**Elham Mirkoohi 1,\*, Jinqiang Ning 1, Peter Bocchini 2, Omar Fergani 3, Kuo-Ning Chiang <sup>4</sup> and Steven Y. Liang <sup>1</sup>**


Received: 12 August 2018; Accepted: 8 September 2018; Published: 12 September 2018

**Abstract:** A physics-based analytical model is proposed in order to predict the temperature profile during metal additive manufacturing (AM) processes, by considering the effects of temperature history in each layer, temperature-sensitivity of material properties and latent heat. The moving heat source analysis is used in order to predict the temperature distribution inside a semi-infinite solid material. The laser thermal energy deposited into a control volume is absorbed by the material thermodynamic latent heat and conducted through the contacting solid boundaries. The analytical model takes in to account the typical multi-layer aspect of additive manufacturing processes for the first time. The modeling of the problem involving multiple layers is of great importance because the thermal interactions of successive layers affect the temperature gradients, which govern the heat transfer and thermal stress development mechanisms. The temperature profile is calculated for isotropic and homogeneous material. The proposed model can be used to predict the temperature in laser-based metal additive manufacturing configurations of either direct metal deposition or selective laser melting. A numerical analysis is also conducted to simulate the temperature profile in metal AM. These two models are compared with experimental results. The proposed model also well captured the melt pool geometry as it is compared to experimental values. In order to emphasize the importance of solving the problem considering multiple layers, the peak temperature considering the layer addition and peak temperature not considering the layer addition are compared. The results show that considering the layer addition aspect of metal additive manufacturing can help to better predict the surface temperature and melt pool geometry. An analysis is conducted to show the importance of considering the temperature sensitivity of material properties in predicting temperature. A comparison of the computational time is also provided for analytical and numerical modeling. Based on the obtained results, it appears that the proposed analytical method provides an effective and accurate method to predict the temperature in metal AM.

**Keywords:** metal additive manufacturing; analytical model; temperature prediction; FEA; melt pool geometry

#### **1. Introduction**

Metal additive manufacturing (AM) is a "process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies" [1]. Additive manufacturing (AM) processes have potential to be the pillar of the next industrial revolution. AM can be used to improve existing manufacturing processes and rapidly introduce new prototypes and products [2,3]. It also offers the potential to spin off entirely new industries and lead to new production methods [4]. AM offers design flexibility, the ability to produce complex parts, and lower cost due to the reduced requirement of materials and decreased lead time.

AM may also hold the potential for the repair and replacement of existing plant components [5,6]. Results from on-line monitoring and complimentary non-destructive evaluation (NDE) inspections can provide indications of component health and enable repair or replacement prior to a forced outage situation. As an example, imaging tools and software can be leveraged to create a digital image of the damaged component which can be used to 3-D print a new one. This can be especially advantageous if the component is no longer in production and/or would require a long-lead time to fabricate.

There are many challenges that necessitate being focused on this field in order to expedite the adoption of AM as an advanced manufacturing technology. The issues in this field can be classified in to: the distortion, fatigue, defects, and residual stress of the manufactured parts [7–9]. The modeling in additive manufacturing technology is a key to the advancement of the field due to obstacles in in-situ measurements of temperature, thermal stress, residual stress, and distortion. The available knowledge and technology to-date on the descriptions and predictions of the metal AM process have been fragmented, mostly driven by phenomenological or numerical observations [10–13], and primarily limited to macroscopic analysis in nature [14,15], thus restricting the full capability and potential of the AM process. Using numerical methods and experiments are not just expensive, but also time-consuming. On the other hand, the physics-based analytical modeling eliminates all the above-mentioned difficulties and can help to better understand the physical aspects of the metal additive manufacturing process.

The most important part of the metal AM process modeling and prediction is the prediction of the temperature induced by laser since the non-uniform temperature will cause the thermal stress to appear in the structure. As a result of thermal stress in the build material, the tensile residual stress on the surface accelerates the crack propagation and growth [16,17]. Several researchers worked on predicting the temperature profile during the additive manufacturing process. Fergani et al. introduced an analytical model to predict the temperature in the direct metal deposition process. They predict the temperature using a moving point heat source analysis. In this work, the effect of material temperature sensitivity is ignored [18]. C.Y. Yap et al. have proposed an analytical model to predict the energy input required to process different metallic materials for selective laser melting (SLM) process. The model holds many assumptions, such as a semi-circular cross-section for melt tracks, temperature-independent specific heat, no heat loss to the surroundings and absorptance of material to laser irradiation based on bulk material properties. The melting, solidification, and solid-state phase change is also not considered in their model. The simplified model is able to predict the required energy input within an order of magnitude and provide researchers with a useful model to estimate the optimal SLM parameters [19].

Predicting the temperature precisely in metal AM is the pillar for predicting the thermal stress, residual stress, and part distortion. The non-uniform heating during AM processes may lead to the thermal stress. The large thermal gradient and cooling rate during the metal AM processes can generate complex microstructures in the build material [20]. Kelly et al. used the temperature in the AM processes in order to predict the microstructure evolution in the build part. In their work, the melting/solidification phase change is not considered [21]. Hoadley and Rappaz introduced a 2D quasi-stationary model to predict the temperature in the laser cladding process. Their research focused on the influence of the laser speed and power on the layer thickness [22]. Toyserkani et al. developed a 3D model, their proposed model tried to solve the heat problem using a coupled multi-physics

system. They have used thermal analysis in order to predict the melt pool shape [23]. Cao and Ayalew have developed a control-oriented multiple input multiple output modeling of the laser-aided powder deposition processes. The objective of their work is to control the height and the temperature of a layer. Their investigation described the essential role of temperature modeling to control the quality of the final part [24]. Hitzler et al. investigated the influence of scan strategy on material characteristics, such as strength, hardness, and young's modulus [25,26]. Rashid et al. worked on the effect of scan strategy on density and metallurgical properties of a build part during the selective laser melting (SLM) process. Their results showed that parts which are made using a single scan have higher levels of hardness than parts that are made by scanning each layer twice [27].

Due to the complexity of the additive manufacturing processes, such as direct metal deposition (DMD), and SLM, not only is it time-consuming to do the experiments in order to capture the physical aspects of the metal AM processes, but it is also expensive. In the past few decades, the numerical simulations appear to be the only effective way to achieve an understanding of metal additive manufacturing processes [28,29]. The numerical methods have low computational efficiency and cannot capture all the physical aspects of the metal AM processes. On the other hand, physics-based analytical models provide a deep understanding of the physical concepts of AM. The analytical solutions have the potential to predict the key AM attributes in ways significantly faster than finite element method (FEM) simulations, by two or more orders of magnitudes [30]. Efficient and accurate predictions are therefore enabled, and the optimization of metal additive manufacturing processes which would be too complicated to cope with by the majority of other studies, who have resorted to empirical and FEM attempts. It also reduces, if not completely eliminates, the need for a costly and lengthy trial and error developmental curve for new material and components [31]. A complete build analysis with high accuracy becomes computationally tractable using the analytical model.

The AM process is a coupling of many physical phenomena such as heat transfer, fluid dynamics, phase transformation and solid mechanics. Moreover, the transient nature of heat transfer phenomena and interaction of layers make it a complicated multi-physics problem. Many researchers tried to predict the temperature in metal additive manufacturing, but each of them has several limitations. For example, not considering the temperature dependent material properties, the melting/solidification phase change, and layering aspect of metal AM. The key advantage of the proposed model is the ability to capture the most physical phenomena in metal AM, which has mostly been ignored in previous works. In this work, all the above-mentioned limitations are considered in the analytical solution of temperature. It is assumed that the thermal properties of material are temperature dependent. The melting, solidification, and solid-state phase change is included by using the modified specific heat, which relates the specific heat and latent heat of fusion. As each layer is deposited, the temperature profile is predicted using the moving heat source analysis. The laser thermal energy deposited into a control volume is absorbed by the material thermodynamic latent heat and conducted through the contacting solid boundaries. The deposited energy on the first layer introduces a thermal profile. The thermal behavior in the second pass of the laser will not be the same as the first pass since the thermal interaction of the successive layers have an influence on heat transfer. The melt pool geometry is well captured based on the proposed model, since it considers most of the previous lacks.

The outline of the paper is as follows. Section 2 presents the mathematical and practical details of the proposed analytical and numerical models. Section 3 presents detail of the experimental work which is used for validation of the proposed model, the results and a detailed discussion about the obtained results. Last but not least, Section 4 presents the conclusion of this research.

## **2. Approach and Methodology**
