Objective Control of microstructure and residual stress, and optimization of process parameters based on the temperature field are important methods to improve mechanical properties of the cladding parts. A relatively accurate laser heat source model should be provided to reconstruct the temperature field during the formation of the cladding layer. First, an accurate mathematical model of hollow ring laser heat source, which should be verified by numerical simulation and the experiment results, needs to be constructed to obtain the fundamentals for a practical temperature field, microstructure distribution, residual stress control, and process parameters optimization. The temperature distribution inside the molten pool could be reflected precisely by an appropriate heat source model to simplify the numerical calculation and deeply explore the influence of temperature gradient and cooling rate on residual stress and actual solidification microstructure during the solidification and cooling process. Since Goldak proposed the classical double ellipsoid heat source in 1984, a wide variety of heat source equations have been generated based on the mathematical modeling evolution of solid heat sources. However, the mathematical model of the hollow ring heat source based on the inside-laser coaxial powder-feeding system was rarely studied. A simple and practical heat source equation should be established to verify the reliability of the model through the combination of experiment and theory analysis.

Methods Based on the finite element software called ABAQUS and the hollow ring laser system, the energy distribution discipline of the hollow ring laser was verified by the probe beam quality analyzer, and a formal hollow ring laser heat source equation was proposed. The energy density determined by numerical analysis was used to establish the final mathematical equation. Response surface method (RSM) was adopted after the hollow ring laser heat source equation was established. The fitting equation of molten pool size under mutative process parameters was conducted, and the three-dimensional size of the heat source model was parameterized through experiments.

Also, a laser-melting deposition model with certain thermal-physical properties was established, and a mobile heat source subroutine was written in Fortran language. The subroutine was called in the ABAQUS software, and the heat source was loaded by the life-and-death element method to simulate the temperature field and thermal history of a single wall under mutative cladding process parameters. To verify the reliability of the heat source model, the actual temperature of the melting pool during the experiment was measured by K-type thermocouple, paperless data recorder (HIOKI LR8501), and infrared camera monitoring equipment (InfraTec, VarioCam^® hr head).

Results and Discussions The energy distribution of the hollow ring laser indicated a Gaussian distribution in the annular region, and the energy in the middle was higher than that on both sides of the annular region, as shown in Fig. 2(a). As a result, a “Gaussian-like” heat source distribution equation was proposed. The hollow ring laser morphology equation was obtained [Eq. (1)] after a Gaussian curve with a certain displacement was biased and rotated around the central axis [Fig. 2(b)]. In Eqs. (2)--(6), the peak power density of the hollow ring laser was calculated step by step through theoretical analysis, and the final mathematical analysis expression was further obtained [Eq. (7)]. In Eq. (7), parameters of multiple molten pools were used. Through response surface method (RSM) and laser cladding experiment, polynomial equations of molten pool width and depth on laser power, scanning speed, and defocus were fitted [Eqs. (9)--(10)]. The processing interval of the response surface method (RSM) is shown in Table 2. The physical model of laser cladding was established by the ABAQUS software, and the heat source equation subroutine was recorded to simulate the temperature field and thermal history in the process of laser cladding. The simulation results are verified in Fig. 7 and Fig. 10. Thermocouple (Fig. 9) and infrared camera were applied for verification, good fitting results were indicated, and the hollow ring laser heat source equation possessed certain accuracy and applicability.

Conclusions The mathematical modeling of the three-dimensional hollow ring laser heat source was constructed to obtain an analytical formula using the idea of the normal distribution of revolution. The morphological parameters of the hollow ring laser heat source were determined by the shape of the molten pool and the value of the energy peak of the laser spot under mutative process parameters. The simulation results of the energy distribution of the heat source model indicated that the peak energy intensity was located at the edge instead of the center, and the vertical cross-section presented a “saddle”-shaped distribution. The measured results agreed well with the theory simulation, which confirmed the validity of the mathematical model.

The finite element simulation of the temperature field of the hollow ring heat source indicated that temperature peak distribution was close to the edge, which correspond to the mathematical model results. Due to the high energy at the edge, it could compensate the weak part of the central heat input to promote the uniform temperature distribution, which was consistent with the theoretical analysis of the energy intensity of the ring light source. The temperature peak continuously reduced, and the temperature trough was increasingly influenced by heat accumulation and conduction. The thermal history was simulated by the ABAQUS software, and its simulation curve was also consistent with the measured value, which confirmed that the hollow ring heat source model possessed good accuracy and applicability.