Numerical-experimental single point incremental forming of thin circular plate

3.1. Formability limit

Based on investigations into fracture behavior and thickness variation in the SPIF process, it has been established that uniform plastic deformation continues throughout the thinning of the formed surface until fracture occurs, without any experimental evidence of necking [39]. This observation suggests that the SPIF technique inherently prevents the formation of a necking region. The absence of necking can be attributed to the nature of the process: necking typically develops along the bending path in conventional forming, but in SPIF, it is suppressed by the continuous pressure applied by the forming tool. The hemispherical tool imposes localized plastic deformation at each forming point, and the surrounding material resists the growth of localized thinning, effectively preventing necking. As a result of these unique deformation characteristics, the conventional Forming Limit Curve (FLC) becomes inadequate for predicting failure in SPIF. Instead, the Fracture Forming Line (FFL) is used to describe the material’s failure behavior under incremental forming conditions. The FFL lies above the FLC, as shown in the Fig. 2, reflecting the enhanced formability of materials in the SPIF process [40]:

In typical SPIF experiments, the ratio of tool radius to sheet thickness ($r_{tool}/t$) commonly falls within the range of 2 to 10. Based on this ratio and the derived relationships, the slope of the corresponding forming limit line lies approximately between (–1 $<m<$ –1.4). This theoretical range is consistent with experimental findings. Accordingly, in the Forming Limit Curve (FLC) specific to the SPIF process, fracture tends to occur along a linear path described by ${\epsilon }^1+{\epsilon }^2=q$, where $q$ is a material-dependent constant. Correspondingly, the thickness strain at fracture can be expressed as ($\epsilon ₜ=-q$). This relationship reinforces the notion that failure in SPIF is governed by a combination of principal strains, and that fracture occurs without the onset of necking, differentiating it from conventional sheet forming methods.

Fig. 2A view of the FLC position in SPIF compared to deep drawing methods [40]

3.2. Stress and strain state

In the incremental forming process, the CDEF element, illustrated in the corresponding Fig. 3 is analyzed to evaluate the vertical forces, shear forces, and bending moments at the contact interface between the sheet and the tool tip [41]:

To derive the equilibrium equations and accurately assess the stress–strain state in the SPIF process, the following assumptions are made [42]:

1) Plane strain conditions are assumed to govern the deformation behavior.

2) The region of interest is located directly beneath the contact point between the forming tool and the sheet.

3) The overall forming process is divided into three distinct deformation zones:

– Deformation of the flat region.

– Deformation of the axisymmetric region, treated under plane strain conditions.

– Deformation of the corner region, considered to undergo biaxial tension.

4) Bending moments are neglected, and the effect of friction is assumed to be negligible.

5) The material is considered to exhibit perfectly plastic behavior, with no strain hardening incorporated into the model.

Fig. 3A view of the stress analysis position for normal and tangential stress [41]

3.3. Tensile test

The tensile test is a standard method used to evaluate a material's mechanical response under axial loading. It provides critical information about tensile properties, including the elastic and plastic deformation ranges, elongation, Young’s modulus, ultimate tensile strength, yield point, and yield strength. In this study, tensile specimens were prepared according to the ASTM E8 standard to ensure consistency and reliability of results. The mechanical testing was conducted using a SANTAM STM-150 universal testing machine, operating at a constant crosshead speed of 10 mm/min. The obtained data were used to determine the yield stress and other relevant mechanical properties of the sheet material, which are presented in the corresponding Table 1. Fig. 4 displays the stress, strain curves generated from the tensile test, illustrating the elastic and plastic behavior of the galvanized sheet under uniaxial tension.

Fig. 5 presents the sample model created using the simulation software, with its corresponding mechanical properties detailed in Table 1.

4.1. Identification of safe zones and stress distribution

Finite Element Analysis (FEA) performed with FormingSuite revealed distinct zones of deformation, categorized into safe, low-stress, and high-stress areas. The novel contribution here lies in the coupled visualization of deformation patterns (tensile-tensile and tensile-compressive), offering a comprehensive understanding of the mechanics at various forming stages. Fig. 6 confirms the spatial stress evolution, especially in curved regions and near the forming edge. The analysis shows that stress accumulation intensifies near the tool's reversal path, where curvature changes are more abrupt. This finding is essential for toolpath optimization and failure risk reduction, particularly in parts requiring complex geometries.

4.2. Thickness strain and material flow analysis

Fig. 7 demonstrates that thickness strain reaches its peak near the initial contact zone, gradually stabilizing with increasing radial distance. The minimum thickness is observed at approximately 90° of curvature, coinciding with the highest formability challenge. Notably, simulation results indicate strain localization without necking, consistent with SPIF’s suppression of classical necking mechanisms. This behavior supports the use of FFL over traditional FLC, emphasizing the uniqueness of SPIF-induced failure dynamics. The presence of a stable deformation gradient affirms the viability of the SPIF process for high-angle forming of circular geometries.

Fig. 6Sheet metal forming: a) stretching, b) qualitative effect in different regions, c) safe margin

a)

b)

c)

Fig. 7Solid mechanics, a) strain percentage of the forming operation, b) Specimen’s thickness, c) maximum principal strain, d) minimum principal strain

a)

b)

c)

d)

4.3. equivalent strain and deformation modes

Figs. 8 and 9 illustrate the equivalent strain and stress distributions. It is observed that regions with highest strain do not always correspond to peak stress, due to the material’s local deformation accommodation. This decoupling is a novel observation and has implications for thickness prediction models and residual stress control in future SPIF strategies. These insights are pivotal for improving simulation fidelity, as they highlight the nonlinear, non-proportional nature of SPIF deformation, especially in ductile, coated materials like galvanized steel.

Fig. 8Maximum equivalent strain

FormingSuite offers the additional capability of measuring both elongation and compression of the workpiece during the forming process. In this study, the FLD generated by FormingSuite was analyzed, as shown in Fig. 10(a). When compared to the FLD obtained from the ISF process in Fig. 10(b), the results highlight the enhanced formability characteristics enabled by the ISF method. The flexibility of the ISF process significantly improves the material’s ability to deform, often surpassing conventional forming methods. The FLD serves as a predictive tool, illustrating the material’s deformation capacity under plane strain conditions by plotting the maximum and minimum principal strains. This diagram also includes the FLC, which delineates two key regions: a safe zone, where deformation occurs without failure, and a failure zone, indicating the threshold beyond which fracture is likely. The FLC obtained through ISF demonstrates superior performance in terms of allowable strain, confirming the method’s effectiveness for developmental and prototype-level shaping applications.

Fig. 9Equivalent stress

Fig. 10Elongation and compression of different points, a) top face, b) 1 – failure, 2 – FLC, 3 – safe, 4 – tension-compression, 5 – tension-tension

a)

b)

Fig. 11Experimentation, applied force, a) mold structure, and b), c) specimen

This technique has significantly enhanced formability, nearly tripling the material’s deformation limit. Fig. 11 is provided to enhance understanding of the forming process and tool interaction. As illustrated in Fig. 11(a), the shaping tool must apply varying levels of force to achieve the desired geometry, depending on the complexity of the mold structure, which is depicted in Fig. 11(d). In the initial stages of the forming process, the tool maintains vertical point contact with the sheet. As the operation progresses and the forming angle approaches approximately 90 degrees, the tool’s interaction becomes more complex, requiring greater force and control, as highlighted in Fig. 11(a). Using simulation outputs, two samples were successfully generated, shown in Fig. 11(b) and 11(c). The geometrical dimensions of these formed samples were then analyzed using the CMM, with the results presented in Fig.11(a) through 11(c). These Figs. 11 and 12 validate the accuracy of the forming process and the consistency between simulated and physical results.

Fig. 12Inverse engineering process, a) dimensions, b) CMM, c) 3D model

a)

b)

c)