**4. Materials**

The initial aluminum tube (Ø25.40 mm × 1.65 mm) is 6061 per ASTM B210 and the initial copper tube (Ø28.58 × 1.65 mm and Ø25.40 mm × 1.65 mm) is C12200 per ASTM B75. The true stress-strain curves of the materials, after initial annealing, are presented in Figure 5. Flow curves were determined per ASTM E8 using the bulk tube as the specimen.

**Figure 5.** True stress-strain curves of annealed copper and aluminum. Material behavior is modeled by the power law.

Before processing, the as-received material is annealed since the aluminum and copper were tempered to T6, and H58, respectively. Both initial annealing cycles were performed as recommended by the Society of Manufacturing Engineers to produce optimally ductile materials as previously described. To confirm the effectiveness of the initial annealing, hardness measurements were taken; the results are shown in Table 5. The copper experienced a 60.2% decrease in hardness and the aluminum experienced a 67.8% decrease. Knoop hardness testing was performed using a 500 g force held for 10 to 15 s where the hardness value was averaged over 10 samples.

**Table 5.** Hardness of mill and annealed material (HK).


The copper and aluminum material behavior are fitted with the power law with an R 2 value of 0.9816, and 0.9893, respectively. Figure 5 shows that the copper and aluminum exhibit strain-hardening which is represented well by the power hardening law shown in Equation (1):

$$
\sigma = \mathbb{K} \mathfrak{e}^n \tag{1}
$$

The strength coefficient, *K*, and the strain hardening exponent, *n*, are presented in Table 6. An important item of note is the dissimilarity between both materials true stress-strain curves. For consistent plasticity to occur within both metals, and to maintain balanced layer formation, similar flow stress behavior across the material is tactically sought; however, the copper has a strength coefficient 99.7% larger than the aluminum, which is expected to influence the difference between extruded layer thicknesses. The values of the strain hardening exponent represent how quickly the material hardens when deforming. A value closer to zero represents a material resisting deformation, while the values closer to one represent a material where true stress and true strain vary proportionality.



#### **5. Finite Element Method-Based Simulations of Extrusion**

The extrusion process is modeled in ANSYS Mechanical (ANSYS 19.1., ANSYS Software Company, Canonsburg, PA, USA) to gain insight on the plastic behavior of the bimetal and to understand the stresses that develop within the die. The geometry of the extrusion model consists of four components: Two metals experiencing extrusion and two workpieces enabling the extrusion. The two metals are referred to as the outer and inner metals, which represent the initial outer, and inner diametrical layers, respectively. For all simulations and experiments, the copper is always the outermost metal tube. The model is axisymmetric and 2-dimensional. Figure 6a shows the ANSYS model of the extrusion process, where axis symmetry is taken about the farthest left edge. The geometry represents a 2-dimensional "slice" of the area-of-interest, which is the lower section of the die and mandrel identified by the red circle in Figure 6b. The model is axisymmetric and 2-dimensional because no 3-dimensional irregularities in stress or strain are expected since the tooling is precision-ground and the design utilizes a self-centering die.

**Figure 6.** (**a**) Finite element geometry which represents the two deformation cases studied: 52% and 68%. Dimension L was adjusted to obtain the two different cases. (**b**) The experimental setup is shown at dead-bottom position, where the simulated area is identified by a red circle.

The geometry of the finite element model consists of the bottom section of the extrusion process. This is the section which contains the extrusion ledge at the bottom of the die. No further geometry is necessary because the model has been iteratively reduced to capture the significant stresses in the die while tolerating manageable convergence duration. The punch is not modeled; instead, an input displacement is utilized.

The model is partitioned into various sub-sections of the original geometry to focus higher element density to the area of interest which is the internal edges of the mandrel

−

and die. The mesh is shown in Figure 7. After performing a mesh sensitivity study, the model includes 26,286 elements with 75,785 nodes.

**Figure 7.** (**a**) The extrusion model meshing, and (**b**) close-up of the meshing, where the metals that will experience extrusion are located at the top of the die ledge at t = 0 s.

− The contact control between the die, mandrel, and the metals experiencing extrusion are controlled with an augmented LaGrange formulation with nodal-normal to target detection method. The augmented LaGrange formulation comes with a computational penalty for longer solve time but controls nodal penetration very well, which is important during sliding-type simulation. An allowance of 1.27 × 10 <sup>−</sup><sup>2</sup> mm penetration was tolerated. The contact between the two workpieces is bonded as a simplification to assist in convergence.

A frictional value of 0.025 is used between all sliding surfaces, which is consistent with the conclusions of [65], but slightly less than the values used in other research [66–68]. For comparison to rolling, this frictional value is less than the "normal" lubrication value of 11 as reported in [69]. Unlike rolling, the frictional value must be as low as possible in practice as metal adhesion is a major failure mode in extrusion and is not easily resolved as in rolling. For this reason, the A2 tool steel of the die and mandrel are polished to a 0.8 µm Ra surface finish after a thermally diffused coating of vanadium carbide is applied. The coating has a hardness of 3400 HV minimum and is very smooth. In addition, lubrication is used during testing to reduce the friction coefficient as modeled.

An input displacement of 7.62 mm is applied to the top surfaces of the metals experiencing extrusion which forces them to interact with the extrusion ledge as shown. The input of 7.62 mm is used, as this is sufficient displacement to achieve steady-state plastic flow during the extrusion simulation.

The die and mandrel are evaluated for yielding in four instances where copper-copper and copper-aluminum bimetals are extruded at 52% and 68% deformation. Results are shown in Figure 8. Peak stress occurs on the 30 ◦ ledge of the die when extruding coppercopper at 68% deformation where a maximum Von Mises stress is found to be 955 MPa. This demonstrates that a factor-of-safety of 1.7 is achieved in the most stressed condition with the yield strength of 1600 MPa estimated for the die material. Because of this, neither mandrel nor the die are expected to yield. As shown in previous research, a 30 ◦ die angle is optimal when compared to 22.5, 45, and 60 ◦ angles of similar die design for reducing peak stresses [31].

**Figure 8.** Von Mises stress within the die and mandrel during extrusion of copper-aluminum at (**a**) 52%, and (**b**) 68% deformation and copper-copper at (**c**) 52%, and (**d**) 68% deformation. The metals that experienced extrusion are omitted. Units of stress are MPa.

Steady-state extrusion begins after 4 mm of punch displacement. The peak input force, found at 4.4 mm, is 167 kN as shown in Figure 9 for the copper-copper simulation. Beyond this peak, the input force decreases linearly with a slight negative slope as less bimetal is within the die causing sidewall friction. As shown, the maximum input force is increased by 60.8% for the copper-aluminum bimetal when increasing the deformation from 52% to 68%, and a 55.7% input force increase is observed for the copper-copper bimetal. The predicted peak input force matches the recorded peak input forces within ~1% when averaged across multiple extrusions, where the multiple recorded peak forces are found contained within 5% of the expected peak force.

**Figure 9.** Force-displacement curve for extrusion at 52% and 68% deformation for copper-copper and copper-aluminum bimetals.

the outer tube's edge is impacted by the extrusion ledge first, which As the bimetals pass the extrusion ledge, a significant amount of plastic deformation occurs. The resultant total plastic deformation is shown in Figure 10 for all four cases. In the steady-state extrusion, the plastic strain varies through the thickness and length of the bimetal. Shown in Figure 11 is the plastic strain variance through the wall thickness. In all cases, a higher plastic strain is found on the innermost edge of the tube and decreases close to linearly to the midpoint. Beyond the midpoint, the plastic strain levels out but then dips close to the outer diameter. The innermost metal exhibits higher strain because the outer tube's edge is impacted by the extrusion ledge first, which forces the inner metal to push ahead of the outer metal causing more strain in the inner metal. This is observed in all four cases, and predominately in the 52% deformation case, where the inner material is drawn past the die and is extruded first (reference Figure 10a). Additionally, the layer thickness varies ± 3% from the nominal thicknesses. Most notable, as observed in Figures 10 and 11, is the drastic increase in plastic strain at the metal-metal interface: 50.1% for copper-copper and 52.6% for copper-aluminum when the deformation is increased from 52% to 68%.

**Figure 10.** Total equivalent plastic strain at 7.62 mm of displacement for copper-aluminum at (**a**) 52%, and (**b**) 68% deformation and copper-copper at (**c**) 52%, and (**d**) 68% deformation. The die and mandrel are omitted. Units of strain are mm/mm.

**Figure 11.** Equivalent plastic strain through the extruded wall where normalized 0.0 is the inner edge of the tubular wall and normalized 1.00 is the outer surface of the tube. Vertical line represents bonding interface location. Plastic strain is taken at the vertical midpoint of the extruded bimetal.

Producing the tubes in this manner requires sacrificing the beginning and end of the extruded tubes. As is evident, the initial section does not fully achieve plastic deformation and the end section is not pressed fully past the extrusion ledge.
