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A new friction-based riveting technique, Rotating Hammer Riveting (RHR), is demonstrated to fully form AZ31 Mg rivet heads in a mere 0.23 s. Heat and pressure generated through severe plastic deformation during the process was sufficient to form the Mg rivet head without the need for a pre-heating operation. Due to preliminary twinning and followed by dynamic recrystallization, AZ31 Mg grains in the rivet head were refined during RHR, which enhance the formability of Mg rivets by triggering grain boundary sliding and reducing plastic anisotropy of Mg. In addition, RHR joints showed a metallurgical bond between the rivet head and top AZ31 Mg sheet, which eliminates a significant pathway for corrosion.

Shear assisted processing and extrusion (ShAPE) was used to extrude aluminum alloy 7075 tubing at speeds up to 12.2 m/min without surface tearing. This work presents the first experimental evidence for high-speed extrusion of 7075, which improves upon conventional extrusion where 2.0 m/min is the limit. The increased speed is primarily attributed to more extensive shear deformation, compared to conventional extrusion, which results in a high density of low angle grain boundaries that facilitate continued deformation and delay the onset of surface tearing. Mechanical testing after heat treating to the T6 condition provided an ultimate tensile strength of 565.3 ± 4.6 MPa, yield strength of 495.7 ± 8.7 MPa, and elongation of 16.4 ± 1.0%. Strength values exceed the ASTM International minimum standard and are on par with American Society for Metals (ASM) typical values, while elongation was substantially improved compared to 7 and 11% for the ASTM and ASM values respectively. It was observed that low temperature extrusion at 341 °C and 40 rpm gave superior material properties in the T6 condition compared to high temperature extrusion at 441 °C and 120 rpm because of variances in nanoscale second phase size and distribution.

A novel single pass friction stir dovetailing (FSD) method was developed to join thick-plate Al-8Zn-2Mg-1.8Cu alloy (AA7099) to Ni-Cr-Mo steel in a lap configuration through the use of an Al-1Mg-0.6Si-0.25Cu alloy (AA6061) butter layer that significantly reduces interfacial failures by inhibiting the formation of brittle Fe–Al intermetallic compounds (IMC). This study evaluates the strength and microstructural evolution of the aluminum/steel joint interface between single pass FSD of AA7099/AA6061(butter)/steel fabricated using three different FSD tool configurations (full thread, half smooth/half thread, no thread), of which the half smooth/half thread tool produced the strongest weld joint due to the absence of a Zn-rich IMC layer. The strength was tested using lap shear tensile testing, and the microstructure was examined using analytical electron microscopy. A discussion of the new FSD technique, joint configurations, and process parameters is provided along with joint microstructural analyses and mechanical performance.

Immiscible alloys, whether in well-mixed or layered forms, are of increasing interest based on their novel structural and functional properties, such as enhanced thermal stability against grain growth or radiation-induced defect trapping at the interfaces. To address the need for new approaches to tailor microstructures, the microstructural development of an immiscible Cu-4 wt.% Nb alloy processed via friction consolidation of elemental powders is investigated. Friction consolidation is a solid phase processing technique that imparts severe plastic strain into a deforming volume resulting in elevated temperatures below the melting temperature of the alloy. Two distinct processing pathways were chosen to understand the effect of thermomechanical conditions on the final microstructure. The microstructure was characterized using scanning electron microscopy, scanning transmission electron microscopy, and X-ray diffraction techniques. Path 1 exhibited larger strain, strain rate, and temperature as compared with path 2. In path 1, agglomerated Nb particles were present in the recrystallized ultrafine-grained Cu matrix, while in path 2 extremely fine and dispersed Nb particles were present in a highly deformed Cu matrix. In both pathways, supersaturation of Cu in Nb lattices was noted, but not vice versa. The asymmetry in mixing is explained based on deformation-based, thermodynamic and kinetic factors. These findings provide a pathway for creation of novel tailored microstructures and improved properties in any number of binary immiscible alloy systems.

Joining of high strength Al alloy (AA7099) to steel (rolled homogeneous armor) with and without AA6061 interlayer was achieved using a friction stir dovetailing (FSD) method. Detailed investigations were conducted to establish correlations between mechanical performance and microstructural evolution of FSD joints. Lap shear tensile testing revealed that incorporation of the AA6061 interlayer combined with tailored processing methodology significantly improved both mechanical interlocking and load transmission across the dovetail interface. Following incorporation of the interlayer, the maximum load carrying capacity and peak extension (of the AA7099/AA6061/steel joint) are 60% and 210% higher, respectively, than results obtained from an AA7099/steel joint without an interlayer. Without an AA6061 interlayer, formation of zinc rich intermetallic compounds led to brittle interfacial failure of the AA7099-steel FSD joint. Incorporation of novel tool and joint configuration eliminated asymmetric joint performance as a function of loading direction, which is generally observed in friction stir lap welding. The efficacy of the use of an AA6061 interlayer in joining AA7099 to steel was demonstrated through mechanical testing as well as high resolution analytical electron microscopy characterization.

Shear-assisted processing and extrusion (ShAPE) is used to consolidate Al-4.2Mg-1.6Zr-0.25Si-0.1Fe (wt.%) powders. The extruded alloy exhibits: (i) near-zero porosity; (ii) an attractive combination of tensile yield strength (~220 MPa), ultimate strength (330 MPa) and elongation (>20%); and (iii) excellent thermal stability at 400 °C. The ultra-fine grain size in the gas-atomized powders – created and stabilized by primary L12-Al3Zr submicron precipitates - is maintained after consolidation and subsequent exposure at 400 °C, contributing twice as much to strength as Mg in solid-solution. Electron microscopy reveals the microstructure of the L12-Al3Zr precipitates, and other precipitates (Mg2Si and Al3Fe) and dispersoids (oxides).

We survey various state-of-the-art methods for welding magnesium alloys and steels using different joint configurations. Microstructural characterizations indicate that four microstructures may form at the Mg/steel interface after welding: unwelded gap, metal oxides, solid solutions, or intermetallic compounds. Reaction products at the Mg/steel interface vary with different welding methods, alloying elements in base materials, interlayers or coatings applied, and preparations of base material before welding. Mechanical property characterizations, (a) lap tensile shear testing for lap-welded and spot-welded joints, (b) tensile testing for butt-welded joints and (c) fatigue properties of lap-welded and spot-welded joints are summarized and compared, separately. Reaction products at the Mg/steel interface are correlated with mechanical properties. Finally, ways to enhance Mg/steel joint strength, such as introducing interlocking features during friction stir lap and butt welding, are discussed.

Friction stir interlocking (FSI), a new derivative friction stir welding, was used to lap join AZ31 magnesium sheet and thermoset carbon fiber reinforced polymer (TS-CFRP) sheet. Instead of directly joining AZ31 and TS-CFRP, a series of magnesium interlocks were used to friction stir weld with AZ31 sheet to enable joining with TS-CFRP. Microstructural characterization of joint cross sections showed that a tool rotation rate of 800 rpm produced incomplete mixing of magnesium interlocks and the top AZ31 sheet, while 1200 rpm achieved sufficient material mixing to avoid unwelded regions. Mechanical property characterization showed that AZ31/TS-CFRP dissimilar joints failed through the magnesium interlock shanks during lap shear tensile testing, with a maximum normalized joint strength reaching ~ 100 MPa. Finite element analysis of the joints helped elucidate the presence of high stress region around the narrow band of the interlock shank providing pathway of fracture.

Accurate temperature measurement is critical to understanding the thermomechanical conditions and microstructural evolution that materials experience during severe plastic deformation (SPD). Others have previously measured the interface between a non-deforming tool and metal workpiece using thermocouples, infrared measurement, and the thermoelectric principle. Measuring temperature within the deforming material itself, especially at a high-shear deforming interface, has proven to be extremely difficult, thus limiting the application of these measurements to fundamental SPD research. In this study, the thermoelectric principle is used to directly measure temperature at the interface between copper and nickel billets undergoing SPD. This is done by utilizing the deforming materials themselves as two halves of a thermocouple junction. The spatial resolution of this measurement is shown via microscopy to be less than 1 micron. Temperature changes are sensed almost instantly, with time delays on the order of the data logger frequency (0.01 seconds). Process variations that are both very brief (

In the present work, we utilized Shear Assisted Processing and Extrusion (ShAPE), a solid-phase processing technique, to extrude hollow tubes of ZK60 Mg alloy. Hot rolling was performed on these as-extruded tubes (after slitting them longitudinally) to thickness reductions of 37%, 68%, and 93% to investigate their viability as rolling feedstock material. EBSD analysis showed the formation of twinned grains in the ShAPE processed material and a gradual re-orientation of the basal texture parallel to the extrusion direction with each rolling step. Moreover, an equiaxed grain size of 5.15 ± 3.39 μm was obtained in the ShAPE extruded material, and the microstructure was retained even after 93% rolling reduction. The rolled sheets also showed excellent tensile strengths and no mechanical anisotropy, a critical characteristic for formability. The unique microstructures developed and their excellent mechanical properties, combined with the ease of scalability of the process, make ShAPE a promising alternative to existing methods for producing rolling feedstock material.

ZK60 Magnesium tubing has been friction extruded from as-cast billets and T5 conditioned bars using Shear Assisted Processing and Extrusion (ShAPE). Tubes having an outer diameter of 50.8 mm and wall thickness of 1.9 mm were extruded with >20 times less ram force compared to conventional extrusion due to the unique shearing conditions and tooling inherent to ShAPE. Microstructures of the as-cast billet and T5 bar feedstock materials were significantly different from each other in terms of grain size, texture, and second phase distribution; yet the resulting microstructures after ShAPE were remarkably similar. An average grain size of 4–5 μm, 20° tilt of basal texture away from the extrusion axis, and refined second phases having a uniform distribution were achieved independent of the feedstock material. Hardness for as-extruded and artificially aged tubes are presented with isotropic behavior explained by detailed microstructural analysis. This work suggests that bulk ZK60 magnesium alloys extrusions may be fabricated in a single step, with microstructures that are unobtainable with conventional extrusion.

Shear-assisted processing and extrusion (ShAPE) experimental setup and tooling were adopted for extruding thin-walled AA7075 aluminum tube from as-cast non-homogenized billet material in a single run. The mechanical and microstructural characterizations were performed on the as-extruded tube through tensile, hardness, electron backscatter diffraction (EBSD), and energy dispersive spectroscopy (EDS) tests. The results showed that the ShAPE process developed a significantly refined microstructure with uniform and almost equiaxed grain structure on both hoop and axial cross-sections of the extrudate as well as through the thickness of the material. The pole figures and inverse pole figures of the EBSD data showed a strong shear texture development, and it was found out that axial shear is the dominant deformation mechanism in the regions near the inner surface of the tube, while combined axial and torsional shears are the two dominant modes of deformation near the outer surface of the extrudate. As for the mechanical properties, there was an increase of 150 and 73% in the yield and ultimate strengths of the tube produced using ShAPE process, respectively, and an 18% decrease in maximum uniform plastic elongation compared to the conventionally extruded AA7075-O tube.

Conventional extrusion of aluminum alloy 7075 is limited to 1–2 m/min in order to avoid surface tearing and cracking. An emerging technique called Shear Assisted Processing and Extrusion (ShAPE) was used to extrude aluminum alloy 7075 tubing at a speed of 7.4 m/min without inducing surface defects. The faster extrusion speed is attributed to the unique flow characteristics inherent to the ShAPE process compared to conventional extrusion. Tubes with inner diameter of 10 mm, outer diameter of 12 mm, and length of 2 m were extruded at temperatures ranging from 340 to 466 °C. Tensile testing was performed per ASTM B557-15 with strain measured using digital image correlation. An ultimate tensile strength of 565 ± 4.6 MPa, yield strength of 496 ± 8.7 MPa, and elongation of 16.4 ± 1.0% were measured for extrusions made at 362 °C and heat treated to a T6 condition with extended aging.

Shear Assisted Processing and Extrusion (ShAPE) was used to fabricate thin-walled AA6063 tubing with improved mechanical properties compared to conventional extrusion. Namely, tensile properties for ShAPE extrusions after T5 heat treatment far exceed conventional T5 properties and are on par with T6 tensile properties. Die rotational speed did not affect tensile properties, although increased extrusion speed did lead to moderate increases in elongation.

Charges of melt-spun AZ91E flake were indirectly extruded into tubes using Shear Assisted Processing and Extrusion (ShAPE™). The effect of instrument parameters and tool features on densification and microstructural evolution was studied. At a constant extrusion ratio, the tool rotational velocity varied from 75 to 300 rpm and was demonstrated to reduce the forge force by a factor of four. Modification of the extrusion die face with successively more aggressive scrolled features was found to enhance material flow into the extrusion orifice which led to a 30% reduction in spindle torque. Microstructure, texture and hardness are reported for the range of rpm and scroll geometries investigated. It was observed that the ShAPE™ process is able to retain the average grain size of the as-spun flake (2.5-4 μm) while simultaneously imparting strong textural alignment in the resultant tube.

A new solid-phase technique called friction stir dovetailing (FSD) has been developed for joining thick section aluminum to steel. In FSD, mechanical interlocks are formed at the aluminum-steel interface and are reinforced by metallurgical bonds where intermetallic growth has been uniquely suppressed. Lap shear testing shows superior strength and extension at failure compared to friction stir approaches where metallurgical bonding is the only joining mechanism. High resolution microscopy revealed the presence of a 40–70 nm interlayer having a composition of 76.4 at.% Al, 18.4 at.% Fe, and 5.2 at.% Si, suggestive of limited FeAl 3 intermetallic formation.

Friction consolidation (FC) is a solid phase processing methodology that densifies a material through high-shear deformation and pressure at elevated temperature. The method has garnered interest in the scientific community because of its ability to produce extremely refined and homogeneous microstructures, off-axis texture development, and improved material properties. This manuscript presents an investigation of Cu and Ni material mixing via evaluation of morphological evolution, grain boundary characterization, and compositional analysis to provide insights on the operational alloying mechanisms occurring under high shear and elevated temperature. Using correlative microscopy techniques, we show alloying progresses via a combination of grain boundary diffusion and interfacial roughening at heterophase boundaries. Evidence supporting Cu infiltration along Ni-Ni grain boundaries along with asymmetric diffusion of Cu into Ni grains is highlighted. The resultant, consolidated microstructure was produced directly from a powder compact in ∼30 s and exhibited a submicrometer, equiaxed grain size.

Shear Assisted Processing and Extrusion (ShAPE) enables the extrusion of many alloys with enhanced properties. In this study, ShAPE was used to extrude tubes of aluminum alloy 6063 measuring 12 mm in diameter at extrusion speeds up to 3.8 m/min, an increase of 10 times over what has previously been reported for ShAPE. Increasing the extrusion speed from 0.7 to 3.8 m/min resulted in using 68% less process energy at steady state without any loss in mechanical properties. As-extruded tubes had ultimate tensile strengths on par with conventional T5 extrusions and double the elongation at break. ShAPE extruded tubes that underwent a T5 heat treatment had yield and ultimate strengths of 198 and 234 MPa, respectively, which is ~30% higher than standard T5 material and comparable to T6 properties. Microstructural analyses were performed on as-extruded and T5 treated tubes. Grain refinement below 20 μm was identified, with no detectable growth of macroscale Mg2Si strengthening precipitates. Nanoscale β″ was not observed in the as-extruded materials but was prominent after T5 heat treatment suggesting that β″ strengthening precipitates were solutionized in situ during the ShAPE process. The ability to perform solution heat treating in situ, rather than post-extrusion, eliminates an energy intensive process step and is applicable to a wide variety of alloys.

Shear Assisted Processing and Extrusion (ShAPE) -a novel processing route that combines high shear and extrusion conditions- was evaluated as a processing method to densify melt spun magnesium alloy (AZ91E) flake materials. This study illustrates the microstructural regimes and transitions in crystallographic texture that occur as a result of applying simultaneous linear and rotational shear during extrusion. Characterization of the flake precursor and extruded tube was performed using scanning and transmission electron microscopy, x-ray diffraction and microindentation techniques. Results show a unique transition in the orientation of basal texture development. Despite the high temperatures involved during processing, uniform grain refinement and material homogenization are observed. These results forecast the ability to implement the ShAPE processing approach for a broader range of materials with novel microstructures and high performance.

A robust, economically viable joining method for Mg/Al and Mg/CFRP could enable multi-material assemblies that decrease vehicle weight while offering more flexibility for designers. However, certain challenges exist for joining Mg/Al and Mg/CFRP. Mechanical joining, such as conventional riveting, clinching and bolting do not form a metallurgical bond between the fastener and metal sheet being fastened. Large differences in physical and mechanical properties of metals and polymers make joining Al or Mg to CFRP challenging via various welding techniques. For Mg/Al pair, solid-phase and fusion-based welding results in rapid formation of brittle intermetallic compounds at the interface leading to premature interfacial fracture under mechanical loading. In this study, a Rotational Hammer Rivet (RHR) technique was developed to fabricate Mg/CFRP and Mg/Al joints. With RHR technique, direct joining between Mg/Al and Mg/CFRP were replaced by joining Mg rivet head and top Mg sheet. Through heat generated by plastic deformation of an Mg rivet, RHR creates a metallurgical bond between rivet head and Mg sheet which seals corrosive electrolyte from penetrating around the rivet head.

Soft magnetic materials are often limited in scalability due to conventional processes that do not retain beneficial microstructures, and their associated physical properties, during densification. In this work, friction consolidation (FC) has been studied to fabricate Fe Si soft magnetic materials from gas-atomized powder precursors. Fe Si powder is consolidated using variable pressure and tool rotation speed in an effort to evaluate this unique densification approach for potential improvements in magnetic properties. FC, due to the high shear deformation involved, is shown to result in uniform gradual grain structure refinement across the consolidated workpiece from the center nearest the tool to the edge. Magnetic properties along different orientations indicate little, if any, textural orientation in the refined grain structure. The effect of annealing on the magnetic properties is evaluated and shown to decrease coercivity. FC processing was able to retain the magnetization of the original gas-atomized powders but further process optimization is needed to reach the optimal coercivity for the soft magnetic materials applications.

Refined grain sizes and texture alignment have been shown to improve transport properties in bismuth-telluride (Bi2Te3) based thermoelectric materials. In this work we demonstrate a new approach, called friction consolidation processing (FCP), for consolidating Bi2Te3 thermoelectric powders into bulk form with a high degree of grain refinement and texture alignment. FCP is a solid-state process wherein a rotating tool is used to generate severe plastic deformation within the Bi2Te3 powder, resulting in a recrystallizing flow of material. Upon cooling, the far-from-equilibrium microstructure within the flow can be retained in the material. FCP was demonstrated on n-type Bi2Te3 feedstock powder having a −325 mesh size to form pucks with a diameter of 25.4 mm and thickness of 4.2 mm. Microstructural analysis confirmed that FCP can achieve highly textured bulk materials, with sub-micrometer grain size, directly from coarse feedstock powders in a single process. An average grain size of 0.8 μm was determined for regions of one sample and a multiple of uniform distribution (MUD) value of 15.49 was calculated for the (0001) pole figure of another sample. These results indicate that FCP can yield ultra-fine grains and textural alignment of the (0001) basal planes in Bi2Te3. ZT = 0.37 at 336 K was achieved for undoped stoichiometric Bi2Te3, which approximates literature values of ZT = 0.4–0.5. These results point toward the ability to fabricate bulk thermoelectric materials with refined microstructure and desirable texture using far-from-equilibrium FCP solid-state processing.

Shear deformation has been highlighted in multiple research efforts for its ability to impart novel microstructures that demonstrate improvements in mechanical properties. When used to process and densify powdered material, these shear-based consolidation techniques are commonly referred to as friction consolidation (FC). In this paper, the microstructural evolution from compacted Cu and Ni powders to a consolidated Cu0.5Ni0.5 alloy is examined. Various stages of porosity reduction and deformation are shown. Deformation was observed to accumulate preferentially in the more ductile material early in the process, leading to the formation of a tortuous microstructural zone. Porosity reduction was extensive, decreasing from ~65% in the pre-compacted state to ~1% in the fully consolidated alloy. The final consolidated alloy showed a ~2× hardness improvement over the unalloyed, compacted material. Unique aspects of this work include demonstration of FC processing to produce an equiaxed, sub-micrometer grain size in samples within a 0.5 to 2 min processing time. The results point to future opportunities to implement shear deformation during powder densification to expand the range of property outcomes in bulk materials.

In this work, we combined phase field method and crystal plasticity to study the effects of initial grain structures, recrystallized grain orientation, and strain rates on dynamic recrystallization and deformation behavior under stress states typical of the Shear Assisted Processing and Extrusion (ShAPE) process. The geometrically necessary dislocation density was used as the nucleation criterion for recrystallization, and the minimization of the deformation energy drove the growth of the recrystallized grains. With the set of material properties and model parameters, the simulation results demonstrate that (1) a polycrystalline structure with the texture observed in ShAPE process has the lowest yield stress under the ShAPE stress conditions, (2) the recrystallized grains with the texture observed in ShAPE process largely soften the materials, and (3) higher shear strain rate or rotation speed results in larger magnitude of material softening.

Friction stir dovetailing (FSD) was used to join 0.5 in. (12.7 mm) AA7099 to 0.5 in. (12.7 mm) Ni-Cr-Mo steel in a lap configuration. Two new FSD approaches are reported that significantly reduce zinc embrittlement of Fe–Al intermetallic compounds (IMCs) which form during conventional friction stir welding (FSW). The first method uses the general FSD approach where a custom designed tool is employed to extrude the AA7099 into the pre-machined dovetail groove of underlying steel by forming mechanical interlocking and metallurgical bonding simultaneously. The second method uses a two-step approach where FSD of AA6061 is first used to form a silicon-rich Fe-Al IMC within the dovetail groove. AA7099 plate is then joined to the AA6061 within the dovetail using conventional FSW. A discussion of the new FSD technique, joint configurations, and process parameters are provided along with joint microstructural analyses and mechanical performance.

Friction stir dovetailing (FSD) is a new dissimilar material joining process that simultaneously forms a mechanical interlock and metallurgical bond at the dissimilar material interface. This work presents development of a modeling and simulation approach to predict mechanical performance of FSD thick section aluminum to steel joints. A finite element analysis (FEA) was carried out in order to predict the load-carrying capacity and failure location for aluminum thicknesses between 12.7 mm and 50.8 mm with different numbers of dovetails. The numerical results and corresponding experimental investigation are in agreement and suggest that the developed methods can be used effectively for design and analysis of FSD joints using standard FEA tools.

Friction extrusion has been used to consolidate and extrude aluminum alloy powder into bulk nanostructured rods. Extrudates exhibited 450 MPa ultimate tensile strength, 380 MPa yield strength, and 15.7% elongation at ambient temperature. Twice the elongation was achieved compared to conventional direct extrusion of the same material, with similar ultimate and yield strengths, and is attributed to extensive reduction of the matrix grain size and refinement and redistribution of nanoscale second phases. Quasicrystalline approximants within the starting powder were not observed in the extrudate, indicating shear processing was effective in solutionizing some fraction of the alloying elements and re-precipitating them as second phase nanoscale intermetallics.

Shear Assisted Processing and Extrusion (ShAPE) has been scaled-up and applied to direct extrusion of thin-walled magnesium tubing. Using ShAPE, billets of ZK60A-T5 were directly extruded into round tubes having an outer diameter of 50.8 mm and wall thickness of 1.52 mm (extrusion ratio of 17.7). Due to material flow effects resulting from the simultaneous linear and rotational shear intrinsic to ShAPE, the ram force and k-factor during extrusion were just 40 kN (9000 lbf) and 3.33 MPa (0.483 ksi) respectively. This represents a >10 times reduction in k-factor, and therefore ram force, compared to conventional extrusion. The severe shearing conditions inherent to ShAPE resulted in microstructural refinement with an average grain size of 3.8 μm measured at the midpoint of the tube wall. Tensile testing per ATSM E-8 on specimens oriented parallel to the extrusion direction gave an ultimate tensile strength of 254.4 MPa and elongation of 20.1%. Specimens tested perpendicular to the extrusion direction had an ultimate tensile strength of 297.2 MPa and elongation of 25.0%.

Solid-state joining of thick section aluminum to steel plate has been achieved using a new process called friction stir dovetailing (FSD). In FSD, a custom designed pin tool is used to flow a lower melting point material (AA6061) into dovetail grooves machined into the surface of an underlying material that has a higher melting point (rolled homogeneous armor [RHA]). Repeating dovetails form a mechanical interlocking structure akin to metallic Velcro. In this study, 38.1 mm (1.5 in.) thick AA6061 was joined to 12.7 mm (0.5 in.) thick RHA plates. The effectiveness of FSD is demonstrated through tensile test data that shows specimens failing in the processed aluminum rather than at the joint interface. Numerical simulations that highlight the importance of optimizing dovetail geometry are presented. The effect of process parameters on joint strength and microstructure also are discussed.