[American Institute of Aeronautics and Astronautics 44th AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics1

MICRO-MECHANICAL MODELING OF THE ADHESIVE INTERFACE IN COMPOSITE-COMPOSITE JOINTS *

M. Lu and C. H. Jenkins (AIAA Member)∗∗ Compliant Structures Laboratory, Mechanical Engineering Department

South Dakota School of Mines and TechnologyRapid City, SD 57701

R. M. WinterChemistry and Chemical Engineering DepartmentSouth Dakota School of Mines and Technology

Rapid City, SD 57701

* Copyright 2003 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.∗∗ Corresponding Author

ABSTRACTThe joining of composite materials is

always an important subject. Although there are many kinds of connection methods, two of the most popular are mechanically connected joints and adhesively bonded joints. Adhesively bonded joints have many advantages, but in the most important situations, people still prefer mechanically connected joints. The main reason is that adhesively bonded joints are relatively difficult to design and to predict accurately their useful life. It is thus necessary to develop more accurate models to simulate adhesively bonded joints. This paper focuses attention on the non-uniform modulus of composite materials. We show that the details of the adherend modulus model have great affect on the calculated maximum stress in adhesively bonded joints. Thus, we introduce a “Layer Model” in order to improve adhesive stress prediction. As the fiber volume fractions and orientation of different layers in a composite are different, the moduli and Poisson’s ratios are also changing from layer to layer. In calculating the moduli and Poisson’s ratios of different layers using finite element analysis (FEA), we established a new “Iterative Method”. In this method the effect of Poisson’s ratio is taken into consideration, and the results are satisfying.

INTRODUCTIONThe bonding of composite materials is

an important subject, because bonding locations are weak points for a composite material system, and the useful life of a system is impacted by its weak points.

There are mainly two kinds of bonding methods for composite materials: mechanically bonded joints and adhesively bonded joints.Mechanically bonded joints use mechanical methods, such as rivets or bolts; adhesively bonded joints use an adhesive layer.

Compared with mechanically bonded joints, adhesively bonded joints have many advantages:

(a) Adhesively bonded joints can greatly reduce stress concentrations.

(b) Adhesively bonded joints have good strength and low weight.

(c) Adhesively bonded joints are easy for manufacture and usually they are low cost.

(d) Adhesively bonded joints provide ductility and damping.

There are many types of adhesively bonded joints. Single lap and double lap are two common examples and are the types focused on in this paper (Figure 1).

(a) single lap:

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1759

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


(b) double lap:

Figure 1. Two Common Examples for Adhesively Bonded Joints.

THE LAYER MODEL FOR COMPOSITE MATERIALS

It is nearly impossible to model the real micro-structure of composite materials directly in finite element analysis.

The classic model to simulate composite materials is a homogenized orthotropic model. It assumes that a composite material is equivalent to a homogenous and orthotropic material.

Although the classic model is widely used and accepted in composite material design, it ignores the effects of adherend interfacial properties on the adhesive stress field. Thus we compared it with a more detailed model: the Layer Model. It assumes that a composite material is equivalent to the total effect of several different layers, each layer being distinctly homogenous and orthotropic (Figure 2).

Classic Model: homogenous and orthotropic.

Layer Model: composite is divided into several different homogenous and orthotropic layers.

Figure 2. Difference of the Classic Model and the Layer Model.

VALIDATION OF THE LAYER MODELIn order to make sure the layer model is

accurate and reliable, we compared the results of a double lap adhesive bonded joint with a closed-form solution (Tong (1997)), with finite element analysis (ABAQUS) based on the classic model, and with finite element analysis (ABAQUS) based on the layer model.

Figure 3 and Figure 4 show that the results of these two methods have similar stress distributions. The max difference between FEA Layer Model results and the closed-form results is about 25%. The main reason is the material properties are not the same. However, the trend

shows that the Layer Model is reasonable.Figure 3. Comparison of Peeling Stress.

Figure 4. Comparison of Shear Stress.

THE NECESSITY OF LAYER MODELThe detailed structure of the adherend,

especially close to the adhesive layer, has a great affect on the maximum stress.

We studied both quarter-symmetry models (Figure 5) and half-symmetry models for a double lap joint using ABAQUS FEA to compare the difference between Layer Model and Homogenous Orthotropic Model.

Figure 5. Quarter Model for Double Lap Adhesively Bonded Joints.

We first compared the FEA stress results using the homogenous adherend (Figure 6) and layered adherend (Figure 7). The Young’s modulus used in Figure 6 is 180 GPa, which is equal to the equivalent Young’s modulus of the composite adherends in the reinforcing direction. In Figure 7, the adherend in the middle is divided into 60 layers, and the laps are divided into 30

-8.00E+04

-6.00E+04

-4.00E+04

-2.00E+04

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

0 0.2 0.4 0.6 0.8 1

Normalized Overlap

S33

in th

e M

iddl

e of

adh

esiv

e

laye

r (P

a)

FEA Laye r M odel Cl osed-form Poly. (FEA Layer M ode l)

0

20000

40000

60000

80000

100000

120000

0 0.2 0.4 0.6 0.8 1

Normalized overlap

S13

in t

he

Mid

dle

of

the

adh

esiv

e la

yer

(Pa)

FEA (Layer Model) Close-form Poly. (FEA (Layer Model))

P/2

L1

L2

Ln

…


layers each. The layer thickness is 0.1 mm. Two different kind of layers (alternately): High Modulus Layer, E = 269.9 GPa; Low Modulus Layer, E = 4.6 GPa. The modulus of adhesive is 2.6 GPa. The max. stresses (absolute value) in the two models are greatly different. The homogenous adherend gives a max. adhesive stress twice that of the layered adherend.

We also compared the stress distribution of different layer orders, as shown in Figure 8.

Figure 6. The Stress Distribution of Tensile Test in Uniform Adherend Model (E=180 Gpa).

Figure 7. The Stress Distribution of Tensile Test in Layered Adherend Model(layer thickness=0.1 mm).

Structure 1 Structure 2

Structure 3 Structure 4A: Adhesive Layer, E = 2.6 GPaL: Low Modulus Layer, E = 4.6 GPaH: High Modulus Layer, E = 269.9 GPaLayer thickness: 0.1 mm

Figure 8. Different Layer Orders.

A comparison of the results is shown in Figure 9.

Figure 9. Comparison of Max Stress for Layer Adherends in Different Layer Orders.

Given that the Layer Model is more accurate for composite adhesive bonded joints, the material properties for each layer of the composite material is now given.

THEORY OF THE ITERATIVE METHOD TO SOLVE THE LOCAL MATERIAL

PROPERTIES IN COMPOSITEThe usual method to calculate the

Young’s modulus is to apply a load in x-direction, y-direction, and z-direction,

respectively, and then using ii

iiiE ε

σ= to calculate

E1 for load in x-direction, E2 for load in y-direction, and E3 for load in z-direction, where σii and εii mean the stress and strain in the i direction, respectively, i = x, y, z. This method assumes that stress is solely in the direction in

A

H

L

H

L H

L

L

H

A

L

H

L

H L

H

L

H

012345678

Stru. 1 Stru. 2 Stru. 3 Stru. 4

Max

Str

ess

(MP

a)


which the load is applied. This assumption is reasonable for homogenous materials. But

composite materials can transfer some load to other directions. For example, if we apply load in the x-direction (Figure 10), the stress in the y-direction is about 10% as much as the stress in the x-direction. In this case to assume the stresses in another direction as zero is no longer reasonable. For this reason, we suggest using the

Iterative Method in solving the equivalent material properties for composite material, including the equivalent material properties for the whole composite, and those for the local layers of composite.

Figure 10. X-Y View of FEA Model for Calculating the Local Material Properties.

The formula we used is stress-strain relationship for orthotropic materials:

−−

−−

−−

=

12

13

23

33

22

11

12

13

23

32

23

1

13

3

32

21

12

3

31

2

21

1

12

13

23

33

22

11

100000

01

0000

001

000

0001

0001

0001

2

2

2

τττσσσ

νν

νν

νν

γγγεεε

G

G

G

EEE

EEE

EEE(1)

In Equation 1, 2

21

1

12

EE

νν = , 3

31

1

13

EE

νν = , and

3

32

2

23

EE

νν = . Equation 1 can be rewritten as

−−

−−

−−

=

12

13

23

33

22

11

12

13

23

32

23

1

13

2

23

21

12

1

13

1

12

1

12

13

23

33

22

11

100000

01

0000

001

000

0001

0001

0001

2

2

2

τττσσσ

νννννν

γγγεεε

G

G

G

EEE

EEE

EEE(2)

Load Cases FOR THE ITERATIVE METHOD

Three load cases are examined:Load Case A: Tensile load in x-direction (11-direction)Data: Average σA11, σA22, σA33, εA11, εA22, εA33 for Layer n .Load Case B: Tensile load in y-direction (22-direction)Data: Average σB11, σB22, σB33, εB11, εB22, εB33 for Layer n .Load Case C: Tensile load in z-direction (33-direction)Data: Average σC11, σC22, σC33, εC11, εC22, εC33 for Layer n .

STEPS OF THE ITERATIVE METHOD

A. Initial Values of Material PropertiesIn this step, we used simplified formulas to

calculate the approximate solution of these material properties. These values are taken as the initial values of material properties, which will be used in the next step.

11

1101

A

AE εσ

= (3)

Equation 3 is based on

332211

11 1

13

1

12

1 AAA

A EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case A and assumes σA22 = σA33 = 0 .

22

2202

B

BE εσ= (4)


3322

1122 2

23

21

12B

BBB EEE

σνσσνε −+−= , which is the

second line of Eqation 2. It is in Load Case B and assumes σB11 = σB33 = 0.

33

3303

C

CE εσ

= (5)

Adhesive

interfaceLayer n

Layer 2

Layer 1

Matrix

Comp.

x

y

Fiber



32

23

1

13 33221133 EEE

CCCC

σσνσνε +−−= , which is the

third line of Equation 2. It is in Load Case C and assumes σC11 = σC22 = 0.

3322

110110

113

CC

CC E

σσεσν

+⋅−= (6)


332211

11 1

13

1

12

1 CCC

C EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case C and assumes ν12 = ν13.

0

33

0

11

0

2222

0

21

223

EE

E

CC

CC

σσσε

ν−−

−= (7)


3322

1122 2

23

21

12C

CCC EEE


second line of Equation 2. It is in Load Case C and assumes ν12 = ν23.

22

1103313110

112

B

BBB E

σεσνσν ⋅−⋅−= (8)


332211

11 1

13

1

12

1 BBB

B EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case B.

B. Iterative Loop to Get the Accurate Material Properties

The following steps use no simplification, all the work is based on the stress-strain relationship for orthotropic materials. Every iterative loop gives more accurate results, converging rapidly.

11

33122111 13121

A

AiAiAiE ε

σνσνσ ⋅−⋅−= −− (9)


332211

11 1

13

1

12

1 AAA

A EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case A.

111

22

33122

1

1223

2

Bi

iB

BiBi

E

Eσνεσνσ

−−

+

−=(10)


3322

1122 2

23

21

12B

BBB EEE


second line of Equation 2. It is in Load Case B.

221

111

33

33

2

23

1

133

Ci

iC

i

iC

Ci

EE

Eσνσνε

σ−− ++

=(11)


32

23

1

13 33221133 EEE

CCCC

σσνσνε +−−= , which is the

third line of Equation 2. It is in Load Case C.

33

1122111 1213

C

CiCiCi

E

σεσνσν ⋅−⋅−= − (12)


332211

11 1

13

1

12

1 CCC

C EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case C.

Figure 11. The Process of the Iterative Method to Calculate the Local Material Properties for a

Composite.

Average σA11, σA22, σA33, εA11, εA22, εA33 from FEA, x-direction loadAverage σB11, σB22, σB33, εB11, εB22, εB33 from FEA, y-direction loadAverage σC11, σC22, σC33, εC11, εC22, εC33 from FEA, z-direction load

Calculate the start values for 6 material properties:E1, E2, E3, ν13, ν23 and ν12 through Equation-3∼ 8

Ccalculate the new values for 6 material properties:E1, E2, E3, ν13, ν23 and ν12 through Equation-9 ∼ 14

To compare the results of the (i)th step with those of the (i-1)th step

Final Results

Acceptable

Not

acc

epta

ble


i

C

i

Ci

i

cC

i

E

EE

2

112

223

33

1112222

σσνσε

ν−

⋅+−=

−(13)

Equation 13 is based

on33

221122 2

23

21

12C

CCC EEE

σνσσνε −+−= , which is

the second line of Equation 2. It is in Load Case C.

22

113311 11312

B

BiBiBi

E

σεσνσν ⋅−⋅−= (14)


332211

11 1

13

1

12

1 BBB

B EEEσνσνσε −−= , which is the

first line of Equation 2. It is in Load Case B.At last we would get 6 engineering materialconstants for Layer n . Repeat this process, but using the stress and strain data averaged for Layer n+1, we can get the material properties for Layer n+1 .

RESULTS OF THE ITERATIVE METHOD TO SOLVE THE LOCAL

MATERIAL PROPERTIES IN A COMPOSITE

(the reinforcing direction is the 3-direction )

Figure 12. Effect of Iteration (Young's Modulus vs. Iteration No. for Layer 6).

Figure 12 shows:(1) This method converges rapidly. The

results of the 3rd loop is pretty close to those of the 4th loop. The max. difference is less than 1%.

(2) The final results for Layer 6, which is pure resin, are very close to the bulk material properties of resin. The error is less than 0.25%.

(3) The results of the 1st loop are the results of the classic method. The classic method has about 10% error compared with the iterative method.

The final results of the local material properties of the composite material are shown in Figure 13 and Figure 14.

All three Young’s moduli keep increasing with the increase of fiber fraction, which is reasonable. The Young’s moduli of the middle layer, which is made up of pure matrix, is exactly equal to the Young’s moduli of matrix resin.

Local Young's Modulus for Composite

0.00E+00

5.00E+04

1.00E+05

1.50E+05

2.00E+05

2.50E+05

3.00E+05

0 0.01 0.02 0.03 0.04 0.05 0.06

Y (mm)

Yo

un

g's

Mo

du

lus

E1 E2 E3 Efiber

Eresin E1composite E2composite E3composite

Figure 13. The Local Young’s Modulus of Composite Material.

Local Poisson's Ratio for Composite

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.01 0.02 0.03 0.04 0.05 0.06

Y (mm)

Po

isso

n's

Rat

io

v13 v23 v12 vfiber

vresin v13composite v23composite v12composite

Figure 14. The Local Poisson’s Ratio of Composite Material.

THE LOCAL SHEAR MODULUS OF COMPOSITE MATERIALS

We already know the relationship among the shear stress, shear strain and shear modulus from the 4th ∼ 6th lines of Equation 2, which are:

12

1212 2G

τγ = (15)

13

1313 2G

τγ = (16)

23

2323 2G

τγ = (17)

Calculation of G12 presents a problem. For a small piece of composite material, as the fiber

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Iteration No.

Yo

un

g's

Mo

du

lus

(MP

a)

E1

E2

E3


shear modulus is much bigger than that of matrix, its boundary would become obviously distorted subjected to the shear load (Figure 15). This situation makes the average shear deformation physically meaningless.

Figure 15. Deformation of Small Composite Piece Subjected to Shear Load.

To solve this problem, in the FE model , we added rigid bodies on the four sides of the composite cell with good results (Figure 16).

Figure 16. FEA Shear Model to Calculate G12.

The results of shear modulus are given in Figure 17. All three shear moduli keep increasing with the increase of fiber fraction, which is reasonable. The shear moduli of the middle layer, which is made up of pure matrix, is exactly equal to the shear moduli of matrix resin.

CONCLUSION

•The detailed structure of an adhesively bonded composite joints, especially at the range close to the interface between composite and adhesive, is very important. For example, the interface layer modulus changes, the max. stress changes greatly. •We suggest a new model (Layer Model) for composite materials, which is closer to the real structure of composite materials than the classic model.

•We established an iterative method in micro-mechanical analysis to simulate the layer properties of composite materials. We found the difference from the classic method to be about 10%.

Figure-17. Local Shear Modulus of Composite Material.

ACKNOWLEDGEMENTThe authors gratefully acknowledge the

support of Dr. Charles Lee and the AFOSR for this work.

REFERRENCE1. Tong, L.; Sheppard, A.; Kelly, D (1995),

"Relationship between surface displacement and adhesive peel stress in bonded double lap joints", International Journal of Adhesion and Adhesives v 15 n 1 Jan 1995. p 43-48

F

rigid

rigid

F

Local Shear Modulus of Composite Material

0

20000

40000

60000

80000

100000

120000

0 0.02 0.04 0.06 0.08 0.1 0.12Y, mm

Sh

ear

Mo

du

lus,

MP

a

Local G12 Local G23 Local G13composite G12 composite G23 composite G13resin fiber

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