Version 4.4 V.1.1 3.12.2012 Theoretical Background of ESAComp Analyses V Joints 1 Adhesive Bonded Joints 1 ADHESIVE BONDED JOINTS Flemming Mortensen and Ole Thybo Thomsen (Aalborg University, Institute of Mechanical Engineering, Denmark, 2000) The method used in ESAComp for engineering analysis of adhesive bonded joints of various complexities is presented. The joints considered are divided in two types: standard and advanced. The standard joints consist of two or three adherends bonded together with a straight continuous adhesive layer parallel to the in-plane direction of the adherends. The advanced joints consist of two adherends bonded together with either a single or double-sided scarfed adhesive interface. The adherends are modelled as beams or plates in cylindrical bending. They are formed from laminates with arbitrary lay-ups using the classical lamination theory (CLT). The adhesive layer is modelled by a two-parameter elastic foundation model, where the adhesive layer is assumed composed of a continuous layer of linear tension/compression and shear springs. Since non-linear effects in the form of adhesive plasticity play an important role in the load transfer, the analysis allows inclusion of non-linear adhesive properties by an iterative method based upon the linear-elastic approach. The load and boundary conditions can be chosen arbitrarily. Approaches for predicting the cohesive failure in the adhesive layers and laminate failure in the joint area are also presented. SYMBOLS i jk A Element of the adherend in-plane stiffness matrix i jk B Element of the adherend coupling stiffness matrix c fi Constant used in system equations C S Constant used in effective stress formulation C V Constant used in effective stress formulation i jk D Element of the adherend bending stiffness matrix E a Adhesive elastic modulus e Effective strain e N Effective strain in the N’th iteration step G a Adhesive elastic shear modulus h fi Constant used in system equations I 1 First invariant of the general strain tensor I 2D Second invariant of the deviatoric strain tensor J 1 First invariant of the general stress tensor
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Version 4.4 V.1.1
3.12.2012 Theoretical Background of ESAComp Analyses
V Joints
1 Adhesive Bonded Joints
1 ADHESIVE BONDED JOINTS
Flemming Mortensen and Ole Thybo Thomsen (Aalborg University, Institute of Mechanical Engineering, Denmark, 2000)
The method used in ESAComp for engineering analysis of adhesive bonded joints of various complexities is
presented. The joints considered are divided in two types: standard and advanced. The standard joints consist of
two or three adherends bonded together with a straight continuous adhesive layer parallel to the in-plane
direction of the adherends. The advanced joints consist of two adherends bonded together with either a single or double-sided scarfed adhesive interface. The adherends are modelled as beams or plates in cylindrical bending.
They are formed from laminates with arbitrary lay-ups using the classical lamination theory (CLT). The adhesive
layer is modelled by a two-parameter elastic foundation model, where the adhesive layer is assumed composed
of a continuous layer of linear tension/compression and shear springs. Since non-linear effects in the form of
adhesive plasticity play an important role in the load transfer, the analysis allows inclusion of non-linear
adhesive properties by an iterative method based upon the linear-elastic approach. The load and boundary
conditions can be chosen arbitrarily. Approaches for predicting the cohesive failure in the adhesive layers and
laminate failure in the joint area are also presented.
SYMBOLS
i
jkA Element of the adherend in-plane stiffness matrix
i
jkB Element of the adherend coupling stiffness matrix
cfi Constant used in system equations
CS Constant used in effective stress formulation
CV Constant used in effective stress formulation
i
jkD Element of the adherend bending stiffness matrix
Ea Adhesive elastic modulus
e Effective strain
eN Effective strain in the N’th iteration step
Ga Adhesive elastic shear modulus
hfi Constant used in system equations
I1 First invariant of the general strain tensor
I2D Second invariant of the deviatoric strain tensor
J1 First invariant of the general stress tensor
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J2D Second invariant of the deviatoric stress tensor
kgi Constant used in system equations
L Length of the overlap zone in the adhesive joint
L1, L2 Length of adherends outside the overlap zone
i
xxM , i
xyM , i
yyM Adherend moment resultants
mfi Constant used in system equations
i
xxN , i
xyN , i
yyN Adherend in-plane stress resultants
i
xxQ , i
xyQ , i
yyQ Adherend shear force resultants
RFadh Reserve factor for cohesive failure of adhesive (linear or non-linear
adhesive model)
RFadh,prop Reserve factor for proportional limit of adhesive (non-linear adhesive
model)
RFFPF Reserve factor for adherend (laminate) first ply failure in the vicinity
of the joint
s Effective stress
*
Ns Calculated stress in the N’th iteration step
sN Experimental stress in the N’th iteration step
sprop Stress proportional limit
DsN Difference between calculated and experimental stress
ti Adherend thickness
ti(x) Adherend thickness as a function of x
ta Adhesive layer thickness
x Adherend in-plane coordinate system in the longitudinal direction
iu0 Longitudinal displacement of the adherend mid-plane (x-direction)
ui Longitudinal displacement of the adherend (x-direction)
iv0 Displacement of the adherend mid-plane in the width direction (y-
direction)
vi Displacement of the adherend in the width direction (y-direction)
wi Transverse displacement of the adherend (z-direction)
a, a1, a2 Transition angles of scarfed adherend
i
xb , i
yb Rotation of mid-plane normal to the adherend
d Weight factor for the change in elastic modulus
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ef Principal strains (f = 1, 2, 3)
l Ratio between compressive and tensile yield stress
sa Adhesive layer out-of-plane normal stress
san Adhesive layer out-of-plane normal stress
sani Adhesive layer out-of-plane normal stress
sc Compressive yield stress
sf Principal stresses (f = 1, 2, 3)
st Tensile yield stress
tax, tay Adhesive layer shear stress
tan, Adhesive layer shear stress
taxi Adhesive layer shear stress
Subscripts
a Adhesive layer
i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)
N Iteration number for non-linear tangent modulus
,x Differentiation with respect to the x-coordinate
,y Differentiation with respect to the y-coordinate
ult Ultimate
Superscripts
i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)
end Adherend end thickness at the overlap zone
end,L Adherend thickness at the left end of the overlap zone
end,R Adherend thickness at the right end of the overlap zone
t Identifier used for non-linear tangent modulus
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1.1 INTRODUCTION
Joining of composite structures can be achieved through use of bolted, riveted or adhesive
bonded joints. The performances of the mentioned joint types are severely influenced by the
characteristics of the layered composite materials, but adhesive bonded joints provide a much
more efficient load transfer than mechanically fastened joints. Accurate analysis of adhesive
bonded joints, for instance by using the finite element method, is an elaborate and
computationally demanding task as described by Crocrombe et al. [3], Harris et al. [11] and
Frostig et al. [5]. Hence, there is an obvious need for analysis and design tools that can
provide accurate results for preliminary design purposes.
This chapter introduces the analysis approach used in ESAComp for determining the stress
and displacement fields in commonly used adhesive bonded joint configurations. The last
sections deal with the handling of plasticity effects in the adhesive layers and failure
prediction of bonded joints.
The bonded joint types considered in ESAComp are:
· Single lap joint (SL)
· Single strap joint (SS)
· Bonded doubler (BD)
· Double lap joint (DL)
· Double strap joint (DS)
· Single sided scarfed lap joint (SSC)
· Double sided scarfed lap joint (DSC)
These joint types are illustrated in Figure 1.1. All the joint configurations can be composed of
similar or dissimilar laminates with an arbitrary lay-up. The joints are subjected to a general
loading condition as shown in Figure 1.2.
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Figure 1.2 Schematic illustration of an adhesive single lap joint subjected to a general loading condition.
According to the complexity of the joints, the lap and strap joints and bonded doublers can be
referred to as standard joints. In these joints, the adhesive layer or layers are parallel to the in-
plane direction of the adherends. Correspondingly, the scarfed joints can be referred to as
advanced joints. The advanced joints are more efficient due to the reduced eccentricity of the
load path, but the advanced joints are also much more expensive to manufacture and they are
therefore only used for high-performance applications.
1.2 STRUCTURAL MODELLING
The structural modelling is carried out by adopting a set of basic restrictive assumptions for
the behaviour of bonded joints. Based on these restrictions, the constitutive and kinematic
relations for the adherends are derived, and the constitutive relations for the adhesive layers
are adopted. Finally, the equilibrium equations for the joints are derived and, by combining all
these equations and relations, the set of governing equations is obtained.
1.2.1 Model dimensions
The adhesive bonded joint configurations were introduced in Section 1.1. The adherend
thicknesses are given by t1 and t2 for all the joints outside the overlap zone. For the double lap
joint, the thickness of the third adherend (lower adherend) is t3. The adherend length outside
the overlap is L1 and L2, and the length of the adhesive layer is L as illustrated in Figure 1.3.
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Figure 1.3 Illustration of adherend lengths and adhesive layer length and thickness.
Inside the overlap zone (0 £ x £ L) the thicknesses are:
· Single lap joint, single strap joint, and bonded doubler:
( ) ( ) 2211 , txttxt == (1.2.1)
· Double lap and double strap joint:
( ) ( ) ( ) 332211 ,, txttxttxt === (1.2.2)
· Single sided scarfed lap joint:
( ) ( ) xL
tttxtx
L
tttxt
endend
end
2222
1111 ,
--=
--= (1.2.3)
Where the superscript end in endt1 and endt2 refers to the thicknesses of the adherends at the free
ends of the overlap, see Figure 1.4.
Figure 1.4 Thicknesses and scarf angle for single sided scarfed lap joint.
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· Double sided scarfed lap joint:
( ) ( )
( ) ( )Lx
xL
tttxtx
L
tttxt
xL
tttxtx
L
tttxt
Rend
b
Lend
bLend
bb
Rend
a
Lend
aLend
aa
Lend
b
Rend
bb
Lend
a
Rend
aa
££
ïïþ
ïïý
ü
--=
--=
--=
--=
0
,
,
,
2
,
2,
22
,
2
,
2,
22
,
2
,
212
11
,
2
,
212
11
(1.2.4)
Where the subscripts a and b and the superscript end in Lend
at ,
2 and Rend
bt ,
2 refer to the thickness
of adherend 2 at the left (L) and right (R) ends of the overlap above and below adherend 1
(Figure 1.5).
Figure 1.5 Thicknesses for double sided scarfed lap joint.
1.2.2 Basic assumptions for the structural modelling
The basic restrictive assumptions for the structural modelling are the following:
Adherends
· The adherends are modelled as beams or plates in cylindrical bending, using ordinary
“Kirchhoff” plate theory (“Love-Kirchhoff” assumptions).
· The constitutive behaviour of the adherends is obtained using the classical lamination
theory (CLT). No restrictions are set on the laminate lay-up, i.e. unsymmetric and
unbalanced laminates can be included in the analysis.
· The laminates are assumed to obey linear-elastic constitutive laws.
· The strains are small and the rotations are very small.
Adhesive layers
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· The adhesive layers are modelled as continuously distributed linear tension/compression
and shear springs.
· Non-linear adhesive properties are considered by using a secant modulus approach for the
non-linear tensile stress-strain relationship in conjunction with a modified Von Mises
yield criterion.
Loads and boundary conditions
· The structural model allows boundary conditions to be chosen arbitrarily as long as the
system is in equilibrium. Sets of prescribed external loads (in-plane and out-of-plane
forces and bending moments) and geometric boundary conditions are defined in
ESAComp to avoid selections of inconsistent boundary conditions, which can lead to
singularity problems in the system of equations.
The system of governing equations is set up for two different cases, i.e. the adherends are
modelled as plates in cylindrical bending or as wide beams. In the following, the case where
the adherends are modelled as plates in cylindrical bending is primarily considered since the
modelling of the adherends as beams is a reduced case of this.
1.2.3 Constitutive relations for adherends modelled as plates
For the purposes of the present investigation, and with references to Figures 1.6 and 1.2,
cylindrical bending can be defined as a wide plate (in the y-direction), where the displacement
field can be described as a function of the longitudinal coordinate only. As a consequence, the
displacement field in the width direction is uniform. Thus, the displacement field can be
described as
( ) ( ) ( )xwwxvvxuu iiiiii === ,, 0000 (1.2.5)
where u0 is the mid-plane displacement in the longitudinal direction (x-direction), v0 is the
mid-plane displacement in the width direction (y-direction), and w is the displacement in the
transverse direction (z-direction). The displacement components u0, v0 and w are all defined
relative to the mid-plane of the laminates, and i = 1, 2, 3 corresponds to the laminates 1, 2 and
3, respectively.
Based on the earlier assumptions, the following holds also true:
0,,,0,0 ==== i
yy
i
y
i
y
i
y wwvu (1.2.6)
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Figure 1.6 Schematic illustration of adhesive single lap joint “clamped” between two vertical laminates, which
prevent the adherends of the single lap joint from moving and rotating freely in the width direction. This
represents the conceptual interpretation of cylindrical bending as defined in the present formulation.
In the concept of “cylindrical bending”, the boundary conditions at the boundaries in the
width direction are not well defined. However, it is assumed that there are some restrictive
constraints on the boundaries, such that the boundaries are not capable of moving freely. It
should be noted that the concept of “cylindrical bending” is not unique, and that other
definitions than the one used in the present formulation can be adopted, see Whitney [20].
Substitution of the quantities in Eq. (1.2.5) into the constitutive relations for a laminated
composite material gives the constitutive relations for a laminate (i) in cylindrical bending
[20]:
i
xx
ii
x
ii
x
ii
xy
i
xx
ii
x
ii
x
ii
xy
i
xx
ii
x
ii
x
ii
yy
i
xx
ii
x
ii
x
ii
yy
i
xx
ii
x
ii
x
ii
xx
i
xx
ii
x
ii
x
ii
xx
w-DvBuBMw-BvAuAN
w-DvBuBMw-BvAuAN
w-DvBuBMw-BvAuAN
,16,066,016,16,066,016
,12,026,012,12,026,012
,11,016,011,11,016,011
,
,
,
+=+=
+=+=
+=+=
(1.2.7)
where i
jkA , i
jkB and i
jkD (j,k = 1,2,6) are the extensional, coupling and flexural rigidities based
on the classical lamination theory (see Part III, Chapter 2). i
xxN , i
yyN and i
xyN are the in-plane
stress resultants i
xxM , i
yyM and i
xyM are the moment resultants. For the joints with scarfed
adherends the rigidities i
jkA , i
jkB and i
jkD (j,k = 1,2,6) within the overlap zone are changed as a
function of the longitudinal coordinate in accordance with their definition, i.e. i
jkA is changed
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linearly, i
jkB is changed parabolically and i
jkD is changed cubically (j,k = 1,2,6) between their
values at the ends of the overlap zone. This is of course an approximation since the actual
stiffnesses of the laminates are changing by changes within the layers as a function of the
longitudinal direction.
1.2.4 Constitutive relations for adherends modelled as beams
Modelling of the adherends as wide beams can be considered as a special case of cylindrical
bending. When the adherends are modelled as beams, the width direction displacements are
not considered, and only the longitudinal and vertical displacements are included. Thus, the
displacement field in Eq. (1.2.5) is reduced to
( ) ( )xwwxuu iiii == ,00 (1.2.8)
For this case the constitutive relations for a composite beam are reduced to
i
xx
ii
x
ii
xx
i
xx
ii
x
ii
xx w-DuBMw-BuAN ,11,011,11,011 , == (1.2.9)
1.2.5 Kinematic relations
From the “Love-Kirchhoff” assumptions, the following kinematic relations for the laminates
in cylindrical bending are derived:
0,, ,0 =-=+= i
y
i
x
i
x
i
x
ii wzuu bbb (1.2.10)
here ui is the longitudinal displacement, iu0 is the longitudinal displacement of the mid-plane,
and wi is the vertical displacement of the i’th laminate.
The kinematic relations of Eq. (1.2.10) are the same for the beam case as for the cylindrical
bending case except that all the variables associated with the width direction are nil.
1.2.6 Constitutive relations for the adhesive layer
The coupling between the adherends is established through the constitutive relations for the
adhesive layer, which as a first approximation is assumed homogeneous, isotropic and linear
elastic. The constitutive relations for the adhesive layer are established by use of a two-
parameter elastic foundation approach, where the adhesive layer is assumed to be composed
of continuously distributed shear and tension/compression springs. The constitutive relations
of the adhesive layer are suggested in accordance with Thomsen [16–17], Thomsen et al. [18]
and Tong [19]:
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( ) ( ) ( )( )( ) ( )( )
( )( )ji
ji
ww
vvvv
xtuxtuuu
ji
t
E
a
ji
t
Gji
t
G
ay
j
xj
ji
xi
i
t
Gji
t
G
ax
a
a
a
a
a
a
a
a
a
a
¹
=
ïþ
ïý
ü
-=
-=-=
---=-=,3,2,1,
00
00
st
bbt (1.2.11)
where i and j are the numbers of the adherends, Ga is the shear modulus, and Ea is the elastic
modulus of the adhesive layer.
The consequence of using the simple spring model approach for the modelling of the adhesive
layers is that it is not possible to satisfy the equilibrium conditions at the (free) edges of the
adhesive. However, in real adhesive joints no free edges are present at the ends of the overlap,
since a fillet of surplus adhesive, a so-called spew-fillet, is formed at the ends of the overlap
zone. This spew fillet allows for the transfer of shear stresses at the overlap ends. Modelling
of the adhesive layer by spring models has been compared with other known analysis methods
such as finite element analysis (Crocrombe et al. [3] and Frostig et al. [5]) and a high-order
theory approach including spew fillets (Frostig et al. [5]). The results show that the overall
stress distribution and the predicted values are in very good agreement.
1.3 EQUILIBRIUM EQUATIONS
The equilibrium equations are derived based on equilibrium elements inside and outside the
overlap zone for each of the considered joint types.
1.3.1 Adherends outside the overlap zone
The equilibrium equations are derived for plates in cylindrical bending since the equilibrium
equations for the beam modelling can be considered as a reduced case of this. The equilibrium
equations outside the overlap zone for each of the adherends, i.e. in the regions -L1 £ x £ 0
and L £ x £ L + L2, are all the same (see Figure 1.2) and are derived based on Figure 1.7:
21
,
,
,
,
,
00
0
0
LLxLandxL
QM
QM
Q
N
N
i
y
i
xxy
i
x
i
xxx
i
xx
i
xxy
i
xxx
+££££-
ïïï
þ
ïïï
ý
ü
=
=
=
=
=
(1.3.1)
where i correspond to the adherends i = 1, 2, 3.
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Figure 1.7 Equilibrium elements of adherend outside the overlap zone; -L1 £ x £ 0 and L £ x £ L + L2.
1.3.2 Single lap and single strap joints
The equilibrium equations inside the overlap zone for the single lap joint and the single strap
are derived based on Figure 1.8. For the single lap joint the adherend thickness’ will remain
the same in the entire overlap zone as specified by Eqs. (1.2.1), thus giving the equations:
Lx
ttQM
ttQM
ttQM
ttQM
QQ
NN
NN
aayyxxy
aayyxxy
aaxxxxx
aaxxxxx
axxaxx
ayxxyayxxy
axxxxaxxxx
££
ïïïï
þ
ïïïï
ý
ü
++=
++=
++=
++=
-==
=-=
=-=
0
2,
2
2,
2
,
,
,
222
,
111
,
222
,
111
,
2
,
1
,
2
,
1
,
2
,
1
,
tt
tt
sstttt
(1.3.2)
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Figure 1.8 Equilibrium element of adherends inside the overlap zone for joints with one adhesive layer and
straight adherends; 0 £ x £ L.
1.3.3 Bonded doubler
Inside the overlap zone for bonded doubler joint the equilibrium equations are derived based
on Figure 1.8. and Eqs. (1.2.1) and yields exactly the same equations as for the single lap joint
(see the previous section).
1.3.4 Double lap and double strap joint
The equilibrium equations inside the overlap zone for the double lap joint and the double strap
are derived based on Figure 1.9:
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Figure 1.9 Equilibrium element of adherends inside the overlap zone for joints with two adhesive layers and
straight adherends; 0 £ x £ L.
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Lx
ttQM
ttQM
Q
N
N
ttQM
ttQM
Q
N
N
ttttQM
ttttQM
Q
N
N
aayyxxy
aaxxxxx
axx
ayxxy
axxxx
aayyxxy
aaxxxxx
axx
ayxxy
axxxx
aay
aayyxxy
aax
aaxxxxx
aaxx
ayayxxy
axaxxxx
££
ïïïïïïïïïïïïïï
þ
ïïïïïïïïïïïïïï
ý
ü
+-=
+-=
=
=
=
++=
++=
-=
=
=
+-
++=
+-
++=
-=
--=
--=
0
2
2
2
2
22
22
23
2
33
,
232
33
,
2
3
,
2
3
,
2
3
,
12
1
22
,
12
1
22
,
1
2
,
1
2
,
1
2
,
21
2
11
1
11
,
21
2
11
1
11
,
21
1
,
21
1
,
21
1
,
t
t
s
tt
t
t
st
t
tt
tt
sstt
tt
(1.3.3)
1.3.5 Single sided scarfed lap joint
The equilibrium equations inside the overlap zone for the single sided scarfed lap joint are
derived based on Figure 1.10. They are different from the earlier ones due to the linear change
of the adherend thicknesses and the sloping bond line:
( ) ( )
( ) ( )
Lx
L
ttN
txtQ
M
L
ttN
txtQ
M
L
ttN
txtQ
M
L
ttN
txtQ
M
QQ
NN
NN
end
xy
aayy
xxyend
xy
aayy
xxy
end
xx
aaxx
xxxend
xx
aaxx
xxx
axxaxx
ayxxyayxxy
axxxxaxxxx
££
ïïïïïïï
þ
ïïïïïïï
ý
ü
--
++
=-
-
++
=
--
++
=-
-
++
=
-==
=-=
=-=
0
2
2
,2
2
2
2
,2
2
,
,
,
221
22
2
,
111
11
1
,
221
22
2
,
111
11
1
,
2
,
1
,
2
,
1
,
2
,
1
,
tt
tt
ss
tttt
(1.3.4)
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where the relationship between tax, sa in Eq. (1.3.4) and tan, san shown in Figure 1.10 is