W. Scherzinger
Sandia National Laboratories,
P.O. Box 5800, MS 0847,
Albuquerque, NM 87185
Mem. ASME
N. Triantafyllidis
Department of Aerospace Engineering,
The University of Michigan,
Ann Arbor, MI 48109-2140
Fellow ASME
Asymptotic Stability Analysis for
Sheet Metal Forming—Part I:
Theory
In this paper is presented a general methodology for predicting puckering instabilities in
sheet metal forming applications. A novel approach is introduced which does not use shell
theory approximations. The starting point is Hill’s stability functional for a three-
dimensional rate-independent stressed solid which is modified for contact. By using a
multiple scale asymptotic technique with respect to the small dimensionless thickness
parameter , one can derive the two-dimensional version of the stability functional which
is accurate up to O
4
, thus taking into account bending effects. Loss of positive defi-
niteness of this functional indicates possibility of a puckering instability in a sheet metal
forming problem with a known stress and deformation state. An advantage of the pro-
posed method is that the puckering investigation is independent of the algorithm used for
calculating the deformed state of the sheet. S0021-89360000804-7
1 Introduction and Motivation
Stamping of sheet metal is one of the most widely used indus-
trial manufacturing processes. There are three major problems that
limit the formability of a stamped part which have to be accounted
for in its design: springback, tearing, and puckering/wrinkling.
Springback is the change in shape of the part that occurs after a
part is removed from the blankholder/die assembly and is due to
the elastic unloading of the part. Tearing is the splitting of the part
in areas of high strain concentrations and is due to the localized
necking of the sheet. Puckering, as defined by Devons 1 is a
waviness of the sheet that is not in contact with the tooling sur-
faces and is a bifurcation buckling phenomenon due to the pres-
ence of compressive in-plane stresses in the sheet. When the same
phenomenon occurs in areas that come into contact with the tool-
ing, usually the flat part of the binder, the surface waviness phe-
nomena is known as wrinkling.
In modeling a tearing problem, the difficulty is in the determi-
nation of the proper constitutive law since the phenomenon is
local in nature. The difficulty in modeling springback is due to the
geometry of the part and in the determination of its prestress state
from which an elastic unloading takes place. Modeling of puck-
ering requires both an accurate description of the constitutive be-
havior of the material and the solution of a boundary value prob-
lem. Moreover, experimental investigation of puckering faces the
difficulty of the determination of the onset of the phenomenon,
since imperfections in the form of minute amounts of surface
waviness are always present in stamped parts.
Of interest here is the modeling of puckering instabilities for
general stamping geometries. The standard approach thus far uses
a shell-type analysis usually in conjunction with a finite element
method code and follows the deformation of the part all the way
to the formation of finite amplitude wrinkles e.g., Taylor et al.
2. More refined analyses use a linearized stability method to
check for bifurcation in a part with a known prestress state ob-
tained by using a shell-type analysis e.g., Neal and Tugcu 3.
The obvious shortcoming of this approach is the stability results’
dependence on the shell theory employed.
To overcome the inconsistencies associated with the use of a
particular shell theory to calculate the onset of bifurcation in shell
buckling problems, Triantafyllidis and Kwon 4 proposed revers-
ing the order of the limiting process in the analysis, by first for-
mulating the stability problem of the three-dimensional solid and
then finding its critical load and buckling mode as the dimension-
less thickness parameter, , goes to zero. This asymptotic meth-
odology has recently been applied by Scherzinger and Triantafyl-
lidis 5 to another similar problem, namely the buckling of
slender beams with arbitrary cross sections there the parameter
is the beam’s slenderness defined as the square root of its sec-
tional area over its length where the interested reader can find
another comprehensive application of the proposed methodology.
The departing point for our analysis is Hill’s 6,7 stability
functional for a three-dimensional elastoplastic solid, properly
modified to account for contact with tooling surfaces. Since sta-
bility against puckering depends on the sign of the functional’s
minimum eigenvalue, the present work consists of a multiple scale
asymptotic analysis to obtain the minimum eigenvalue and the
corresponding eigenmode in terms of . The multiple scale analy-
sis is a finite strain adaptation of the methodology proposed by
Destuynder 8 for the consistent derivation of linear elastic shell
theories from the corresponding three-dimensional equations of
elasticity. The present method results in the calculation of the
stability functional of a prestressed stamped sheet that is accurate
to O(
4
). The functional includes bending stiffness effects and
only requires a two-dimensional stress state and eigenmode the
degrees-of-freedom are the displacements of the midsurface. No
shell theory assumptions are required and normality of plane sec-
tions and the plane stress assumption arise naturally as a part of
the analysis.
The outline of this work is as follows: The presentation begins
with a description of the kinematics for a shell of arbitrary shape
in Section 2.1. The treatment of contact, essential for the stability
of sheet metal forming problems, is presented in Section 2.2 fol-
lowed by the statement of the variational problem to find the
minimum eigenvalue in Section 2.3. The asymptotic analysis of
the problem is presented in Section 3. Expansion of field quanti-
ties are given and substituted into the stability functional. The
stability functional is evaluated in Section 3.2 and its two-
dimensional form, suitable for sheet metal forming, is found ac-
curate to O(
4
).
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF
MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED
MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Sept.
24, 1999; final revision, Jan. 30, 2000. Associate Technical Editor: S. Kyriakides.
Discussion on the paper should be addressed to the Technical Editor, Professor
Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston,
Houston, TX 77204-4792, and will be accepted until four months after final publi-
cation of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Copyright © 2000 by ASME Journal of Applied Mechanics DECEMBER 2000, Vol. 67 Õ 685