Formation mechanism and modeling of surface waviness in incremental sheet forming

Abstract

Improving and controlling surface quality has always been a challenge for incremental sheet forming (ISF), whereas the generation mechanism of waviness surface is still unknown, which impedes the widely application of ISF in the industrial field. In this paper, the formation mechanism and the prediction of waviness are both investigated through experiments, numerical simulation, and theoretical analysis. Based on a verified finite element model, the waviness topography is predicted numerically for the first time, and its generation is attributed to the residual bending deformation through deformation history analysis. For more efficient engineering application, a theoretical model for waviness height is proposed based on the generation mechanism, using a modified strain function considering deformation modes. This work is favorable for the perfection of formation mechanism and control of surface quality in ISF.

Key words： surface waviness; incremental sheet forming; numerical simulation; formation mechanism; deformation history

Acknowledgements

The authors thank the support from the National Natural Science Foundation of China (Grant Nos. 51575028 and 51975328), and the Fundamental Research Funds for the Central Universities of China (Grant No. YWF-18-BJ-J-75). Also, the authors thank the suggestions of editors and reviewers.

Nomenclature

Abbreviations
BS	Bending and stretching strain model
FE	Finite element
FM	Full model
ISF	Incremental sheet forming
MBS	Modified bending and stretching strain model
PM	Partial model
TPISF	Two-point incremental sheet forming
Variables
F, G, H, Q,M, and N	Material coefficients of Hill48 yield function
F_{\mathrm{A}}, F_{\mathrm{B}}, F_{\mathrm{C}}	Tangential forces per unit width in Regions A, B, and C, respectively
F_{\mathrm{d}\mathrm{c}\mathrm{f}}	Tension force in friction experiment
F_{\mathrm{f}}	Frictional force in friction experiment
F_{\mathrm{n}}	Compressive force in friction experiment
H_{\mathrm{w}}	Waviness height
K	Material coefficient of Ludwik constitutive function
l	Forming depth
L	Width size of partial FE model
n	Material hardening index of Ludwik constitutive function
r	Distance to the spherical centre
R	Tool radius
R_z	Surface roughness of maximum peak to valley height
sz	Step size
t_0	Initial sheet thickness
V	Feed rate in friction experiment
\alpha	Forming angle
{\alpha }_{\mathrm{f}}	Actual contacting angle
{\alpha }_{\mathrm{t}}	Actual forming angle after incremental forming
\beta	Residual forming angle
{\sigma }_0	Yield stress
{\sigma }_{\mathrm{t}\mathrm{a}\mathrm{n}}, {\sigma }_{\mathrm{c}\mathrm{i}\mathrm{r}}, {\sigma }_{\mathrm{t}\mathrm{h}\mathrm{i}}	Normal stresses in tangential, circumferential, and thickness directions, respectively
{\sigma }_{\tan }^{\mathrm{A}}, {\sigma }_{\mathrm{t}\mathrm{h}\mathrm{i}}^{\mathrm{A}}	Tangential stress and thickness stress at Region A, respectively
{\sigma }_{\tan }^{\mathrm{B}}, {\sigma }_{\mathrm{t}\mathrm{h}\mathrm{i}}^{\mathrm{B}}	Tangential stress and thickness stress at Region B, respectively
{\sigma }_{\mathrm{t}\mathrm{h}\mathrm{i},r=R}^{\mathrm{B}}	Contact stress along the thickness direction at Region B
{\sigma }_{xx}, {\sigma }_{yy}, {\sigma }_{zz}	Normal stresses along the X (rolling direction), Y (transverse direction), and Z (thickness direction) directions, respectively
\overline{\sigma }	Equivalent stress
\phi	Current forming angle
\mathrm{d}\varphi	Increment of tangential length
\mathrm{d}\theta	Increment of circumference width
{\epsilon }_{\mathrm{B},\mathrm{B}\mathrm{S}}, {\epsilon }_{\mathrm{S},\mathrm{B}\mathrm{S}}	Bending strain and stretching strain in a bending and stretching strain model, respectively
{\epsilon }_{\mathrm{B},\mathrm{M}\mathrm{B}\mathrm{S}}, {\epsilon }_{\mathrm{S},\mathrm{M}\mathrm{B}\mathrm{S}},	Bending strain and stretching strain in MBS, respectively
{\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{P}\mathrm{M}}, {\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{F}\mathrm{M}}	Max principal strain of partial and full model at specific forming depth of l, respectively
{\epsilon }_{\mathrm{p}}	Predeformation strain
{\epsilon }_{\mathrm{t}\mathrm{a}\mathrm{n}}, {\epsilon }_{\mathrm{c}\mathrm{i}\mathrm{r}}, {\epsilon }_{\mathrm{t}\mathrm{h}\mathrm{i}}	Normal strains in the tangential, circumference, and thickness directions, respectively
{\epsilon }_{\tan ,\mathrm{B}\mathrm{S}}	Tangential strain evolution in BS
{\epsilon }_{\tan ,\mathrm{M}\mathrm{B}\mathrm{S}}	Tangential strain evolution in MBS
{\epsilon }_{xx}, {\epsilon }_{yy}, {\epsilon }_{zz}	Normal strain along rolling direction, transverse direction and thickness direction, respectively
\overline{\epsilon }	Equivalent strain
\mu	Friction coefficient
{\delta }_i	Strain error

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