<p>This study aims to precisely predict the onset of fracture during the single-point incremental forming (SPIF) of extra deep drawing (EDD) steel sheets using the Bao–Wierzbicki (BW) ductile damage model, incorporating anisotropy of the sheet metal in the analytical formulation. In this regard, a fresh attempt was made to optimize the theoretical BW fracture locus by optimizing the central hole (CH) fracture specimen geometry. Subsequently, four different CH specimens, namely CHD0, CHD2.5, CHD5, and CHD6, were considered by varying the hole-to-ligament width ratios within a range of 0-0.3. The CH specimen geometry was optimized by comparing the evolution of effective plastic strain with respect to stress triaxiality <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11665_2025_11794_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\left(\eta \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>η</mi> </mfenced> </math></EquationSource> </InlineEquation> and the Lode angle parameter <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11665_2025_11794_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\left(\theta \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>θ</mi> </mfenced> </math></EquationSource> </InlineEquation>. It was found that the CHD5 specimen experienced a purely uniaxial stress state with a <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11665_2025_11794_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11665_2025_11794_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> value almost equal to 0.33 and 1.0, respectively. Afterward, the BW damage model was calibrated using the four CH specimens, and different fracture loci were generated. Subsequently, the analytical fracture curves were validated with the safe and failed experimental strain data obtained through the SPIF of variable wall angle cone (VWAC) and variable wall angle pyramid (VWAP) cups. It was observed that the fracture locus obtained using CHD5 geometry accurately predicted the onset of fracture for SPIF cups. Further, the four distinct fracture loci were integrated separately into the finite element (FE) simulation of the SPIF process linked with the Hill48 anisotropic material model to estimate the formability. The error in dome height prediction was observed as 6.97% and 6.35% for VWAC and VWAP cups, respectively. It was concluded that the BW fracture locus calibrated using the CHD5 geometry was the optimized fracture locus. Furthermore, the surface strain distribution was predicted, incorporating the best-predicted BW fracture locus into the FE simulation. A decent agreement was observed between FE and experimental values.</p>

Fracture Identification during the Incremental Forming Process Using a Calibrated Damage Model with Optimized Sample Geometries

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发表日期：

2025年8月21日

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Abdul Samad,
Shamik Basak

摘要

This study aims to precisely predict the onset of fracture during the single-point incremental forming (SPIF) of extra deep drawing (EDD) steel sheets using the Bao–Wierzbicki (BW) ductile damage model, incorporating anisotropy of the sheet metal in the analytical formulation. In this regard, a fresh attempt was made to optimize the theoretical BW fracture locus by optimizing the central hole (CH) fracture specimen geometry. Subsequently, four different CH specimens, namely CHD0, CHD2.5, CHD5, and CHD6, were considered by varying the hole-to-ligament width ratios within a range of 0-0.3. The CH specimen geometry was optimized by comparing the evolution of effective plastic strain with respect to stress triaxiality $\left(\eta \right)$ $(η)$ and the Lode angle parameter $\left(\theta \right)$ $(θ)$ . It was found that the CHD5 specimen experienced a purely uniaxial stress state with a $\eta$ $η$ and $\theta$ $θ$ value almost equal to 0.33 and 1.0, respectively. Afterward, the BW damage model was calibrated using the four CH specimens, and different fracture loci were generated. Subsequently, the analytical fracture curves were validated with the safe and failed experimental strain data obtained through the SPIF of variable wall angle cone (VWAC) and variable wall angle pyramid (VWAP) cups. It was observed that the fracture locus obtained using CHD5 geometry accurately predicted the onset of fracture for SPIF cups. Further, the four distinct fracture loci were integrated separately into the finite element (FE) simulation of the SPIF process linked with the Hill48 anisotropic material model to estimate the formability. The error in dome height prediction was observed as 6.97% and 6.35% for VWAC and VWAP cups, respectively. It was concluded that the BW fracture locus calibrated using the CHD5 geometry was the optimized fracture locus. Furthermore, the surface strain distribution was predicted, incorporating the best-predicted BW fracture locus into the FE simulation. A decent agreement was observed between FE and experimental values.