Navigating the Extremes of Cold Forming: Numerical Comparative Deep Drawing Analysis of Copper, Aluminum, and Nickel Alloys

This deep-drawing process has become a significant element of the growing trend in metallurgical research, driven by the ever-increasing need for lightweight, high-reliability components in new production. In current research, there is growing hope for a combination of robust finite element analysis (FEA) and lab validation, with the aim of establishing the correct sequence of complex form parameters for most high-performance metals, especially aluminum, copper, and nickel alloys.

Shamso et al. [1] examined the deep drawing process of aluminum blanks by combining LS-DYNA finite element simulations with Taguchi and response surface approaches. The statistical analysis they conducted found punch velocity to be the key factor in critical deformation events, such as maximum thinning, forming force, and effective plastic strain. At the same time, the bulk thickness of initial blanks has a secondary effect. Bahanan et al. [2] conducted an in-depth experimental and computational study of the cold deep drawing of 3104-H32 aluminum alloys, with particular emphasis on the geometric effects of the quarter-corner plunger design and on different tribological settings. Using their FEA, they suggested a plunger radius of 3 mm and a coefficient of friction of 0.1 as a good fit to achieve homogeneous strain and address localized thinning. A research study conducted by Pan and Cai [3] on hydromechanical deep drawing of the alloy (AA6013) cylindrical components in the shape of a cylinder was carried out. A response surface design was used in the numerical model to optimize the selection of many important variables of the work. The authors concluded that the quality curve is highly sensitive to the combination of fluid pressure rate, interfacial friction coefficient, and die fillet radii, and identified a set of parameters that minimized the maximum thinning rate

Leelaseat et al. [4] proposed a new formability measurement method that uses the Q-value, the optimized formability value based on thinning limit diagrams, to optimize deep drawing of automotive aluminum alloy sheets. They showed that manufacturers can greatly reduce sidewall wrinkling and localized material thinning by simply varying initial blank sizes and redesigning tool shapes, e.g., by adding optimized semicircular draw beads. In a study by Khazaal and Szabolcs [5], the tribological and mechanical effects of using punches made from a hybrid iron-based metal matrix composite (MMC) produced by powder metallurgy during deep drawing were examined. Highly comparative tests conducted by the authors of the current paper showed that conventional steel tooling produced a constant effect: an increase in the required pulling force and a decrease in the total energy required to create, as set by the MMC punch nose. The study by Ulibarri et al. [6] examined the mechanics of deep drawing of Inconel 718 superalloy, with particular attention to how intermediate annealing treatments affect the material's structural response. Together with experimental validation, their numerical simulations showed that strategic thermal processing can reduce degradation of elastic modulus and greatly enhance overall formability.

Kotkunde et al. [7] studied the warm deep-drawing behavior of Inconel 625 by combining the constitutive equations of Sellars with the improved anisotropic yield criteria, namely, the Hill 48 and Barlat 89 formulations. Their detailed finite element simulations and tensile tests have shown that the limiting drawing ratio (LDR) increases significantly with rising forming temperature, and, at the same time, these experiments have reduced the forces required for drawing. Ulibarri et al. [8] conducted a detailed study of multi-stage deep-forming operations at room temperature to determine the harsh conditions for post-forming springback and the formability of the Inconel 718 alloy. The researchers concluded that it is important to apply intermediate heat treatments between sequential forming steps to mitigate residual stresses and alter the material's hardening behavior. Morchhale et al. [9] examined the formability of the IN625 superalloy by analyzing the relationships between critical process parameters (i.e., temperature, punch velocity, and blank-holding pressure) and its deep-drawing behavior. They have plotted operating limits for safe forming, and they have suitably observed the thermo-mechanical conditions for safe forming beyond which they have transitioned to wrinkling or fracture flaws. Finally, the study provides a valuable parameter to consider when tweaking the production of Inconel 625 parts to prevent structural failures, whether due to complex drawing procedures or material breakdown. To achieve active product formation in the deep drawing of copper-nickel alloy sheets, Deng et al. [10] developed a simple friction model for ultrasonically vibration-assisted deep drawing. By combining the dynamic interactions between the slide's velocity and vibration amplitude, their numerical experiments in ABAQUS and subsequent experimental tests showed that ultrasonic assistance significantly reduces interfacial friction. Kosti et al. [11] conducted a systematic review of the formability of pure copper sheets, using a combination of hydraulic press tests and FEA in Abaqus. Their tests set a maximum LDR of 1.82 for 1 mm-thick blanks, indicating that they can readily be formed with good structural formability. Phanitwong and Thipprakmas [12] have addressed the restriction in the formability of materials due to their anisotropy by designing a new Multi-Draw Radius (MDR) die. Their specialized design, using larger draw radii perpendicular to the rolling direction and smaller radii perpendicular to the 0° and 90° directions, was effective in reducing non-axisymmetric material flow and minimizing defects in the earnings by approximately 40%. The great interest in the article by Qin et al. [13] lies in the microstructural mechanics of the C10200 copper plate, arising from changes in crystallographic texture, and in the deep-drawability of the material developed during rolling reductions and annealing. The researchers showed that creating a dominant Cube texture (001)100 induces extreme planar anisotropy and promotes an imbalanced strain distribution, leading to catastrophic fracture during forming. The paper by Chen et al. [14] presents an advanced 3D FEA using an updated Lagrangian formulation to simulate micro deep drawing of square-hole flanges in copper sheets. By applying a specialized model for micro-elastoplastic materials, the researchers showed that a further increase in the punch fillet radius dilutes the peak stress and strain localized in the workpiece during deformation. Gong et al. [15] made a contribution to the sphere of micro-manufacturing by experimentally testing the deep drawing of C1100 pure copper (50-micron-thick) to produce miniature conical-cylindrical cups. Using a specially designed multi-operation compound mold, the team produced multi-part complex micro-elements with internal bottom diameters as small as 0.4 mm.

The direct intent of this work is thus to compare the deep drawing behavior of the three alloys—pure copper, AA6061 aluminum, and Monel 400, as these three classes of alloy are deep-drawn under the same geometric conditions. Most of the literature that examines the effect of a draw punch would deal with a single material, but the novelty of this paper is the quantification of the responses of two completely different crystallographic structures to the same macroscopic tooling geometry (a 40 mm diameter draw punch and a 1 mm blank thickness). The process input parameters include non-linear elastoplastic material models, defined differential friction boundaries, and mathematically tuned blank-holder forces. The main output parameters being measured are temporal reaction forces/press tonnage, von Mises equivalent stress topologies, and the spatial distribution of equivalent plastic and elastic strain.

The finite element model (FEM) consists of 160325 elements and 37539 nodes. To obtain the hexahedral layers, the sheet was discretized using a strict sweep with an element size of 2 mm to avoid shear locking [18]. This results in a very fine 2 mm mesh capturing – extremely localized – strain and stress gradients at the die and punch corners. The rigid punch, die, and blank holder were discretized at 3 mm using the ANSYS Automatic Meshing algorithm. This larger mesh can efficiently represent surface curvature for contact algorithms without over-consuming computational power since there will be no internal stresses to resolve in rigid bodies. A mesh convergence study was conducted to ensure that this particular sizing choice achieved grid independence and was mathematically optimal. The deep drawing simulation was performed 10 times sequentially, each time reducing the element size by 0.5 mm (blank 0.5-10 mm and tooling 1-11 mm). The maximum von Mises stress was used to determine the optimum mesh size, as the output magnitudes flattened, resulting in a low variance of 2–5% between iterations. The results of these convergence tests are summarized in Table 1.

The mesh convergence study is a trade-off between numerical accuracy and computational efficiency. Large-scale stress singularities resulting from penalty-based contact were prevented using Peak Punch Force as the global convergence metric. As shown in Table 1 and Figure 2, a size of 2–3 mm is optimal, with a blank of 2 mm and tooling of 3 mm. As shown in Table 1, only 160,325 elements are needed, which is approximately 84% fewer than the number required to be left at extreme refinement (0.5-1.5 mm). In so doing, the number of nodal degrees of freedom is drastically diminished while accuracy of the nodal structure is preserved. Moreover, this chosen mesh size serves as an additional mechanical filter that attenuates high-frequency contact noise while still clearly displaying plastic flow or local thinning at the die radius. Finally, a Grid Convergence Index (GCI) was computed using Richardson Extrapolation (mathematically detailed in Appendix A) to rigorously confirm the solution's grid independence.

Table 1. The simulation model mesh independence test results

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Figure 2. Simulation model von Mises stress convergence with the mesh element size

The ANSYS Explicit Dynamics solver was used to simulate the deep-drawing process using a modified Lagrangian approach, in which the mesh moves through the deforming material continuously. Computational stability depends on the ability to solve nonlinear partial differential equations at each time step. At the macro scale, the primary factor governing macroscopic deformation is the dynamic equation of linear momentum conservation, which can be written as [19].

$∇. \sigma+\rho b=\rho$      (1)

In this context, (σ) denotes the Cauchy stress tensor, (ρ) the density, (b) the body force, and (u) the acceleration. The equations of elastoplasticity relate external strain to internal stress, as the Plasticity created by punch harvesting is irreversible. Simulation results show the real-time plastic start based on J2 theory and the von Mises criterion. The von Mises equivalent stress (σ_vm), based on deviatoric stress components (S_ij), is defined as [19]:

$\sigma_v m=\sqrt[2]{\frac{3}{2} S_{i j} S_{i j}}$           (2)

According to von Mises' criterion, yielding occurs when the energy of deformation in the material reaches a critical value equal to the yield strength in plain tension. Such a phenomenological model works perfectly well for ductile metals such as aluminum, copper, and nickel alloys, since it assumes that these metals cannot undergo plastic deformation under hydrostatic pressure. In contrast, shear stresses cause the material to flow. The solver uses an isotropic hardening rule when deforming the material (which is often considered a power law or prescribed by the power law: σ = K εN). This guarantees that the equivalent plastic strain can continuously increase the yield surface as the strain increases dynamically, thereby continually forecasting the additional force required to draw the strain-hardened cup walls. The equations governing the resistance to the flow of material over the die corner and under the blank holder are determined by interfacial friction. This interaction is developed based on a modified Coulomb friction model, where the shear frictional stress (Tf) due to friction between two bodies at the interface is [20]:

$T_f=\mu p$              (3)

where, µ is the coefficient of friction and p is the normal contact pressure exerted by the rigid tooling. In the formulation of the contact penalty employed in this work, the normal pressure is not a constant value. Still, a dynamic penalty reaction is given by p = kn δ, where kn is the normal contact stiffness, and δ is the penetration depth between the finite elements. If the calculated shear stress exceeds a critical shear yield limit (often exceeded by localized thinning), the software switches to a sticking-friction model. It is this mathematical boundary condition that will ultimately determine what happens to the blank: whether it will be drawn into the die cavity without catastrophically fracturing under tensile stress due to excessive frictional drag. The geometric deformation during deep drawing is extremely high, rendering traditional small-strain engineering formulas invalid. ANSYS Explicit solver employs a large-strain (finite-strain) formulation based on the deformation gradient tensor. The instantaneous rate of deformation represents the spatial rate of deformation of the material, the spatial rate of deformation tensor D_ij, which is the symmetric part of the velocity gradient [21]:

$D_{i j}=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$           (4)

where, ∂vi and ∂vj represent the velocity vectors of the material points, x_i and x_j are their spatial coordinates in the current configuration. The updated Lagrangian formulation, in contrast to linear static analyses, which quantify the strain with respect to the original undeformed shape, is a dynamic reference frame. That is, mathematically, integrating the time-dependent version of the tensors D_ij yields the actual (logarithmic) strain in the material. The derivation is key to predicting localized necking and sheet thinning because it accounts for the material matrix's constant reorientation due to bending around the die corner and stretching of the cup wall. Whereas the von Mises yield criterion is applicable to determine when the material will yield, the plastic flow rule is applicable to determine how the material will behave once the yield point has been surpassed. The associated flow rule (also known as the von Mises equation in isotropic materials) is used in the simulation, which is mathematically formulated as [21]:

$d \varepsilon_{i j}^p=d \lambda \frac{\partial f}{\partial \sigma_{i j}}$           (5)

where, $d \varepsilon_{i j}^p$ denotes the plastic strain increment, dλ is the plastic multiplier (which can be either positive or negative), and f is the yield function. The rationale of the yield surface in the stress space is denoted by the term ∂f/(∂σ_ij). The related flow rule mathematically requires that the orientation of plastic strain increments be strictly perpendicular (normal) to the expanding yield surface. In a deep drawing scenario, where aluminum and copper blanks are made to deform by approaching the blank holder in a hydrostatically applied pressure fluid, this equation demonstrates the fact that the hydrostatic pressure between the material and the blank holder will not cause the material to permanently compress the volume of the material (i.e., plastic deformation is a volume-preserving process, $d \varepsilon_{i j}^p$ = 0). The entire plastic energy is instead used solely to form the blank into a cylindrical cup. The deformation or plastic flow of the elemental nodes causes an even greater resistance to further deformation as microscopic dislocations are generated in the metal lattice. To correctly model this macroscopic strengthening, a strain-hardening isotropic model, typically a power law, is implemented in the solver [22]:

$\sigma_y=\sigma_o+K\left(\bar{\varepsilon}^p\right)^n$             (6)

εp is the accumulated plastic strain, n is the strain-hardening exponent, K is the strength coefficient, and σ_o is the initial yield stress. Evidently, these are constitutive parameters, which govern the growth of the von Mises yield surface. In a metallurgical sense, the high strain-hardening exponent (n = 0.45) of copper results in the maximum of dynamic load redistribution. AA6061 Aluminum is intermediate-hardened (n = 0.20), preventing stable flow. Finally, Monel 400 is mechanically very resistant, with a high initial yield strength (σ_o = 280 MPa) and strength coefficient (K = 950 MPa), which means it will require very high blank-holder pressure. The thermomechanical and elastoplastic properties of the three investigated materials used in this simulation are listed in Table 3 below.

Table 3. Thermomechanical and elastoplastic constitutive material properties inputted into the finite element solver

This equation is a dynamic form for calculating the von Mises yield surface radius at all integration points. As the cup wall is extended downward, the punch is pushed, and the wall accumulates the cup's strain, as shown below. As a consequence, the local yield strength σ_y mathematically rises. Without this hardening equation, the solver would predict a perfectly plastic material, and the simulation would fail (artificial tearing) immediately and catastrophically at the die corner. It is actually the parameter n that determines the extent to which the sheet can be punched before localized necking occurs. The solver employs a purely mathematical Penalty Method that aims to prevent the cold flow of the blank's deformable mesh through the rigid tooling mesh. The normal contact force (fn) of a cutting node can be obtained as [22]:

$F_n=K_n \cdot g_n \cdot n$              (7)

In which the penalty contact stiffness factor is given as K_n, the geometric penetration depth (calculated) as (g_n), and the outward normal unit vector of the target surface as n. The penalty method is a numerical approach, mathematically akin to contact mechanics, that uses a series of very small virtual springs at the interface between the instrument and the sheet to enforce kinematic constraints and prevent interpenetration. As the software detects that a node on the aluminum sheet has violated the punch or die surface (g_n > 0), it immediately activates this "spring". The penetration distance is multiplied by the stiffness, K_n, which creates a large, highly localized repulsive force, F_n, that propels the node into the surface boundary. The subtly associated physics of deep drawing simulation is that the stiffness of the material is selected to ensure that the sheet merges realistically with the tooling; if the stiffness is too low, the sheet will become excessively merged; if the stiffness is too high, the mesh will create numerical stress singularities and high-frequency noise that have been studied in the mesh independence experiment.

7.1 The three models of equivalent stress analysis

The von Mises equivalent stress (σ_vm) is used to compare the blank response from multi-axial stresses–flange tension/compression, die radius bending, and cup-wall tension, etc.–to a single scalar value that is based on distortion energy. The resulting stress distribution and localization are shown in Figures 5(A), (B), and (C).

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Figure 5. Deep drawing process von Mises stress distribution map: (A) AA6061, (B) copper, and (C) Monel 400 alloy

In Figure 5(A), the deformation of the AA6061 aluminum alloy is highly localized, with the maximum equivalent stress occurring at the transition between the punch corner and the cup wall. The strain-hardening exponent of AA6061 is moderate, leading to high stress gradients and stress concentrations. There is a small process window: small increases in friction/tightening force would cause localized "necking" and "cup-bottom" tearing, with critical factors affecting the desired qualities including optimized tribology.

In contrast, Figure 5(B) shows the wider and more consistent stress distribution of the vertical wall of the cup made of the copper alloy. Due to its high strain-hardening exponent [23], the yielding areas will quickly strain-harden, thereby transferring mechanical loading to less-hard parts of the wall. The yielding and hardening of the material in this dynamic process enable the avoidance of extreme localized reductions, reduce peak stresses by involving a greater volume of the geometry, and, with a mathematical argument, justify copper's better macroscopic drawability, despite sacrificing dimensional stability [24]. Finally, Figure 5(C) shows the very elastoplastic behavior of the Monel 400 superalloy. It is characterized by a high initial yield strength, thereby inducing excessively high stresses in the cup flange and the clamped flange. Drawing forces are significantly greater, and blank-holder pressures are enormous when forming Monel, because compressive hoop stresses counteract wrinkling tendencies. This results in high frictional drag and excessive bending stresses in the die profile, which are close to the alloy's plasticity limits. In addition, significant residual stresses indicate a high level of elastic springback after tool removal. Therefore, hot forming or special lubrication is required for successful deep drawing in Monel 400. Finally, treatment of areas of extreme von Mises stress is an important step in diagnosing specific kinematic and metallurgical failure modes. Determining stress extrema can identify the location of the maximum plastic strain energy relative to the remaining undamaged areas [25]. ANSYS numerical probes in Figures 6(A), 6(B), and 6(C) enable a highly accurate comparative analysis of the stress loci of the three alloys.

Figure 6(A) shows that the maximum point of the equivalent stress of the AA6061 simulation is 292.31 MPa, which occurs at the punch profile radius. This area is subjected to traction forces required for flaring and is a critical site for cup bottom tear. The flange lip, where the minimum stress occurs (15,155 MPa), simply pulls inward with minimal severe friction/bending.

By contrast, the highest stress region (262.82 MPa) on copper is observed on the upper part of the cylindrical wall, with the lowest stress (25.777 MPa) being seen along the center of the cup bottom (mechanical dead zone), where there is a great deal of friction and therefore not much flow. However, due to the unusual strain-hardening property of copper, the deformation zone is "moving" dynamically upwards, avoiding the localization of deformation at the punch corner and redistributing the load and consequently enhancing formability [26]. The stress topology differs most markedly in the Monel 400 superalloy (Figure 6(C)). However, because it is so rigid, it requires very high blank-holder pressure and therefore produces a high peak stress of 568.16 MPa, with a high concentration of stress around the die profile radius, along with large cyclic bending and frictional drag forces. Like Cu, it has a minimum stress at the bottom of the cup (30.902 MPa). It is a severe stress gradient, indicating a high potential for elastic springback and sidewall fracture in the absence of lubrication [27]. Finally, color maps of the time-dependent evolution of von Mises stresses during the punch stroke (showing the dynamic strain-hardening laws of each material) are shown in Figure 7. Monel 400 has the highest and steepest trajectory, driven by its solid-solution strengthening and tremendous initial yield strength (σ0), and responds very strongly to the initiation of plastic flow [28].

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Figure 6. Deep-drawing maximum and minimum stress locations: (A) AA6061, (B) copper, and (C) Monel 400 alloy

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Figure 7. The time-dependent stress curves

Temporal changes in the von Mises equivalent stress, usually proportional to the punch's downward movement over time, provide a dynamic visualization of the constitutive hardening laws of each alloy. Although spatial stress contours determine the location of a material's failure, time-dependent stress contours in Figure 7 show how the material's internal resistance to plastic flow varies as macroscopic deformation develops. Based on the unique metallurgical characteristics of these alloys, the time-dependent stress curves indicate three distinct strain-hardening and flow stress responses during the deep-drawing sequence. Monel 400 nickel-copper superalloy always takes the highest-magnitude trajectory in a time-dependent stress distribution. Since this kind of material is characterized by solid-solution strengthening, it naturally has a very high initial yield strength (σ0). As initial contact is made with the punch and time passes, the Monel 400 curve rises steeply. The material vehemently resists the onset of plastic flow [28].

After yielding, Monel 400's time-stress curve is steep and continues to climb due to extreme drawing resistance during radial drawing and strong microscopic dislocation entanglement. This action demands an ever-growing compressive force during the stroke, driving the alloy into catastrophic frictional drag and tensile failure, as shown by the pronounced stress peak [29].

By contrast, the AA6061 aluminum alloy exhibits an intermediate energy profile. It is a steeply curving line with a much more rapid leveling off. AA6061 has moderate strain-hardening capacity and does not tend to distribute strain energy evenly, causing a mechanical bottleneck at the punch corner. The material here takes in all the tensile load shown on the drawing and quickly approaches its breaking strength. This also means that the stable processing window is very small and the alloy is susceptible to cup-bottom fractures when not processed with ideal lubrication. Lastly, pure copper is highly efficient in its plastic flow, which arises from a mobile FCC crystal structure. It shows a continuous rise in the stress-time curve, with no sudden upsurge as in Monel, and an early exhaustion of Aluminum. These regions of localized yielding with an impressive strain-hardening exponent (n) immediately develop hardening and transfer mechanical loads to softer, unyielded regions. This temporal stabilization will enable the copper to distribute and absorb the deformation energy throughout the whole punch stroke, leading to a high LDR and a stable cup wall thickness of copper [30].

7.2 The three models of elastic strain analysis

The Equivalent Elastic Strain reflects dimensional accuracy of the final part, while the limiting value of plastic strain defines the formability limits. In Phase 3 (tool retraction), the entire accumulated elastic (recoverable) strain was released, resulting in the macroscopic observation of residual-stress recovery (springback) [31]. Elastic strain profiles were created to assess the stability of the structures of all three alloys after forming, as shown in Figures 8(A), 8(B), and 8(C). The severe bending and unbending phenomenon under high tensile load increases the equivalent elastic strain in AA6061 aluminum (Figure 8(A)) and is concentrated precisely at the transition radii (punch corner and die profile). For 6061, a medium-strength alloy, the work hardening is weak, and a large amount of deformation energy is stored elastically in the crystal lattice at such radii. These localized strain bands will serve as mechanical hinges, and as the tooling is removed, the cup walls will tend to flare outwards, and the flange will bend.

For 6161 aluminum, the elastic strain at the transition radius is very high, whereas at the cup bottom it is close to zero, which means manufacturers will have to resort to die compensation or even over-bending to compensate for springback. On the other hand, the elastic strain profile of pure copper is much more uniform along the drawn wall, a feature related to the material's dynamic strain hardening (Figure 8(B)). Since little energy is stored in the radii as recoverable energy, the effect of springback is minimal, and the dimensional integrity after bending is very good, which is desirable for micro-manufacturing with copper. Monel 400, a superalloy used in the aerospace industry, is difficult to form. It has very high flow resistance, which requires very high tooling forces and results in a very high elastic strain at the die profile radius. The large amount of stored energy may result in high elastic recovery forces upon removal of the tool, which can cause serious shape faults such as sidewall curling and flange deformations. Thus, for successful cold forming of Monel 400, intermediate thermal annealing, springback compensation, and die design may need to be aggressive. The total strain applied can, in theory, be divided into permanent plastic strain and recoverable elastic strain, that is, potential energy stored temporarily in the atomic bonds [32]. This interplay of absorption and dissipation of elastic energy into plastic during the punch stroke is shown as a function of time in Figure 9. This accumulated elastic strain is the starting point of the recovery of the residual stress in Phase 3 when mechanical loads are released - this is macroscopically seen as dimensional springback.

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Figure 8. Deep-drawing elastic strain profile and the maximum and minimum strain locations: (A) AA6061, (B) copper, and (C) Monel 400 alloy

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Figure 9. The time-dependent elastic strain curves

For AA6061 aluminum, the elastic strain at the transition radius is very high, whereas at the cup bottom it is close to zero, which means manufacturers will have to resort to die compensation or even over-bending to compensate for springback. On the other hand, the elastic strain profile of pure copper is much more uniform along the drawn wall, a feature related to the material's dynamic strain hardening (Figure 8(B)). Since little energy is stored in the radii as recoverable energy, the effect of springback is minimal, and the dimensional integrity after bending is very good, which is desirable for micro-manufacturing with copper. Monel 400, a superalloy used in the aerospace industry, is difficult to form. It has very high flow resistance, which requires very high tooling forces and results in a very high elastic strain at the die profile radius. The large amount of stored energy may result in high elastic recovery forces upon removal of the tool, which are responsible for serious shape faults such as sidewall curling and flange shape deformations. Thus, for successful cold forming of Monel 400, intermediate thermal annealing, springback compensation, and die design may need to be aggressive. The total strain applied can, in theory, be divided into permanent plastic strain and recoverable elastic strain, that is, potential energy stored temporarily in the atomic bonds [32]. This interplay of absorption and dissipation of elastic energy into plastic during the punch stroke is shown as a function of time in Figure 9. This accumulated elastic strain is the starting point of the recovery of the residual stress in Phase 3 when mechanical loads are released - this is macroscopically seen as dimensional springback.

7.3 The three models of plastic strain analysis

Equivalent Plastic Strain (EPS) quantifies plastic deformation on a macroscopic scale that can accumulate, is irreversible, and is not elastic. On these maps of the extremes of plastic strain, shown in Figures 10(A), 10(B), and 10(C), the area of severe permanent strain can be identified. Finally, these ANSYS topologies highlight important areas (where thinning may occur) where localized ductile fracture is likely, illustrating that these three different alloys have distinct material flow mechanisms [33].

Figure 10(A) further shows that the AA6061 aluminum alloy is subjected to a very strong plastic strain gradient at the punch profile radius. Due to its moderate strain-hardening ability, it is unable to distribute deformation energy; therefore, this area is significantly affected under biaxial stretching, creating a high-risk zone for localized thinning and ductile cup-bottom tearing. In contrast, Figure 10(B) shows an efficient distribution of plastic strain in pure copper. Regions of yielding work harden instantly due to an outstanding strain-hardening exponent (n), and mechanically load-bearing properties are dynamically transferred to softer regions. Such equidistribution imposes no additional strain on any localized region, thus giving copper much better formability and the ability to withstand much deeper draws [34]. In both metals, the surface at the bottom of the cup is kept within a frictional "dead zone," resulting in minimal strain. The Monel 400 model (see Figure 10(C)) provides a clear example of the very high mechanical property requirements of hot- and cold-formed high-strength superalloys. It has very high plastic flow resistance; therefore, very high blank-holding forces are needed to prevent flange wrinkling. During the dragging of this stiff material, extreme cyclic bending and shear friction occur, and as such, high plastic strains will be focused at the die profile radius. This acute localization indicates the onset of sidewall shear fractures and must be hot-formed or lubricated to avoid. Figure 10 shows the spatial distribution of permanent deformation, whereas Figure 11 shows the temporal (dynamic) accumulation of Equivalent Plastic Strain (εp). It is a volume-preserving curve that correlates strain increments over time in continuum mechanics. It explicitly accounts for the kinetic pressure energy that is lost to permanently change the macroscopic geometry of the material, as different from that of recoverable elastic strain energy [35].

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Figure 10. Deep-drawing plastic strain profiles and the locations of maximum and minimum strain: (A) AA6061, (B) copper, and (C) Monel 400 alloy

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Figure 11. The time-dependent plastic strain curves

Pure copper exhibits a continuous, consistent, time-dependent plastic strain trajectory. With high ductility and a very good dynamic strain-hardening exponent (n), it can balance mechanical loads smoothly during the punching stroke. During constant yielding, constant hardening occurs, thereby avoiding local thinning and providing the material with excellent drawability and strength. On the other hand, the elasto-plastic range of the AA6061 aluminum alloy is very small. It has limited strain-hardening capacity, resulting in significant accumulation of erroneous plastic strain early in deformation. This peak indicates a high level of biaxial stretching at a narrow radius in the punch profile, where the material reaches its load limit rapidly. Hence, AA6061 has a very narrow stable forming window, and any delay from the rapid accumulation period leads to localized necking and catastrophic ductile fracture (cup-bottom tearing) [36].

Finally, the Monel 400 superalloy exhibits high thermodynamic stability owing to its high initial yield strength (σ0). Analysis of its temporal shape shows none of the time is being spent where it really counts - within the limits of elasticity of the rigid matrix. Plastic strain increases suddenly at the die profile radius, once the process begins to yield under very high pressure, applied in the tooling process. This sudden strain accumulation at high velocity indicates the severe bending, unbending, and worst of all, frictional shear of the superalloy, tilting it right on the edge of the ability to retain a plastic hinge, making it extremely prone to what could be called a "sidewall shear rupture.

7.4 The reaction force analysis

The reaction force model direction for the object being analyzed must be set, and the reaction force direction at each point must be defined before the object's time load profile is analyzed. Concentric deep drawing is used to determine uniaxial deformation energy (Y-axis: vertical punch stroke), as illustrated in Figure 12. This opposing Reaction Force is a global stress vector representing 3 mechanical and thermodynamic resistances: the undeformed sheet flange, the instant yield strength, and the geometry-wide yield strength. The resistance to material flow is greatly influenced by the material's strain hardening, clamping forces, and friction drag at the die profile. Hence, the Y-axis reaction force is a measure of the net or ultimate out-of-work mechanical energy required by the press to make the cup. The internal resistance to plastic flow over time is mapped in Figure 13 and is directly proportional to the stroke depth, as the punch velocity is constant. These distinct force trajectories were found to reflect distinct kinetic flow processes in the three alloys [37]. Notably, the upper limit of the chart is the Monel 400 superalloy, which requires the most energy. The Monel 400 is solid-solution-strengthened and, as such, has tremendous yield strength—so much so that it will require a great deal of kinetic energy to exceed its elastic limit (as evidenced by the extreme curvature of the Y-axis reaction energy forces). The force curve increases rapidly with punch depth due to the alloy's high strain hardening and the need to press the blank at high force to avoid flange wrinkling. Die-radius frictional shearing and structural resistance require an exponential increase in die-pressing tonnage. Therefore, the use of cold-formed Monel 400 will cause machinery to really take its licking and potentially accelerate tool deterioration and sidewall shear fractures [38].

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Figure 12. Reaction force direction in the three deep-drawing models

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Figure 13. Reaction force as a function of time

This type of reaction force curve, as seen for the AA6061 aluminum alloy, is elastoplastic in shape, characteristic of precipitation-hardened light metals. The curve initially rises but soon levels off, reflecting the alloy's arming behavior, while at the punch corner, the deformation is severely localized. This plateau is a precursor to macroscopic thinning; any further increase initiates ductile fracture (cup-bottom tearing). By contrast, a copper that is 100% pure has the most stable, continuous, and lowest absolute level of force curve. Copper has a FCC crystal structure, which is highly mobile and exhibits good dynamic strain hardening. It evenly transfers drawing force to gentler areas without 'load spikes' or 'plateaus'. That stability enables copper to be drawn at a highly aggressive draw ratio, resulting in reduced press tonnage and reduced tool wear. As shown in Figure 13, the numerical values of the final reaction forces are very high in the main Y direction (the punch stroke axis). Physically and metallurgically, this is predicted because three large thermodynamic and mechanical obstacles to be surmounted by the cumulative Y-direction force are envisioned:

Initial Yield Strength: It must be greater than the material's structural yield strength to be shaped over and through the deformation zone; a large amount of kinetic energy is required to deform high-strength materials such as Monel 400.

Dynamic Work Hardening must be continually overcome by strain hardening. The number of dislocations accumulated and entangled during severe plastic deformation increases exponentially with each millimeter of stroke, and so does the force required for the drawing.

Tribological resistance must withstand extreme friction forces that pull the outer flange against the blank holder and the tremendous clamping forces of the blank holder that tear the sheet around a die corner.

A better understanding of elastoplastic behavior under multiaxial loading is required to optimize the sheet metal forming process. Table 5 lists a comparative study of finite element simulations of aluminum, pure copper, and Monel 400.

Deep drawing success is strictly related to the alloy, as this comparison table shows. Copper provides the best of all these benefits with dynamic strain hardening, achieving more uniform wall thickness, virtually zero springback, and drawability – without failure – thanks to Copper's mobile FCC lattice. In contrast, AA6061 aluminum has a narrow temperature range over which it can be used. The small work-hardened zone also causes high stresses and plastic strain to concentrate at the punch profile radius, resulting in high tearing forces that lead to localized necking and cup-bottom tearing. Finally, there is the very high tonnage and blank-holder pressure required for the extremely stiff Monel 400 superalloy. This produces an excessive amount of die-radius frictional shear and elastic energy buildup, which requires special tooling or heat treatments to prevent denuding shear and sidewall fracture and to reduce substantial springback.

Table 5. Comparison of elastoplastic and kinematic behaviors during deep drawing of AA6061, copper, and Monel 400 alloys

In summary, this investigation demonstrates that the enhanced Lagrangian finite element framework can correct and realistically predict the different macro failure modes, press tonnage, localized strain, and elastic recovery behavior as it relates to different alloys; and this framework can be used as a powerful tool for optimization of deep drawing parameters on the shop floor:

•Pure Copper: Exhibits excellent drawability and stability. Has a high dynamic cross-over, strain-hardening exponent, so dynamically redistributes all mechanical loads. This uniform flow helps to avoid thinning in local spots, reduces springback, and allows minimal tonnage on the press for aggressive geometries.

•AA6061 Aluminum has a very limited operating range. It has a low work-hardening rate, so energy is not properly dispersed, generating very high equivalent stress and plastic strain at the punch profile radius. This kinematic issue makes it highly susceptible to ductile tearing instantaneously under non-optimal conditions.

•Monel 400: Taps the very limits of cold-forming. Its very high initial yield strength and flow resistance require an exponential increase in the blank-holder and press forces. This rigidity creates very high frictional shear stress at the die radius and will cause large elastic energy to be stored, leading to high post-forming distortion (springback) if warm forming or die compensation is not used.

These validated baseline models will provide an excellent basis for systematically sweeping blank holder forces, friction coefficients, and die radii in future Design of Experiments (DOE) parametric studies to extrapolate the ranges of wrinkling and thinning that are not adversely affected by these parameters.