**1. Introduction**

Metal additive manufacturing (AM) technologies are gaining increasing attention from both academia and industry because of the advantages they offer over conventional metal manufacturing techniques. These AM technologies enable the manufacture of near net shape complex structures and functionally graded components, which are impossible to fabricate through conventional methods.

In metal AM, an energy source (e.g., laser, electron beam, or plasma arc) irradiates and melts the surface of the substrate while filler material is added, building three-dimensional functional parts layer by layer. The advantages of using laser-based AM compared to conventional metallurgy and subtractive manufacturing were listed by Yan et al. [1], including a finer grain size, a small heat-affected zone, the possibility to process difficult-to-machine or refractory materials, as well as to combine materials, among others. Regarding the mechanical properties, Attar et al. [2] performed a comparative study of commercially pure titanium manufactured by laser-based AM processes and obtained comparable or even better mechanical properties than cast material. Much research has been devoted to deepening the understanding of laser-based metal AM by combining experimentation and modeling. In such works, the full characterization of the manufactured parts in terms of both mechanical and thermal properties is highly important.

Numerous investigations have focused on the mechanical properties of AM parts. For instance, Zhong et al. [3] presented an analysis of the mechanical properties and internal defects encountered in AM Inconel 718. Zhang et al. [4] studied the microstructures that developed in di fferent zones of the substrate and deposited material, and Sun et al. [5] investigated the importance of stress-relieving treatments of AM AISI 4340 steel.

In addition, several works have modeled laser-based AM processes and studied their heat transfer characteristics. Roberts et al. [6] reported that the thermal conductivity value depends on factors such as the porosity of the material. Many models of laser-based powder bed fusion have been proposed to define the relationship between the thermal conductivities of powder-shaped and solid metals. All authors agree that the e ffective thermal conductivity of loose metallic powder is defined by the gas in the pores. Rombouts et al. [7] studied the e ffective thermal conductivity of the powder bed and concluded that its value is almost independent of the material composition and depends mainly on the size and morphology of the particles and the void fraction. Wei et al. [8] investigated five metal powders for powder bed additive manufacturing (Inconel 718, 17-4 stainless steel, Inconel 625, Ti-6Al-4V and AISI 316L) and concluded that the pressure and composition of the gas between the metallic particles have a significant e ffect on the thermal conductivity of the powder. Cernuschi et al. [9] calculated the thermal conductivity and density of the porous material using the Maxwell model.

Foteinopoulos et al. [10] reported an increase in the accuracy of the thermal model by assuming that the material's thermal properties, including thermal conductivity, are temperature dependent. In the same direction, Li et al. [11] found that, by neglecting the temperature dependence of the material properties, the size and volume of the melt pool were overpredicted. Although both Foteinopoulos et al. and Li et al. have considered the thermal conductivity reduction in the metallic powder (where the influence of the porosity is considered), once the material is melted, the tabulated value of the thermal conductivity of the cast material is used.

To the best of our knowledge, none of the previously published works considered that the thermal conductivity of AM parts may di ffer from that of the cast material. However, factors such as the existence of pores and micro-cracks a ffect the e ffective thermal conductivity of the manufactured part. Furthermore, the high cooling rate of laser processes (104–107 <sup>K</sup>·s<sup>−</sup>1), in comparison with the much lower rate in casting (1–10 <sup>K</sup>·s<sup>−</sup>1), influences the grain size that develops. In fact, Zhang et al. [12] reported that the grain size is much finer in parts produced by laser processes than that by casting, thus impacting the e ffective thermal conductivity of the AM part.

Therefore, this paper presents a study of the e ffective thermal conductivity of laser-deposited AISI H13 tool steel as a function of depth. In this work, the thermal di ffusivity of the deposited material is measured experimentally. The accuracy of the methods employed to measure this thermal property enables the calculation of thermal conductivity from the di ffusivity data. In addition, the impact of this issue on industrial applications is illustrated by means of a case study of the hot stamping process.

Hot stamping, also known as press hardening, is a process in which an ultra-high-strength steel blank is simultaneously formed and quenched. For this purpose, the blank is heated above the austenitic temperature, approximately 950 ◦C, and cooled at rates above 27 K·s<sup>−</sup><sup>1</sup> to ensure a complete martensitic phase transformation.

As stated by Shan et al. [13], the cooling of the blank consumes almost 30% of the total cycle time required to form and quench the material. According to Chen et al. [14], one approach to reducing the cycle time is to increase the thermal conductivity of the hot stamping tools, because it ensures quick heat transfer between the tools and the stamped part. In this direction, Directed Energy Deposition (DED) has arisen as an alternative to enhance the cooling performance of the tools. On the one hand, DED enables the manufacture of conformal cooling channels that can follow the surface of the tools, therefore avoiding hot spots. On the other hand, DED allows the combination of di fferent materials to produce bimetallic tools. Materials with high thermal conductivity are used in the core of the tools, and high-resistance tool steel is used as a coating to withstand the high pressures and temperatures that are reached in the hot stamping process.

In the cooling stage of the hot stamping process, the thermal conductivity of the hot stamping tools is an important factor in the prediction of the temperature distribution within the blank and the cooling rate. This fact was highlighted by Karbasian and Tekkaya [15]. Different software programs have been developed to model the hot stamping process, such as LS-DYNA, Auto-Form, and PamStamp; however, all of these require the material data as inputs. Therefore, if DED-manufactured tools are to be used in hot stamping, their effective thermal properties must be known. Hence, defining the thermal conductivity of the DED-manufactured material is essential for determining the cooling capability of the tools. Many authors have developed diverse models of different additive manufacturing processes. Denlinger et al. [16] developed a thermomechanical model of electron beam deposition aimed at large parts, while Mukherjee et al. [17] studied the mitigation of thermal distortion during AM using a numerical heat transfer and fluid flow model. Peyre et al. [18] developed an analytical and numerical model of laser-aided DED, and Shi et al. [19] proposed a three-dimensional finite element model to investigate the effects of laser processing parameters on the thermal behavior and melting/solidification mechanism during selective laser melting. Regardless of the modeled process, all of them consider that properties such as thermal conductivity and specific heat are temperature dependent. However, all of them consider the thermal conductivity of only the cast material, neglecting possible variations caused by the manufacturing process to which the tool material has been subjected. This omission motivated the selection of the hot stamping process as the case study presented here.

## **2. Materials and Methods**
