## Abstract

In any pulsed and repetitive laser process a part of the absorbed laser energy is thermalized and stays in the material as residual heat. This residual heat is accumulating from pulse to pulse, continuously increasing the temperature, if the time between two pulses does not allow the material to sufficiently cool down. Controlling this so-called heat accumulation is one of the major challenges for materials processing with high average power pulsed lasers and repetitive processing. Heat accumulation caused by subsequent pulses (HAP) on the same spot and heat accumulation caused by subsequent scans (HAS) over the same spot can significantly reduce process quality, *e.g.*, when the temperature increase caused by heat accumulation exceeds the melting temperature. In both cases, HAS and HAP, it is of particular interest to know the limiting number of pulses or scans after which the heat accumulation temperature exceeds a critical temperature and a pause has to be introduced. Approximation formulas for the case, where the duration of the heat input is short compared to the time between two subsequent heat inputs are derived in this paper, providing analytical scaling laws for the heat accumulation as a function of the processing parameters. The validity of these approximations is confirmed for HAP with an example of surface ablation of CrNi-steel and for HAS with multi-scan cutting of carbon fiber reinforced plastics (CFRP), both with a picosecond laser at an average power of up to 1.1 kW. It is shown that for the important case of 1-dimensional heat flow the limiting number of heat inputs decreases with the inverse of the square of the average laser power.

© 2017 Optical Society of America

## 1. Introduction

Short- and ultra-short pulsed lasers promise high-quality processing of virtually any material with superior quality presuming the correct processing parameters [1–5]. This is in particular also the case for materials which absorb in the surface layer and which have high thermal conductivity such as metals. However, in any laser material interaction at least a part of the absorbed laser energy is e.g. not removed together with ablated material, is thermalized and remains in the workpiece as residual heat. In this paper, the residual heat after the laser-material interaction and after all phase transitions back to solid state is of particular interest. The fraction *η _{Heat}* of this residual heat relative to the absorbed energy depends on the material properties, the process parameters (such as pulse energy pulse duration, fluence, intensity, laser frequency, intensity distribution), and the process itself (energy coupling, ablation, plasma formation), especially in the case of ultra-short pulsed laser processing with high intensities >10

^{12}W/cm

^{2}[6]. During repetitive processing this residual heat becomes significant if the material has not enough time to cool down sufficiently before the next heat input is applied. The importance of this so-called heat accumulation effect was reported in [7–11] where the consequence of heat accumulation between subsequent pulses on the same spot on the processed surface was identified as one of the major mechanisms that significantly decreased the resulting process quality. Figure 1 shows the typical evolution of the temperature at the origin of the beam on the surface of the processed material (CrNi-Steel, 300 kHz laser frequency, 3 mJ pulse energy, 37% absorptance, 34% of residual heat, and 1-dimensional heat flow) when heat accumulation occurs.

The time between the first and the second pulse is too short to allow the material to cool down to the initial temperature (left arrow). The resulting continuous increase of the remaining temperature when the following pulses arrive (marked with squares in Fig. 1) leads to a significant change of the process, and usually to a reduction in processing quality, when the local temperature exceeds a critical temperature *T _{crit}*, such as for example the melting temperature in metals (Fig. 1, right arrow), or the matrix evaporation temperature in carbon fiber reinforced plastics (CFRP) [7].

#### Heat accumulation scenarios

Two different heat accumulation scenarios have to be distinguished when considering process strategies.

- (a) For short-pulse materials processing working with a static beam when percussion drilling or a quasi-static beam when trepanning very small holes, the heat accumulation effect is caused by the heat input from subsequent laser pulses on the same spot (HAP). In the case of percussion drilling, the total number of pulses required to complete the processing task is determined by the depth of the hole which has to be drilled, and the energy input required to remove the material.
- (b) For processing with a moving beam, such as surface structuring or multi-scan cutting, the HAP effect is caused by the finite number
*N*of subsequent pulses on the same spot of the moving beam. This number of pulses per spot is approximately given by_{Spot}*N*≅□_{Spot}*d*, where_{Spot}· f_{Laser}/ v_{Feed}*d*is the diameter of the laser beam on the surface,_{Spot}*f*is the pulse repetition rate, and_{Laser}*v*is the velocity of the moving beam on the surface. This means that HAP usually can be avoided by moving the laser beam fast enough across the surface, i.e. with a feed rate of_{Feed}*v*≳_{Feed}*d*In addition to HAP, however, the same basic heat accumulation mechanism also occurs for the heat input resulting from subsequent scans over the same spot (HAS) during multi-scan processing [9]. Again, the total number of scans required to complete the process is given by the depth of the structure which has to be achieved or by the material thickness which has to be cut, which defines the total energy input required to remove the material. Hence, depending on the applicable pulse energy, the total number of scans is inherently given for any specific application and cannot be reduced. This might be up to 2100 scans as reported in [10] for cutting of 2 mm thick CFRP. In this case HAS becomes the dominant thermal effect reducing the process quality [9]._{Spot}· f_{Laser}.

To maintain process quality it is crucial to know when the temperature increase reaches its critical value due to heat accumulation in both cases, (a) and (b). For a given average laser power (i.e. pulse energy times laser frequency) it is therefore of particular interest to know the maximum acceptable number of laser pulses or scans (in the following in a more general way called “maximum number of heat inputs”, *N _{HeatInp,maxt}*) after which the temperature increase in the interaction region has reached the specific, material-dependent temperature limit. If this limit is reached, a processing pause must be introduced in order to allow the material to cool down and to avoid detrimental effects resulting from heat accumulation.

#### Heat accumulation equation

Heat accumulation was modelled analytically in detail in [7]. These calculations serve as basis for the following considerations of the limitations imposed by heat accumulation. The model presented in [7] is based on the formalism of Rykalin, described in [12,13] which assumes, that the energy is applied in an infinitesimal short time and volume. This holds for pulsed laser processing of metals and CFRP when the considered time scale is much larger than the pulse duration. When considering heat accumulation the relevant time scales typically are in the range of Microseconds for lasers with MHz repetition rates up to Milliseconds for lasers with KHz repetition rates or multi-scan processing, which both are orders of magnitude larger than the pulse duration of fs-, ps-, or ns-lasers. Furthermore the model presumes processing of materials with a high absorptivity, where the optical penetration depth is much smaller than the thermal diffusion length between two subsequent heat inputs. With increasing time the heat conduction into the material leads to a temperature field which strongly depends on the geometry of the heat flow. Typical examples for heat flow in one dimension (1D) with a plane heat source at the surface, two dimensions (2D) with a line source inside the workpiece, and three dimensions (3D) with a point source on the surface are shown in Fig. 2. The 1D case is for example of interest for large laser beams and for long processing times in thin sheets as will be explained later. The 2D case might apply for drilling of deep holes and the 3D case for small laser beams and long processing times in bulk material. The arrows sketch the directions of the heat flow.

The resulting temperature fields of each pulse can be summed up in time and space when the material parameters are taken as constant. This summing up results in the mechanism of heat accumulation as shown as an example in Fig. 1 for 1D-heat flow. The squares mark the remaining heat-accumulation temperature in the origin just before the next heat input occurs, which is of particular interest in this paper. The evolution of this heat-accumulation temperature is described by the general, so-called “heat accumulation equation” [7]. For a dimensionality *nD* ∈ {1,2,3} it reads

*T*is the temperature increase due to heat accumulation caused by a number of

_{HA,nD}*N*heat inputs in a temporal δ-peak at the origin of the heat source

_{HeatInp}*Q*and with the repetition rate

_{nD}*f*. In contrast to the notation in [7], we modify the term “pulses” into the more general term “heat inputs” to account for the both heat accumulation mechanisms, HAP and HAS.

_{HeatInp}*ρ*is the mass density of the solid or liquid material,

*c*its specific heat capacity,

_{p}*κ*=

*λ*(

_{th}/*ρ c*) the thermal diffusivity,

_{p}*λ*the heat conductivity, and

_{th}*t*is time. In this solution, the material properties

*ρ*,

*c*, and

_{p}*λ*have to be taken as constant with respect to temperature. The heat sources

_{th}*Q*

_{1}_{,}

*Q*

_{2}_{,}and

*Q*

_{3}_{,}for 1-dimensional (

*1D*), 2-dimensional (

*2D*) and 3-dimensional (

*3D*) heat flow are

*A*and ℓ are the area and the length where the energy

*Q*is deposited in the case of

_{Heat}*1D*and

*2D*heat flow, respectively. The factor

*σ*depends on the geometry under consideration.

*σ*= 1 when the deposited heat can flow in the complete surrounding of the source, and

*σ*= 2 when the hat can only flow in one half space. In the examples shown in Fig. 2,

*σ*= 2 in the

*1D*,

*σ*= 1 in the

*2D*, and again

*σ*= 2 in the

*3D*case.

Equations (1) and (2) allow to conveniently calculate the temporal evolution of the heat-accumulation temperature resulting from a sequence of energy inputs. In order to be able to solve Eq. (1) for the number *N _{HeatInp}* of heat which yields a certain temperature increase Δ

*T*, approximation formulas are derived in the following. This will then allow to formulate analytically expressed scaling laws for the maximum number of energy inputs or the maximum average laser power allowed to keep the temperature increase caused by heat accumulation below a given value.

_{HA,nD}## 2. Approximations for large numbers of heat inputs

Explicit analytical expressions for the sum term in (1) were found by approximating the sum with the corresponding integral, plus a constant *C _{nD}*, which depends on the heat flow geometry, i.e. by setting

*C*= −1.46,

_{1}*C*= 0.58, and

_{2}*C*= 2.61 were found numerically by setting

_{3}*N*→□∞ and rounded to two digits for convenience. The relative accuracy of these approximations for the sum is shown in Fig. 3.

_{HeatInp}The relative deviation of the analytical approximation (right expressions in Eqs. (4)) from the exact values (left of Eq. (3)) is < 10% for all three dimensionalities, when *N _{HeatInp}* > 3. For

*N*> 100 the error becomes negligibly small. Inserting the Eqs. (4) into (1) yields the following analytical approximations for the temperature increase due to heat accumulation for the three dimensionalities:

_{HeatInp}## 3. Heat accumulation limits

When the heat inputs are incident at a repetition rate *f _{HeatInp}*, and with an absorbed part

*η*of the average incident power

_{abs}*f*·

_{HeatInp}*E*=

_{Inc,}*P*, each single residual heat input remaining in the workpiece is given by

_{Inc,av}*η*is a function of both, the laser parameters (such as pulse duration, fluence above threshold, intensity distribution, and wavelength) and the processing parameters (such as material, structure size, and structure depth) it has usually to be determined experimentally. It is noted that the assumption of a constant

_{Heat}*η*might restrict the parameter range where the model is valid, especially in the case of ultrashort-pulse laser processing. Furthermore it is noted that

_{Heat}*P*equals the incident average laser power only in the case of HAP. For HAS

_{Inc,av}*P*is the time average of the laser power that is incident on the processed contour and is smaller than the average laser power in the case of open contours due to the positioning time.

_{Inc,av}The thermo-mechanical properties of the processed material as well as *η _{Heat}*, and

*η*, can be combined in the expression

_{abs}*C*decreases with increasing

_{Mat,nD}*η*and

_{Heat}*η*and increases with increasing thermal conductivity

_{abs}*λ*. It is further seen that${C}_{Mat,1}\propto \sqrt{{c}_{p}\rho}$,

_{th}*C*is independent of

_{Mat,2}*ρ*and

*c*, and ${C}_{Mat,3}\propto 1/\sqrt{{c}_{p}\rho}$.

_{p}The limiting average incident power *P _{Inc,Limit,nD}* at which a given maximum acceptable temperature increase ∆

*T*is reached for a given total number

_{Limit}*N*of heat inputs, is found by inserting Eqs. (2a-c), (6), and (7) into Eqs. (5a-c), setting

_{tot}*N*=

_{HeatInp}*N*, and solving for the average incident power, yielding

_{tot}If, however, the applied average incident power *P _{Inc,av}* >

*P*, pauses have to be introduced after a certain number

_{Inc,Limit,nD}*N*of heat inputs to keep the temperature increase below the acceptable limit. Hence, for given processing parameters

_{Limit}*N*is the number of heat inputs, at which the temperature increase accumulated until just before the next heat input reaches ∆

_{Limit}*T*.

_{Limit}*N*is found from Eqs. (8a-c) by replacing

_{Limit}*P*with

_{Inc,Limit,nD}*P*and

_{Inc,av}*N*with

_{tot}*N*, and solving for

_{Limit}*N*, yielding

_{Limit}*P*as given in Eqs. (8). With this assumption the duration of these pauses is

_{Inc,Limit,nD}It should be pointed out that in practice *N _{tot}* is the total number of energy inputs which is required for the desired application, e.g. the total number of pulses needed for drilling a hole, or the total number of scans needed to cut the material. It is given by

*V*is the volume of the processed material,

_{Proc}*h*is the volume specific energy for the desired process and

_{Proc}*η*is the process efficiency (fraction of incident laser energy that contributes to the process) as defined in [14]. Replacing

_{Proc}*P*with

_{Inc,av}*P*in (12), inserting this into (8), and solving for

_{Inc,Limit,nD}*P*therefore allows to calculate the maximum average incident power that is applicable to prevent a temperature increase from exceeding ∆

_{Inc,Limit,nD}*T*for any given process. Hence, with materials properties, geometry, volume specific process energy, and process efficiency known and with a given frequency of the heat inputs this uniquely defines the maximum applicable average incident laser power allowed for the process to keep the temperature increase below ∆

_{Limit}*T*.

_{Limit}## 4. Frequency of heat inputs and average incident power

In the case of HAP, the frequency of the heat inputs and the average incident laser power simply equal the laser pulse frequency and the average laser power, respectively, i.e.

In the case of HAS, when processing contours, the frequency of the heat inputs is given by*f*= (

_{HeatInp}*t*+

_{Scan}*t*)

_{Pos}^{−1}, where

*t*is the positioning time between subsequent scans, if the contour is not closed, and

_{Pos}*t*= ℓ

_{Scan}*/*

_{Contour}*v*is the time required to scan the contour given by the length of the contour ℓ

_{Feed}*, and the feed velocity*

_{Contour}*v*, i.e.The average incident power in the case of HAS is

_{Feed}*P*=

_{Inc,av}*f*·

_{HeatInp}*E*, where

_{Contour}*E*=

_{Contour}*E*ℓ

_{Pulse}·f_{Laser}·*is the total energy incident during one scan along the complete contour, yielding an average incident power*

_{Contour}/ v_{Feed}*P*equals

_{Inc,av}*P*for closed contours, i.e when

_{Laser}*t*= 0. It is noted that for other process geometries such as ablation of areas with special hatching patterns, modified expressions might apply for Eqs. (14) and (15).

_{Pos}Two experimental examples to illustrate the validity of the above equations and assumptions are given in the following.

## 5. Materials and methods

For experimental verification of the model the kW-class ultrafast laser of the IFSW [16] was used to ablate grooves on CrNi-steel (1.4310) and cut 2 mm thick CFRP plates. The CFRP consisted of Toray T700S-12k carbon fibers with a RTM 6 matrix, which is a monocomponent resin. The carbon fibers were arranged in different layers with the orientation [0/90, −45/+45, 90/0, 0/90, −45/+45, 90/0]. The volume fraction of carbon fibers was 50%. The laser parameters used for the experiments are summarizes in Table 1.

## 6. Surface ablation of CrNi-steel as an example for 1D HAP

Surface ablation of CrNi-steel as an example for heat accumulation between pulses (HAP) with one-dimensional (1D) heat flow was investigated experimentally using the laser parameters listed in Table 1. The beam was moved with a fast scanner over the surface with the velocity *v _{Feed}* in order to create shallow grooves in the surface of the samples. The large focus diameter allows to assume 1D heat flow as shown in Fig. 2 (left) as long as

*d*is larger than the thermal diffusion length

_{Focus}*v*with a number of pulses per spot according to

_{Feed}*N*the laser power was increased until a liquid surface in the grooves could clearly be identified by visual inspection. Figure 4 (left and center) shows two microscope images of typical results obtained with single scans with two different average laser powers and

_{Pulses}= d_{Focus}·f_{Laser}/ v_{Feed}*v*= 1 m/s, which corresponded to 39 pulses per spot. The shiny surface of solidified liquid inside the groove that was produced with an average laser power of 610 W and the much deeper groove seen in the corresponding height-plot gives evidence that the process has significantly changed compared to the one observed with 306 W of laser power. Figure 4 (right) shows the minimum number of pulses at which a liquid surface is observed as a function of the average laser power. The number of pulses incident on one point in the groove was varied by changing the feed rate

_{Feed}*v*. For a given number of pulses per spot on the grove, the data points show the mean value between the largest average power

_{Feed}*P*applied that did not lead to a noticeable solidified liquid and deep grooves and the smallest average power

_{NL}*P*that clearly produced a visible solidified liquid and a deep groove. The error bars extend from

_{L}*P*to

_{NL}*P*. The line shows the theoretically expected relation according to 1D-model given by Eq. (9a) using the material parameters

_{L}*ρ*= 7900 kg/m

^{3},

*λ*= 25 W/mK (@800K),

_{th}*c*= 559 J/kgK (@800K), and

_{p}*η*= 37% yielding

_{abs}*C*= 7.2·10

_{Mat,1}^{5}J/s

^{0.5}/m

^{2}/K. The limiting increase of temperature was set to Δ

*T*= 1500 K, and σ = 2 as the heat source is on the surface. The only fit parameter was

_{Limit}*η*= 14%, which was chosen to yield the best fit to the data points.

_{Heat}The good agreement between the theoretical curve given by Eq. (9a) and the experimental findings confirm, firstly, the predominantly one-dimensional heat flow during the relevant time scale, secondly, the assumption of constant material properties, and thirdly, the assumption of a constant fraction *η _{Heat}* of the absorbed energy being converted to residual heat for at least the range of parameters considered in this experiment.

## 7. Multi-scan cutting of thin sheets as an example for 1D HAS

As an example for heat accumulation caused by multiple scans (HAS), fast multi-scan ultra-short-pulsed laser cutting of a thin sheet is now considered as sketched in Fig. 5. This is an important application for generating cuts with very high quality in difficult materials such as thin copper sheets with plastic layers or carbon fiber reinforced plastics (CFRP). Multi-scan cutting is suffering in particular from HAS, when a large number of passes (>2000) over the same contour is required to cut through the whole material [10]. Provided that the following preconditions apply, it can again be assumed that the heat flow away from the cutting kerf into the thin sheet is predominantly 1D.

#### 1D approximation precondition

One condition to be able to apply the approximation of 1-dimensional heat flow is that the temperature gradient along the contour $\overrightarrow{\nabla}{T}_{Cont}$ is small due to the high scan speed, i.e. when the focus diameter ${d}_{Focus}\cdot {v}_{Feed}>>\kappa $ [15]. Furthermore also the temperature gradient $\overrightarrow{\nabla}{T}_{z}$from the top to the bottom of the material needs to be small, which can be assumed to be the case after a “large number” of scans, i.e. a “long” processing time *t _{Proc}* =

*N*·

_{Scans}*t*, and the bottom of the sheet is not in contact with a heat sink. More precisely this can be assumed to be the case when the thermal diffusion length ℓ

_{Scan}*significantly exceeds the material thickness*

_{Diff}*d*,

_{Mat}In the geometry shown in Fig. 5, σ = 1 as the heat flows into both sides of the heated area. For metals, *κ* is in the range of about 10^{−4} m^{2}/s to 10^{−5} m^{2}/s, whereas for CFRP it is in the range of about 10^{−4} m^{2}/s (along the fibers) to 10^{−6} m^{2}/s (perpendicular to the fibers and through the matrix). In order to ensure 1D heat flow according to the above approximations, the processing time for e.g. a 2 mm thick sample must therefore be larger than about 10 ms to 100 ms for metals and about 1 s for CFRP. For comparison, the duration for 100 passes over a 100 mm long contour at the scanning speed of 10 m/s is 1 s. To ensure small temperature gradients along the contour, the scan speed ${v}_{Feed}>>\kappa /{d}_{Focus}$should amount to ${v}_{Feed}>>0.1\text{\hspace{0.17em}}\text{m/s}$ for *κ* = 10^{−6} m^{2}/s and to ${v}_{Feed}>>1\text{\hspace{0.17em}}\text{m/s}$ for *κ* = 10^{−4} m^{2}/s, assuming a focus diameter of 100 µm. In [11] it was shown that these assumptions hold and that calculating the 1D-heat flow using averaged material properties leads to results that are in excellent agreement with experimental data when cutting CFRP with a cw laser.

#### Limits of average power and maximum allowed number of scans

For the process geometry described in Fig. 5, the Eqs. (8a) and (9a) resulting from the model with 1D heat flow and Eqs. (14) and (15) for the multi-scan heat input apply. The area *A* over which the power is dissipated is given by *A* = ℓ* _{Contour}* ·

*d*and the process volume is about

_{Mat}*V*≅

_{Proc}*d*ℓ

_{Focus}·*Inserting this and Eqs. (12), (14), (15) with*

_{Contour}· d_{Mat}.*P*replaced by

_{Laser}*P*and σ = 1 into Eq. (8a) yields the limit

_{Inc,Limit,1}*T*. The maximum allowed number of scans for a given laser power is found accordingly from Eq. (9a)

_{Limit}*P*, i.e. if

_{Inc,Limit,1}*N*<

_{Scans,Limit,1}*N*, the duration and number of the necessary processing pauses can be determined with Eqs. (10) and (11), respectively.

_{tot}#### HAS scaling laws for cutting of thin sheets

As *C _{1}*/2 ≅ 1 and both, C

_{1}≪ $2\sqrt{{N}_{Tot}}$and

*N*≫ 1 usually apply when the abovementioned preconditions that lead to 1D heat flow are fulfilled, simple HAS scaling laws can be derived from (18) and (19). For a constant process efficiency and closed contours, where

_{Scans,Limit,1}*t*= 0, and neglecting

_{Pos}*C*

_{1}one finds

#### Multi-scan cutting of CFRP with a 1.1 kW ps-laser

The validity of the derived approximations for *N _{Limit,1}* due to HAS were verified with multi-scan cutting of circles with a diameter of 50 mm in 2 mm thick CFRP [10] described in the section “materials and methods” using the processing parameters summarized in Table 1. The beam was moved with a fast scanner with a maximum scan speed of

*v*30 m/s. To avoid HAP the minimum scan speed was set to 3 m/s corresponding to 10 pulses per spot.

_{Feed}=The values of *N _{Scans,Limit,1}* were experimentally determined by increasing the number of scans at each processing parameter set, until a clear increase of the damage of the matrix material was visible in cross-sections inspected with an optical microscope. The measured

*N*as a function of the feed rate and as a function of the average power are shown as squares in Fig. 6 left and right, respectively. The solid lines show the calculated relations as given by Eq. (19) assuming ∆

_{Scans,Limit,1}*T*= 400 °C for the damage by evaporation of the matrix material and using the identical constant

_{Limit}*C*= 9.0·10

_{Mat,1}^{3}J/s

^{0.5}/m

^{2}/K for both graphs. This value for the material constant

*C*is motivated by assuming

_{Mat,1}*η*= 90%,

_{Abs}*η*= 90%,

_{Heat}*ρ*= 1610 kg/m

^{3}and

*c*= 906 J/kg/K (the average of the material values according to the volume fraction), and

_{p}*λ*= 2.9 W/m/K (the geometrical average of the thermal conductivity of 50 W/m/K, 5 W/m/K, and 0.1 W/m/K for Carbon parallel to the fibers, Carbon perpendicular to the fibers, and plastic, respectively (see also [11] for comparison).

_{th}Strictly speaking and in view of the many uncertainties concerning the actual material parameters, *C _{Mat}*

_{,1}here has the role of a fitting parameter. Nonetheless, the good agreement between the theoretical model and the experimental results shown in Fig. 6 confirm the basically inverse quadratic scaling of the maximum allowed number of scans with respect to the average power (left) and the predominantly linear scaling of the maximum allowed number of scans with the feed rate (right) as explicitly seen from the approximation given in Eq. (21). It should be mentioned, that

*C*

_{Mat}_{,1}is identical for the calculated limits in both, Fig. 6 (a) and (b). Furthermore, the large error bars have to be taken into account to not be misled by the apparently constant experimental values between about 8 m/s and 20 m/s in Fig. 6 (b).

Using the above material constants, *h _{Proc}* = 45 J/mm

^{3},

*η*= 10%, and the contour length of 157 mm, the upper limit of the average laser power given by Eq. (20) yields 24.8 W, which applies when the whole sample is to be cut through without pauses. By inserting this average power into Eq. (11) and Eq. (10) one finds 38 pauses with a duration of 15 are required, if the available laser power of 1.1 kW shall be applied instead. In the experiments that were performed before developing the present theory, very high quality cuts were achieved by inserting pauses with a duration of 60 s after every 200 passes [10].

_{Proc}## 8. Conclusion

The approximation formulas derived in this paper for the three different dimensionalities *nD* ∈ {1,2,3} of the heat flow allow to predict the maximum allowed average power *P _{Limit,nD}* and the corresponding maximum number of heat inputs

*N*that can be applied for a given process before the local temperature increase due to heat accumulation reaches a specified limit ∆

_{Limit,nD}*T*.

_{Limit}The approximation formulas show that the maximum number *N _{Limit,1}* of consecutively applicable heat inputs in the important case of 1D heat flow unfavorably decreases with the inverse of the square of the average laser power, but increases linearly with the feed rate and the contour length. If more than

*N*scans are required to produce a certain structure, pauses have to be introduced into the process to avoid thermal damage. The duration and the number of these pauses can be calculated with Eqs. (10) and (11), respectively.

_{Limit,1}These findings agree very well with experimental results: Two examples of practical relevance, surface ablation of CrNi-steel as an example for heat accumulation due to pulses (HAP) and cutting of thin CFRP-sheets as an example for heat accumulation due to multiple scans (HAS) were experimentally investigated, and an excellent agreement of the scaling of the limiting number of heat inputs with respect to average power and feed rate was found.

A major and important task for future work will be to model and determine *C _{Mat,nD}*, and in particular

*η*, in order to specify the basic physical limits for the maximum possible number of heat inputs and the maximum average power.

_{Heat}## Funding

German Research Foundation (DFG) (grant GR 3172/17-1); Russian Foundation of Basic Research (grant 15-02-91347); and the BMBF (grant 13N13931).

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