Adhesive Joints – Theory (and
use of innovative joints)
ERIK SERRANO STRUCTURAL MECHANICS, LUND UNIVERSITY
Wood and Timber – Why I’m intrigued
From this…
…to this!
…via this…
Fibre deviation close to knots
…and this…
Deviation in fibre direction
Outline
• Wood adhesive bonds
• Basic behaviour (equations)
• Volkersen theory
• Fracture mechanics approaches
• Equivalent elastic layer approach
• A few words on testing
• Conclusions
An adhesive bond
Thin or thick bond lines?
10 m
0.0001 m
Thin adhesive bond lines
• A thin layer where the ”most important” stress
components do not vary in the thickness direction
Material models
•
Commonly used models
Stress
Stress
Strength
Stiffness
Deformation
Deformation
Material models
• Less commonly used models
Stress
Softening behaviour
Deformation
Softening behaviour
Stress
Strength
Fracture energy
Shape
Stiffness
Deformation
Basics
Luftfahrtforschung, Vol. 15, 1938
• Simplest possible analysis – 1D-shear lag (Volkersen)
» Constant shear across adhesive layer thickness (t3)
» Pure axial action in adherends and pure shear in adhesive
» Linear elastic behaviour of materials
P1
P2
E1, t1, b1
P3
P4
G3, t3, b3
E2, t2, b2
Basic variables
A1
x
N1(0)
N1(L)
t3
N2(0)
N2(L)
A2
L
b
Cross section
dx
N1
N1+dN1
t3
N2
N2+dN2
Horizontal equilibrium
 dN1  t 3b dx  0

dN 2  t 3b dx  0
1



1
 

 2 

b
t3  0
A1
b
t3  0
A2
 1
 A
1
1

 A2
 1, 2 
dN1 b
 t3  0
dx A1
dN 2 b

t3  0
dx A2
Axial stress in adherends
Axial displacement of adherends
 3  (u 2  u1 ) / t3

 1  u1
  u 
2
 2
t 3  G3 3

 1  E1 1
  E 
2 2
 2
2a
Assumes constant shear strain
in adhesive layer
2b
2c
3
Assumes linear elastic materials
Derivation twice of 2a
we obtain, using
3
and using
and
2b
and then 2c
1 :
4
t 3   2t 3  0
5
G3b  1
1 


 

t3  A1 E1 A2 E2 
with the definition of
2
Stiffness ratio shear/axial
Solution of governing equation
• The solution of
t 3   2t 3  0
is given by
t  C1 cosh(x)  C2 sinh(x)
where constants C1 and C2 are determined by the
boundary conditions
6
Load case “pull-pull”
P
E1, t1, b1
P
G3, t3, b3
E2, t2, b2

PG3 
1
1


C1 

t3  E1 A1 tanh(L) E 2 A2 sinh(L) 
PG3 1
C2  
t3 E1 A1
7
Example 1
• Assume wood-wood joint
• Thicknesses 30-30 (mm)
• E1-E2-G3 12 000 -12 000 -1 000 (MPa)
• L: 10, 20, 50 and 100 mm, t3: 0.1 or 1.0 mm
10 mm
100 mm
Example 1 – Results
• Symmetric stress distribution
• Influence of L and stiffness ratio (G3/t3) / (EA)
Relative shear stress in bond line
L=100
Different y-axis scales!!!
Example 2
• Assume glass-wood-adhesive
• Thicknesses 8-30-1 (mm)
• E1-E2-G3 70 000-12 000-1 000 (MPa)
• L: varying 10, 20, 50 and 100 mm
• Non-uniform stress distribution
Relative shear stress in bond line
Conclusions – so far …
• Stress distribution depends on stiffness ratios expressed
through joint parameter L
G3b  1
1 


 

t3  A1 E1 A2 E2 
2
• If L
<< 1 (small) => uniform stress distribution
>> 1 (large) => non-uniforms stress distribution
Conclusions – so far …
• Parameter L
<< 1 (small)
– small overlap length
– low bond line stiffness in relation to axial stiffness of
adherends
>> 1 (large)
– large overlap length
– high bond line stiffness in relation to axial stiffness of
adherends
Joint load-bearing capacity
• Assume joint capacity is reached when max shear
stress equals bond line shear strength, then from 6
Pmax

t3 
1
1


t f

G3  A1 E1 tanh(L) A2 E 2 sinh(L) 
1
Where t f is the bond line shear strength
(Assumes
A2 E 2
A1 E1
is chosen such that
max stress occurs at x=0)
A2 E 2
1
A1 E1
for which case
7
Example: Influence of overlap length
Conclusions – so far …
• Joint capacity (N) depends on adhesive shear strength
and parameter L
Pmax  t f

t3 
1
1



G3  A1 E1 tanh(L) A2 E2 sinh(L) 
1
G3b  1
1 


 


t3  A1 E1 A2 E2 
2
• Pmax thus depends on geometry, strength AND stiffness
parameters
Stress/strength analyses (using FEM)
• Conventional strength analysis:
– Linear elastic material response
– Mostly stressed point governs failure
– Sharp corners can give high stress ( )
– For brittle/stiff joints (i.e. traditional wood adhesive
joints)
» Depicts the stress distribution at low load levels (?)
» Difficult (impossible) to use for prediction of joint
capacity?
Stress/strength analyses using FEM
• Elasto-plastic analysis
– Elasto-plastic material
– Mostly stressed point governs failure
– Sharp corners can give high strains ( )
– Depicts the stress distribution in ductile joints at low
load levels
– Can be used for capacity prediction of ductile/flexible
joints
Fracture mechanics-based analysis
• Linear elastic fracture mechanics (LEFM)
– Assumes a brittle joint/bond line
– Assumes stress singularities (sharp corner/ existing
crack)
– Can be used for capacity prediction of brittle joints
Linear elastic fracture mechanics
• Stress intensity approach
– Assume a small crack exists
– Calculate the stress intensity (stress concentration
factor)
• Crack propagation approach (compliance method)
– Assume an existing crack propagates
– Calculate the change of compliance (flexibility) of the
joint as the crack propagates
• J-integral
– Calculate the value of a path-independent integral of
stress close to the crack tip
Example – LEFM (compliance method)
Finger joints
Aicher & Radovic (1999)
Non-linear fracture mechanics-based analyses
• Assume a non-linear material (bond line) behaviour
including softening (Non-linear fracture mechanics=NLFM)
• Can be used for
– Any brittleness of the joint
– Any geometry
– Can be used for prediction of joint capacity in “all”
cases – from brittle to ductile joints
Softening behaviour
Stress
Strength
Fracture energy
Shape
Stiffness
Deformation
Bonded-in rods
Influence of rod length
Initial softening
Stress distributions
Linear elastic
At max load
Shear
Normal stress (peel)
NLFM Approach – Softening Bonds
Shear
Tension perp.
Serrano, E. Adhesive Joints in Timber Engineering – Modelling and Testing of Fracture
Properties. PhD thesis, Report TVSM-1012, Lund University 2000.
Result presentation NLFM
• Joint brittleness ratio is given by
– Material
» Bond line fracture energy and strength
» Adherend material stiffness
» Shape of softening curve
– Geometry
» Joint shape and absolute size of joint
• Normalised strength at failure is given by:
”some stress measure” /”material strength”
Example: for a beam in bending:
𝑀𝑚𝑎𝑥 ⋅6
/𝑓𝑚
𝑏ℎ2
Normalised ultimate strength
Influence of joint brittleness ratio
Perfectly plastic
NLFM
LEFM
Joint brittleness ratio
E. Serrano. “Glued-in Rods for Timber Structures. A 3D Model and Finite Element Parameter
Studies”. International Journal of Adhesion and Adhesives. 21(2) (2001) pp.115-127.
Equivalent
Elastic layer
Comparison with tests – Pull-out
FEM
Test
E. Serrano and P. J. Gustafsson. “Fracture mechanics in timber engineering – Strength
analyses of components and joints”. Materials and Structures (2006) 40:87–96.
A generalised method
• Equivalent elastic fracture layer method
– Assume simplest possible stress-deformation behaviour
– Adapt stiffness (reduce it) in order to take into account
fracture energy
– Perform linear elastic analysis
– Failure criterion: maximum stress in one point
Stress(MPa)
Material strength, ft
Gf
Displacement (mm)
A generalised method
• How come this works?
t (MPa)
x (mm)
Linear elastic solution (standard elastic stiffness used)
A generalised method
• How come this works?
t (MPa)
x (mm)
Linear elastic solution (standard elastic stiffness used)
NLFM solution
A generalised method
• How come this works?
t (MPa)
x (mm)
Linear elastic solution (standard elastic stiffness used)
NLFM solution
Equivalent elastic layer (adapted stiffness)
Bonded-in rods – Calculation Results
Vessby, J., Serrano, E., Enquist, B. Materials and Structures (2010) 43:1085–1095
Analyses by LEFM and NLFM
Influence of lamination thickness
on beam behaviour
Serrano, E., Larsen, H.J. ASCE Journal of Structural Engineering (1999) 125:740-745.
Bending strength (MPa)
Results – NLFM and LEFM
Compliance method (analytical)
Compliance method (analytical)
Lamination thickness (mm)
Obstacles (at least some of them…)
• Material behaviour
– Time (creep)
– Moisture (hygroscopic materials)
– nonlinear (plasticity, damage, cracking)
• Experiments…?
Testing – Parameters needed
• Bond line strength
– Local strength of the bond line at a “material point”
level (not joint “strength”)
• Stiffness
• Failure strain
• Fracture energy
• Shape of response curve, e.g. softening behaviour
Testing for local strength
• How to test for local strength?
– Small specimen
»Uniform stress distribution
»Small amount of energy released at failure
– Large specimen
»Non-uniform stress distribution
»Large amount of energy released at failure
Example – Stress distributions
Stiff/brittle
Soft/ductile
Standard test specimen (not very useful)
Shear stress
Normal stress (peel stress)
Linear elastic
At max load
Shear stress
Brittle adhesive
Peel stress
Semi ductile
adhesive
Ductile adhesive
Brittle adhesive
Semi-Ductile adhesive
Ductile adhesive
DIC-measurements
Serrano, E., Enquist, B. Holzforschung,
Vol. 59, pp. 641–646, 2005
Brittle adhesive
Ductile adhesive
Fracture mechanics tests (softening)
• Capture the complete response, including softening
• Deformation controlled testing
• Stiff test arrangement
– Testing machine
– Load cell
– Specimen and grips
• Fast response of control system
Stress
Softening behaviour
Deformation
Fracture mechanics tests (softening)
• The energy released during softening (diminishing load at
increasing deformation) must be dissipated by the failure
process in the bond line
• Test arrangements with high stiffness release small
amounts of energy  stable test performance can be
achieved
• Small specimens required (3–5 mm bond line length in
shear)
Serrano, E. Adhesive Joints in Timber Engineering – Modelling and Testing of Fracture
Properties. PhD thesis, Report TVSM-1012, Lund University 2000.
Serrano, E. Adhesive Joints in Timber Engineering – Modelling and Testing of Fracture
Properties. PhD thesis, Report TVSM-1012, Lund University 2000.
Conclusions
• Adhesive joint capacity (in N or MPa) is determined by
– Local strength of the bond line
– Material stiffness(es)
– Fracture energy of the bond line
– Shape of the softening curve of the bond line
– The geometry of the joint
– The absolute size of the joint
• Large adhesive joints need soft/ductile bond lines to be
efficient
• NLFM can be considered a general theory for brittle to
ductile joints
A few other factors affecting... (Marra,1992)
Thanks for the attention…questions?