*in f* /(1/2*blbt*) ≈ 0.96 and *Al*.*sup*/(1/2*blbt*) ≈ 0.97. Therefore, a further increase in *ntr* or decrease in *dc*/*d<sup>t</sup>* can increase the ratios *Al*/(1/2*blbt*) only up to 4%. To increase the cross-sectional area of the laser-processed metal, *A<sup>l</sup>* , it is better to increase the width, and hence the depth, of the laser-processed track *d<sup>t</sup>* than to increase *ntr* or decrease *dc*/*d<sup>t</sup>* .

The considerations given above concerning the ratios *Al*/(1/2*bldt*), *Al*,*in f* /(1/2*bldt*), *Al*,*sup*/(1/2*bldt*), *Al*,*rem*/*Al*,*sup*, *Aov*/*Atr*,1 and the corresponding cross-section areas *Al*,*in f* , *Al*,*sup*, (1/2*bldt*), *Aov*, *Atr*,1, *Al*,*upp* = 1/2*bld<sup>t</sup>* are also valid for the corresponding ratios of the volumes *Vl*/*Vl*,*upp*, *Vl*,*in f* /*Vl*,*upp*, *Vl*,*sup*/*Vl*,*upp*, *Vl*,*rem*/*Vl*,*sup*, *Vov*/*Vtr*,1 and the corresponding volumes *Vl*,*in f* , *Vl*,*sup*, *Vov*, *Vtr*,1, *Vl*,*upp* of the laser-processed metal, where *V* stands for the volume of the laser-processed metal and its indexes "*l*, *in f* ", "*l*,*sup*", "*l*, *upp*", "*Vl*,*rem*", "*ov*", and "*tr*, 1" means the same as the corresponding indexes of the cross-sectional areas denoted by *A*.

Since the above analysis, results, conclusions and recommendations are expressed in relative terms, then these considerations are also valid for other cases of plates, laser-processed areas, their widths *b<sup>l</sup>* , the laser-processed track widths *d<sup>t</sup>* , the distances between the track centers *dc*, the number of laser tracks *ntr* and so on.

### 4.4.2. Axial Stiffness and Force—Strain Behavior of the Laser-Processed Plate

The dependencies of the ratios of the axial forces of the laser-processed plates (see Figure 1, cases II, III and IV) to the unprocessed plate, case I: *Ni*/*N*Case I, *i* ∈ {CaseII, Case III, Case IV} on strain ∈ [0, *ul*] = h 0, 302 · <sup>10</sup>−<sup>3</sup> i are shown in Figure 14. The axial forces *N<sup>i</sup>* , *i* ∈ {Case I, . . . , Case IV} were calculated according to Equations (1) and (2), using the properties of the base and laser-processed metals given in Table 7. The cross-sectional areas of the laser-processed metal of plate *A<sup>l</sup>* were calculated by the proposed Equations, (3), (7) and (11) when the number of the laser-processed sides *n<sup>s</sup>* = 2 and the number of the laser-processed tracks of one side of the plate, corresponding to cases I, II, III and IV *ntr* ∈ {0, 14, 17, 38} is given in Table 5. The following cross-sectional areas were obtained: *A<sup>l</sup>* = 0 for case I, *<sup>A</sup><sup>l</sup>* = 1.1504 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>m</sup><sup>2</sup> for case II, *<sup>A</sup><sup>l</sup>* = 6.5423 <sup>×</sup> <sup>10</sup>−<sup>6</sup> <sup>m</sup><sup>2</sup> for case III, *<sup>A</sup><sup>l</sup>* = 5.3878 · <sup>10</sup>−<sup>6</sup> <sup>m</sup><sup>2</sup> for case IV. The cross-sectional areas of the base metal of the plate corresponding to cases I–IV, *A<sup>b</sup>* = *Apl* − *A<sup>l</sup>* , where *<sup>A</sup>pl* = *<sup>b</sup>pl* = <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>m</sup><sup>2</sup> is the cross-sectional area of the plate; see Figure 1b.

**Figure 14.** The dependencies of the ratios of the axial forces of the laser-processed plates, cases II, III and IV, to the unprocessed plate, case I: *Ni*/*NCase I*, *i* ∈ {CaseI, Case III, Case IV} on strain ∈ [0, *ul*] = h 0, 302 · <sup>10</sup>−<sup>3</sup> i on strain .

As we can see from Figure 14, the influence of laser processing is almost infinitesimal when ∈ h 0, 1,*<sup>b</sup>* i = h 0, 1.28 <sup>×</sup> <sup>10</sup>−<sup>3</sup> i ; see section AB in Figure 14. The maximum ratio, maxn *Ni*/*NCase I*, *i* ∈ {CaseII, Case III, Case IV}, ∈ h 0, 1,*<sup>b</sup>* io = 1.0015. If the axial stiffness of the cross-section of the plate is *B* = *N*/, then we can say that laser processing does not have any

influence on the axial stiffness of the plate when ∈ h 0, 1,*<sup>b</sup>* i . However, the influence of laser processing on the ratio of the axial forces *Ni*/*NCase I* increases with increasing strain within the interval ∈ h 1,*<sup>b</sup>* , 1,*<sup>l</sup>* i = h 1.28 <sup>×</sup> <sup>10</sup>−<sup>3</sup> , 1.962 <sup>×</sup> <sup>10</sup>−<sup>3</sup> i and remains constant when ∈ h 1,*<sup>l</sup>* , *ul*<sup>i</sup> = h 1.962 <sup>×</sup> <sup>10</sup>−<sup>3</sup> , 302 <sup>×</sup> <sup>10</sup>−<sup>3</sup> i , where hereafter *ul* = *ul*,*<sup>b</sup>* = *ul*,*<sup>l</sup>* = <sup>302</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> ; see sections BC and CD in Figure 13.

As shown in Figure 14, laser processing increases the axial forces *N<sup>i</sup>* and hence the axial stiffness *Bi* , *i* ∈ {Case II, Case III, Case IV}, about 17.5% for case II, 9.9% for case III, and 8.2% for case IV. For practical structural calculations, it is reasonable to assume that the plate's bearing capacity corresponds to the axial force *N<sup>i</sup>* at = 1,*<sup>l</sup>* , since the increase of the ratio *Ni*/*NCase I* within the interval h 1,*<sup>l</sup>* , *ul*<sup>i</sup> is very low: (1.179–1.175) = 2.5% for case II; 1.102 − 1.009 = 0.9% for case III; and 1.083 − 1.082 = 0.1% for case IV.

The dependencies of the ratios of the axial forces η,*Al*/*A<sup>b</sup>* = *N*(*Al*/*A<sup>b</sup>* , )/*N*(0, ), at ∈ n 1,*<sup>b</sup>* , 1,*<sup>l</sup>* , *ul*<sup>o</sup> with respect to the ratio of the cross-sectional areas of the laser-processed and base metals *Al*/*A<sup>b</sup>* ∈ [0, 0.538] of the plate shown in Figure 1 are shown in Figure 15. In the situation when *A<sup>l</sup>* = 0, since *Al*/*A<sup>b</sup>* = 0 also corresponds to the laser-unprocessed plate, i.e., case I, then *N*(0, ) = *N*Case I. In these calculations, it was assumed that the cross-sectional area of the laser-processed metal attains values from the interval: *A<sup>l</sup>* ∈ h 0, *<sup>A</sup>l*,*up*<sup>i</sup> = [0, 2 × 1/2*bld<sup>t</sup>* ] = h 0, 1.4 <sup>×</sup> <sup>10</sup>−<sup>5</sup> i . Since *A<sup>b</sup>* = *Apl* − *A<sup>l</sup>* , where *<sup>A</sup>pl* = *<sup>b</sup>pl* <sup>×</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> = <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> , then *max*{*Al*/*A<sup>b</sup>* } = 1.4/(4 − 1.4) = 0.5385. Thus, *max*{*Al*/*A<sup>b</sup>* } = 0.5385 corresponds to the upper bound of the cross-sectional area of the laser processed metal when *<sup>A</sup>l*,*up* = *<sup>n</sup>s*1/2*bld<sup>t</sup>* = <sup>20</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>×</sup> 0.7 <sup>×</sup> <sup>10</sup>−<sup>3</sup> = 1.4 <sup>×</sup> <sup>10</sup>−<sup>3</sup> of the two-side laser-processed plate; see Figure 1b. Other parameters used to draw Figure 15 were the same as in Figure 14; see the explanations given at the beginning of the present subchapter.

**Figure 15.** The dependencies of the ratios of the axial forces η,*Al*/*A<sup>b</sup>* = *N*(*Al*/*A<sup>b</sup>* , )/*N*(0, ), at ∈ n 1,*<sup>b</sup>* , 1,*<sup>l</sup>* , *ul*<sup>o</sup> with respect to the ratio of the cross-section areas of the laser-processed and base metals *Al*/*A<sup>b</sup>* ∈ [0, 0.538] of the plate shown in Figure 1.

As we can see from Figure 15, the ratios η,*Al*/*A<sup>b</sup>* on *Al*/*A<sup>b</sup>* do not change according to the linear law, as in the case of the dependence of the *N* on *A<sup>l</sup>* when *A<sup>b</sup>* is constant, according to Equation (2). This happens since the cross-sectional area of the base metal *A<sup>b</sup>* = *Apl* − *A<sup>l</sup>* , i.e., *A<sup>b</sup>* also changes when *A<sup>l</sup>* changes. Also, it is clearly visible that the ratio *Al*/*A<sup>b</sup>* affects the ratios η,*Al*/*A<sup>b</sup>* , especially when ∈ h 1,*<sup>b</sup>* , 1,*<sup>l</sup>* i . Thus, when ∈ n 0, 1,*<sup>b</sup>* o , then the relative difference <sup>∆</sup>*max*η,*Al*/*A<sup>b</sup>* <sup>=</sup> *max*<sup>n</sup> η,*Al*/*A<sup>b</sup>* − η,0, *Al*/*A<sup>b</sup>* ∈ {0, 0.538} o = η,0.538 − η,0 = 1.7%, while when ∈ n 1,*<sup>b</sup>* , 1,*<sup>l</sup>* o , then ∆*max*η,*Al*/*A<sup>b</sup>* = 21.3%; and when ∈ n 1,*<sup>b</sup>* , *ul*<sup>o</sup> , then ∆*max*η,*Al*/*A<sup>b</sup>* = 21.8%. When the

strain increases from 1,*<sup>l</sup>* to *ul* and *Al*/*A<sup>b</sup>* = 0.538, then the relative difference ∆*max*η,*Al*/*A<sup>b</sup>* = η*ul*,0.538 − η*<sup>l</sup>* ,0.538 = 0.5%. From Figure 15, we can see that the biggest possible increase of the axial force is when *Al*/*A<sup>b</sup>* = *Al*,*sup*/ *<sup>A</sup>pl* <sup>−</sup> *<sup>A</sup>l*,*sup* = 0.538 is η*ul*,0.538 − η*<sup>l</sup>* ,0.538 = 1.213 − 1.017 = 19.6%.
