BENDING OF A BONDED BEAM AS A TEST METM01) FOR ADHESIVE PROPERTIES by ...

109 Pages·2013·2.24 MB·English

BENDING OF A BONDED BEAM AS A TEST METM01)
FOR ADHESIVE PROPERTIES
by
Eric Moussiaux
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
MASTER OF SCTENCE
in
Engineering Mechanics
APPROVED:
O
H. F. Brinson, Chairman
1D.
A. Dillard J. . Grant
D. Frederick
June 1987
Blacksburg, Virginia
BENDING OF A EONDED BEAM AS A TEST
METHOD FOR ADHESIVE PROPERTIES
by
Eric Moussiaux
Committee Chairman, Hal F. Brinson
Engineering Science and Mechanics
(ABSTRACT)
A strength of materials type solution is obtained for the shear
stress state in the adhesive layer of a bonded cantilever beam
subjected to an end load.
The shear stress is constant through the thickness of the
adhesive layer and varies from zero at the fixed end to a maximum
value at the free end. This maximum value can, under certain condi-
tions, be calculated from knowledge of the load and the beam geometry
only. The adhesive's shear modulus can then be determined from a
measurement of the shear strain in the adhesive layer.
An expression for the beam deflection is also obtained. It
contains a coefficient of adhesion which is potentially useful to
evaluate surface treatments or other factors leading to different
states of adhesion.
Fracture mechanics application of the specimen, nonlinear and
viscoelastic adhesive behavior are briefly mentioned.
ACKNOWLEDGEMENTS
I'm grateful to the members of my committee, Dr. D. A. Dillard
and Dr. J. W. Grant and especially to my advisor, Dr. Hal F. Brinson,
with whom many pleasant hours of discussion were Spent.
Prof. G. Moussiaux, Dr. A. Cardon and S. Roy are gratefully
acknowledged for their valuable help through different stages of
this work.
iii
TABLE OF CONTENTS
Page
ABSTRACT........ ................... ii
ACKNOWLEDGEMENTS....................... iii
TABLE OF CONTENTS ......................
iv
LIST OF FIGURES....................... vii
LIST OF TABLES ........................ X
CHAPTER
1. INTRODUCTION .................... 1
2. STRESSES AND DISPLACEMENTS IN AN ADHESIVELY BONDED
CANTILEVER BEAM SUBJECTED TO AN END LOAD ...... 6
2.1 Introduction .................. 6
2.2 Geometry and Notations .............
7
2.3 Shear Stress Distribution in the Adhesive
Layer ..................... 9
2.3.1 Deflection and Continuity Equations . . . 9
2.3.2 Differential Equation for the Shear
Stress ................. 15
2.3.3 Shear Stress Distribution ........ 17
2.3.4 Limit Cases ............... 19
2.3.5 Discussion of the Shear Stress in the
Adhesive Layer ............. 20
2.3.6 Experimental Aspects .......... 27
2.4 Deflection of the Bonded Cantilever Beam .... 3Q
2.4.1 Integration of the Deflection Equation . 3Q
2.4.2 Analysis of the End Deflection
Expression ............... 33
2.4.3 Experimental Aspects .......... 37
2.5 Conclusions .................. 44
iv
4
Eiä
3. STRESS FUNCTION SOLUTION .............. 45
3.1 Introduction .................. 45
3.2 Stress State in the Beam ............ 46
3.2.1 Stress Function Analysis ........ 46
3.2.2 Choice of the Stress Function Consider-
ing the Basic Assumptions ........ 51
3.2.3 Boundary Conditions ........... 53
3.2.4 Shear Stress in the Adhesive Layer . . . 62
3.3 Deflection of the Cantilever ......... 64
3.4 Comparison of Both Solutions .......... 65
3.5 Conclusions .................. 69
4. NUMERICAL EVALUATION OF THE STRENGTH OF MATERIALS
SOLUTION ...................... 70
4.1 Introduction .................. 70
4.2 Shear Stress in the Adhesive Layer ....... 70
4.3 Effect of the Loading Mode ........... 77
4.4 Deflection of the Beam ............. 80
4.5 Conclusions .................. 85
5. RECOMMENDATIONS FOR FUTURE WORK ........... 86
5.1 Introduction .................. 86
5.2 Nonlinear Adhesive Behavior .......... 86
5.3 Viscoelastic Adhesive Behavior ......... 89
5.4 Fracture Mechanics Application ......... 90
5.5 Conclusions .................. 93
6. CONCLUSIONS ..................... 95
v
Page
REFERENCES .......................... 97
VITA ............................. 99
vi
Figufé Pagg
LIST OF FIGURES
1.1 Short beam shear test for composites ......... 3
1.2 Three-point bending of a bonded beam ......... 5
—
2.1 Three—point bending cantilever beam analogy .... 3
2.2 Geometry of the cantilever beam ........... 3
2.3 Cantilever beam cut along the midplane of the
adhesive layer .................... 1Q
2.4 Relative displacement due to bending ......... 12
2.5 Relative displacement due to shear deformation of
the adhesive ..................... 12
2.6 Normal deformation of a beam in tension ....... 14
2.7 Relative displacement due to normal deformation of
the adherends .................... lg
2.8 Parameter E ..................... 22
2.9 Shear stress in an isotropic cantilever beam
subjected to an end load ............... 23
2.10 Shear stress variation in the adhesive layer along
the length of the beam ................ 25
2.11 Maximum shear stress in the adhesive layer as a
function of adhesive deformability .......... 26
2.12 Maximum shear strain in the adhesive layer as a
function of adhesive deformability .......... 29
2.13a Dependence of the end deflection of the cantilever
on the adhesive deformability for various t/h .... 35
2.13b Dependence of the end deflection of the cantilever
on the adhesive deformability for various 1/h .... 36
2.14 Dependence of the dimensions of a test beam on the
absolute value of the thickness of the adhesive
layer (scale 1/1) .................. 39
— vii
G
Figure Page
2.15 Graphical solution for Ga from a deflection
measurement ..................... 4l
2.16 Coefficient of adhesion B .............. 43
3.1 Definition of h for both methods ........... 47
3.2 Geometry of the bonded beam for the stress function
solution ....................... 48
3.3 Stresses in a two—dimensional problem ........ 49
3.4 Deformation of an element of adhesive in the
adhesive layer .................... 56
3.5 Comparison of the shear stress in the adhesive for
both solutions .................... 67
3.6 Comparison of the end deflection of the beam for both
solutions ...................... 68
4.1 Discretization of the beam .............. 71
4.2 Comparison of numerically and theoretically obtained
shear stress ..................... 73
4.3 Variation of shear stresses over the thickness of the
adhesive layer for various positions along the
x-axis ........................ 75
4.4 Comparison of numerically and theoretically obtained
maximum shear stress values in the adhesive layer . . 76
4.5 Effect of loading mode ................ 78
4.6 Shear stresses and normal vertical stresses in the
adhesive layer for one—sided loading ......... 79
4.7 Comparison of the end deflection of the beam
obtained by finite element analysis and by the
strength of materials solution ............ 81
4.8 Comparison of the deflection of the cantilever beam
obtained by finite elements and by the strength of
materials solution .................. 82
viii
4
Figure Page
4.9 Deflection of a steel—rubber beam, compared to the
cases of "perfect adhesion" and "no adhesion" (E =
30 msi (207 GPa), Ga = 357 psi (2.46 MPa); plane
lines: theoretical values; slashed line: finite
element values) . . . . . . . . . . . . . ...... 83
4.lO Deflection of an aluminum—epoxy beam, compared to
the cases of "perfect adhesion" and "no adhesion"
(E = l0 msi (69 GPa); Ga = 38.5 ksi (0.265 GPa);
plane lines: theoretical values; slashed line:
finite element values) ,,,,,,,,,,,,,,, 84
5.l Creep loading .................... 91
5.2 Double cantilever beam test ............. 92
5.3 Bonded cantilever subjected to an end load ..... 92
5.4 Bonded cantilever beam in mixed mode loading .... 94
ix
LIST OF TABLES
Table Page
2.1 Common adhesive shear moduli ............ gg
4.1 Overview of numerical study (the adherends are
aluminum, E = 107 psi) ............... 72
x