Analytical Modeling of Residual Stress in Laser Powder Bed Fusion Considering Volume Conservation in Plastic Deformation

Abstract

Residual stress (RS) is the most challenging problem in metal additive manufacturing (AM) since the build-up of high tensile RS may influence the fatigue life, corrosion resistance, crack initiation, and failure of the additively manufactured components. While tensile RS is inherent in all the AM processes, fast and accurate prediction of the stress state within the part is extremely valuable and results in optimization of the process parameters to achieve a desired RS and control of the build process. This paper proposes a physics-based analytical model to rapidly and accurately predict the RS within the additively manufactured part. In this model, a transient moving point heat source (HS) is utilized to determine the temperature field. Due to the high temperature gradient within the proximity of the melt pool area, the material experiences high thermal stress. Thermal stress is calculated by combining three sources of stresses known as stresses due to the body forces, normal tension, and hydrostatic stress in a homogeneous semi-infinite medium. The thermal stress determines the RS state within the part. Consequently, by taking the thermal stress history as an input, both the in-plane and out of plane RS distributions are found from the incremental plasticity and kinematic hardening behavior of the metal by considering volume conservation in plastic deformation in coupling with the equilibrium and compatibility conditions. In this modeling, material properties are temperature-sensitive since the steep temperature gradient varies the properties significantly. Moreover, the energy needed for the solid-state phase transition is reflected by modifying the specific heat employing the latent heat of fusion. Furthermore, the multi-layer and multi-scan aspects of metal AM are considered by including the temperature history from previous layers and scans. Results from the analytical RS model presented excellent agreement with XRD measurements employed to determine the RS in the Ti-6Al-4V specimens.

On the Applicability of Approximate Rolling and Sliding Contact Algorithms in Anisothermal Problems with Thermal Softening

“…Secondly, the loading they prescribe to the half-space's surface cannot be reconstructed easily, since it results from extensive cutting force models that usually require in-process measurement data as input. However, parameters for the Johnson-Cook law are provided in the recent study of Mirkoohi et al [17], corresponding to h ≈ 650 GPa and n = 0.45 within our notation, and indeed, highly oscillatory residual stress approximations are reported even though the thermomechanical properties, the temperature, and the elastic input stresses evidently change smoothly along the depth. It should be noted that Mirkoohi et al [17] use a slightly modified version of McDowell's hybrid algorithm; however, their formulation lacks a consistent treatment of thermal softening as proposed in this paper (which is also the case in other earlier studies, e.g., [10,11]).…”

“…However, parameters for the Johnson-Cook law are provided in the recent study of Mirkoohi et al [17], corresponding to h ≈ 650 GPa and n = 0.45 within our notation, and indeed, highly oscillatory residual stress approximations are reported even though the thermomechanical properties, the temperature, and the elastic input stresses evidently change smoothly along the depth. It should be noted that Mirkoohi et al [17] use a slightly modified version of McDowell's hybrid algorithm; however, their formulation lacks a consistent treatment of thermal softening as proposed in this paper (which is also the case in other earlier studies, e.g., [10,11]). Thus, they effectively set h T = 0 within our notation, in opposition to the Johnson-Cook model they consider, which at least for our parameter set noticeably impacts the agreement of the stress response with control results obtained by the mechanically equivalent FE analyses described in Section 4.1.…”

“…Indeed, this assessment seems consistent with the choice of a number of authors [9,11,13,23] to employ the JS method for manufacturing applications in which elevated temperatures occur, while we had difficulties finding literature that relies on the method of MM for similar problems. However, other authors [10,[14][15][16][17]] have used the hybrid scheme proposed by McDowell [8], with some even employing a similar Johnson-Cook-type hardening law [10]. This hybrid scheme essentially blends between the MM and JS methods using a scalar blending factor 0 ≤ Ψ ≤ 1 to formally "unify" both based on purely heuristic considerations.…”

“…Ulutan et al [9] and Liang and Su [10] later published similar applications using the algorithms of Jiang and Sehitoglu [7] and McDowell [8], respectively, while applying analytic cutting force models to estimate the surface loads' magnitudes. Since then, a multitude of methodologically rather similar studies have been carried out for various manufacturing processes such as turning [11,12], milling [13][14][15], or additive manufacturing [16,17], which differ w.r.t. the surface load modeling but all rely on the same mechanical baseline assumptions and the algorithms of Jiang and Sehitoglu [7] or McDowell [8] to attempt estimating residual stresses depth profiles.…”

“…• While the strains generally do not have to be computed explicitly during this first part of the scheme, which is entirely formulated in terms of stresses, all strain components can easily be updated at the end of each increment based on the discretized version of Equation (13). However, doing so is only strictly necessary for ε ξξ , since the only strain component required in the overall scheme is ε ξξ , for which Equation (17) has to be checked prior to potentially invoking the relaxation step.…”