A closed-form solution for the thermal stress distribution in rectangular
metal/composite bonded joints
A. Deheeger
a,
Ã
, J.D. Mathias
b
, M. Gre
´
diac
a
a
Laboratoire de Me
´
canique et Inge
´
nieries, Institut Franc-ais de Me
´
canique Avance
´
e-Universite
´
Blaise Pascal Clermont II, Campus des Ce
´
zeaux, BP 265, 63175 Aubie
`
re Cedex, France
b
Laboratoire d’Inge
´
nierie pour les Syste
`
mes Complexes, CEMAGREF, Campus des Ce
´
zeaux, 24 avenue des Landais, BP 50085, 63172 Aubie
`
re Cedex, France
article info
Article history:
Accepted 20 October 2008
Available online 24 December 2008
Keywords:
S-composite
Me-stress distribution
Mi-joint design
Composite patch
abstract
The aim of this paper is to provide a closed-form solution for the thermal stress distribution in a
rectangular metal/composite bonded joint. This distribution depends on two directions, so it is suitable
for studying bidimensional structures such as metallic plates reinforced with a rectangular composite
patch. The solution is obtained using Fourier series. The relevance of the approach proposed in this
paper is highlighted through various examples that clearly show that some coupling effects between the
two directions of the problem take place. The influence of the nature of the composite plate used is
analysed and discussed.
& 2008 Elsevier Ltd. All rights reserved.
1. Introduction
The accurate calculation of the stress distribution in bonded
joints is a key issue in various engineering problems. The solution
for this stress distribution was proposed first under some simple
assumptions within the framework of the so-called shear lag
model [1,2]. This model was then refined to account for some
additional phenomena such as a spew fillet [3], large deflections
[4] or a possible elastic–plastic response of the adhesive [5,6] for
instance. It must be emphasized that the above theories are
unidirectional, thus meaning that the adherends are subjected to
unidirectional loadings only. In this case, the problem consists in
finding the stress distribution in the main direction of the
adhesive. These theories are, however, not suitable for finding
the stress distribution in 2D-problems. Such problems occur when
composites are bonded on metallic substrates, for example in
initial design of modern aircrafts [7,8] or when damaged
aeronautical components are repaired with composite patches
[9]. In this case, the composite plates are considered as additional
components that reinforce the metallic substrate. This type of
heterogeneous structure often exhibits a difference in coefficient
of thermal expansion (CTE) which causes thermally induced
stresses. For instance, the temperature is about 120
C when the
adhesive is allowed to cure. Some residual stresses are therefore
expected to take place in the structure after cooling. Such stresses
may also appear when the bonded joint is subjected to a thermal
cycling in service. The resulting thermal stresses after cure in the
metal substrate are generally positive because the CTE of the
composite is generally lower than the CTE of the metal involved in
the bonded joint. In case of cracked substrate repaired with a
composite patch, these tensile residual stresses may increase the
maximum stress intensity factor after repair and may also
enhance fatigue crack growth rate [9]. Finally, it must be pointed
out that thermally induced stresses are added to the stresses
caused by a mechanical load. They can therefore lead to early
failure in certain cases even if their amplitude is small. Conse-
quently, estimating these stresses is an important issue which has
already been addressed in the literature. For instance, solutions
for the thermal residual stress distribution in symmetric or one-
sided repairs with a circular reinforcement have been proposed in
Refs. [10,11]. In these cases, the composite is considered either as
isotropic or orthotropic. Several studies have also been carried out
on the determination of the stress distribution in polygonal-
shaped patches. They rely on the so-called inclusion method
presented in Ref. [12] and on Rodin’s computational algorithm for
polygonal inclusions[13]. The residual stresses in a composite
patch bonded on a cracked structure have been evaluated by
increasing step by step the complexity of the models, first for a
two-sided polygonal-shaped patch repair [14], then for a one-
sided repair [15]. Some other features have been taken into
account such as large deflections [16], thermo-mechanical loading
[17] and tapered bonded joints [18]. The above studies either deal
with cracked or uncracked substrates. Finding the stress distribu-
tion in the first case is relevant even if cracked substrates are
studied since the solution for the stress distribution in this case
generally relies on two steps, one of them being the stress
determination in case of uncracked substrate. More details are
available in Ref. [19].
This paper deals with the calculation of thermal stresses in
case of thin rectangular composite plates bonded on a metallic
ARTICLE IN PRESS
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ijadhadh
International Journal of Adhesion & Adhesives
0143-7496/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijadhadh.2008.10.004
Ã
Corresponding author. Tel.: +33 473 28 80 06; fax: +33 4 73 28 80 27.
E-mail address: antoine.deheeger@ifma.fr (A. Deheeger).
International Journal of Adhesion & Adhesives 29 (2009) 515–524