A numerical study of the hygro-mechanical deformation
of two cardboard layups
*, M.L. Dano
, A. Cloutier
Department of Mechanical Engineering, GIREF Research Center, Universite Laval, Quebec, Canada G1K 7P4
Department of Wood and Forest Sciences, GIREF Research Center, Universite Laval, Quebec, Canada G1K 7P4
Received 18 November 2002; received in revised form 23 June 2003; accepted 24 June 2003
A ﬁnite element and a semi-analytical model have been set up to study the hygro-mechanical deformation of two diﬀerent three-
layer square cardboard sheets. Both models are based on a linear constitutive equation and nonlinear von Karman strain–dis-
placement relationships. Results corresponding to steady-state deformations reached after a moisture content change of 5% are
presented. Both models agree on the ﬁnal deformation shapes as well as on the magnitude of the maximal deformation reached.
Results show that diﬀerent equilibrium conﬁgurations can be obtained depending on the layer mechanical and expansion proper-
ties, the magnitude of the change in ambient conditions, and the inplane dimensions of the cardboard sheet.
# 2003 Elsevier Ltd. All rights reserved.
Keywords: Cardboard; C. Deformation; C. Computational simulation; C. Finite element analysis (FEA); C. Laminates
Over the last 10 years, sheet curl has become a sig-
niﬁcant issue for the printing industry and other paper
producers. Curl (deviation from a ﬂat shape) of paper
and cardboard is caused by the combined eﬀect of
changes in ambient conditions of humidity or tempera-
ture and the unsymmetrical nature of the cardboard
layered construction. This situation causes serious diﬃ-
culties during the printing process.
Several researchers have already studied the hygro-
mechanical deformation of paper. Carlson  used
classical lamination theory (CLT) to study the defor-
mation of a sheet of paper subjected to a change in
humidity content. His results showed that due to the
large magnitude of the displacements, several times the
thickness of the sheet, CLT is not adequate. Hyer  has
proposed a nonlinear model for the prediction of the
shape of unsymmetric cross-ply laminates. His model
has then been used by Nordstrom et al.  to study the
deformation of paper subjected to a change in humidity.
Their results indicate that the correct equilibrium form
can be predicted and that the magnitude of deformation
can be compared to results obtained by ﬁnite elements.
No work on cardboard has been found in the literature.
In order to study the tendency of cardboard to curl due
to changes in ambient humidity conditions, two models
have been set up. Both are based on a linear constitutive
model and the equations of equilibrium. The kinematics
of the deformation is governed by the von Karman
nonlinear strain displacement relationships. One model
is based on a semi-analytical approach proposed by
Hyer  and the other one is a ﬁnite element plate
model set up in the ABAQUS  ﬁnite element software.
The objective of the study is to compare the behavior
of two diﬀerent cardboard layups when subjected to a
5% change in moisture content. The paper begins by a
description of the problem along with the properties of
both cardboard sheets. Next, the models that have been
set up are described. Finally, results corresponding to
diﬀerent humidity levels and sheet sizes are presented
and discussed. The paper ends with several concluding
2. Problem statement
Two square pieces of cardboard subjected to uni-
formly distributed changes in moisture content have
0266-3538/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.
Composites Science and Technology 64 (2004) 619–627
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E-mail address: email@example.com (G. Gendron).