1. Introduction
A major area of focus in the field of additive manufacturing (AM) is understanding pore formation, and process optimization with the goal of creating fully dense, defect-free components. Studying the effect of pores on metal failure, along with the development of failure models based on theory and experiments has a long and continued history for conventionally processed ductile metals due to the importance of avoiding failure in load bearing components or during metal forming. For example, in 1977 Gurson developed a model in terms of a yield function and microstructurally informed void volume fraction to understand void growth and ductile failure using simplified spherical and cylindrical void models. By combining the existing frameworks for fracture modeling with the unique manufacturing capabilities of AM, new insight into the effect of pores on ductile failure is possible.
Laser powder bed fusion (L-PBF) AM is a process for building metallic components layer-by-layer using a focused laser heat source to melt a selected 2-dimensional pattern in a thin layer (10–100 µm) of powder feedstock to a baseplate, lowering the baseplate by a prescribed layer thickness, spreading on a new layer, scanning the next layer pattern, which fuses to the previous layer, and repeating until a final 3-dimensional (3D) component is completed. In AM, there are numerous processing parameters that dictate a completed component's quality. One primary evaluation metric for optimizing a parameter set for a given alloy is component density, or the reduction of pores. Pores in AM parts can be formed via different mechanisms; two of the primary mechanisms are: (1) gas entrapment during melting and solidification of the meltpools, analogous to that seen in casting and welding, which generates spherical pores and (2) irregular morphology lack-of-fusion (LoF) pores, which are formed due to incomplete fusion between layers or adjacent meltpools on the same layer along the heat source scanning path, and which can have sharp features. LoF pores are of primary interest in AM because of their significantly more detrimental influence on material ductility compared to spherical pores.
In addition to pores having a harmful effect on the fracture behavior of ductile metals, stress state is also known to play an important role in fracture. Stress state can be defined using the two parameters stress triaxiality (η) and Lode angle parameter (θ¯). The stress triaxiality is the ratio of the mean stress (σm) and von Mises equivalent stress (σ¯VM): equation (1) where I1 = σkk is the first invariant of the stress tensor, σ, and J2=1/2sijsij is the second invariant of the deviatoric stress tensor, s. The normalized Lode angle parameter is a function of the third invariant of the deviatoric stress tensor, J3=det(sij), and is defined as: equation (2).
Increased stress triaxiality is known to accelerate the void nucleation and growth process in ductile metals, resulting in reduced ductility even in fully dense samples. Designing load bearing components to be safe under the spatially varying stress state is an important consideration for engineers in the design against failure.
Fracture models that describe the effects of stress state on ductile failure have primarily been presented in mixed stress-strain space, meaning equivalent plastic strain to failure versus η and θ¯ (referred to here as equivalent strain space) because in ductile fracture, the resolution in strain is typically much larger than the resolution in stress, that is, large differences in strain result in relatively minor differences in stress due to the fact that the elastic contribution to failure is negligible compared to the plastic contribution. However, for alloys with little to no plastic deformation to failure, describing failure in terms of strain to failure becomes more challenging. This is shown schematically in Figure 1, which compares the engineering stress-strain curves for the two materials in this study—Ti-6Al-4V and 316L stainless steel (316L).
In this study, the effect of pores relative to the behavior of dense samples was assessed in two different alloys using well-known fracture models calibrated in both equivalent stress and strain space. Data over a wide range of stress states from previous studies by the authors on L-PBF stainless steel 316L and L-PBF Ti-6Al-4V that included intentionally manufactured, penny-shaped pores of varying diameter were used to calibrate fracture models for each pore size in stress triaxiality versus Lode angle parameter versus equivalent stress space (Haigh-Westergaard space, referred to here as equivalent stress space) and equivalent strain space. By comparing the ductile 316L alloy (>60% engineering strain to failure under uniaxial tension) to the less ductile Ti-6Al-4V (<10% engineering strain to failure under uniaxial tension) in both equivalent stress and strain space as a function of pore size, an assessment of dominant fracture mechanism changes, if any, can be made for each material. Additionally, the appropriateness of fracture models in equivalent stress space versus equivalent plastic strain space for capturing the effect of pore size on the failure behavior of both materials is discussed.
2. Materials and Methods
The complete description of the manufacturing process, samples, experimental set-up, and simulations are described in extensive detail for the 316L builds in Refs. and for the Ti-6Al-4V builds in Refs.. The following is an overview of information that is most pertinent to the current study.
2.1. Experimental Methods
Samples used in this study for both alloys were manufactured on a 3DSystems, Inc. ProX 320 L-PBF AM machine (3DSystems, Rock Hill, SC, USA). No post-processing heat treatment was used for the 316L builds, while a post-processing stress relief heat treatment of 650 °C for 3 h in an argon environment was used for the Ti-6Al-4V builds. Data obtained using five different sample geometries, selected as they have a pre-determined failure initiation location, and corresponding to five unique stress states, were used in the current study: pure shear, equibiaxial tension (punch tests), and round notched tension with three different notch radii (3 mm (R3), 5 mm (R5), and 12 mm (R12)), as shown in Figure 2. The three notched tension geometries each had a minimum cross-sectional diameter of 6 mm. The gauge regions of all samples were fabricated using computer numeric control (CNC) machining (Lynx 220L, Doosan Machine Tools Co., Ltd., Pine Brook, NJ, USA).
The five sample geometries for both materials were evaluated in the dense condition, that is using process parameters optimized to build each material. Only the R3, R5, and R12 tests selected in the current study had single, penny shaped pores of varying diameters that were directly designed into the CAD files for the samples, and therefore, included at the center of each sample during the AM fabrication. The pore diameters in µm (and their percentage of the cross-sectional area of the notched tension samples) evaluated in this study were: 300 (0.3%), 600 (1%), 1200 (4%), and 2400 (16%). The four pore sizes were chosen in the current analysis because they interrogated a wide range of percentage of the cross-sectional area that led to varied mechanical response relative to dense samples; additionally, all four pore sizes were studied in each alloy, allowing for a direct comparison. In the 316L material, each of the pores were designed to be 180 µm or 3 layers tall cylinders with the varying diameters, while in the Ti-6Al-4V samples, the internal pores were designed with a conical feature on the top surface after 180 µm vertical walls to prevent pore closure due to dross formation in this alloy system. As discussed in Ref., a study on the effect of pore height performed for Ti-6Al-4V indicated that the pore height did not impact the measured strength or ductility in uniaxial tension specimens.
For all tests at least two repetitions were completed for each combination of stress state and pore size, including the dense samples. Tests were performed to failure on their respective load frames. Displacement and strain, using a virtual extensometer, were measured using digital image correlation (DIC), a non-contact surface strain field measurement technique. For each test, two different definitions of failure were examined, displacement to maximum force and displacement to material separation, and were used to inform the simulation data in Section 2.1 and for calibration of the equivalent stress and strain fracture models, respectively, in Section 2.3.
2.2. Finite Element Analysis Simulations
Calibrated plasticity models for 316L and Ti-6Al-4V were developed previously by the authors and used in simulations of each dense geometry using the finite element method in the commercial software Abaqus; the complete model details for each fracture geometry are provided in Refs.. Finite element analysis (FEA) simulations of each dense geometry were used to probe, as a function of applied displacement, the components of the Cauchy stress tensor (σ), stress state (η and θ¯), and von Mises equivalent stress (σ¯VM) and equivalent plastic strain (ε¯) in the centermost element of each sample geometry, which is where failure is assumed to initiate. Simulations were performed to mimic the experimental displacement to catastrophic failure of the dense specimens, as measured via the virtual extensometer in experiments and an equivalent extensometer in each of the FEA models. The average stress triaxiality and Lode angle parameter up to the experimentally measured displacement to failure for each test condition (i.e., pore size and sample geometry) were calculated using: equation (3) and: equation (4)
The equivalent stress from the FEA model was recorded at the displacement corresponding to the average experimental displacement to maximum force for every test condition. The equivalent plastic strain from the FEA model was recorded at the displacement corresponding to the average experimental displacement to catastrophic failure, or material separation, for every test condition.
To summarize, for both materials the data sets were analyzed in two different 3D spaces (ηav, θ¯av, σ¯VM), or equivalent stress space, and (η(ε¯), θ¯(ε¯), ε¯f), or equivalent strain space. For each 3D space there were five data points for the dense condition (pure shear, punch, R3, R5, and R12) and three additional data points for each pore size (R3, R5, and R12). These data will be referred to in subsequent sections in the calibration of the fracture models in equivalent stress and strain spaces.
2.3. Fracture Models
In this study, three fracture models were calibrated in the 3D equivalent stress space of (ηav, θ¯av, σ¯VM) and two fracture models in the 3D equivalent strain space of (η(ε¯), θ¯(ε¯), ε¯f), which are transformed versions of two of the stress-based models using assumed plasticity frameworks as described in Refs.. Each model was calibrated for the dense material and each pore size. For the fracture surface calibrations for dense material, the dense pure shear, punch, R3, R5, and R12 data were used. For each subsequent pore size, five tests were also used in calibration: the dense pure shear and punch data were used as "anchor points" as the pore configurations being considered in this study (with the cylinder axis oriented parallel to the vertical build direction) are assumed to have relatively negligible impact on ductility in those stress states, and the three notched tension data for the corresponding pore size.
2.3.1. Equivalent Stress versus Stress Triaxiality and Lode Angle Space
Maximum Stress Failure Criterion
The maximum principal stress fracture criterion is based on the premise that a crack or defect will grow in a direction perpendicular to the maximum principal stress, resulting in failure when the maximum principal stress reaches a critical value. This model is given as: equation (5) where σ1, σ2, and σ3 are the principal stress components, and σc is the critical maximum principal stress resulting in fracture. For each notched tension geometry, the average σc values from the three notched tensions tests for the dense data, as well as each pore size, were calculated and are given in Table 1 and Table 2 for both materials. To plot the maximum stress fracture loci for dense material and each pore size, von Mises equivalent stress, η, and θ¯ were calculated and plotted as a function of the three principal stress components with σc held constant. The discrete points were then interpolated to generate a fracture locus using only a single input, σc.
The Mohr-Coulomb (MC) fracture criterion is a classical stress-based criterion that has its origins in describing brittle failure (e.g., rocks and ceramics). The model is phenomenological in that it describes fracture as occurring when a combination of normal stress (the intermediate principal stress is ignored) and shear stress reach a critical value.